On solutions of quadratic integral equations

On solutions of quadratic integral equations Ph.D Thesis Written at The Faculty of Mathematics and Computer Science, Adam Mickiewicz University Poznań, Poland, 2013 By MOHAMED METWALI ATIA METWALI Under the guidance of Dr. hab. Mieczyslaw Cichoń Department of Differential Equations, Faculty of Mathematics and Computer Science, Adam Mickiewicz University In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics Acknowledgments First of all, I am thankful to Allah for all the gifts has given me. I would like to express my gratitude and thanks to my advisor and my professor, Dr. hab. Mieczyslaw Cichoń, Department of Differential Equations, Faculty of Mathematics and Computer Science, Adam Mickiewicz University for his help and valuable advice in the preparation of this dissertation. I am thankful to my family (my wife and my lovely sons ”Basem and Eyad”) for their support, encouragement and standing beside me during my stay in Poland. ii Contents 1 Preliminaries 1.1 Introduction . . . . . . . . . . . . . 1.2 Notation and auxiliary facts . . . . 1.2.1 Lebesgue Spaces . . . . . . 1.2.2 Young and N-functions . . . 1.2.3 Orlicz spaces . . . . . . . . 1.3 Linear and nonlinear operators. . . 1.3.1 The superposition operators. 1.3.2 Fredholm integral operator. 1.3.3 Volterra integral operator. 1.3.4 Urysohn integral operator. . 1.3.5 The multiplication operator. 1.4 Monotone functions. . . . . . . . . 1.5 Measures of noncompactness. . . . 1.6 Fixed point theorems. . . . . . . . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monotonic integrable solutions for quadratic integral a half line. 2.1 Motivations and historical background. . . . . . . . . . 2.2 Introduction. . . . . . . . . . . . . . . . . . . . . . . . 2.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . equations on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 On some integrable solutions for quadratic functional integral equations 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The existence of L1 -solution . . . . . . . . . . . . . . . . . . . 3.2.2 The existence of Lp -solution p > 1 . . . . . . . . . . . . . . . . 3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii 4 4 4 5 5 6 7 7 9 10 11 12 14 17 19 23 23 25 26 29 31 31 33 33 38 42 4 Functional quadratic integral half line 4.1 Introduction . . . . . . . . . 4.2 Main result . . . . . . . . . 4.3 Examples . . . . . . . . . . equations with perturbations on a 45 . . . . . . . . . . . . . . . . . . . . . . . 45 . . . . . . . . . . . . . . . . . . . . . . . 46 . . . . . . . . . . . . . . . . . . . . . . . 54 5 On quadratic integral equations in Orlicz spaces 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . 5.2 The case of operators with values in L∞ (I). . . . 5.2.1 The case of W1 = L∞ (I). . . . . . . . . . 5.2.2 The case of W2 = L∞ (I). . . . . . . . . . 5.3 The existence of Lp -solution. . . . . . . . . . . . . 5.3.1 Remarks and examples. . . . . . . . . . . 5.4 A general case of Orlicz spaces. . . . . . . . . . . 5.4.1 The case of N satisfying the ∆′ -condition. 5.4.2 The case of N satisfying the ∆3 -condition. 5.4.3 Remarks on classes of solutions. . . . . . . 5.5 Conclusions and fixed point theorems. . . . . . . 5.5.1 A fixed point theorem. . . . . . . . . . . . iv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 55 58 58 59 64 67 68 69 74 78 80 82 INTRODUCTION Linear and nonlinear integral equations form an important class of problems in mathematics. There are different motivations for their study. Some equations describe mathematical models in physics, engineering or biology. There are also such equations whose interest lies in other branch of pure mathematics. Bearing in mind both mentioned aspects we are interested on a special class of integral equations, namely on quadratic ones. In this case the unknown function is treated by some operators, then a pointwise multiplication of such operators is applied. The study of such a kind of problems was begun in early 60’s due to mathematical modeling of radiative transfer (Chandrasekhar [44], Crum [49]). From the mathematical point of view they are interesting because of lack of compactness for considered operators. Thus some of the classical methods for proving existence theorems are not allowed. There is one more reason which makes this type of equations interesting. In contrast to the case of standard integral equations only continuous solutions were considered. It seems to be strange from application point of view (as will be described below) as well as it prevents a common treatment for both quadratic and non-quadratic equations. This dissertation is devoted to study quadratic integral equations and different classes of their solutions. We concentrate on the aspect of possible discontinuity of solutions and the best possible assumptions ensuring the existence of solutions. This leads us to the Lebesgue spaces and some class of Orlicz spaces. Our approach allow, for the first time, to consider simulatneously quadratic and ”classical” integral equations. We stress on strongly nonlinear problems, which leads to mentioned function spaces, but require a new method of the proof. We prove several results for such a class of equations (existence, monotonicity) on finite and infinite intervals, including functional-integral problems. Let us begin a prototype for this theory, that is the Chandrasekhar integral equation (Chandrasekhar [44, 43]) Z 1 t ψ(s)x(s) ds. x(t) = 1 + x(t) 0 t+s It describe a scattering through a homogeneous semi-infinite plane atmosphere. In particular, solutions for this equations need not to be continuous. Nevertheless, till now only such a kind of solutions was investigated. Note, that for non-homogeneous problems only approximated methods is known (Hollis and Kelley [74]). Different types of quadratic integral equations will be described later (applicable in plasma corners [40, 86], kinetic theory of gases, the theory of neutron transport, in the traffic theory or in mathematical biology). We need only to stress, that the continuity of 1 considered solutions does not follow from applicability of problems (on the contrary), but only from some mathematical unsolved questions (operators in functions spaces and their properties, fixed point theorems). In particular, we solve a problem from [40]. Quadratic integral equations was investigated by many authors. Initial studies by Chandrasekhar ([44] in 1947, cf. also a book [43] in 1960) form only a beginning for this theory, mainly made by astrophysicists. Then research was conducted by mathematicians. They found some interesting open questions in this theory. Let us mention some papers by Anichini, Conti [6], Cahlon, Eskin [42], Banaś, Argyros [11], Caballero, Mingarelli, Sadarangani [40, 41], Nussbaum [96], Gripenberg [72], Mullikin [94, 95], Rus [107], Shrikhant, Joshi [113], Schillings [111] and many others. In the first Chapter we collect all necessary definitions and theorems. We present some function spaces, linear and nonlinear operators and their properties are described. Among others, we present some new studies on a.e. monotonic functions in Lebesgue and Orlicz spaces. Our Chapter 2 is devoted to study quadratic integral equations on unbounded intervals. We present our motivations for the study of presented equation and preceded by a historical background we investigate the quadratic integral equation on a half-line. Here we are looking for a.e. monotonic locally integrable solutions. An illustrative example completes this Chapter (and all others). In the next Chapter 3, we study some functional integral equations of quadratic type. This aspect of the theory is not sufficiently investigated due some restrictions on functional part. Since we try to unify both quadratic and non-quadratic cases, we need to investigate functional equations. Here we study a.e. monotonic L1 and Lp solutions for the considered problem. In the Chapter 4 we unify our research by considering quadratic functional integral equations on a unbounded intervals. Note, that in this Chapter we do not assume, that the operators preserve monotonicity properties. A different method of the proof is then used, which allow us to locate our results among earlier ones. The last Chapter 5 contains our main theorems and conclusions. We consider strongly nonlinear functions, which lead us to solutions in Orlicz spaces. This is well-known for classical (non-quadratic) equations, but it is completely new in the context of quadratic integral equations. We study the pointwise multiplication in Orlicz spaces and in a class of such spaces we solve considered equations. By reducing our problem to an operator equation we present an existence result for a large class of function spaces. The idea of the proof is not only to prove our theorems but also to fully cover the theory for classical equations, which was impossible in the 2 previous approach. We need to stress, that this allow us to prove new fixed point theorem for product of two operators. 3 Chapter 1 Preliminaries 1.1 Introduction This chapter is devoted to recall some notations and known results that will be needed in the sequel. We start in section 1.2 by setting basic notations and definitions that are observed throughout the work. In section 1.3 we study some important linear and nonlinear operators of various types and the multiplication of the operators. We discuss the monotonicity of the functions in section 1.4. Section 1.5 deals with the strong and weak measure of noncompactness. We end this chapter by section 1.6 in which we introduce some important fixed point theorems. 1.2 Notation and auxiliary facts Let R be the field of real numbers, R+ be the interval [0, ∞) and by I = [a, b] denotes an interval subset R. Assume that (E, k · k) is an arbitrary Banach space with zero element θ. Denote by Br (x) the closed ball centered at x and with radius r. The symbol Br stands for the ball B(θ, r). When necessary we will also indicate the space by using the notation Br (E). If X is a subset of E, then X̄ and convX denote the closure and convex closure of X, respectively. We denote the standard algebraic operations on sets by the symbols k · X and X + Y . 4 1.2.1 Lebesgue Spaces Define Lp = Lp (I), 1 ≤ p < ∞ be the space of Lebesgue integrable functions (equivalence classes of functions) on a measurable subset I of R, with the norm ||x||Lp (I) = Z  p1 |x| dt . p I For p = ∞, L∞ (I) denotes the Banach space of essentially bounded functions on I with the norm kxkL∞ = ess sup |x(t)| < ∞. t∈I Recall that the essential supermum is defined as ess sup |x(t)| = inf{a : t∈I the set {t : |x(t)| > a} has measure 0}. Let L1 (I) denote the space of Lebesgue integrable functions on the fixed interval I ⊂ R, bounded or not. Further, denote by BC(R+ ) the Banach space of all real functions defined, continuous and bounded on R+ . This space is furnished with the standard norm kxk = sup{|x(t)| : t ∈ R+ }. Let us fix a nonempty and bounded subset X of BC(R+ ) and a positive number T . For x ∈ X and ε ≥ 0 let us denote by ω T (x, ε) the modulus of continuity of the function x, on the closed and bounded interval [0, T ] (cf. [35]) defined by ω T (x, ε) = sup{|x(t2 ) − x(t1 )| : t, s ∈ [0, T ], |t2 − t1 | ≤ ε}. 1.2.2 Young and N -functions A function M : [0, +∞) → [0, +∞) is called a Young function if it has the form Z u M(u) = a(s)du for u ≥ 0, 0 where a : [0, +∞) → [0, +∞) is an increasing, left-continuous function which is neither identically zero nor identically infinite on [0, +∞). In particular, if M is finite-valued, where limu→0 Mu(u) = 0, limu→∞ Mu(u) = ∞ and M(u) > 0 if x > 0 (M(u) = 0 ⇐⇒ u = 0), then M is called an N−function. 5 The functions M and N are called complementary N-functions. If N(x) = sup(xy − M(x)). y≥0 Further, the N− function M satisfies the ∆2 -condition, i.e. (∆2 ) there exist ω, t0 ≥ 0 such that for t ≥ t0 , we have M(2t) ≤ ωM(t). Let us observe, that an N-function M(u) = exp u2 − 1 satisfies this condition, while the function M(u) = exp |u| − |u| − 1 does not. An N-function M is said to satisfy ∆′ -condition if there exist K, t0 ≥ 0 such that for t, s ≥ t0 , we have M(ts) ≤ KM(t)M(s). If the N-function M satisfies the ∆′ -condition, then it also satisfies ∆2 -condition. α Typical examples: M1 (u) = |u|α for α > 1, M2 (u) = (1 + |u|) ln (1 + |u|) − |u| or √ M3 (u) = |u|α(| ln |u|| + 1) for α > 3 + 25 . The last class of N-functions, interesting for us, consists of functions which increase more rapidly than power functions. An N-function M is said to satisfy ∆3 -condition if there exist K, t0 ≥ 0 such that for t ≥ t0 , we have tM(t) ≤ M(Kt). 1.2.3 Orlicz spaces The Orlicz class, denoted by OP , consists of measurable functions x : I → R for which Z ρ(x; M) = M(x(t))dt < ∞. I We shall denote by LM (I) the Orlicz space of all measurable functions x : I → R for which Z    x(s) M ds ≤ 1 . kxkM = inf λ>0 λ I p The N-function M(u) = |u|p , 1 < p < ∞ leads to the classical Lebesgue space Lp (I) with the norm mention before. Let EM (I) be the closure in LM (I) of the set of all bounded functions. Note that EM ⊆ LM ⊆ OM . The inclusion LM ⊂ LP holds if, and only if, there exists positive constants u0 and a such that P (u) ≤ aM(u) for u ≥ u0 . An important property of EM spaces lies in the fact that this is a class of functions from LM having absolutely continuous norms. Moreover, we have EM = LM = OM if M satisfies the ∆2 -condition. Sometimes, we will use more general concept of function spaces i.e. ideal spaces. 6 Definition 1.2.1. [118] A normed space (X, k·k) of (classes of ) measurable functions x : I → U (U is a normed space) is called pre-ideal if for each x ∈ X and each measurable y : I → U the relation |y(s)| ≤ |x(s)| (for almost all s ∈ I) implies y ∈ X and kyk ≤ kxk. If X is also complete, it is called an ideal space. Ideal spaces are a very general class of normed spaces of measurable functions, which includes Lebesgue, Orlicz, Lorentz, and Marcinkiewicz spaces as well as weighted and combined forms of these spaces. Sometimes these spaces are also called Banach function spaces or (normed) Köthe spaces. 1.3 Linear and nonlinear operators. In this section we define and discuss some properties of the nonlinear superposition operators and many integral operators that are needed throughout this dissertation such as Fredholm, Volterra and Urysohn operators in Lp (I), p ≥ 1, L∞ (I) and LM (I) spaces. We will distinguish between two different cases: when the operators take their values in Lebesgue (Orlicz) spaces Lp (I) (LM (I)) or in a space of essentially bounded functions L∞ (I). 1.3.1 The superposition operators. One of the most important operator studied in nonlinear functional analysis is the so-called superposition operator [10]. Definition 1.3.1. Assume that a function f : I × R → R satisfies the Carathéodory conditions i.e. it is measurable in t for any x ∈ R and continuous in x for almost all t ∈ I. Then to every function x(t) being measurable on I we may assign the function Ff (x)(t) = f (t, x(t)), t ∈ I. The operator Ff in such a way is called the superposition (Nemytskii) operator generated by the function f . Furthermore, for every f ∈ L1 and every φ : I → I we define the superposition operator generated by the functions f and φ, Fφ,f : L1 (I) → L1 (I) as Fφ,f (t) = f (t, x(φ(t))) , t∈I Lemma 1.3.1. ([10, Theorem 17.5]) Assume that a function f : I × R → R satisfies Carathéodory conditions. Then the superposition operator F transforms measurable functions into measurable functions. 7 Lemma 1.3.2. ([84, Lemma 17.5] in S and [102] in LM ) Assume that a function f : I × R → R satisfies Carathéodory conditions. The superposition operator F maps a sequence of functions convergent in measure into a sequences of functions convergent in measure. We will be interested in the case when F acts between some Lebesgue (Orlicz) spaces. In Lp (I) we have the ”automatic” continuity of the Nemytskii operator ([10, 81]): Theorem 1.3.1. Let f satisfies the Carathéodory conditions. The superposition operator F generated by the function f maps continuously the space Lp (I) into Lq (I) (p, q ≥ 1) if and only if p (1.1) |f (t, x)| ≤ a(t) + b · |x| q , for all t ∈ I and x ∈ R, where a ∈ Lq (I) and b ≥ 0. This theorem was proved by Krasnoselskii [81] in the case when I is a bounded interval. The generalization to the case of an unbounded interval I was given by Appell and Zabrejko [10]. Remark 1.3.1. It should be also noted that the superposition operator F takes its values in L∞ (I) iff the generating function f is independent of x (cf. [10, Theorem 3.17]). Lemma 1.3.3. ([83, Theorem 17.5]) Assume that a function f : I × R → R satisfies Carathéodory conditions. Then M2 (f (s, x)) ≤ a(s) + bM1 (x), where b ≥ 0 and a ∈ L1 (I), if and only if the superposition operator F acts from LM1 (I) to LM2 (I). In Orlicz spaces there is no automatic continuity of superposition operators like in Lp spaces, but the following lemma is useful (remember, that the Orlicz space LM is ideal and if M satisfies ∆2 condition it is also regular cf. [7, Theorem 1]): Lemma 1.3.4. ([118, Theorem 5.2.1]) Let f be a Carathéodory function, X an ideal space, and W a regular ideal space. Then the superposition operators F : X → W is continuous. Let us note, that in the case of functions of the form f (t, x) = g(t)h(x), the superposition operator F is continuous from the space of continuous functions C(I) into LM (I) even when M does not satisfies ∆2 condition ([7]). Since EM (I) is a regular part of an Orlicz space LM (I) (cf. [119, p.72]), in the context of Orlicz spaces, we will use the following (see also Lemma 1.3.3): 8 Lemma 1.3.5. Let f be a Carathéodory function. If the superposition operator F acts from LM1 (I) into EM2 (I), then it is continuous. The problem of boundedness of such a type of operators will be described in the proofs of our main results. Remark 1.3.2. Let us recall, that the acting condition from Lemma 1.3.3 is not sufficient for taking EM1 (I) into EM2 (I) (cf. [10, p.95]), especially for the continuity of this operator. For the case considered when M1 = M2 we can put, for example, f (t, x) = x to fulfil this requirement. But this is true also for an arbitrary Carathéodory function f when M1 satisfies the ∆2 -condition. For a general result of this type see [83, Th. 17.7]. Remark 1.3.3. Let X, Y be ideal spaces. A superposition operator F : X → Y is called improving if it takes bounded subsets of X into the subsets of Y with equiabsolutely continuous norms. The following two theorems ”Lusin and Dragoni” [56, 109], which explain the structure of measurable functions and functions satisfying Carathéodory conditions, where D c denotes the complement of D and the symbol meas(D) stands for the Lebesgue measure of the set D. Theorem 1.3.2. Let m : I → R be a measurable function. For any ε > 0 there exists a closed subset Dε of the interval I such that meas(Dεc ) ≤ ε and m|Dε is continuous. Theorem 1.3.3. Let f : I×R → R be a function satisfying Carathéodory conditions. Then for each ε > 0 there exists a closed subset Dε of the interval I such that meas(Dεc ) ≤ ε and f |Dε ×R is continuous. 1.3.2 Fredholm integral operator. Assume that k : I × I → R be measurable with respect to both variables. For an arbitrary x ∈ Lp (I) let Z (K0 x)(t) = k(t, s)x(s) ds, t ∈ I. (1.2) I This operator K is linear and is called Fredholm integral operator (cf. [84, 122]). The next theorem gives a sufficient conditions which which ensure that K maps from Lp into Lq and is continuous. Theorem 1.3.4. [122] Let k : I × I → R be measurable with respect to both variables. Let the linear integral operator K0 with kernel k(t, s) map Lp into Lq . Then it is continuous. 9 Lemma 1.3.6. [78] Let k : I ×I → R be measurable with respect to both variables. Let the linear integral operator K0 with kernel k(·, ·) maps Lp (I) into L∞ (I) i.e. either Z b  p1 p <∞ essupt∈[a,b] |K(t, s)| ds a or Z a Then it is continuous. b essups∈[a,b] |K(t, s)|
p  p1 < ∞. dt The necessary results concerning the properties of such a kind of operators in Orlicz spaces can be found in [83], let we mention Zaanen’s theorem [83] which shows that the operator (1.2) acts between Orlicz spaces. Let, the N-functions M1 and M2 are the complementary functions to the Nfunctions N1 and N2 respectively. Lemma 1.3.7. Suppose the kernel k(x, y) satisfies either one of the following two conditions: (a) for almost all t ∈ I the kernel k(t, s), as a function of s, belongs to the space LN1 , where the function ϕ(t) = kk(t, s)kN1 belongs to the space LM2 , (b) for almost all s ∈ I the kernel k(t, s), as a function of t, belongs to the space LM2 , where the function Ψ(s) = kk(t, s)kM2 belongs to the space LN1 . Then the operator (1.2) maps LM1 into LM2 and is continuous. 1.3.3 Volterra integral operator. Suppose k : ∆ → R is a given function and measurable with respect to both variables where ∆ = {(t, s) : 0 ≤ s ≤ t ≤ ∞}. For an arbitrary function x ∈ L1 (R+ ) define Z t (V x)(t) = k(t, s)x(s) ds, 0 t ≥ 0. The linear integral operator defined above is the well known linear Volterra integral operator (cf. [84, 122]). When we consider this operator on the space Lp ([a, b]), then it is a special case of the Fredholm operator investigated in previous section, where Z b (V x)(t) = χ[0,t] k(t, s)x(s) ds, t ≥ 0. a 10 1.3.4 Urysohn integral operator. The most important nonlinear integral operators are the Urysohn operators [122]: Z U(x)(t) = u(t, s, x(s)) ds. (1.3) I Here, the kernel u : I × I × R → R satisfies Carathéodory conditions i.e. it is measurable in (t, s) for any x ∈ R and continuous in x for almost all (t, s) ∈ I × I. Moreover, for arbitrary fixed s ∈ I and x ∈ R the function t → u(t, s, x(s)) is integrable. A particular case of a Urysohn operator (1.3) is the Hammerstein integral operator H = K0 ◦ F : H(x)(t) = Z k(t, s)f (s, x(s)) ds. (1.4) I Note, that for Urysohn operators the continuity is not ”automatic” as in the case of superposition operators (for Nemytskii operators see Theorem 1.3.1). Let us recall an important sufficient condition: Theorem 1.3.5. [84, Theorem 10.1.10] Let u : I ×I ×R → R satisfies Carathéodory conditions i.e. it is measurable in (t, s) for any x ∈ R and continuous in x for almost R all (t, s) ∈ I × I. Assume that U(x)(t) = I u(t, s, x(s))ds maps Lp (I) into Lq (I) (q < ∞) and for each h > 0 the function Rh (t, s) = max |u(t, s, x)| |x|≤h is integrable on s for a.e. t ∈ I. If moreover for each h > 0 this operator satisfies Z lim sup k u(t, s, x(s))dskLq (I) = 0 meas(D)→0 |x|≤h D and for arbitrary non-negative z(t) ∈ Lp (I) Z lim sup k u(t, s, x(s)) dskLq (I) = 0, D→0 |x|≤z D then U is a continuous operator. The first two conditions are satisfied when R I Rh (t, s)ds ∈ Lq (I), for instance. We will use also the majorant principle for Urysohn operators (cf. [84, Theorem 10.1.11]. The following theorem which is a particular case of much more general result ([84, Theorem 10.1.16]), will be very useful in the proof of the main result for operators in L∞ (I): 11 Theorem 1.3.6. [84] Let u : I × I × R → R satisfies Carathéodory conditions i.e. it is measurable in (t, s) for any x ∈ R and continuous in x for almost all (t, s). Assume that |u(t, s, x)| ≤ k(t, s) · (a(s) + b · |x|), where the nonnegative function k is measurable in (t, s), a is a positive integrable function, b > 0 and such that the linear integral operator with the kernel k(t, s) maps L1 (I) into L∞ (I). Then the operator U maps L1 (I) into L∞ (I). Moreover, if for arbitrary h > 0 Z lim k max |u(t, s, x1 ) − u(t, s, x2 )| dskL∞ (I) = 0, δ→0 D |xi |≤h,|x1−x2 |≤δ then U is a continuous operator. We mention also that some particular conditions guaranteeing the continuity of the operator U may be found in [116, 122]. 1.3.5 The multiplication operator. We need to describe the multiplication operator which is the key point of our work. We will denote the pointwise multiplication operator by A(x)(t) of the form: A(x)(t) = F (x)(t) · U(x)(t), where U(x) is a Urysohn integral operator (1.3), in some chapters replaced by the the Hammerstein integral operator H = K0 F , where K0 is the linear integral operator and F as in definition 1.3.1. Generally speaking, the product of two functions x, y ∈ Lp (I)[LM (I)] is not in Lp (I) [LM (I)]. However, if x and y belongs to some particular Lebesgue (Orlicz) spaces, then the product x· y belong to a third Lebesgue (Orlicz) space. Let us note, that one can find two functions belonging to Lebesgue (Orlicz) spaces: u ∈ Lp (LU ) and v ∈ Lp (LV ) such that the product uv does not belong to any Lebesgue (Orlicz) space (this product is not integrable). We will use the technique of factorization for some operators acting on Lebesgue (Orlicz) spaces through another Lebesgue (Orlicz) spaces. We can mention, that by using in this place different ideal spaces it is possible to obtain some extensions of our results and then we try to facilitate this approach. To stress the connection of our results with the growth condition we restrict ourselves to the case of Lebesgue (Orlicz) spaces. 12 Remark 1.3.4. For the so-called pre-ideal spaces (cf. [118]) if x ∈ E and y ∈ L∞ implies that xy ∈ E and kxykE ≤ kxkE kykL∞ i.e. the elements from L∞ are pointwise multipliers for E. For more details in L1 space see [46]. Nevertheless, we have: Lemma 1.3.8. ([83, Lemma 13.5]), [92, Theorem 10.2] Let ϕ1 , ϕ2 and ϕ are arbitrary N-functions. The following conditions are equivalent: 1. For every functions u ∈ Lϕ1 (I) and w ∈ Lϕ2 , u · w ∈ Lϕ (I). 2. There exists a constant k > 0 such that for all measurable u, w on I we have kuwkϕ ≤ kkukϕ1 kwkϕ2 .
≤ 3. There exists numbers C > 0, u0 ≥ 0 such that for all s, t ≥ u0 , ϕ st C ϕ1 (s) + ϕ2 (t). 4. lim supt→∞ −1 ϕ−1 1 (t)ϕ2 (t) ϕ(t) < ∞. Let us recall the following simple sufficient condition for the above statements hold true. Lemma 1.3.9. ([83, p. 223]) If there exist complementary N-functions Q1 and Q2 such that the inequalities Q1 (αu) < ϕ−1 [ϕ1 (u)] Q2 (αu) < ϕ−1 [ϕ2 (u)] hold, then for every functions u ∈ Lϕ1 (I) and w ∈ Lϕ2 , u · w ∈ Lϕ (I). If moreover ϕ satisfies the ∆2 -condition, then it is sufficient that the inequalities Q1 (αu) < ϕ1 [ϕ−1 (u)] Q2 (αu) < ϕ2 [ϕ−1 (u)] hold. An interesting discussion about necessary and sufficient conditions for product operators can be found in [83, 92]. Remark 1.3.5. An ideal space E is called regular if for every x ∈ E we have limmesD→0 kx · χD kE = 0. A set of all elements x with this property is called a regular part of E. Thus this a set of all x ∈ E with absolutely continuous norm. A space is called perfect if the Fatou lemma holds for E. 13 1.4 Monotone functions. Let S = S(I) denote the set of measurable (in Lebesgue sense) functions on I and let meas stand for the Lebesgue measure in R. Identifying the functions equals almost everywhere the set S furnished with the metric d(x, y) = inf [a + meas{s : |x(s) − y(s)| ≥ a}] a>0 becomes a complete space. Moreover, the space S with the topology convergence in measure on I is a metric space, because the convergence in measure is equivalent to convergence with respect to d (cf. Proposition 2.14 in [119]). For σ-finite subsets of R we say that the sequence xn is convergent in finite measure to x if it is convergent in measure on each set T of finite measure. The compactness in such spaces we will call a ”compactness in measure” and such sets have important properties when considered as subsets of some Orlicz spaces (ideal spaces). Let us recall, in metric spaces the set U0 is compact if and only if each sequence from U0 has a subsequence that converges in U0 (i.e. sequentially compact). In this dissertation, we need to investigate some properties of sets and operators in such a class of spaces instead of the space S. Some of them are obvious, the rest will be proved. We are interested in finding of almost everywhere monotonic solutions for our problems. We will need to specify this notion in considered solution spaces. Let X be a bounded subset of measurable functions. Assume that there is a family of subsets (Ωc )0≤c≤b−a of the interval I such that meas(Ωc ) = c for every c ∈ [0, b − a], and for every x ∈ X, x(t1 ) ≥ x(t2 ), (t1 ∈ Ωc , t2 6∈ Ωc ). It is clear, that by putting Ωc = [0, c) ∪ Z or Ωc = [0, c) \ Z, where Z is a set with measure zero, this family contains nonincreasing functions (possibly except for a set Z). We will call the functions from this family ”a.e. nonincreasing” functions. This is the case, when we choose a measurable and nonincreasing function y and all functions equal a.e. to y satisfies the above condition. This means that such a notion can be also considered in the space S. Thus we can write, that elements from L1 (I), LM (I) belong to this class of functions. Further, let Qr stand for the subset of the ball Br consisting of all functions which are a.e. nonincreasing on I. Functions a.e. nondecreasing are defined by similar way. It is known, that such a family constitute a set which is compact in measure in S. We are interested, if the set is still compact in measure as a subset of subspaces of S. In general, it is not true, but for the case of Lebesgue spaces L1 (I), Lp (I), p > 1 and Orlicz spaces LM (I), we have the following: 14 Due to the compactness criterion in the space of measurable functions (with the topology of the convergence in measure) (see Lemma 4.1 in [18]) we have a desired theorem concerning the compactness in measure of a subset X of L1 (I) (cf. Corollary 4.1 in [18] or Section III.2 in [60]). Theorem 1.4.1. Let X be a bounded subset of L1 (I) consisting of functions which are a.e. nonincreasing (or a.e. nondecreasing) on the interval I. Then X is compact in measure in L1 (I). In the following theorems, denote by E the spaces Lp (I), p ≥ 1 or LM (I) (cf. [47, 48]). We have a new characterization of compactness in measure for subspaces of S. Lemma 1.4.1. Let X be a bounded subset of E consisting of functions which are a.e. nondecreasing (or a.e. nonincreasing) on the interval I. Then X is compact in measure in E. Proof. Let r > 0 be such that X ⊂ Br ⊂ E. It is known (cf. [84, 18]), that X is compact in measure as a subset of S. By taking an arbitrary sequence (xn ) in X we obtain that there exists a subsequence (xnk ) convergent in measure to some x ∈ S. Since Orlicz spaces are perfect (cf. [118]), the balls in E are closed in the topology of convergence in measure. Thus x ∈ Br ⊂ E and then x ∈ X. Remark 1.4.1. The above lemma remains true for subsets of arbitrary perfect ideal spaces ([118]). If we consider the set of indices c ≥ 0 in the definition of the family of a.e. nonincreasing functions, we are able to extend this result for the space L1 (R+ ). For simplicity, we will denote such a space by L1 . Due to some results of Väth we are able to extend the desired result from the interval I = [a, b] into the σ-finite subsets of R and the topology of the convergence in finite measure. Theorem 1.4.2. Let X be a bounded subset of L1 (R+ ) consisting of functions which are a.e. nonincreasing (or a.e. nondecreasing) on the half-line R+ . Then X is compact in finite measure in L1 (R+ ). Proof. If we consider the space L1 (T ) for σ-finite measure space T , then there is some equivalent finite measure ν (ν(R+ ) < ∞) (Proposition 2.1. in [119] or Corollary 2.20 in [119]). Then the convergence of sequences in S are the same for the metric d and for dν (x, y) = inf [a + ν{s : |x(s) − y(s)| ≥ a}] a>0 (Proposition 2.2 in [117]). Take an arbitrary bounded sequence (xn ) ⊂ X. As a subset of a metric space X = (L1 (R+ ), dν ) the sequence is compact in this metric 15 space (Theorem 1.4.1). Then there exists a subsequence (xnk ) of (xn ) which is convergent in the space X to some x i.e. k→∞ dν (xnk , x) −→ 0. As claimed above the two metrics have the same convergent sequences, then k→∞ d(xnk , x) −→ 0. This means that X is compact in L1 (R+ ). We have also an important Lemma 1.4.2. (Lemma 4.2 in [15]) Suppose the function t → f (t, x) is a.e. nondecreasing on a finite interval I for each x ∈ R and the function x → f (t, x) is a.e. nondecreasing on R for any t ∈ I. Then the superposition operator F generated by f transforms functions being a.e. nondecreasing on I into functions having the same property. We will use the fact, that the superposition operator takes the bounded sets compact in measure into the sets with the same property. Thus we can prove the following (cf. [46, Proposition 4.1]): Proposition 1.4.1. Assume that a function f : I × R → R satisfies Carathéodory conditions and the function t → f (t, x) is a.e. nondecreasing on a finite interval I for each x ∈ R and the function x → f (t, x) is a.e. nondecreasing on R for any t ∈ I. Assume, that F : LM (I) → EM (I). Then F (V ) is compact in measure for arbitrary bounded and compact in measure subset V of LM (I). Proof. Let V be a bounded and compact in measure subset of LM (I). By our assumption F (V ) ⊂ EM (I). As a subset of S the set F (V ) is compact in measure (cf. [18]). Since the topology of convergence in measure is metrizable, the compactness of the set is equivalent with the sequential compactness. By taking an arbitrary sequence (yn ) ⊂ F (V ) we get a sequence (xn ) in V such that yn = F (xn ). Since (xn ) ⊂ V , as follows from Lemma 1.3.2 F transforms this sequence into the sequence convergent in measure. Thus (yn ) is compact in measure, so is F (V ). For the integral operator (1.2), we have the following theorem due to Krzyż ([85, Theorem 6.2]): Theorem 1.4.3. The operator K0 preserve the monotonicity of functions iff Z b Z b k(t1 , s) ds ≥ k(t2 , s) ds 0 0 for t1 < t2 , t1 , t2 ∈ I and for any b ∈ I. 16 1.5 Measures of noncompactness. Now we present the concept of a regular measure of noncompactness (or of weak noncompactness ), we denote by ME the family of all nonempty and bounded subsets of E and by NE , NEW its subfamily consisting of all relatively compact and weakly relatively compact sets, respectively. The symbol X̄ W stands for the weak closure of a set X while X̄ denotes its closure. Definition 1.5.1. [23] A mapping µ : ME → [0, ∞) is said to be a measure of noncompactness in E if it satisfies the following conditions: (1) the family kerµ = {X ∈ ME : µ(X) = 0} is nonempty and kerµ ⊂ NE , where kerµ is called the kernel of the measure µ. (2) X ⊂ Y ⇒ µ(X) ≤ µ(Y ). (3) µ (convX) = µ (X) (4) µ [ λ X + (1 − λ) Y ] ≤ λ µ(X) + (1 − λ) µ(Y ), λ ∈ [0, 1]. (5) If Xn ∈ ME , Xn = X̄n and Xn+1 ⊂ Xn for n = 1, 2, . . . and if lim µ(Xn ) = 0, then X∞ = n→∞ ∞ \ n=1 Xn 6= φ. Definition 1.5.2. [29] A mapping γ : ME → [0, ∞) is said to be a measure of weak noncompactness in E if it satisfies conditions (2)-(4) of definition 1.5.1 and the following two conditions (being counterparts of (1) and (5)) hold: (1’) the family kerγ = {X ∈ ME : γ(X) = 0} is nonempty and kerγ ⊂ NEW , where kerγ is called the kernel of the measure γ. (5’) If Xn ∈ ME , Xn = X̄n W and Xn+1 ⊂ Xn for n = 1, 2, . . . and if lim γ(Xn ) = 0, then X∞ = n→∞ ∞ \ n=1 Xn 6= φ. In addition the measure of noncompactness µ (or of weak noncompactness γ) is called S • Measure with maximum property if µ(X Y ) = max [ µ(X), µ(Y ) ]. • Homogeneous measure if µ (λX) = |λ| µ(X), λ ∈ R. • Subadditive measure if µ (X + Y ) ≤ µ (X) + µ (Y ). • Sublinear measure if it is homogeneous and subadditive. • Complete (or full) if ker µ = NE (ker γ = NEW ) . • Regular measure if it is full, sublinear and has a maximum property. 17 An classical example of measure of noncompactness is the following: Definition 1.5.3. [23] Let X be a nonempty and bounded subset of E. The Hausdorff measure of noncompactness βH (X) is defined as βH (X) = inf{ǫ > 0 : Xcan be covered with a finite number of balls of a radii less than ǫ} It is worthwhile to mention that the first important example of measure of weak noncompactness has been defined by De Blasi [52] by: β(X) = inf {r > 0 : there exists a weakly compact subset W of E such that x ⊂ W +Br }. Both the Hausdorff measure βH and the De Blasi measure β are regular in the sense of the above definitions. Another regular measure of noncompactness was defined in the space L1 (I) (cf. [28]). For any ε > 0, let c be a measure of equiintegrability of the set X (the so-called Sadovskii functional [10, p. 39]) i.e. Z c(X) = lim sup{sup{sup[ |x(t)| dt, D ⊂ I, meas(D) ≤ ε]}}. ε→0 x∈X D Restricted to the family compact in measure subsets of this space it forms a regular measure of noncompactness (cf. [66]). However, by considering this measure of noncompactness instead of usually considered ones based on Kolomogorov or Riesz criteria of compactness (cf. [23]) we are able to examine by the same manner the case of Lp (I) spaces, where χD denotes the characteristic function of D. Let us also denote by c a measure of equiintegrability of the set X in an Orlicz space LM (I) (cf. Definition 3.9 in [119] or [67, 66]): c(X) = lim sup ε→0 sup sup kx · χD kLM (I) , meas D≤ε x∈X where χD denotes the characteristic function of D. Then we have the following theorem, which clarify the connections between different coefficients in Orlicz spaces. Since Orlicz spaces LM (I) are regular, when M satisfies ∆2 condition, then Theorem 1 in [66] read as follows: Proposition 1.5.1. Let X be a nonempty, bounded and compact in measure subset of an ideal regular space Y . Then βH (X) = c(X). 18 As a consequence, we obtain that bounded sets which are additionally compact in measure are compact in LM (I) iff they are equiintegrable in this space (i.e. have equiabsolutely continuous norms cf. [5]). The contraction of the measure of weak non compactness is a bit more complicated when I is an unbounded interval. Let us, we recall the following criterion for weak noncompactness due to Dieudonné [55, 60], which is of fundamental importance in our subsequent analysis. Theorem 1.5.1. A bounded set X is relatively weakly compact in L1 (R+ ) if and only if the following two conditions are satisfied: R (a) for any ε > 0 there exists δ > 0 such that if meas(D) < δ then D |x(t)|dt ≤ ε for all x ∈ X, R∞ (b) for any ε > 0 there is T > 0 such that T |x(t)|dt ≤ ε for any x ∈ X. Now, for a nonempty and bounded subset X of the space L1 (R+ ) let us define: Z c(X) = lim{sup{sup [ |x(t)| dt, D ⊂ R+ , meas(D) ≤ ε]}}, (1.5) ǫ→0 x∈X and D Z d(X) = lim {sup[ T →∞ Put ∞ T |x(t)| dt : x ∈ X]}. γ(X) = c(X) + d(X). (1.6) (1.7) Then we have the following theorem, which clarify the connections between these two measures βH (x) and γ(x) ([22]). Theorem 1.5.2. Let X be a nonempty, bounded and compact in measure subset of L1 (R+ ). Then βH (x) ≤ γ(x) ≤ 2βH (x). 1.6 Fixed point theorems. Fixed point theorems have always a major role in various fields, specially, in fields of differential, integral and functional equations. Fixed point theorems constitute a topological tool for the qualitative investigations of solution of linear and nonlinear equations. The theory of fixed points is concerned with the conditions which guarantee that a map T : X → X of a topological space X into it self admits one or more fixed points, that is, points x of X for which x = T x. Here we give a brief history of fixed point theorems. The following definition states some types of mapping in a metric space (X, ρ) [71]. 19 Definition 1.6.1. Let (X, ρ) be a metric space. The mapping T : X → X is called Lipschitz map, if there exist a number γ ≥ 0, such that ρ(T x, T y) ≤ γ ρ(x, y), ∀ x, y ∈ X. The mapping T is called contraction if γ γ ≤ 1. Furthermore, T is contractive if < 1, and is called non expensive, if ρ(T x, T y) < γ ρ(x, y), ∀ x 6= y. Problems concerning the existence of fixed point for Lipschitz map have been of considerable interest in non linear operator theory. In 1922, the so-called Banach contraction mapping principle was given to obtain solutions for several problems. Theorem 1.6.1. (Banach contraction mapping principle, [71]) Let X be a complete metric space and let T : X → X be a contraction map. Then T has a unique fixed point in X. Moreover, for any x0 ∈ X, the sequence {T n (x0 )}∞ n=0 converges to the fixed point. This theorem is perhaps the most useful fixed point theorem, which is involved in many of the existence and uniqueness proofs in ordinary differential equations. The mapping T is the Banach contraction mapping principle still has a unique fixed point in any closed subset M of X. There are some conditions for a continuous mapping T in X, that guarantee the existence of a unique fixed point, such as the contraction of T n or if there exist a function φ : X → R+ , such that for all x ∈ X, ρ (T x, T y) = φ(x) φ(T x). In a normed space, the next fixed point theorem, is concerned with continuous mapping and has an advantage over Banach Contraction Mapping Principle in that is applied to a large class of functions. Theorem 1.6.2. (Brouwer [71]) Let Q be a nonempty, convex, closed and bounded subset of a finite dimensional Banach space En and let T : Q → Q be continuous. Then T has at least one fixed point in the set Q. A generalization of Brouwer’s result to any Banach space was due to Schauder. Theorem 1.6.3. (Schauder, [71]) Let Q be a convex subset of a Banach space X, and T : Q → Q is compact, continuous map. Then T has at least one fixed point in Q. Next, we need the following definition. 20 Definition 1.6.2. [71] A mapping H : E → E is called completely continuous if H is continuous and H(Y ) is relatively compact for every bounded subset of Y . Theorem 1.6.4. (Schauder-Tychonoff, [71]) Let C be a nonempty, convex, closed and bounded subset of a Banach space E. Let H : C → C be a completely continuous mapping. Then H has at least one fixed point in C. When the concept of measure of noncompactness appeared, some fixed point theorems based on such measure were given. Among these is the Darbo fixed point theorem. Such theorem is used for a contraction mapping with respect to the Hausdorff measure of non compactness, that is, there exist a constant α ∈ (0, 1), such that χ(HX) ≤ α χ(X), for any nonempty bounded subset X of G. An importance of such a kind of functions can be clarified by using the contraction property with respect to this measure instead of compactness in the Schauder fixed point theorem. Namely, we have a theorem ([23]). Theorem 1.6.5. (Darbo, [50]) Let Q be a nonempty, bounded, closed and convex subset of E and let V : Q → Q be a continuous transformation which is a contraction with respect to the measure of noncompactness µ, i.e. there exists k ∈ [0, 1) such that µ(V (X)) ≤ kµ(X), for any nonempty subset X of E. Then V has at least one fixed point in the set Q and the set F ixV of all fixed points of V satisfies µ(F ixV ) = 0. Emmanuele gives the corresponding version of Darbo fixed point theorem in the weak sense. Theorem 1.6.6. (Emmanuele, [65] ) Let Q be a nonempty, closed, convex and bounded subset of a Banach space E, Assume that F : Q → Q be a weakly continuous operator having the proprty that, there is a constant α ∈ (0, 1), such that β (F (X)) ≤ α β(X), for any nonempty subset X of Q, where β(X) is the measure of noncompactness. Then F has at least one fixed point in the set Q. Theorem 1.6.7. [82] Let M be a nonempty, closed, and convex subset of E. Suppose, that A, B be two operators such that i) A(M) + B(M) ⊆ M, 21 ii) A is a contraction mapping, iii) B(M) is relatively compact and B is continuous. Then there exists a y ∈ M with Ay + By = y. Next we state a nonlinear alternative of Leray-Schauder type fixed point theorem (cf. [51]). Theorem 1.6.8. (the Leray-Schauder alternative) Let C be an open subset of a convex set Q in a Banach space E. Assume 0 ∈ C and the map T : C̄ → Q is continuous and compact. Then either (i) T has a fixed point in C̄, or (ii) there exist λ ∈ (0, 1) and u ∈ ∂C such that u = λT u, where ∂C is a boundary of U. The relative compactness for a subset in Lp (0, 1) can be proved by a several methods, among these, Kolmogorov compactness criterion stated in the following theorem [59]. Theorem 1.6.9. (the Kolmogorov compactness criterion) Let Ω ⊆ Lp (0, 1), 1 ≤ p < ∞. If (i) Ω is bounded in Lp (0, 1), (ii) xh → x as h → 0 uniformly with respect to x ∈ Ω, then Ω is relatively compact in Lp (0, 1), where Z t+h 1 xh (t) = x(s) ds. h t Theorem 1.6.10. (the Arzela-Ascoli theorem , [80]) Let E be a compact metric space and C(E) be the Banach space of real or complex valued continuous functions normed by kf k = sup | f (t) |. t ∈ E If A = {fn } is a sequence in C(E) such that fn is uniformly bounded and equicontinuous, then Ā is compact. Theorem 1.6.11. (the Lebesgue dominated convergence theorem, [80]) Let {fn } be a sequence of functions converging to a limit f on A, and suppose that | fn (t) | ≤ φ(t), t ∈ A, n = 1, 2, . . . , where φ is integrable on A. Then f is integrable on A and Z Z lim fn (t) dµ = f (t) dµ. n → ∞ A A 22 Chapter 2 Monotonic integrable solutions for quadratic integral equations on a half line. 2.1 Motivations and historical background. This Thesis is devoted to study so-called quadratic integral equations. This is a kind of problems of the form Z β x(t) = g(t) + F (x)(t) · u(t, s, x(s)) ds, α where t ∈ I ⊂ R+ and F is an operator. Some generalizations for the presented equations are also considered. Such a kind of problems is of mathematical and practical interests and has a long history. A classical theory of Urysohn integral equations does not include the above problem. Since for equations of this type an approach via the Schauder fixed point theorem is not useful and the Banach contraction principle is too restrictive in many applications, we need to investigate such equations very carefully. The first considered equation of this type is the Chandrasekhar equation Z 1 t ϕ(s)x(s) ds. x(t) = 1 + x(t) 0 t+s It is an important example, because it show some of our motivations. This equation describe a radiative transfer through a homogeneous stellar atmosphere. It was investigated by many authors. The solutions was considered only in the space C(I) or in Banach algebras (cf. [34]). However, such a class of solutions seems to be inadequate for integral problems and leads to several restrictions on functions. In order to apply earlier results we have to impose an additional condition that the so-called 23 ”characteristic” function ψ is a polynomial (as in the book of Chandrasekhar [43, Chapter 5]) or at least continuous (cf. [40, Theorem 3.2]). This function is immediately related to the angular pattern for single scattering and then our results allow to consider some peculiar states of the atmosphere. In astrophysical applications of R1 the Chandrasekhar equation the only restriction, that 0 ψ(s) ds ≤ 1/2 is treated as necessary (cf. [40, Chapter VIII; Corollary 2 p. 187] or [69]). The continuity assumption for ψ implies the continuity of solutions for the considered equation (cf. [40]) and then seems to be too restrictive even from the theoretical point of view. About nonhomogeneous (discontinuous ”characteristic” ψ in the Chandrasekhar equation) stellar atmosphere: it is only a discretization for the equation (Hollis and Kelley 1986 [74]) - till now there is no analytical methods (unless our results). An interesting discussion about the continuity of solutions for the Chandrasekhar equation and the relation between the kernel of an integral operator can be found in [114, Proposition 4.1, Theorem 4.3] - cf. also [69]. More general problem (motivated by some practical interests in plasma physics (cf. Stuart [114]) was investigated in [86] Z 1 J 2 2 K(t, s, x(t), x(s)) ds. x (t) = t − 4π 0 Let us list some of considered previously particular cases of quadratic integral equations with their applications: a) biology: model of spread of a disease (epidemic model) (Gripenberg [72] )   Z t Z t x(t) = k P − A(t − s)x(s)ds · a(t − s)s(s)ds, −∞ −∞ b) physics: kinetic theory of gases (Hu, Khavanin and Zhuang [75])   Z ∞ Z ∞ x(t) = a(t) + f (t, x(t)) + g(t, s)x(s)ds · h(t, s)K(s, x(s))ds, 0 0 c) physics: statistical mechanics, the Percus-Yevick equation (Nussbaum [96], Wertheim [121], Pimbley [101], Ramalho [104], Rus [107]) Z 1 x(t) = 1 + λ x(s) · x(s − t)ds, t c) the Chandrasekhar equation: in astrophysics (Chandrasekhar, Fox, Argyros, Crum, Cahlon, Rus, Shrikhant, Joshi, Schillings, Leonard and Mullikin [94, 95, 88], Stuart [114] and many others): Z 1 t x(t) = 1 + x(t) ψ(s)x(s) ds. 0 t+s It is worthwhile to mention, that our equation cover as special cases among others the following ones: 24 1. f1 (t, x) = g(t), f2 (t, x) = λ the functional Urysohn integral equation ([14, 15, 24]), 2. f1 (t, x) = g(t), f2 (t, x) = x, φ2 (t) = t the functional-integral equation ([91]), 3. f2 (t, x) = 0 the abstract functional equation ([15], for instance), 4. for continuous solutions with φ1 (t) = φ2 (t) = t and u1 (t,s,x) u(t, s, x) = Γ(α)·(t−τ see [35, 61], )1−α 5. f2 (t, x) = λ the functional integral equation (for continuous solutions see [3, 21, 54]), 6. f2 (t, x) = x the quadratic (functional) Urysohn integral equation ([27, 26], for instance). Note, that the choice of spaces allow us to consider less restrictive growth conditions, which will be clarified in next chapters. 2.2 Introduction. In this chapter we study the following functional integral equation   Z β x(t) = g(t) + f t, x(t) · u(t, s, x(s)) ds . (2.1) α The particular cases of our equation, were investigated for existence for both continuous (cf. [6, 32, 35, 62] and integrable solutions ([20, 30, 31]). The existence of different subclasses of solutions were proved (nonnegative functions, monotone, having limit at infinity etc.). Let us note, that the problem is investigated for finite or infinite intervals. We extend the existing results dealing the monotonicity problem in a half-line for the most complicated problem of the Urysohn operators. For continuous solutions such a property was recently investigated in [62], for instance. By applying Darbo fixed point theorem associated with the measure of noncompactness, we obtain the sufficient conditions for the existence of monotonic solutions of equation (2.1), which are integrable. The results presented in this chapter are motivated by the recent works of Banaś and Chlebowicz [20], Banaś and Rzepka [32, 33] and extend these papers in many ways. 25 2.3 Main result Denote by L1 for L1 (R+ ) and H the operator associated with the right hand side of equation (2.1) which takes the form x = Hx, where  Z (Hx)(t) = g(t) + f t, x(t) · β  u(t, s, x(s)) ds , α t ≥ 0. The operator H will be written as the product Hx(t) = g(t) + F Kx(t) where F (x)(t) = f (t, x(t)), Kx(t) = x(t) · U(x)(t) and U(x) is the Urysohn integral operator of the form Z β (Ux)(t) = u(t, s, x(s)) ds. α Thus equation (2.1) becomes x(t) = g(t) + F Kx(t). (2.2) We shall treat the equation (2.1) under the following assumptions which are listed below. (i) g ∈ L1 (R+ ) and is a.e. nonincreasing on R+ . (ii) f : R+ × R → R satisfies Carathéodory conditions and there are a positive function a ∈ L1 and a constant b ≥ 0 such that |f (t, x)| ≤ a(t) + b |x|, for all t ∈ R+ and x ∈ R. Moreover, f (t, x) ≥ 0 for x ≥ 0 and f is assumed to be nonincreasing with respect to both variable t and x separately. (iii) u : R+ × R+ × R → R satisfies Carathéodory conditions i.e. it is measurable in (t, s) for any x ∈ R and continuous in x for almost all (t, s). The function u is nonincreasing with respect to each variable, separately. Moreover, for arbitrary fixed s ∈ R+ and x ∈ R the function t → u(t, s, x(s)) is integrable. (iv) There exists a measurable function k such that: |u(t, s, x)| ≤ k(t, s) 26 for all t, s ≥ 0 and x ∈ R. A measurable nonnegative function k : R+ → R+ is supposed to be nonincreasing with respect to each variable separately and such that the linear integral operator K0 with kernel k(t, s) maps L1 into L∞ . Moreover, for each non-negative z ∈ L1 let Z lim sup k u(t, s, x(s))dskL∞ = 0 meas(D)→0 |x|≤z D and assume that for arbitrary h > 0 (i = 1, 2) Z lim k max |u(t, s, x1 ) − u(t, s, x2 )| ds kL∞ = 0. δ→0 D |xi |≤h ,|x1 −x2 |≤δ (v) b · kK0 k∞ < 1. Then we can prove the following theorem. Theorem 2.3.1. Let the assumptions (i) - (v) be satisfied. Then the equation (2.1) has at least one solution a.e. nonincreasing on R+ which is locally integrable. Proof. First of all observe that by Assumption (ii) and Theorem 1.3.1 F is a continuous operator from L1 into itself. Moreover, by (iv) U is a continuous operator from L1 into L∞ (see Theorem 1.3.6) and then by Hölder inequality the operator K maps L1 into itself. Finally, for a given x ∈ L1 the function Hx belongs to L1 and is continuous. Using (2.2) together with assumptions (iii) and (iv), we get kHxk ≤ ≤ ≤ ≤ = kgk + kF Kx(t)k Z ∞ Z β kgk + [a(t) + b|x(t)| |u(t, s, x(s))| ds ]dt 0 α Z ∞ Z β kgk + kak + b |x(t)| [ k(t, s) ds ]dt 0 α Z ∞ kgk + kak + b [|x(t)| · kK0 (t)k∞ ]dt 0 kgk + kak + b · kK0 k∞ · kxk. From the above estimate it follows, that there is a constant r > 0 such that H maps the ball Br into itself. Indeed, by (v) we get kHxk ≤ kgk + kak + b · kK0 k∞ · kxk ≤ kgk + kak + b · kK0 k∞ · r and then we obtain that H(Br ) ⊂ Br , where r = kgk + kak . 1 − bkK0 k∞ 27 Further, let Qr stand for the subset of Br consisting of all functions which are a.e. t nonincreasing on R+ . This set is nonempty (x(t) = e r ∈ Br ∩ Qr , for instance), bounded (by r), convex (direct calculation from the definition) and closed in L1 (R+ ) similarly as claimed in [17]. To prove the last property, let (yn ) be a sequence of elements in Qr convergent in L1 to y. Then the sequence is convergent in finite measure and as a consequence of the Vitali convergence theorem and of the characterization of convergence in measure (the Riesz theorem) we obtain the existence of a subsequence (ynk ) of (yn ) which converges to y almost uniformly on R+ . Moreover, y is still nonincreasing a.e. on R+ which means that y ∈ Qr and so the set Qr is closed. Now, in view of Theorem 1.4.1 the set Qr is compact in measure. To see this it suffices to put Ωc = [0, c] \ P for any c ≥ 0, where P denotes a suitable set of with meas(P ) = 0. Now, we will show, that H preserve the monotonicity of functions. Take x ∈ Qr , then x(t) is a.e. nonincreasing on R+ and consequently Kx(t) is also of the same type in virtue of the assumption (iii) and Theorem 1.4.2. Further, F Kx(t) is a.e. nonincreasing on R+ thanks for assumption (ii). Moreover, assumption (i) permits us to deduce that Hx = g(t) + F Kx(t) is also a.e. nonincreasing on R+ . This fact, together with the assertion H : Br → Br gives that H is also a self-mapping of the set Qr . From the above considerations it follows that H maps continuously Qr into Qr . From now we will assume that X is a nonempty subset of Qr and the constant ǫ > 0 is arbitrary, but fixed. Then for an arbitrary x ∈ X and for a set D ⊂ R+ , meas(D) ≤ ǫ we obtain Z D |(Hx)(t)|dt ≤ = Z [ |g(t)| + a(t) + b · |x(t)| · Z β |u(t, s, x(s))| ds ] dt Z β kgkL1 (D) + kakL1 (D) + b · kxkL1 (D) · k k(t, s) dskL∞ D α α ≤ kgkL1 (D) + kakL1 (D) + b · kK0 k∞ · kxkL1 (D) . Hence, taking into account the obvious equality Z Z lim {sup [ |g(t)| dt + a(t) dt : D ⊂ R+ , meas(D) ≤ ǫ]} = 0 ǫ→0 D D and by the definition of c(X) (cf. Section 1.5) we get c(HX) ≤ b · kK0 k∞ · c(X). 28 (2.3) Furthermore, fixing T > Z ∞ Z ∞ |(Hx)(t)|dt ≤ T T Z ∞ ≤ ZT∞ ≤ T 0 we arrive at the following estimate Z β [ |g(t)| + a(t) + b|x(t)| |u(t, s, x(s))| ds ] dt α Z β [ |g(t)| + a(t) + b|x(t)| k(t, s) ds ] dt α Z ∞ Z ∞ |g(t)| dt + a(t) dt + bkK0 k∞ |x(t)| dt. T T As T → ∞, the above inequality yields d(HX) ≤ b · kK0 k∞ · d(X), (2.4) where d(X) has been also defined in Section 1.5. Hence, combining (2.3) and (2.4) we get γ(HX) ≤ b · kK0 k∞ · γ(X), where γ denotes our measure of noncompactness defined in Section 1.5. The inequality obtained above together with the properties of the operator H and the set Qr established before allow us to use Theorem 1.5.2 and as a consequence, apply Theorem 1.6.5. This completes the proof. Remark 2.3.1. If we assume that the functions g and t → u(t, s, x) are a.e. nondecreasing and negative then applying the same argumentation, we can show that there exists a solution of our equation being a.e. negative and nondecreasing. Moreover, let us remark, that the monotonicity conditions in the main theorem (and examples given below) seems to be restrictive, but they are necessary as claimed in ([32] Example 2). 2.4 Examples We need to show two examples of problems for which our main result is useful and allow to extend the existing theorems. Let us recall, that we are looking for monotonic solutions for the considered problems in a half-line. Let us start with a classical Chandrasekhar integral equation. R1 Example 2.4.1. In the case g(t) = 1 and f (t, x) = x(t) 0 t +t s φ(s)x(s) ds, equation (2.1) takes the form Z 1 t x(t) = 1 + x(t) φ(s)x(s) ds. (2.5) t + s 0 29 Equation (2.5) is the famous quadratic integral equation of Chandrasekhar which is considered in many papers and monographs (cf. [11, 26, 43, 75] for instance). In this case we have k(t, s) = t +t s φ(s) and as k(·, s) is increasing, we can put m(s) = φ(s) and then for some sufficiently good functions φ our result applies (φ(s) = e−s , for instance). In order to illustrate the results proved in Theorem 2.3.1, let us consider the following examples Example 2.4.2. Let us consider the following equation Z β t −t ds. x(t) = e + x(t) 2 2 t + s + (x(s))2 α (2.6) By putting g(t) = e−t , f (t, x) = x and u(t, s, x) = t2 +st2 +x2 it is easy to see, that u is nonincreasing with respect to each variable separately and the integrability condition is also satisfied (Assumptions (i),(ii) and (iii) are satisfied). 1 We have the following functions: k(t, s) = t2 +s 2 and since Z β α β α k(t, s) ds = arctan − arctan , t t ⇒| Z β α k(t, s) ds| ≤ |β − α|. Thus the expected property (Assumption (v)) for K0 holds (for sufficiently small parameter b dependent on α and β). Moreover, given arbitrary h > 0 and |x2 − x1 | ≤ δ we have |u(t, s, x1 )−u(t, s, x2)| ≤ | 2htδ t(x22 − x21 ) ≤ 2 2 2 2 2 2 2 2 (t + s + x1 )(t + s + x2 ) (t + s + x21 )(t2 + s2 + x22 ) and the Assumption (iv) is satisfied. Taking into account all the above observations and Theorem 2.3.1 we conclude that the equation (2.6) has at least one solution x = x(t) defined, integrable and a.e. nonincreasing on R+ . 30 Chapter 3 On some integrable solutions for quadratic functional integral equations 3.1 Introduction The object of this chapter is to study the solvability of a nonlinear Urysohn functional integral equation Z 1 x(t) = f1 (t, x(φ1 (t))) + f2 (t, x(t)) · u(t, s, x(φ2(t))) ds, t ∈ I. (3.1) 0 Special cases for considered equation (quadratic integral equations) were investigated in connection with some applications of such a kind of problems in the theories of radiative transfer, neutron transport and in the kinetic theory of gases (cf. [12, 26, 40, 43]). More general problem (motivated by some practical interests in plasma physics) was investigated in [86]. The existence of continuous solutions for particular cases of the considered problem was investigated since many years (see [33, 79] or a very recent paper [4]). On the other hand, different kind of integral equations (including quadratic integral equations) should be investigated in different function spaces. This was remarked, for instance, in [86, Theorem 3.14] for the case of Lp (I)-solutions, for the Hammerstein integral equation see also [79, 93] for Lp -solutions or [13, 64, 108] for integrable solutions. A very interesting survey about different classes of solutions (not only in C(I) or Lp (I), but also in Orlicz spaces Lϕ (I) or even in ideal spaces) for a class of integral equations related to our equation can be found in [8]. Next, let us recall that the equations involving the functional dependence have still growing number of applications (cf. [73]). We try to cover the results of this type. Let us mention, for example, the results from [14, 24]. 31 We are interested in monotonic solutions of the above problem. The considered problem can cover, for instance, as particular cases: 1. f1 (t, x) = g(t), f2 (t, x) = λ the functional Urysohn integral equation ([14, 15, 24]), 2. f1 (t, x) = g(t), f2 (t, x) = x, φ2 (t) = t the functional-integral equation ([91]), 3. f2 (t, x) = 0 the abstract functional equation ([15], for instance), 4. for continuous solutions with φ1 (t) = φ2 (t) = t and u1 (t,s,x) see [35, 61], u(t, s, x) = Γ(α)·(t−τ )1−α 5. f2 (t, x) = λ the functional integral equation (for continuous solutions see [3, 21, 54]), 6. f2 (t, x) = x the quadratic (functional) Urysohn integral equation ([27, 26], for instance). Our problem, as well as, the particular cases was investigated mainly in cases when the solutions are elements of the space of continuous functions. Thus the proofs are based on very special properties of this space (the compactness criterion, in particular), cf. [35, 89]. On the other hand, by the practical interest it is worthwhile to consider discontinuous solutions. Here we are looking for integrable solutions. Thus the operators F1 , F2 and U should take their values in the space L1 (I). Let us recall that we are interested in finding monotonic solutions (a.e. monotonic in the case of integrable solutions). In such a case discontinuous solutions are expected even in a simplest case i.e. when ( 0 t is rational, f1 (t, x) = h(t) = t t is irrational An interesting example of discontinuous solutions for integral equations is taken from [86, Example 3.5]: χ[1/2,1] (t) · (2t − 1) · x(t) + χ[0,1/2] (t) · (1 − 2t) · (x(t) − 1) Z 0 1 (1 − x(s)) ds = 0. In contrast to the previous chapter, we extend the earlier result by considering functional integral equation in a more general form. Moreover, we prove the existence of solutions in some subspaces of L1 (0, 1). Let us add a few comments about functional dependence, i.e. functions ψ1 and ψ2 . Our set of assumptions is based on the paper [24]. Functions of the form ψi (t) = tα (α > 0) or ψi (t) = t − τ (t) with some set of assumptions for τ are most 32 important cases covered in our chapter. Let us note that functional equations with state dependent delay are very useful in many mathematical models including the population dynamics, the position control or the cell biology. A very interesting survey about such a theory and their applications can be found in [73]. The last aspect of our results is to investigate the monotonicity property of solutions. This is important property and there are many papers devoted to its study. Let us note some recent ones [27, 28, 46, 61], for instance. The results obtained in the current chapter create some extensions for several known ones i.e. in addition to those mentioned previously also for the results from earlier papers or books ([10, 15, 38, 51, 76, 97, 99, 122], for example). 3.2 Main result Denote by H the operator associated with the right hand side of equation (3.1) which takes the form x = H(x), where H(x)(t) = f1 (t, x(φ1 (t))) + f2 (t, x(t)) · Z 1 u(t, s, x(φ2 (s)))ds. (3.2) 0 This operator will be written as H(x) = Fφ1 ,f1 (x) + A(x), A(x)(t) = Ff2 (x)(t) · U(x)(t) = Ff2 (x)(t) · Z 1 u(t, s, x(φ2(s))), 0 and the superposition operator F as in Definition 1.3.1. Thus equation (3.1) becomes x(t) = Fφ1 ,f1 (x)(t) + A(x)(t). 3.2.1 The existence of L1-solution We shall treat the equation (3.1) under the following assumptions listed below (i) fi : I × R → R satisfies Carathéodory conditions and there are a positive integrable on I functions ai and constants bi ≥ 0 such that |fi (t, x)| ≤ ai (t) + bi |x| , i = 1, 2, for all t ∈ [0, 1] and x ∈ R. Moreover, fi (t, x) ≥ 0 for x ≥ 0 and fi is assumed to be nonincreasing with respect to both variable t and x separately for i = 1, 2. 33 (ii) u : I × I × R → R satisfies Carathéodory conditions i.e. it is measurable in (t, s) for any x ∈ R and continuous in x for almost all (t, s). The function u is nonincreasing with respect to each variable, separately. Moreover, for arbitrary fixed s ∈ I and x ∈ R the function t → u(t, s, x(s)) is integrable. (iii) Assume that |u(t, s, x)| ≤ k(t, s)(a3 (s) + b3 |x|), for all t, s ≥ 0 and x ∈ R, where the function k is measurable in (t, s), a3 ∈ L1 (I) and a constant b3 > 0. Assume that the linear integral operator K0 with the kernel k(t, s) maps L1 (I) into L∞ (I). Moreover, assume that for arbitrary h > 0 (i = 1, 2) Z lim k max |u(t, s, x1 ) − u(t, s, x2 )| dskL∞ (I) = 0. δ→0 D |xi |≤h,|x1−x2 |≤δ (iv) φi : I → I are increasing, absolutely continuous functions (for i = 1, 2). Moreover, there are constants Mi > 0 such that φ′i ≥ Mi a.e on (0, 1) (for i = 1, 2). Rb Rb (v) 0 k(t1 , s) ds ≥ 0 k(t2 , s) ds for t1 , t2 ∈ I with t1 < t2 and for any b ∈ [0, 1]. q 4b2 b3 ||K0 ||L∞ (I) (ka1 k1 + kK0 kL∞ (I) ka2 k1 ka3 k1 ), where (vi) let W > M2 W =( b1 b3 + kK0 kL∞ (I) ka2 k1 + b2 kK0 kL∞ (I) ka3 k1 ) − 1 M1 M2 and let R denotes a positive solution of the quadratic equation b2 b3 ||K0||L∞ (I) 2 ·t M2 b1 b3 + kK0 kL∞ (I) ka2 k1 + b2 kK0 kL∞ (I) ka3 k1 )] · t − [1 − ( M1 M2 + (ka1 k1 + kK0 kL∞ (I) ka2 k1 ka3 k1 ) = 0. Then we can prove the following theorem. Theorem 3.2.1. Let the assumptions (i) - (vi) be satisfied. Put L=[ b1 b3 + b2 kK0 kL∞ (I) [ka3 k1 + R]]. M1 M2 If L < 1, then the equation (3.1) has at least one integrable solution a.e. nonincreasing on I. 34 Proof. First of all observe that by the assumption (i) and Theorem (1.3.1) implies that Fφ1 ,f1 and Ff2 are continuous mappings from L1 (I) into itself. By assumption (iii) and Theorem 1.3.6 we can deduce that U maps L1 (I) into L∞ (I). From the Hölder inequality the operator A maps L1 (I) into itself continuously. Finally, for a given x ∈ L1 (I) the function H(x) belongs to L1 (I) and is continuous. Thus kH(x)k1 ≤ kFφ1 ,f1 xk1 + kAxk1 Z 1 ≤ [a1 (t) + b1 |x(φ1 (t))|]dt 0 Z 1 Z 1 + [a2 (t) + b2 |x(t)|] |u(t, s, x(φ2(s))| ds dt 0 0 Z 1 ≤ [a1 (t) + b1 |x(φ1 (t))|]dt 0 Z 1 Z 1 + [a2 (t) + b2 |x(t)|] k(t, s)[a3 (s) + b3 |x(φ2 (s))|] ds dt 0 0 Z 1 b1 |x(φ1 (t))|φ′1 (t)dt ≤ ka1 k1 + M1 0 Z 1Z 1 + k(t, s)a2 (t)[a3 (s) + b3 |x(φ2 (s))|]dsdt 0 0 Z 1Z 1 + b2 k(t, s)|x(t)|[a3 (s) + b3 |x(φ2 (s))|]dsdt 0 0 Z φ1 (1) b1 ≤ ka1 k1 + |x(u)|du M1 φ1 (0) Z 1 Z 1 + [a3 (s) + b3 |x(φ2 (s))|] k(t, s)a2 (t)dtds 0 0 Z 1 Z 1 + b2 [a3 (s) + b3 |x(φ2 (s))|] k(t, s)|x(t)| dtds 0 0 Z 1 b1 |x(t)|dt ≤ ka1 k1 + M1 0 Z 1 +kK0 kL∞ (I) ka2 k1 [a3 (s) + b3 |x(φ2 (s))|]ds 0 Z 1 +b2 kK0 kL∞ (I) kxk1 [a3 (s) + b3 |x(φ2 (s))|]ds 0 b1 kxk1 ≤ ka1 k1 + M1 1 b3 |x(φ2 (s))|φ′2 (s)]ds M 2 0 Z 1 b3 +b2 kK0 kL∞ (I) kxk1 [a3 (s) + |x(φ2 (s))|φ′2 (s)]ds M 2 0 +kK0 kL∞ (I) ka2 k1 Z [a3 (s) + 35 b3 b1 kxk1 + kK0 kL∞ (I) ka2 k1 [ka3 k1 + kxk1 ] M1 M2 b3 +b2 kK0 kL∞ (I) kxk1 [ka3 k1 + kxk1 ] M2 b3 b1 + kK0 kL∞ (I) ka2 k1 = ka1 k1 + kK0 kL∞ (I) ka2 k1 ka3 k1 + [ M1 M2   b2 b3 ||K0 ||L∞ (I) +b2 kK0 kL∞ (I) ka3 k1 ] · kxk1 + · (kxk1 )2 . M2 ≤ ka1 k1 + By our assumption (vi) , it follows that there exists a positive constant R being the positive solution of the equation from the assumption (vi) and such that H maps the ball BR into itself. Further, let QR stand for the subset of BR consisting of all functions which are a.e. nonincreasing on I. Similarly as claimed in [17] we are able to show that this set is nonempty, bounded (by R), convex and closed in L1 (I). Only the last property needs some comments. Let (yn ) be a sequence of elements in QR convergent in L1 (I) to y. Then the sequence is convergent in measure and as a consequence of the Vitali convergence theorem and of the characterization of convergence in measure (the Riesz theorem) we obtain the existence of a subsequence (ynk ) of (yn ) which converges to y almost uniformly on I. Moreover, y is nonincreasing a.e. on I which means that y ∈ QR and so the set QR is closed. Now, in view of Theorem 1.4.1 the set QR is compact in measure. To see this it suffices to put Ωc = [0, c] \ P for any c ≥ 0, where P denotes a suitable set with meas(P ) = 0. Now, we will show that H preserve the monotonicity of functions. Take x ∈ QR , then x(t) and x(φi (t)) are a.e. nonincreasing on I and consequently each fi is also of the same type by virtue of the assumption (i) and Theorem 1.4.2. Further, Ux(t) is a.e. nonincreasing on I due to assumption (ii). Moreover, Fφ1 ,f1 , A(x)(t) are also of the same type. Thus we can deduce that H(x) = Fφ1 ,f1 + A(x) is also a.e. nonincreasing on I. This fact, together with the assertion H : BR → BR gives that H is also a self-mapping of the set QR . From the above considerations it follows that H maps continuously QR into QR . From now we will assume that X is a nonempty subset of QR and the constant ε > 0 is arbitrary, but fixed. Then for an arbitrary x ∈ X and for a set D ⊂ I, meas(D) ≤ ε we obtain 36 Z D |(H(x))(t)|dt ≤ Z [a1 (t) + b1 |x(φ1 (t))|]dt Z Z 1 + [a2 (t) + b2 |x(t)|] |u(t, s, x(φ2 (s))| ds dt D 0 Z b1 |x(φ1 (t))|φ′1 (t)dt ≤ ka1 χD k1 + M1 D Z Z 1 + k(t, s)a2 (t)[a3 (s) + b3 |x(φ2 (s))|]dsdt D 0 Z Z 1 + b2 k(t, s)|x(t)|[a3 (s) + b3 |x(φ2 (s))|]dsdt D D ≤ ka1 χD k1 + 0 b1 kxχD k1 M1 b3 R] M2 b3 + b2 kK0 kL∞ (I) kxχD k1 [ka3 k1 + R]. M2 + kK0 kL∞ (I) ka2 χD k1 [ka3 k1 + Hence, taking into account the equalities Z lim{sup[ ai (t) dt : D ⊂ I, meas(D) ≤ ε]} = 0, i = 1, 2, ε→0 D and by the definition of c(X) (cf. Section 1.2) we get   b3 b1 + b2 kK0 kL∞ (I) [ka3 k1 + R] · c(X). c(H(X)) ≤ M1 M2 (3.3) Recall that L = Mb11 + b2 kK0 kL∞ (I) (ka3 k1 + Mb32 R) < 1 and then the inequality obtained above together with the properties of the operator H and since the set QR is compact in measure we are able to apply Theorem 1.6.5 which completes the proof. Remark. Let us recall that in the proof we utilize the following fact: U maps L1 (I) into L∞ (I) and F2 maps L1 (I) into itself. This allows us to use the Hölder inequality. In this situation, we prove the existence of a.e. monotonic solutions which are integrable. Sometimes we need more information about the solution, namely if a solution is in some subspace of L1 (I) (the space Lp , for instance). In such a case we are able to use also the same type of inequality. Namely we need only to modify the growth conditions and consequently the spaces in which our operators act. As claimed in the introductory part of our chapter we can repeat our proof with appropriate changes of domains for considered operators: F2 maps Lp (I) into Lq (I) and U maps Lp (I) into Lr (I), where 1r + 1q = p1 . Whence we obtain an existence result for Lp -solutions. 37 3.2.2 The existence of Lp -solution p > 1 It should be noted that in some papers, their authors consider the existence of solutions in Lp spaces simultaneously for p ≥ 1. As claimed above it cannot be done for quadratic equations. Here is a version for p > 1. An interesting (and motivating) remark about the solutions in Lp spaces for integral equations (by using similar method of the proof) can be found in [64, page 93]. However, by considering the measure of noncompactness c(X) = lim supmeas(D)→0 {supx∈X kxχD kLp (I) } introduced by Appll and De Pascale [9] (cf. also Erzakova [66]) (restricted to the family of sets compact in measure) instead of usually considered ones based on Kolomogorov or Riesz criteria of compactness (cf. [23]) we are able to examine by the same manner the case of Lp (I) spaces. Assume that p > 1 and p11 + p12 = 1p . Denote by q the value min(p1 , p2 ) and by r the value max(p1 , p2 ). This implies, in particular, that q ≤ 2p. We shall treat the equation (3.1) under the following set of assumptions presented below. (i)’ Assume that functions fi : R+ × R → R satisfy Carathéodory conditions and there are positive constants bi (i = 1, 2) and positive functions a1 ∈ Lp (I), a2 ∈ Lq (I) such that |f1 (t, x)| ≤ a1 (t) + b1 |x|, p |f2 (t, x)| ≤ a2 (t) + b2 |x| q , for all t ∈ I and x ∈ R. Moreover, fi (i = 1, 2) are assumed to be nonincreasing with respect to both variable t and x separately. (ii)’ u : R+ × R+ × R → R satisfies Carathéodory conditions. The function u is nonincreasing with respect to each variable, separately. Suppose that for arbitrary non-negative z(t) ∈ Lq (I) Z lim sup k u(t, s, x(s)) dskLr (I) = 0 D→0 |x|≤z D and that p |u(t, s, x)| ≤ k(t, s)(a3 (s) + b3 |x| q ), for all t, s ≥ 0 and x ∈ R, where the function k is measurable in (t, s), a3 ∈ Lq (I) and a constant b3 > 0. Assume that the linear integral operator K0 with the kernel k(t, s) maps Lq (I) into Lr (I). (iii)’ φi : I → I are increasing, absolutely continuous functions (for i = 1, 2). Moreover, there are constants Mi > 0 such that φ′i ≥ Mi a.e on (0, 1) (for i = 1, 2). 38 (iv)’ Rb 0 k(t1 , s) ds ≥ Rb 0 k(t2 , s) ds for t1 , t2 ∈ I with t1 < t2 and for any b ∈ [0, 1]. (v)’ Assume, that the following equation b2 b3 kK0 k 1/q M2 2p + [b2 ka3 kLq )I) + ·tq b3 ka2 kLq (I) 1/q M2 p ]kK0 kt q + ( b1 1/p M1 + (ka1 kLp (I) + kK0 kka2 kLq (I) ka3 kLq (I) ) = 0 − 1) · t has a solution in (0, 1] and denote by s its positive solution. By L′ we will denote a number b1 1/p M1 + b2 s p −1 q kK0 k ka3 kLq (I) + b3 1/p M2 s p q ! . Theorem 3.2.2. Let the assumptions (i)’ - (v)’ be satisfied. If L′ < 1, then the equation (3.1) has at least one Lp (I)-solution a.e. nonincreasing on I. Proof. By assumption (i) and Theorem 1.3.1 implies that Fφ1 ,f1 maps Lp (I) into it self continuously and Ff2 maps Lp (I) into Lq (I) and is continuous. By assumption (ii) and Theorem 1.3.5 we can deduce that U is a continuous map from Lp (I) into Lr (I). From the Hölder inequality the operator A maps Lp (I) into itself continuously. Finally, for a given x ∈ Lp (I) the function H(x) belongs to Lp (I) and is continuous. Thus for 1q + q1′ = 1 kH(x)kp ≤ kFφ1 ,f1 x(t)kp + kAx(t)kp ≤ ka1 + b1 |x(φ1 )kp + kFf2 xkq kUxkr Z 1  p1 ≤ ka1 kp + b1 |x(φ1 (t))|p dt 0 Z 1 p p k(t, s)[a3 (s) + b3 |x(φ2 (s))| q ]kr +ka2 + b2 |x| q kq · k 0 b1 ≤ ka1 kp + 1 M1p Z 0  1p 1 |x(φ1 (t))|p φ′1 (t)dt p p +[ka2 kq + b2 kxkpq ] · kkk(t, ·)kq′ [ka3 kq + b3 k|x(φ2 )| q kq ]kr , where p q kx kq = Z 1  0 |x(s)| p q q ds  1q p = kxkpq and kK0 kr,q′ ≡ kt → kk(t, ·)kq′ kr . 39 kH(x)kp ≤ ka1 kp + b1 Z 1 p M1 φ1 (1) |x(v)|p dv φ1 (0) ! p1 p q + [ka2 kq + b2 kxkp ] · kK0 kr,q′ [ka3 kq + b3 ≤ ka1 kp + b1 1 p M1 Z 1 0 |x(v)|p dv  p1 + [ka2 kq + b2 kxkp ] · kK0 kr,q′ [ka3 kq + ≤ ka1 kp + b1 1 p M1 + [ka2 kq + b2 kxkp ] · kK0 kr,q′ [ka3 kq + ≤ ka1 kp + 1 p M1 1 p M1 1 0  1q   p q ′ |x(φ2 (s))| q φ2 (s)ds ] Z 1 q φ2 (1) φ2 (0) M2  |x(v)| p q q dv ! 1q ] kxkp + [ka2 kq + b2 kxkp ] · kK0 kr,q′ [ka3 kq + b1 1 M2q b3 b3 p q ≤ ka1 kp + Z  q1   p q |x(φ2 (s))| q ds ] kxkp p q b1 0 b3 p q 1 Z 1 q M2 p q Z 1  0 |x(v)| p q q kxkp + [ka2 kq + b2 kxkp ] · kK0 kr,q′ [ka3 kq + ≤ ka1 kp + kK0 kr,q′ ka2 kq ka3 kq + + kK0 kr,q′ [b2 ka3 kq + b3 ka2 kq 1 M2q b3 ka2 kq 1 q 1 M1p p b1 1 p b3 1 q M2 ] p kxkpq ] kxkp ] · kxkpq + ≤ ka1 kp + kK0 kr,q′ ka2 kq ka3 kq + + kK0 kr,q′ [b2 ka3 kq + b1 dv  q1 b2 b3 kK0 kr,q′ 1 M2q 2p kxkpq R′ M1 p ]R′ q + M2 b2 b3 kK0 kr,q′ 1 q M2 R′ 2p q ≤ R′ . From assumption (v)’ it follows there exits a positive solution R′ of the equation, which implies that H maps the ball BR′ into it self. As before in Theorem 3.2.1, we can construct a subset QR′ of BR′ is nonempty, bounded, convex and closed in Lp (I) consisting of all functions which are a.e. nonincreasing on I. Such that H maps QR′ into it self. From now we will assume that X is a nonempty subset of Q′R and the constant ε > 0 is arbitrary, but fixed. Then for an arbitrary x ∈ X and for a set D ⊂ I, 40 meas(D) ≤ ε we obtain kH(x)χD kp ≤ k[a1 + b1 |x(φ1 )]χD kp + kFf2 xχD kq kUxχD kr b1 ≤ ka1 χD kp + 1 kxχD kp M1p Z 1 p p q + [ka2 χD kq + b2 kxχD kq ] · k k(t, s)[a3 (s) + b3 |x(φ2 (s))| q ]kr 0 ≤ ka1 χD kp + b1 1 M1p kxχD kp p + ≤ + ≤ + p [ka2 χD kq + b2 kxχD kqq ]k · kk(t, ·)kq′ [ka3 kq + b3 k|x(φ2 )| q kq ]kr b1 ka1 χD kp + 1 kxχD kp M1p p p b3 [ka2 χD kq + b2 kxχD kpq ] · kK0 kr,q′ [ka3 kq + 1 kxkpq ] M2q b1 ka1 χD kp + 1 kxχD kp M1p p p b3 −1 [ka2 χD kq + b2 R′ q kxχD kp ] · kK0 kr,q′ [ka3 kq + 1 R′ q ]. M2q Since a1 ∈ Lp and a2 ∈ Lq , lim{sup[ka1 χD kp : D ⊂ I, meas(D) ≤ ε]} = 0 ε→0 and lim{sup[ka2 χD kq : D ⊂ I, meas(D) ≤ ε]} = 0. ε→0 Then by the definition of c(x) in Lp space, we have  c(H(X)) =  Recall that L′ = [ b1 1 M1p b1 1 p M1 +b2 s p + b2 s q −1 kK0 kr,q′ (ka3 kq + p −1 q kK0 kr,q′ (ka3 kq + b3 1 M2q p q b3 1 q M2 p  s q ) · c(X). s )] < 1 and then the inequality obtained above together with the properties of the operator H and since the set QR′ is compact in measure we are able to apply Theorem 1.6.5 which completes the proof. Let us note, that in the assumption (v)’ we consider the equation of the type p 2p A+Bt+Ct q +Dt q = t. The case p = q leads to the quadratic equation (considered in our first theorem). Although the case p < q seems to be more complicated, it 41 should be noted that since pq < 1 and 2p < 2 this equation has a solution in (0, 1]. In q some papers the assumption of this type is described by using auxiliary functions. In such a formulation the problem of existence of functions is unclear. Let us note, that for arbitrary pair of spaces Lp (I) and Lq (I) we are able to solve our problem. p 2p ≥ 1, then for t ∈ I we have A + Bt + Ct q + Dt q ≤ A + Bt + C + Dt Indeed, if 2p q A+C < 1. In the case 2p < 1, and our inequality has a solution in (0, 1] whenever 1−B−D q p 2p we have the following estimation: A + Bt + Ct q + Dt q ≤ A + Bt + C + D and then A+C+D < 1 form a sufficient condition for the existence of solutions of our inequality 1−B in (0, 1]. Thus the set of functions satisfying our assumptions is nonempty (cf. also some interesting Examples in [16]). Let us remind that the first case is considered in the chapter. We would like to pay attention, that the condition (ii)’ implies that the kernels k(t, s) are of Hille-Tamarkin classes i.e. kkk(t, ·)k′q kr and kkk(·, s)kq kr′ are finite being at the same time the upper bounds for kK0 k, where q ′ and r ′ are conjugated with q and r, respectively. Moreover, it is worthwhile to note that by the same manner we can extend our main result for other subspaces of L1 (I) for which we are able to check the required properties of considered operators (some Orlicz spaces, for instance) cf. [47]. Remark 3.2.1. Till now, we are interested in finding monotonic solutions of our problem. Assume that we have the decomposition of the interval I into the disjoint subsets T1 and T2 with T1 ∪ T2 = I, such that fi (·, x) are a.e. nondecreasing on T1 and a.e. nonincreasing on T2 . By an appropriate change of the monotonicity assumptions we are able to prove the existence of solutions belonging to the class of functions described above (similarly like in [15]). In such a case we need to consider the operators preserving this property, too. 3.3 Examples We need to show an example for which our main result is useful and allow to extend the existing theorems. Let us recall that we are looking for monotonic solutions for the considered problems in the interval I. But first, let us recall that the quadratic equations have numerous applications in the theories of radiative transfer, neutron transport and in the kinetic theory of gases [12, 26, 40, 43]. In order to apply earlier results of the considered type, we have to impose an additional condition that the so-called ”characteristic” function ψ is continuous (cf. [40, Theorem 3.2]) or even Hölder continuous ([12]). In the theory of radiative transfer this function is immediately related to the angular pattern 42 for single scattering and then our results allow to consider some peculiar states of the atmosphere. In astrophysical applications of the Chandrasekhar equation R1 R1 t ψ(s)x(s) ds the only restriction that 0 ψ(s) ds ≤ 1/2 is x(t) = 1 + x(t) 0 t+s treated as necessary (cf. [40, Chapter VIII; Corollary 2 p. 187]. An interesting discussion about this condition and the applicability of such equations can be found in [40]. Recall that to ensure the existence of solutions normally one assumes that ψ(t) is an even polynomial (as in the book of Chandrasekhar [43, Chapter 5]) or continuous ([40]). The using of different solution spaces in the current chapter allow us to remove this restriction and then we give a partial answer to the problem from [40]. The continuity assumption for ψ implies the continuity of solutions for the considered equation (cf. [40]) and then seems to be too restrictive even from the theoretical point of view. Let us consider now the following integral equation 2 − ln(1 + x2 ( 3t + t2 )) (3.4) x(t) = a(t) + 3+t  Z 1 1 + h(x) 1 x(s) λ + arctan √ [√ ] ds, + 2 2 t+2 s + 1 1 + x2 (s) 0 t +s where a(t) = ( 0 t is rational, , h(x) = 1 − t t is irrational ( 0 sin x 1+ex for x ≤ 0 for x > 0. It can be easily seen that equation (3.4) is a particular case of the equation (3.1), where   2 − ln(1 + x2 ( 3t + t2 )) 1 + h(x) f1 (t, x) = a(t) + , f2 (t, x) = arctan √ 3+t t+2 and u(t, s, x) = t2 x(s) 1 λ [√ ]. + 2 +s s + 1 1 + x2 (s)   √ < 1+h(x) , In view of the inequalities ln(1 + x ) ≤ x (x > 0) and arctan t+2 the functions f1 , f2 and u are nonincreasing in each variable separately. Moreover, 1 + 31 h(x) and |f1 (t, x)| ≤ a(t) + 14 |x|, |f2 (t, x)| ≤ √t+2 2 |u(t, s, x)| ≤ t2 1+h(x) √ t+2 1 λ 1 [√ + |x|], 2 +s s+1 2 1 with a1 (t) = a(t), a2 (t) = √t+2 , a3 (s) = 1 1 constants b1 = 4 , b2 = 3 and b3 = 21 . √1 s+1 43 and k(t, s) = λ . t2 +s2 Here we have the R1 λ 1 Since 0 t2 +s 2 ds = λ arctan t , |k(t, s)| ≤ λ. Thus the expected property for the operator K0 holds true. Moreover, given arbitrary h > 0 and |x2 − x1 | ≤ δ we have x1 (1 + x22 ) − x2 (1 + x21 ) 1 | | t2 + s2 (1 + x21 )(1 + x22 ) (x1 − x2 ) + x1 x2 (x2 − x1 ) 1 | | = 2 2 t +s (1 + x21 )(1 + x22 ) 1 δ(1 + h2 ) ≤ 2 . t + s2 (1 + x21 )(1 + x22 ) |u(t, s, x1) − u(t, s, x2 )| = Put φ1 (t) = t 3 + t2 2 and φ2 (t) = t, then φ′1 (t) = 1 1 + t > = M1 3 3 and φ′2 (t) = 1 > 1 = M2 2 . Thus our assumptions (i)-(iv) are satisfied. Since 34 + 13 λ π2 (1 + R) < 1 for small λ > 0, assumption (v) holds true for sufficiently small λ. Taking into account all the above observations we are able to deduce from Theorem 3.2.1 that for sufficiently small λ the equation (3.4) has at least one integrable solution x which is a.e. nonincreasing on I. 44 Chapter 4 Functional quadratic integral equations with perturbations on a half line 4.1 Introduction We study the solvability of the following functional integral equation. Let t ∈ R+   Z t x(t) = g(t, x(ϕ3 (t))) + f1 t, f2 (t, x(ϕ2 (t))) · u(t, s, x(ϕ1 (s))) ds . (4.1) 0 This equation has been studied for non quadratic integral equation in [19] with g = 0, f2 = 1 using Schauder fixed point theorem and in [115] with a perturbation term. In [24], it was checked the existence of monotonic solutions, where g(t, x(t)) = h(t), f2 = 1, improved also by Emmanuele (cf. [63]). The authors used the general Krasnoselskii fixed point theorem to obtain the existence result (cf. [57, 87, 115]). In [124] the author studied a special case of our equation in a general Banach space X by using the classical Krasnoselskii fixed point theorem. In the case of the sum of two sufficiently regular operators the contraction condition is easily verified. Nevertheless, a construction for the set M make the above theorem more restrictive. The presence of the perturbation term g(t, x(t)) in the integral equation make the Schauder fixed point theorem unavailable. Given operators A and B, it may be possible to find the bounded domains MA and MB in such a way that A : MA → MA and B : MB → MB , but it is often impossible to arrange matters so that MA = MB = M and Ax + By ∈ MA for x, y ∈ M. Actually, the Krasnoselskii fixed point theorem allow us to avoid these problems for obtaining the result of the solution. The results presented in this chapter are motivated by extending the recent results to the functional quadratic integral equation with a perturbation term by using 45 classical Krasnoselskii fixed point theorem and the measure of weak noncompactness. Let us stress, that we will dispense the monotonicity assumptions that mentioned in the previous chapters, so we need to use different method of the proof. 4.2 Main result Equation (4.1) takes the following form x = Ax + Bx, where Ax(t) = Fϕ3 ,g x(t) − g(t, 0), (Bx)(t) = =  f1 t, f2 (t, x(ϕ2 (t))) · Ff1 (Kx)(t) + g(t, 0), Z t  u(t, s, x(ϕ1 (s))) ds + g(t, 0) 0 Kx(t) = Fϕ2 ,f2 x · Ux(ϕ1 )(t) and Ux(t) = Z t u(t, s, x(s)) ds. 0 We shall treat the equation (4.1) under the following assumptions listed below, where L1 = L1 (R+ ). (i) g, fi : R+ × R → R satisfies Carathéodory conditions and there are positive functions ai ∈ L1 and constants bi ≥ 0 for i = 1, 2, 3. such that |fi (t, x)| ≤ ai (t) + bi |x|, i = 1, 2, and |g(t, 0)| ≤ a3 (t), for all t ∈ R+ and x ∈ R. Moreover, the function g is assumed to satisfy the Lipschitz condition with constant b3 for almost all t: |g(t, x) − g(t, y)| ≤ b3 |x − y|. (ii) u : R+ × R+ × R → R satisfies Carathéodory conditions i.e. it is measurable in (t, s) for any x ∈ R and continuous in x for almost all (t, s). Moreover, for arbitrary fixed s ∈ R+ and x ∈ R the function t → u(t, s, x(s)) is integrable. (iii) There exists a function k(t, s) = k : R+ ×R+ → R which satisfies Carathéodory conditions such that: |u(t, s, x)| ≤ k(t, s) 46 for all t, s ≥ 0 and x ∈ R, such that the linear integral operator K0 with kernel k(t, s) maps L1 into L∞ . Moreover, assume that for arbitrary h > 0 (i = 1, 2) Z lim k max |u(t, s, x1 ) − u(t, s, x2 )| ds kL∞ = 0. δ→0 D |xi |≤h ,|x1 −x2 |≤δ (iv) ϕi : R+ → R+ is increasing absolutely continuous function and there are positive constants Mi such that ϕ′i ≥ Mi a.e. on R+ for i = 1, 2, 3.   2 b1 kK k < 1, (v) q = Mb33 + bM 0 ∞ 2 (vi) p = b3 M3 < 1. Then we can prove the following theorem. Theorem 4.2.1. Let the assumptions (i) - (vi) are satisfied, then equation (4.1) has at least one integrable solution on R+ . Proof. The proof will be given in six steps. • Step 1. The operator A : L1 → L1 is a contraction mapping. • Step 2. We will construct the ball Br such that A(Br ) + B(Br ) ⊆ Br , where r will be determined later. • Step 3. We will proof that µ(A(Q) + B(Q)) ≤ qµ(Q) for all bounded subset Q of Br , where µ as in Definition 1.5.2. • Step 4. We will construct a nonempty closed convex weakly compact set M in on which we will apply fixed point theorem to prove the existence of solutions. • Step 5. B(M) is relatively strongly compact in L1 . • Step 6. We will check out the conditions needed in Krasonselskii’s fixed point theorem are fulfilled. Step 1. From assumption (i), we have ||g(t, x)| − |g(t, 0)|| |g(t, x)| − a3 (t) ≤ ≤ |g(t, x) − g(t, 0)| ≤ b3 |x| b3 |x| ⇒ |g(t, x)| ≤ a3 (t) + b3 |x|. The inequality obtained above with Theorem 1.3.1 permits us to deduce that the operator A maps L1 into itself. 47 Now, Z ∞ 0 |(Ax)(t) − (Ay)(t)|dt = Z ∞ 0 ≤ b3 Z 0 |g(t, x(ϕ3(t))) − g(t, y(ϕ3(t)))|dt ∞ |x(ϕ3 (t)) − y(ϕ3 (t))|dt Z ∞ b3 ≤ |x(ϕ3 (t)) − y(ϕ3 (t))|ϕ′3 (t)dt M3 0 Z ϕ3 (∞) b3 |x(v) − y(v)|dv = M3 ϕ3 (0) Z ∞ b3 ≤ |x(v) − y(v)|dv, M3 0 which implies that kAx − AykL1 ≤ b3 kx − ykL1 . M3 (4.2) Assumption (vi) permits us to deduce that the operator A is a contraction mapping. Step 2. Let x and y be arbitrary functions in Br ⊂ L1 (R+ ). In view of our assumptions we get a priori estimation Z ∞ ≤ |g(t, x(ϕ3(t)))|dt kAx + BykL1 0   Z ∞ Z t + |f1 t, f2 (t, y(ϕ2(t))) · u(t, s, y(ϕ1(s))) ds |dt 0 0 Z ∞ ≤ [a3 (t) + b3 |x(ϕ3 (t))| ]dt 0 Z ∞ Z t + [a1 (t) + b1 · |f2 (t, y(ϕ2 (t)))|. |u(t, s, y(ϕ1(s)))| dsdt 0 0 Z ∞ ≤ [a3 (t) + b3 |x(ϕ3 (t))| ]dt 0 Z ∞ Z t + [a1 (t) + b1 · (a2 (t) + b2 |y(ϕ2(t))|). k(t, s) dsdt 0 0 Z ∞ b3 |x(ϕ3 (t))|ϕ′3 (t)dt ≤ ka1 kL1 + ka3 kL1 + M3 0 Z ∞ (a2 (t) + b2 |y(ϕ2(t))|)dt + b1 .kK0 kL∞ 0 Z ϕ3 (∞) b3 ≤ ka1 kL1 + ka3 kL1 + |x(v)|dv M3 ϕ3 (0) Z ∞ b2 |y(ϕ2(t))|ϕ′2 (t)dt ] + b1 .kK0 kL∞ [ka2 kL1 + M2 0 48 Z ∞ b3 |x(t)|dt ≤ ka1 kL1 + ka3 kL1 + M3 0 Z ∞ b2 + b1 .kK0 kL∞ [ka2 kL1 + |y(ϕ2 (t))|ϕ′2 (t)dt ] M2 0 b3 ≤ ka1 kL1 + ka3 kL1 + kx(t)kL1 M3 b2 + b1 ka2 kL1 .kK0 kL∞ + kK0 kL∞ · kykL1 M2 b3 ·r ≤ ka1 kL1 + ka3 kL1 + M3 b1 b2 kK0 kL∞ · r ≤ r. + b1 ka2 kL1 .kK0 kL∞ + M2 From the above estimate, we have that A(Br ) + B(Br ) ⊆ Br provided r= ka1 kL1 + ka3 kL1 + b1 ka2 kL1 .kK0 kL∞ > 0. 1 b2 ) 1 − ( Mb33 + bM kK k 0 L ∞ 2 Step 3. Take an arbitrary number ε > 0 and a set D ⊂ R+ such that meas(D) ≤ ε. For any x, y ∈ Q, we have Z Z Z |Ax(t) + By(t)|dt ≤ |Ax(t)|dt + |By(t)|dt D D D Z Z = |Fg,ϕ3 x(t)|dt + |Ff1 Ky(t)|dt D D Z [a3 (t) + b3 |x(ϕ3 (t))|]dt ≤ D Z Z t + [a1 (t) + b1 |f2 (t, y(ϕ2(t)))| |u(t, s, y(ϕ1(s)))| ds]dt D 0 Z ≤ [a3 (t) + b3 |x(ϕ3 (t)))|]dt D Z Z t [a1 (t) + b1 · (a2 (t) + b2 |y(ϕ2(t)))|). + k(t, s) ds]dt D 0 Z Z Z b3 |x(ϕ3 (t))|ϕ′3 (t)dt ≤ a1 (t)dt + a3 (t)dt + M 3 D D Z D + b1 kK0 kL∞ [a2 (t) + b2 |y(ϕ2(t)))|]dt D Z Z Z b3 ≤ a1 (t)dt + a3 (t)dt + |x(v)|dv M3 ϕ3 (D) D D Z b2 + b1 kK0 kL∞ [a2 (t) + |y(ϕ2(t))|ϕ′2 (t)]dt M2 D 49 Z b3 |x(v)|dv ≤ a1 (t)dt + a3 (t)dt + M3 ϕ3 (D) D D Z Z b2 + b1 kK0 kL∞ [ a2 (t)dt + |y(v)|)dv] M2 ϕ2 (D) D Z Z Z b3 ≤ a1 (t)dt + a3 (t)dt + |x(v)|dv M3 ϕ3 (D) D D Z Z b2 + b1 kK0 kL∞ [ a2 (t)dt + |y(v)|)dv], M2 ϕ2 (D) D Z Z where the symbol kK0 kL∞ (D) denotes the norm of the operator K0 acting from the space L1 (D) into L∞ (D). Now, using the fact that Z lim sup[ ai (t)dt : D ⊂ R+ , m(D) ≤ ε] = 0, for i = 1, 2, 3. ε→∞ D From Definition 1.5 it follows that c(A(Q) + B(Q)) ≤ [q = ( b3 b1 b2 + kK0 kL∞ )]c(Q). M3 M2 (4.3) For T > 0 and any x, y ∈ Q,we have Z ∞ Z ∞ Z ∞ Z ∞ b3 |x(v)|dv |Ax(t) + By(t)|dt ≤ a1 (t)dt + a3 (t)dt + M3 ϕ3 (T ) T T T Z ∞ Z ∞ b2 |y(v)|dv], + b1 kK0 kL∞ [ a2 (t)dt + M2 ϕ2 (T ) T where ϕi (T ) → ∞ as T → ∞ for i = 1, 2. Then as T → ∞ and by the Definition 1.6 we get b1 b2 b3 + kK0 kL∞ )]d(Q). M3 M2 By combining equation 4.3 and 4.4 and Definition 1.7, we have d(A(Q) + B(Q)) ≤ [q = ( µ(A(Q) + B(Q)) ≤ [q = ( (4.4) b3 b1 b2 + kK0 kL∞ )]µ(Q). M3 M2 Step 4. Let Br1 = Conv(A(Br ) + B(Br )), where Br is defined in step 1, Br2 = Conv(A(Br1 ) + B(Br1 )) and so on. We then get a decreasing sequence {Brn }, that is Brn+1 ⊂ Brn for n = 1, 2, · · · Obviously all sets belonging to this sequence are closed and convex, so weakly closed. By the fact proved in step 2. That µ(A(Q) + B(Q)) ≤ qµ(Q) for all bounded subset Q of Br , we have µ(Brn ) ≤ q n µ(Br ), which yields that limn→∞ µ(Brn ) = 0. n Denote M = ∩∞ n=1 Br , and then µ(M) = 0. By the definition of the measure of weak 50 noncompactness we know that M is nonempty. From the definition of the operator A, we can deduce that B(M) ⊂ M. M is just nonempty closed convex weakly compact set which we need in the following steps. Step 5. Let {xn } ⊂ M be arbitrary sequence. Since µ(M) = 0, ∃ T, ∀n, the following inequality is satisfied: Z T ∞ ε |xn (t)|dt ≤ . 4 (4.5) Considering the function fi (t, x) on [0, T ], (i = 1, 2), u(t, s, x) on [0, T ]×[0, T ]×R, and k(t, s) on [0, T ] × [0, T ], in view of Theorem 1.3.3 we can find a closed subset Dε of the interval [0, T ], such that meas(Dεc ) ≤ ε, and such that fi |Dε ×R (i = 1, 2), u |Dε ×Dε ×R , and k |Dε ×[0,T ] are continuous. Especially k |Dε ×[0,T ] is uniformly continuous. Let us take arbitrary t1 , t2 ∈ Dε and assume t1 < t2 without loss of generality. For an arbitrary fixed n ∈ N and denoting Hn (t) = (Fϕ2 ,f2 xn ).(Uxn ))(t) we obtain: |Hn (t2 ) − Hn (t1 )| = − ≤ + − ≤ + − + − t2 Z |f2 (t2 , xn (ϕ2 (t2 ))) u(t2 , s, xn (ϕ1 (s)))ds 0 Z t1 f2 (t1 , xn (ϕ2 (t1 ))) u(t1 , s, xn (ϕ1 (s)))ds| 0 Z t2 |f2 (t2 , xn (ϕ2 (t2 ))) − f2 (t1 , xn (ϕ2 (t1 )))| |u(t2, s, xn (ϕ1 (s)))|ds 0 Z t2 |f2 (t1 , xn (ϕ2 (t1 ))) u(t2 , s, xn (ϕ1 (s)))ds 0 Z t1 u(t1 , s, xn (ϕ1 (s)))ds| f2 (t1 , xn (ϕ2 (t1 ))) · 0 Z t2 |u(t2, s, xn (ϕ1 (s)))|ds |f2 (t2 , xn (ϕ2 (t2 ))) − f2 (t1 , xn (ϕ2 (t1 )))| 0 Z t2 |f2 (t1 , xn (ϕ2 (t1 ))) u(t2 , s, xn (ϕ1 (s)))ds 0 Z t1 f2 (t1 , xn (ϕ2 (t1 ))) u(t2 , s, xn (ϕ1 (s)))ds| 0 Z t1 u(t2 , s, xn (ϕ1 (s)))ds |f2 (t1 , xn (ϕ2 (t1 ))) 0 Z t1 f2 (t1 , xn (ϕ2 (t1 ))) · u(t1 , s, xn (ϕ1 (s)))ds| 0 51 Z ≤ t2 |f2 (t2 , xn (ϕ2 (t2 ))) − f2 (t1 , xn (ϕ2 (t1 )))| k(t2 , s)ds 0 Z t2 |u(t2 , s, xn (ϕ1 (s)))|ds + |f2 (t1 , xn (ϕ2 (t1 )))| t1 Z t1 + |f2 (t1 , xn (ϕ2 (t1 )))| |u(t2 , s, xn (ϕ1 (s))) − u(t1 , s, xn (ϕ1 (s)))|ds 0 Z t2 ≤ |f2 (t2 , xn (ϕ2 (t2 ))) − f2 (t1 , xn (ϕ2 (t1 )))| k(t2 , s)ds 0 Z t2 k(t2 , s)ds + [a2 (t1 ) + b2 |xn (ϕ2 (t1 ))|] + [a2 (t1 ) + b2 |xn (ϕ2 (t1 ))|] t1 Z t1 × |u(t2 , s, xn (ϕ1 (s))) − u(t1 , s, xn (ϕ1 (s)))|ds. 0 Then we have |Hn (t2 ) − Hn (t1 )| ≤ ω T (f2 , |t2 − t1 |)T k̃ + [a2 (t1 ) + b2 |xn (ϕ2 (t1 ))|](t2 − t1 )k̃ + [a2 (t1 ) + b2 |xn (ϕ2 (t1 ))|]T ω T (u, |t2 − t1 |), (4.6) where ω T (f2 , ·) and ω T (u, ·) denotes the modulus continuity of the functions f2 and u on the sets Dε × R and Dε × Dε × R respectively and k̃ = max{|k(t, s)| : (t, s) ∈ Dε × [0, T ]}. The last inequality (4.6) is obtained since M ⊂ Br . Taking into account the fact that µ({xn }) ≤ µ(M) = 0, we infer that the number t2 − t1 is small enough, then the right hand side of (4.6) tends to zero independently of xn as t2 − t1 tends to zero. We have {Hn } is equicontinuous in the space C(Dε ). Moreover, |Hn (t)| ≤ |f2 (t, xn (ϕ2 (t))| · Z t |u(t, s, xn (ϕ1 (s)))|ds Z t ≤ [|a1 (t)| + b2 |xn (ϕ2 (t))|] · k(t, s)ds 0 0 ≤ k̃T [d1 + b2 d2 ], where |a1 (t)| ≤ d1 , |xn (ϕ2 (t))| ≤ d2 for t ∈ Dε . From the above, we have that {Hn } is equibounded in the space C(Dε ). Next, let us put Y = sup{|Hn (t)| : t ∈ Dε , n ∈ N}. Obviously Y is finite in view of the choice of Dε . Assumption (i) conclude that the function f1 |Dε ×[−Y,Y ] is uniformly continuous. So {B(xn )} = {Ff1 Hn + g(t, 0)} 52 is equibounded and equicontinuous in the space C(Dε ). Hence, by Ascoli-Arzéla theorem [80], we obtain that the sequence {B(xn )} forms a relatively compact set in the space C(Dε ). Further observe that the above reasoning does not depend on the choice of ε. Thus we can construct a sequence Dl of closed subsets of the interval [0, T ] such that meas(Dlc ) → 0 as l → ∞ and such that the sequence {B(xn )} is relatively compact in every space C(Dl ). Passing to subsequences if necessary we can assume that {B(xn )} is a Cauchy sequence in each space C(Dl ), for l = 1, 2, · · · . In what follows, utilizing the fact that the set B(M) is weakly compact, let us choose a number δ > 0 such that for each closed subset Dδ of the interval [0, T ] such that meas(Dδc ) ≤ δ, we have Z ε (4.7) |Bx(t)|dt ≤ , 4 Dδc for any x ∈ M. Keeping in mind the fact that the sequence {Bxn } is a Cauchy sequence in each space C(Dl ) we can choose a natural number l0 such that meas(Dlc0 ) ≤ δ and meas(Dl0 ) > 0, and for arbitrary natural numbers n, m ≥ l0 the following inequality holds ε |(B(xn ))(t) − (B(xm ))(t)| ≤ (4.8) 4meas(Dl0 ) for any t ∈ Dl0 . Now use the above facts together with (4.5), (4.7), (4.8) we obtain Z 0 ∞ |(Bxn )(t) − (Bxm )(t)|dt = Z T ∞ |(Bxn )(t) − (Bxm )(t)|dt + − (Bxm )(t)|dt + Z Dlc 0 Z Dl0 |(Bxn )(t) |(Bxn )(t) − (Bxm )(t)|dt ≤ ε, which means that {B(xn )} is a Cauchy sequence in the space L1 (R+ ). Hence we conclude that the set B(M) is relatively strongly compact in the space L1 (R+ ). Step 6. We now can show that: (1) From step 4, we obtain that A(M) + B(M) ⊆ M, where M is the set constructed in step 3. (2) Step 1 allow us to know that A is a contraction mapping. (3) By step 5, B(M) is relatively compact and by assumptions (i), (iii) B is continuous. 53 We can apply Theorem 1.6.7, and have that equation (4.1) has at least one integrable solution in R+ . 4.3 Examples We need to show an example for which our main result is useful and allow to extend the existing theorems. Example 4.3.1. Consider the following quadratic integral equation. Let t ∈ R+  2 Z t t cos(ts) 1 t t2 1 t2 1 −t + x( + )+arctan [e + x(t + )] · ds . (4.9) x(t) = 2 1 + t2 4 3 2 6 2 0 1 + (x(s)) It can be seen that equation (4.9) is a particular case of equation (4.1), where g(t, x) = 1 1 1 + x, f2 (t, x) = e−t + x, f1 (t, x) = arctan x2 ≤ 2x 2 1+t 4 6 and u(t, s, x) = t cos(ts) . 1 + (x(s))2 Let us note that |u(t, s, x)| ≤ t cos(ts), Rt since 0 t cos(ts) ds = sin t2 , then | 0 k(t, s) ds| ≤ 1, which implies that kK0 kL∞ ≤ 1. Rt Moreover, given arbitrary h > 0 and |x2 − x1 | ≤ δ we have x22 − x21 | (1 + x21 )(1 + x22 ) 2thδ |. ≤ | (1 + x21 )(1 + x22 ) |u(t, s, x1 ) − u(t, s, x2 )| ≤ |t cos(ts)|| In view of Theorem 4.2.1, we can deduce • g, f1 , f2 satisfy assumption (i) with a1 (t) = 0, a2 (t) = e−t , a3 (t) = with constants b1 = 2, b2 = 16 and b3 = 41 . 1 1+t2 and • Assumptions (ii), (iii) are satisfied, • ϕ1 = t, ϕ2 = (t + M1 = M2 = 1, M3 = 21 , • p = 1 2 < 1, q = 1 2 + t2 ), 2 1∗2 6 ϕ3 = ( 2t +
= 5 6 t2 ) 2 satisfied assumption (iv) with < 1. Thus all the assumptions of Theorem 4.2.1 are satisfied so the quadratic functional integral (4.9) has at least one integrable solution in R+ . 54 Chapter 5 On quadratic integral equations in Orlicz spaces 5.1 Introduction The chapter is devoted to study the following quadratic integral equation Z b x(t) = g(t) + G(x)(t) · λ K(t, s)f (s, x(s)) ds. (5.1) a We will deal with problems in which either the growth of the function f or the kernel K is not polynomial. This is motivated, for instance, by some mathematical models in physics. An interesting discussion about such a kind of problems can be found in [45] or [120]. The considered thermodynamical problem lead to the R integral equation x(t) + I k(t, s) exp x(s) ds = 0 and thus the integral equations with exponential nonlinearities turn out important from an application point of view. Let us also note, that such a kind of problems can be applied for integral equations associated (making use of the Green kernel) for the operator −∆u + exp u on a bounded regular subsets of R2 (see [37]) and that the solutions in Orlicz spaces are also sometimes studied in partial differential equations ([36], for instance). Our theorems allow to consider the cases of integral equations when the kernel function K is more singular than in previously considered cases. Moreover, we are able to consider strongly nonlinear functions f . Both extensions seems to be important from the applications point of view (cf. [36, 37, 45, 120], for instance). Let us note, that our results are motivated by the paper of Cheng and Kozak [45]. We generalize some of their assumptions and we consider more complicated quadratic integral equations. An operator G is supposed to be continuous on a required space of solutions. The problem is modeled on some quadratic integral equations (G is usually identity operator or the Nemytskii superposition operator), but our approach allows to include 55 also standard integral equations. For standard integral equations different classes of solutions are considered and this is mainly dependent on growth restrictions for f and K. Similar investigations for quadratic integral equations relate mainly to continuous solutions. The key point is to ensure, that an operator of pointwise multiplication is well-defined and has some compactness properties. We propose an approach to this problem allowing us to consider a wide class of integral equations with solutions in some spaces of discontinuous functions. We are interested in a different class of solutions then the earlier papers. Such a kind of integral equations was investigated in spaces of continuous or integrable functions. Similar problems are also important when Lp -solutions are checked. The currently considered case is less restrictive and include large class of real problems. Whenever one has to deal with some problems involving strong nonlinearities (of exponential growth, for instance), it is a useful device to look for solutions not in Lebesgue spaces, but in Orlicz spaces. In the literature, mostly solutions of integral equations are sought in C[0, 1] and Lp [0, 1] with p > 1. The results obtained for Lp [0, 1] invariably assume a polynomial growth (in x) on the nonlinearity f (t, x). On the other hand, seeking solutions in other Orlicz spaces will lead to restrictions that are not of polynomial type, and hence will allow us to consider new classes of equations. All very basic types of integral or differential equations were satisfactory examined (cf. [83, 98, 100, 103, 112]). Some additional properties of solutions (in a simplest case of the ∆2 -condition) are also investigated (constant-sign solutions, for instance [1, 2]). An interesting discussion about advantages of integral equations in Orlicz spaces can be found in [84, Section 40] (see also [7]). Nevertheless, for the quadratic integral equations the operators generated by the right-hand side of the equation are more complicated and was not investigated in this case. Let us note, that in this case the methods based on properties of some Banach algebras are usually applied (cf. [12, 34, 89, 90]). This approach seems to be strictly related with continuous or Hölder continuous solutions (the product is an inner operation in the Banach algebra of continuous functions and at the same time is an operator used in the integral equation). Let us note, that this method is dependent on some properties of C(0, 1) and cannot be easily applied to different classes of spaces. This suggest an operator oriented approach. In a class of Orlicz spaces we consider spaces associated with growth conditions for G and f . For a moment denote by X an Orlicz space of solutions for our problem and by F the Nemytskii superposition operator generated by f . Thus we have G : X → W1 , F : X → U and finally the linear integral operator H with a kernel K is acting from U into 56 W2 . The space U is depending on some growth assumptions of f - not necessarily of polynomial type. In a typical case of quadratic problems the spaces W1 and W2 are supposed both to be the space of continuous functions and then some properties of this Banach algebra allow to solve the problem. Unfortunately, this is really restrictive assumption. We started to replace this assumption by considering X = L1 (I) and W2 = L∞ (I) ([46]). Here we present a complete theory for such problems. In general, allowing U be an Orlicz space depending on f we consider the triples of Orlicz spaces (not: Banach algebras) for which the pointwise multiplication take a pair of functions from W1 and W2 into X. For a case of Lp -solutions we propose to use a factorization and we will assume the Hammerstein integral operator which is multiplied by the function G(x) has values in the space conjugated to the space of solutions. Such an idea was used by Brézis and Browder for Hammerstein integral equations ([39]) by considering conjugated Lebesgue Lp spaces. We extend this procedure for Orlicz (or: ideal) spaces and for a triple of spaces (two for Hammerstein operator and one more for a multiplication operator). This allows us to prove the existence theorems under much more general conditions than previously considered ones. Our method leads to extensions for both types of results (quadratic and non-quadratic ones). We concentrate on the property of monotonicity of solutions for the equation (5.9). This notion is broaden to some function spaces and the basic properties of families of such functions are investigated. Especially, the quadratic integral equation of Chandrasekhar type Z 1 t ϕ(s)x(s) ds x(t) = 1 + x(t) 0 t+s can be very often encountered in many applications (cf. [11, 12, 31, 26, 40, 43]). The results of this chapter are divided into a few parts. This is because the proofs are depending on the choice of considered spaces. We stress on the ”size” of solution spaces and we try to relate the growth assumptions of functions and the expected space of solutions. Since for equations of this type an approach via the Schauder fixed point theorem is not useful and the Banach contraction principle is too restrictive in many applications, we prefer to investigate the properties of operators with respect to the topology of convergence in measure. We check this topology on considered Orlicz spaces and then we use the Darbo fixed point theorem for proving main results. To show a detailed theory we need to consider a few different cases (for different classes of Orlicz spaces). The theorems proved by us extend, in particular, that presented in [6, 11, 27, 33] considered in the space C(I) or in Banach algebras (cf. [34]). In the 57 context of non-quadratic integral equations in Orlicz spaces, which are also cover by our theorems, let us mention the papers [1, 2, 98, 100, 103, 112]. 5.2 The case of operators with values in L∞ (I). This part will be devoted to present some new results, which are related to our theorems from previous chapters. Nevertheless, we are looking for solutions in some Orlicz spaces instead of Lebesgue ones. Functional dependence is not considered here, but our current approach is presented in such a way to cover non-quadratic integral equations as particular cases. Earlier results for quadratic integral equations cannot be compared with standard ones. Denote B the operator associated with the right-hand side of the equation (5.2) Z b x(t) = g(t) + λf1 (t, x(t)) K(t, s)f2 (s, x(s)) ds. (5.2) a i.e. B(x) = g + U(x), where U(x)(t) = F1 (x)(t) · A(x)(t) such that Z F2 (x)(t) = f2 (t, x(t)), A = H ◦ F2 , H(x)(t) = λ K(t, s)x(s) ds, F1 = f1 (s, x(s)). I By considering different spaces of solutions (i.e. different growth conditions) we need to investigate pair of spaces constituting the domain and the range for the operator U. To facilitate the reading of (technical) assumptions we will consider different pairs of spaces, including most typical ones. We stress on applicability of our results. The general case will presented in an abstract form in the next Section of this chapter. Let F1 : V1 → W1 and A : V1 → W2 , where V1 , W1 and W2 are some function spaces. Since U stands for the pointwise multiplication operator and the range of U should be in a space of solutions W , we need to investigate pair of spaces (W1 , W2 ) such that x(t) · y(t) ∈ W for all t ∈ [a, b] and x ∈ W1 , y ∈ W2 . We need to consider the case when either W1 or W2 is a space of all essentially bounded functions L∞ [a, b]. 5.2.1 The case of W1 = L∞ (I). The case of W1 = L∞ (I) is trivial. The unique situation in which the Nemytskii operator takes values in this space is when f1 is autonomous. Namely, we have: Lemma 5.2.1. [10, Theorem 3.17] Let p ≥ 1. The superposition operator F1 generated by f1 maps Lp (I) into L∞ (I) if and only if |f (s, u)| ≤ a(s) for u ∈ R 58 for some a ∈ L∞ (I), in this case, F1 is always bounded; F1 is continuous if and only if F1 is constant (i.e. f1 does not depend on u). Let us briefly discuss the case of Orlicz spaces. We have the following lemma Lemma 5.2.2. Let LM (I) be an Orlicz space. Then F1 generated by a Carathéodory function f1 is continuous from LM (I) into L∞ (I) if and only if F1 is constant (i.e. f1 does not depend on u). Proof. Since we consider a finite measure, an arbitrary function x from LM can be obtained as a pointwise limit of a sequence of simple functions, a fortiori this sequence is convergent in measure. Simple functions belong to EM and then LM is a quasi-regular ideal space (cf. [10, Theorem 2.8, Theorem 3.17]). On the other hand the space L∞ (I) is completely irregular i.e. the regular part of the space consists only {θ}. Since both spaces are ideal, by [10, Theorem 2.8] F1 is continuous if and only if F1 is constant. Note that a domain for F1 cannot be arbitrarily small. Some assumptions on the domain of superposition operators are always expected (the identity on L∞ (I) is continuous, for instance). Thus, this is the standard non-quadratic case and can be easily reduced to known theorems (cf. [100, 103, 112] for most interesting results). 5.2.2 The case of W2 = L∞ (I). We will consider now the new case when W2 = L∞ (I). It is more interesting and we stress on growth conditions for both f1 and f2 allowing to have nonlinear growth. This is closely related to the choice of an intermediate space for A i.e. the domain of H. It can be made because of the assumptions on f2 or the kernel K depending on the practical problem described by our equation. Thus the choice of this space allows to consider more general growth conditions for f2 or K. Theorem given below is formulated for arbitrary Lebesgue space Lp (I) (p ≥ 1) taken as an intermediate space. Theorem 5.2.1. Let 1 p + 1 q = 1 and assume, that ϕ is an N-function and that: (C1) g ∈ Eϕ (I) is nondecreasing a.e. on I, (C2) fi : I × R → R satisfies Carathéodory conditions and fi (t, x) is assumed to be nondecreasing with respect to both variable t and x separately, for i=1, 2. 59 (C3) kF1 (x)kϕ ≤ G1 (kxkϕ ), and kF2 (x)k|p ≤ G2 (kxkϕ ), for all x ∈ Eϕ (I), where Gi are positive, continuous and nondecreasing for i = 1, 2. Moreover there exist γ > 0 such that G1 (u) ≤ γ|u| for |u| ≤ 1. Assume, that the superposition operator F1 generated by f1 is acting from Eϕ (I) into itself. (C4) Assume that the function K is measurable in (t, s) and assume that the linear integral operator H with kernel K(t, s) maps Lp (I) into L∞ (I), s 7→ |K(·, s)| ∈ Lq (I), k(t) = |K(t, ·)| ∈ L∞ (I) and H is continuous with a norm kK0 kL∞ := esssupt∈I (K1) R K(t1 , s) ds ≥ I R I Z I |K(t, s)|q ds  1q . K(t2 , s) ds for t1 , t2 ∈ I with t1 < t2 . Assume, that there exists a positive number r ≤ 1 such that kgkϕ + λkK0 kL∞ G1 (r) · G2 (r) ≤ r. (5.3) Then there exists a number ρ > 0 such that for all λ ∈ R with |λ| < ρ and for every g ∈ Eϕ , there exists a solution x ∈ Eϕ (I) of (5.2) which is a.e. nondecreasing on I. Proof. We need to divide the proof into a few steps. I. The operator B is well-defined from Lϕ (I) into itself. In particular, for operator H we need to check the properties of this operator. II. We will construct an invariant ball Br for B in Lϕ (I). III. We construct a subset Qr of this ball which contains a.e. nondecreasing functions and we investigate the properties Qr . IV. We check the continuity and monotonicity properties of B in Qr , so B : Qr → Qr . V. We prove that B is a contraction with respect to a measure of noncompactness. VI. We use the Darbo fixed point theorem to find a solution in Qr . I. First of all observe that by the assumptions (C2) and (C3) and Lemma 1.3.5 implies that F2 is continuous mappings from Lϕ (I) into Lp . Note, that Lp can be p treated as an Orlicz space LMp for Mp (x) = |x|p . It is clear, that this space satisfies the ∆2 -condition and therefore it is a regular space. Recall, that by Lemma 1.3.2 F2 60 is sequentially continuous with respect to convergence in measure and continuous by Lemma 1.3.5. For the operator F1 the situation is described in Remark 1.3.2. By assumption (C4) the operator A maps Eϕ (I) into L∞ (I) and is continuous. F1 maps Eϕ (I) into itself continuously (thanks to Assumption (C3) and Lemma 1.3.5). Thus U is a continuous mapping from Eϕ (I) into itself. Finally, Assumption (C1) permits us to deduce that the operator B maps Eϕ (I) into itself and is continuous. II. If ϕ satisfies the ∆2 -condition then F1 is bounded in Lϕ (I). In a general case we need restrict this operator to the (regular) space Eϕ (I). We will prove the boundedness of the operator B on this space, namely we will construct the invariant set for this operator. Let Γ be a set of all positive numbers λ such that kgkϕ + λkK0 kL∞ G1 (r) · G2 (r) ≤ r.   Put ρ = min sup Γ, γkK0 kL1 G2 (1) and fix λ with |λ| < ρ. ∞ Let r be a positive number r ≤ 1 such that kgkϕ + |λ|kK0kL∞ · G1 (r) · G2 (r) ≤ r. (5.4) Since we provided a continuity of operators F1 and B only in Eϕ (I), as a domain for the operator B we will consider the ball Br (Eϕ (I)) in this space. Recall, that Lϕ (I) is an ideal space, F1 (x) ∈ Lϕ (I) and A(x) ∈ L∞ (I), so U(x) ∈ Lϕ (I) and kU(x)kϕ ≤ kF1 (x)kϕ kA(x)k∞ . We have kB(x)kϕ ≤ kgkϕ + kU(x)kϕ = kgkϕ + kF1 (x) · A(x)kϕ ≤ kgkϕ + |λ| · kF1 (x)kϕ · k Z I K(t, s)|f2 (s, x(s))|ds k∞ . But by (C4) H is continuous and then the norm of H(x) is estimated by (cf. [78, Theorem XI.1.6]) kH(x)k∞ ≤ |λ|kK0kL∞ kxkp . Thus for x ∈ Eϕ (I) with kxkϕ ≤ r kB(x)kϕ ≤ kgkϕ + kF1 (x)kϕ · |λ|kK0kL∞ · kF2 (x)kp ≤ kgkϕ + |λ|kK0 kL∞ G1 (kxkϕ ) · G2 (kxkϕ ) ≤ kgkϕ + |λ|kK0 kL∞ G1 (r) · G2 (r) ≤ r. Then we have B : Br (Eϕ (I)) → Br (Eϕ (I)). Moreover, B is continuous on Br (Eϕ (I)) (see the part I of the proof). 61 III. Let Qr stand for the subset of Br (Eϕ (I)) consisting of all functions which are a.e. nondecreasing on I. Similarly as claimed in [17] this set is nonempty, bounded (by r), convex (direct calculation from the definition) and closed in Lϕ (I). To prove the last property, let (yn ) be a sequence of elements in Qr convergent in Lϕ (I) to y. Then the sequence is convergent in measure and as a consequence of the Vitali convergence theorem for Orlicz spaces and of the characterization of convergence in measure (the Riesz theorem) we obtain the existence of a subsequence (ynk ) of (yn ) which converges to y almost uniformly on I (cf. [100]). Moreover, y is still nondecreasing a.e. on I which means that y ∈ Qr and so the set Qr is closed. Now, in view of Lemma 1.4.1 the set Qr is compact in measure. IV. Now, we will show, that B preserve the monotonicity of functions. Take x ∈ Qr , then x is a.e. nondecreasing on I and consequently F1 and F2 are also of the same type in virtue of the assumption (C2) and Theorem 1.4.2. Further, A(x) = H ◦ F2 (x) is a.e. nondecreasing on I (thanks for the assumption (K1)). Since the pointwise product of a.e. monotone functions is still of the same type, the operator U is a.e. nondecreasing on I. Moreover, the assumption (C1) permits us to deduce that B(x)(t) = g(t) + U(x)(t) is also a.e. nondecreasing on I. This fact, together with the assertion that B : Br (Eϕ (I)) → Br (Eϕ (I)) gives us that B is also a self-mapping of the set Qr . V. We will prove that B is a contraction with respect to a measure of strong noncompactness. Since Z Z |A(x)(t)| = |λ K(t, s)f2 (s, x(s))ds| ≤ |λ| |K(t, s)||f2(s, x(s))|ds IZ I ≤ |λ| · |K(t, s)||F2(x)(s)|ds I ≤ |λ| · kK(t, ·)kq · |F2 (x)|p ≤ |λ| · kK(t, ·)kq · G2 (kxkϕ ) for a.e. t ∈ I, whence kA(x)χD kL∞ ≤ |λ|kK0 kL∞ · G2 (kxkϕ ). Assume that X is a nonempty subset of Qr and let the fixed constant ε > 0 be arbitrary. Then for an arbitrary x ∈ X and for a set D ⊂ I, meas(D) ≤ ε and t ∈ D we have F1 (x)(t) = F1 (xχD )(t) and for t 6∈ D F1 (x)(t) · χD (t) = 0 and F1 (xχD )(t) = f1 (t, 0). Whence |F1 (x)(t)χD (t)| = |f1 (t, x(t))| = |f1 (t, x(t)) − f1 (t, 0) + f1 (t, 0)| ≤ |f1 (t, x(t)) − f1 (t, 0)| + |f1 (t, 0)| ≤ |F1 (xχD )(t) − F1 (0)(t)χD (t)| + |F1 (0)(t)χD (t)|. 62 Thus for all t ∈ I we have |F1 (x)(t)χD (t)| ≤ |F1 (xχD )(t) − F1 (0)(t)χD (t)| + |F1 (0)(t)χD (t)|. Finally kF1 (x) · χD kϕ ≤ kF1 (xχD ) − F1 (0)kϕ + kF1 (0) · χD kϕ ≤ kF1 (xχD )kϕ + 2kF1 (0) · χD kϕ . For the operator B we get the following estimation kB(x) · χD kϕ ≤ kg · χD kϕ + kU(x) · χD kϕ = kg · χD kϕ + k F1 (x) · A(x) · χD kϕ = kg · χD kϕ + k (F1 (x) · χD ) · A(x)kϕ ≤ kg · χD kϕ + k F1 (x) · χD kϕ · kA(x)kL∞ ≤ kg · χD kϕ + (kF1 (xχD )kϕ + 2kF1 (0) · χD kϕ )|λ|kK0 kL∞ G2 (kxkϕ ) ≤ kg · χD kϕ + (G1 (kxχD kϕ ) + 2kF1 (0) · χD kϕ ) |λ|kK0kL∞ G2 (kxkϕ ) ≤ kg · χD kϕ + (γ · kxχD kϕ + 2kF1 (0) · χD kϕ ) |λ|kK0 kL∞ G2 (kxkϕ ). Since g ∈ Eϕ and F1 (0) ∈ Eϕ , lim sup ε→0 meas(D)≤ε lim sup ε→0 meas(D)≤ε kg · χD kϕ = 0, kF1 (0) · χD kϕ = 0. Thus by definition of c(x) and by taking the supremum over all x ∈ X and all measurable subsets D with meas(D) ≤ ε we get c(B(X)) ≤ γ|λ|kK0kL∞ · G2 (r) · c(X). Since X ⊂ Qr is a nonempty, bounded and compact in measure subset of an ideal regular space Eϕ (I), we can use Proposition 5.5.1 and get βH (B(X)) ≤ γ · |λ| · kK0 kL∞ · G2 (1) · βH (X). The constant in the above inequality is smaller that 1, so the properties of the operator B and assumption (C4) allow us to apply the Darbo Fixed Point Theorem 1.6.5, which completes the proof. Remark 5.2.1. In this chapter we consider continuous linear operators of the form H acting on Lp (I) with values in the space L∞ (I). It is known, that the continuity 63 property is depending on the kernel K. In a particular case of Riemann-Liouville 1 (t − s)α−1 χ[a,t] (s) fractional integral operators i.e. for K(t, s) = Γ(α) 1 Jα x(s) = Γ(α) Z a s (s − t)α−1 x(t)dt s ∈ [a, b] , is not continuous from Lp (I) into L∞ (I) when p = continuous for p < α1 (cf. also [7]). 1 α ([70, Remark 4.1.2]), but Remark 5.2.2. Let us add some comments about assumption (C3). Our acting condition from Lemma 1.3.3 has the form |f2 (t, x)|p ≤ a2 (t)+b2 ϕ(x). This pointwise estimation implies our assumption. Indeed, this implies that for x ∈ Lϕ (I) with kxkϕ ≤ 1 Z Z p |f2 (t, x(t))| dt ≤ ka2 k1 + b2 ϕ(x(t))dt I I kF2 (x)kpp ≤ ka2 k1 + b2 kxkϕ 1 and then kF2 (x)kp ≤ (ka2 k1 + b2 kxkϕ ) p , so our assumption holds true for a special 1 case of G2 (t) = (ka2 k1 + b2 · t) p which is used in [1], for instance. The growth restrictions for G1 result from necessary and sufficient conditions for continuity of F1 (see [10]) and since L∞ (I) is completely irregular i.e. θ is the only singleton with absolutely continuous norm. The boundedness of the Nemytskii operators on ”small” balls was firstly proved by Shragin in [110] and then used to investigate the Hammerstein integral equations in Orlicz spaces by Vainberg and Shragin. 5.3 The existence of Lp-solution. Let us present a special case of Lp -solutions. This will be still more general result than the earlier ones. Theorem 5.3.1. Assume, that p ≥ 1 and 1 p + 1 q = 1. (C1’) g ∈ Lp (I) is nondecreasing a.e. on I, (C2’) fi : I × R → R satisfies Carathéodory conditions and fi (t, x) is assumed to be nondecreasing with respect to both variable t and x separately, for i=1, 2. p (C3’) |f1 (t, x)| ≤ a1 (t) + b1 |x|, and |f2 (t, x)| ≤ a2 (t) + b2 |x| q , for all t ∈ I and x ∈ R, where a1 ∈ Lp (I), a2 ∈ Lq (I) and some constants bi ≥ 0 for i = 1, 2. 64 (C4’) Assume that the function K is measurable in (t, s) and that the linear integral operator K0 with kernel K(·, ·) maps Lq (I) into L∞ (I), essupt∈[a,b] b Z q a |K(t, s)| ds  q1 <∞ and is continuous. Rb Rb (K1) a K(t1 , s) ds ≥ a K(t2 , s) ds for t1 , t2 ∈ [a, b] with t1 < t2 . Assume, that there exists a positive number r < 1 such that kgkp + |λ|kK0 kL∞ · [ ka1 kL1 + b1 · r ][ ka2 kL1 + b2 · r p/q ] ≤ r and choose λ in such a way to get kK0 kL∞ < 1 p |λ|·b1 b2 ·r q (5.5) . Then there exists a number ρ > 0 such that for all λ ∈ R with |λ| < ρ and for every g ∈ Lp (I), there exists a solution x ∈ Lp (I) of (5.2) which is a.e. nondecreasing on I. Proof. I. First of all observe that by assumptions (C2’), (C3’) and Theorem 1.3.1 we have that F1 maps Lp (I) into itself and F2 maps Lp (I) into Lq (I) continuously. By assumption (C4’) we can deduce that A maps Lp (I) into L∞ (I) and is continuous. From the Hölder inequality the operator U maps Lp (I) into itself continuously. Finally, assumption (C1’) permits us to deduce that B maps Lp (I) into itself and is continuous. To prove step II. Put ρ= 1 b1 b2 kK0′ kL∞ ka1 kLp · ka2 kLq . Fix λ with |λ| < ρ. Choose a positive number R in such a way that p kgkLp + |λ| · [ k a1 kLp + b1 · R ] · kK0′ kL∞ [ ka2 kLq + b2 · R q ] ≤ R. As a domain for the operator B we will consider the ball BR (Lp (I)). kB(x)kLp ≤ kgkLp + kUxkLp = kgkLp + kF1 (x) · A(x)kLp ≤ kgkLp + kF1 (x)kLp · kA(x)kL∞ Z ≤ kgkLp + |λ|kf1(t, x)kLp k K(t, s)|f2 (s, x(s))|dskL∞ I 65 (5.6) ≤ kgkLp + |λ| · k a1 + b1 |x(s)|kLp Z p × k K(t, s) [ a2 (s) + b2 |x(s)| q ]ds kL∞ I p ≤ kgkLp + |λ|[ka1 kLp + b1 kxkLp ]kK0′ kL∞ [ka2 kLq + b2 kxkLq p ], where p p kx q kLq = kxkLq p . p kB(x)kLp ≤ kgkLp + |λ| · [ka1 kLp + b1 · R] · kK0′ kL∞ [ka2 kLq + b2 · R q ] ≤ R. Then we have B : BR (Lp (I)) → BR (Lp (I)). Moreover, B is continuous on BR (Lp (I)) (see the part I of the proof). Step III and Step IV are similar as in Theorem 5.2.1. V. We will prove that B is a contraction with respect to a measure of strong noncompactness. Assume that X is a nonempty subset of QR and let the fixed constant ǫ > 0 be arbitrary . Then for an arbitrary x ∈ X and for a set D ⊂ I, meas(D) ≤ ǫ we obtain kB(x) · χD kLp ≤ kg · χD kLp + kUx · χD kLp ≤ kg · χD kLp + k F1 (x) · χD kLp · kA(x) · χD kL∞ Z K(t, s)f2 (s, x(s))dskL∞ = kg · χD kLp + |λ| · kf1 (t, x) · χD kLp k D ≤ kg · χD kLp + |λ| · k [ a1 (s) + b1 |x(s)| ] · χD kLp Z p × k K(t, s)[ a2 (s) + b2 |x(s)| q ]ds kL∞ D ≤ kg · χD kLp + |λ| · k [ ka1 · χD kLp + b1 kx · χD kLp ] Z Z p × k K(t, s) a2 (s) ds + b2 K(t, s)|x(s)| q ds kL∞ D I ≤ kg · χD kLp + |λ| · k [ ka1 · χD kLp + b1 kx · χD kLp ] p × [ kK0′ kL∞ ka2 · χD kLp + b2 kK0′ kL∞ kx(s)kLq p ] × kK0′ kL∞ [ ka2 · χD kLp + b2 R q ]. ≤ kg · χD kLp + |λ| · k [ ka1 · χD kLp + b1 kx · χD kLp ] p Since g ∈ Lp , a1 ∈ Lp and a2 ∈ Lq , then we have lim { sup [sup{kg · χD kLp }]} = 0, lim { sup [sup{ka1 · χD kLp }]} = 0 ε→0 mes D≤ε x∈X ε→0 66 mes D≤ε x∈X and lim { sup [sup{ka2 · χD kLq }]} = 0. ε→0 mes D≤ε x∈X Thus by definition of c(x), p c(B(X)) ≤ b1 b2 |λ| · kK0′ kL∞ R q · c(X). Since X ⊂ Qr is a nonempty, bounded and compact in measure subset of an ideal regular space Lp , we can use Proposition 5.5.1 and get p βH (B(X)) ≤ b1 b2 |λ| · kK0′ kL∞ R q · βH (X). The inequality obtained above together with the properties of the operator B and the set QR established before and assumption (C’4) allow us to apply the Darbo fixed point theorem (see [23]), which completes the proof. 5.3.1 Remarks and examples. We need to stress on some aspects of our results. First of all we can observe, that our solutions are not necessarily continuous as in almost all previously investigated cases. In particular, we need not to assume, that the Hammerstein operator transforms the space C(I) into itself. Our treatment allows to solve problems when function f1 doesn’t have sublinear growth. In this case it is sufficient to take a little bit ”nicer” kernel K, but this is still weaker assumption than in the case of continuous solutions. We need to stress, that quadratic equations are also strictly related to problems of the type ′  x(t) − g(t) = f2 (t, x(t)) x(0) = 0, f1 (t, x) where f1 : I × R → R \ {0}. It is well-known that under typical assumptions this problem is equivalent to the integral equation  Z b  g(0) x(t) = g(t) + f1 (t, x(t)) · χ[0,t] (s)f2 (s, x(s)) ds − . f1 (0, 0) a Nevertheless, when we are looking for continuous solutions for integral equation, we obtain classical solutions for differential one i.e. x is continuously differentiable. This seems to be too restrictive in many applications. In our case we investigate Caratéodory solutions for the Cauchy problem. Another typical example is a boundary value problem (f1 : I × R → R \ {0}, f2 and g satisfy some regularity conditions)  ′′ x(t) − g(t) = f2 (t, x(t)) x(0) = 0 , x′ (0) = 0. (5.7) f1 (t, x) 67 In this case we can consider an equivalent integral problem with a kernel G (an appropriate Green function)  Z b g(0) (5.8) x(t) = g(t) + f1 (t, x(t)) · G(t, s)f2 (s, x(s)) ds − f1 (0, 0) a !! 1 (0, 0) g(0) · ∂f g ′(0) ∂t + t· −t· , f1 (0, 0) (f1 (0, 0))2 where f1 : I × R → R \ {0}. In this case when we are looking for continuous solutions for the quadratic integral equations, the solutions for the above differential problems are classical. Our approach allows to investigate weaker types of solutions (in Orlicz-Sobolev spaces). 5.4 A general case of Orlicz spaces. Here we will present a detailed theory of quadratic integral equations of the form: Z b x(t) = g(t) + G(x)(t) · λ K(t, s)f (s, x(s)) ds (5.9) a in Orlicz spaces. Denote by B the operator associated with the right-hand side of the equation (5.9) i.e. B(x) = g + U(x), where U(x)(t) = G(x)(t) · λ Z b K(t, s)f (s, x(s)) ds. a Rb Thus B = g + G · A = g + G · H ◦ F , where H(x)(t) = λ a K(t, s)x(s) ds and F = f (t, x(t)). We will try to choose the domains of operators defined above in such a way to obtain the existence of solutions in a desired Orlicz space Lϕ (I). We stress on conditions allowing us to consider strongly nonlinear operators. Let us note, that our assumptions on G are referred to the case of quadratic integral equations (i.e. G(x)(t) = q(t) · x(t)). We need to distinguish two different cases. This allow us to obtain more general growth conditions on f (cf. [100, 103, 112, 98] for non-quadratic equations). In every case we need to describe some assumptions on ”intermediate” spaces being the images of Lϕ (I) for G and F (Lϕ1 (I) and LM (I), respectively) and the range for H (i.e. Lϕ2 (I)). This approach is based on a classical (non-quadratic) case as in [100, 103, 112]) and seems to be important in view of optimality of assumptions for every considered case. 68 The case of N satisfying the ∆′-condition. 5.4.1 Theorem 5.4.1. Assume, that ϕ, ϕ1 , ϕ2 are N-functions and that M and N are complementary N-functions. Moreover, put the following set of assumptions: (N1) there exists a constant k1 > 0 such that for every u ∈ Lϕ1 (I) and w ∈ Lϕ2 (I) we have kuwkϕ ≤ k1 kukϕ1 kwkϕ2 , (C1) g ∈ Eϕ (I) is nondecreasing a.e. on I, (C2) f : I × R → R satisfies Carathéodory conditions and f (t, x) is assumed to be nondecreasing with respect to both variable t and x separately, (C3) |f (t, x)| ≤ b(t) + R(|x|) for t ∈ I and x ∈ R, where b ∈ EN (I) and R is nonnegative, nondecreasing, continuous function defined on R+ , (C4) Let N satisfies the ∆′ -condition and suppose that there exist ω, γ, u0 ≥ 0 for which N(ω(R(u))) ≤ γϕ2 (u) ≤ γM(u) for u ≥ u0 , (G1) G : Lϕ (I) → Lϕ1 (I), takes continuously Eϕ (I) into Eϕ1 (I) and there exists a constant G0 > 0 such that kG(x)kϕ1 ≤ G0 kxkϕ and that G takes the set of all a.e. nondecreasing functions into itself, (K1) s → K(t, s) ∈ LM (I) for a.e. t ∈ I, (K2) K ∈ EM (I 2 ) and t → K(t, s) ∈ Eϕ2 (I) for a.e. s ∈ I with kKkM < (K3) , Rb a K(t1 , s) ds ≥ Rb a 1 2k1 · |λ| · G0 · R(1) K(t2 , s) ds for t1 , t2 ∈ [a, b] with t1 < t2 . Then there exists a number ρ > 0 such that for all λ ∈ R with |λ| < ρ and for all g with kgkϕ < 1 there exists a solution x ∈ Eϕ (I) of (5.9) which is a.e. nondecreasing on I. Proof. We need to divide the proof into a few steps. I. The operator B is well-defined from Lϕ (I) into itself and continuous on a domain depending on the considered case. II. We will construct an invariant ball Br for B in Lϕ (I). III. We construct a subset Qr of this ball which contains a.e. nondecreasing functions and we investigate the properties Qr . 69 IV. We check the continuity and monotonicity properties of B in Qr , so U : Qr → Qr . V. We prove that B is a contraction with respect to a measure of noncompactness. VI. We use the Darbo fixed point theorem to find a solution in Qr . I. First of all observe that under the assumptions (C2) and (C3) by Lemma 1.3.3 the superposition operator F acts from Lϕ (I) to LN (I). In this case we will prove, that U is a continuous mapping from the unit ball in Eϕ (I) into the space Eϕ (I). Let us recall, that x ∈ Eϕ (I) iff for arbitrary ε > 0 there exists δ > 0 such that kxχT kϕ < ε for every measurable subset T of I with the Lebesgue measure smaller that δ (i.e. x has absolutely continuous norm). First, let us observe that in view of Lemma 1.3.8, it is sufficient to check this property for the operator A = H ◦ F . Since N is an N-function satisfying ∆′ -condition and by (C3), we are able to use [83, Lemma 19.1]. From this there exists a constant C (not depending on the kernel) such that for any measurable subset T of I and x ∈ Lϕ (I), kxkϕ ≤ 1 we have kA(x)χT kϕ2 ≤ CkKχT ×I kM . (5.10) Now, by the Hölder inequality and the assumption (C2) we get |K(t, s)f (s, x(s))| ≤ kK(t, s)k · |f (s, x(s))| ≤ kK(t, s)k · | (b(s) + R(|x(s)|)) | for t, s ∈ I. Put k(t) = 2kK(t, ·)kM for t ∈ I. As K ∈ EM (I 2 ) this function is integrable on I. By the assumptions (K1) and (K2) about the kernel K of the operator H (cf. [112]) we obtain that kA(x)(t)k ≤ k(t) · (kbkN + kR(|x(·)|)kN ) for a.e. t ∈ I. Whence for arbitrary measurable subset T of I and x ∈ Eϕ (I) kA(x)χT kϕ2 ≤ kkχT kϕ2 · (kbkN + kR(|x(·)|kN ) . Finally if t is such that K(t, ·) ∈ EM (I) and x ∈ Eϕ (I) we have Z kK(t, s)f (s, x(s))k ds ≤ 2kK(t, ·)χT kM · (kbkN + kR(|x(·)|)kN ) for a.e. t ∈ I. T From this it follows that A maps B1 (Eϕ (I)) into Eϕ2 (I). We are in a position to prove the continuity of A as a mapping from the unit ball B1 (Eϕ (I)) into the space Eϕ2 (I). Let xn , x0 ∈ B1 (Eϕ (I)) be such that kxn −x0 kϕ → 0 as n tends to ∞. Suppose, contrary to our claim, that A is not continuous and 70 the kA(xn ) − A(x0 )kϕ2 does not converge to zero. Then there exists ε > 0 and a subsequence (xnk ) such that kA(xnk ) − A(x0 )kϕ2 > ε for k = 1, 2, ... (5.11) and the subsequence is a.e. convergent to x0 . Since (xn ) is a subset of the ball the Rb sequence ( a ϕ(|xn (t)|)dt) is bounded. As the space Eϕ2 (I) is regular the balls are Rb norm-closed in L1 (I) so the sequence ( a |xn (t)|dt) is also bounded. Moreover, by (C3) and (C4) there exist ω, γ, u0 > 0, s.t. (cf. [83, p. 196]) 1 kωR(|x(·)|)kN ω Z  1 ≤ inf N(ωR(|x(t)|)/r)dt ≤ 1 ω r>0   Z b 1 1+ N(ωR(|x(t)|))dt ≤ ω a   Z b 1 1 + N(ωR(u0 )) · (b − a) + γ ϕ2 (|x(t)|)dt , ≤ ω a kR(|x(·)|)kN = whenever x ∈ Lϕ (I) with kxkϕ ≤ 1. Thus Z kK(t, s)f (s, xn (s))k ds ≤ 2kK(t, ·)χT kM · (kbkN + kR(|xn (·)|)kN ) T ≤ 2kK(t, ·)χT kM · (kbkN   Z b 1 1 + N(ωR(u0 )) · (b − a) + γ ϕ2 (|xn (t)|)dt ) + ω a and then the sequence (kK(t, s)f (s, xn (s))k) is equiintegrable on I for a.e. t ∈ I. By the continuity of f (t, ·) we get limk→∞ K(t, s)f (s, xnk (s)) = K(t, s)f (s, x0 (s)) for a.e. s ∈ I. Now, applying the Vitali convergence theorem we obtain that lim A(xnk )(t) = A(x0 )(t) for a.e. t ∈ I. k→∞ But the equation (5.10) implies that A(xnk ) is a subset of Eϕ2 (I) and then lim A(xnk )(t) = A(x0 )(t) k→∞ which contradicts the inequality (5.11). Since A is continuous between indicated spaces, By our assumption (G1) the operator G is continuous from B1 (Eϕ (I)) into Eϕ1 (I) and then by (N1) the operator U has the same property and then U is a continuous mapping from B1 (Eϕ (I)) into the space Eϕ (I). Finally, by the assumption (C1) B maps B1 (Eϕ (I)) into Eϕ (I) continuously. 71 II. We will prove the boundedness of the operator U, namely we will construct the invariant ball for this operator. By B we will denote the right-hand side of our integral equation i.e. B = g + U. Set r ≤ 1 and let ρ= 1 − kgkϕ . 2k1 · C · G0 · kKkM Let x be an arbitrary element from B1 (Eϕ (I)). Then by using the above consideration, the assumption (C3), the formula (5.10) and Proposition 1.3.8 for sufficiently small λ (i.e. |λ| < ρ) we obtain kB(x)kϕ ≤ kgkϕ + kUxkϕ = kgkϕ + kG(x) · A(x)kϕ ≤ kgkϕ + k1 kG(x)kϕ1 · kA(x)kϕ2 Z b = kgkϕ + k1 |λ| · G0 · kxkϕ · k K(·, s)f (s, x(s)) dskϕ2 a ≤ kgkϕ + 2k1 · |λ| · C · G0 · kxkϕ · kKkM ≤ kgkϕ + 2k1 · |λ| · C · G0 · r · kKkM ≤ kgkϕ + 2k1 · ρ · C · G0 · kKkM ≤ r whenever kxkϕ ≤ r. Then we have B : Br (Eϕ (I)) → Br (Eϕ (I)). Moreover, B is continuous on Br (Eϕ (I)) (see the part I of the proof). III. Let Qr stand for the subset of Br (Eϕ (I)) consisting of all functions which are a.e. nondecreasing on I. Similarly as claimed in [17] this set is nonempty, bounded (by r) and convex (direct calculation from the definition). It is also a closed set in Lϕ (I). Indeed, let (yn ) be a sequence of elements in Qr convergent in Lϕ (I) to y. Then the sequence is convergent in measure and as a consequence of the Vitali convergence theorem for Orlicz spaces and of the characterization of convergence in measure (the Riesz theorem) we obtain the existence of a subsequence (ynk ) of (yn ) which converges to y almost uniformly on I (cf. [100]). Moreover, y is still nondecreasing a.e. on I which means that y ∈ Qr and so the set Qr is closed. Now, in view of Lemma 1.4.1 the set Qr is compact in measure. IV. Now, we will show, that B preserve the monotonicity of functions. Take x ∈ Qr , then x is a.e. nondecreasing on I and consequently F (x) is also of the same type in virtue of the assumption (C2) and Lemma 1.4.2. Further, A(x) = H ◦F (x) is a.e. nondecreasing on I thanks for the assumption (K3). Since the pointwise product of a.e. monotone functions is still of the same type and by (G1), the operator U is a.e. nondecreasing on I. 72 Moreover, the assumption (C1) permits us to deduce that Bx(t) = g(t) + U(x)(t) is also a.e. nondecreasing on I. This fact, together with the assertion that B : Br (Eϕ (I)) → Br (Eϕ (I)) gives us that B is also a self-mapping of the set Qr . From the above considerations it follows that B maps continuously Qr into Qr . V. We will prove that B is a contraction with respect to the measure of noncompactness µ. Assume that X is a nonempty subset of Qr and let the fixed constant > 0 be arbitrary. Then for an arbitrary x ∈ X and for a set D ⊂ I, meas(D) ≤ ε we obtain kB(x) · χD kϕ ≤ kgχD kϕ + kU(x) · χD kϕ = kgχD kϕ + kG(x) · A(x)χD kϕ ≤ kgχD kϕ + k1 kG(x)χD kϕ1 · kA(x) · χD kϕ2 Z = kgχD kϕ + k1 · |λ| · kG(x)χD kϕ1 · k K(·, s)f (s, x(s)) dskϕ2 D Z ≤ kgχD kϕ + k1 |λ|G0kxχD kϕ k |K(·, s)|(b(s) + R(|x(s)|)) dskϕ2 D ≤ kgχD kϕ + k1 · |λ| · G0 · kxχD kϕ · 2kKkM k[bχD + R(r))]kN ≤ kgχD kϕ + 2k1 · |λ| · G0 · kxχD kϕ · kKkM [ kbχD kN + R(1)]. Hence, taking into account that g ∈ Eϕ , b ∈ EN lim { sup [sup{kgχD kϕ }]} = 0 and lim { sup [sup{kbχD kN }]} = 0. ε→0 ε→0 mes D≤ε x∈X mes D≤ε x∈X Thus by definition of c(x) and by taking the supremum over all x ∈ X and all measurable subsets D with meas(D) ≤ ε we get c(B(X)) ≤ 2k1 · |λ| · G0 · kKkM · R(1) · c(X). Since X ⊂ Qr is a nonempty, bounded and compact in measure subset of an ideal regular space Eϕ , we can use Proposition 5.5.1 and get βH (B(X)) ≤ 2k1 · |λ| · G0 · kKkM · R(1) · βH (X). The inequality obtained above together with the properties of the operator B and the set Qr established before and the inequality from the Assumption (K2) allow us to apply the Darbo Fixed Point Theorem 1.6.5, which completes the proof. 73 The case of N satisfying the ∆3-condition. 5.4.2 Let us consider the case of N-functions with the growth essentially more rapid than a polynomial. In fact, we will consider N-functions satisfying ∆3 -condition. This is very large and important class, especially from an application point of view (cf. [36, 37, 106, 120]). An extensive description of this class can be found in [106, Section 2.5]. Recall, that an N-function M determines the properties of the Orlicz space LM (I) and then the less restrictive rate of the growth of this function implies the ”worser” properties of the space. By ϑ we will denote the norm of the identity operator from Lϕ (I) into L1 (I) i.e. sup{kxk1 : x ∈ B1 (Lϕ (I))}. For the discussion about the existence of ϕ which satisfies our conditions see [84, p. 61]. Theorem 5.4.2. Assume, that ϕ, ϕ1 , ϕ2 are N-functions and that M and N are complementary N-functions and that (N1), (C1), (C2), (C3), (G1), (K1) and (K3) hold true. Moreover, put the following assumptions: (C5) 1. N satisfies the ∆3 -condition, 2. K ∈ EM (I 2 ) and t → K(t, s) ∈ Eϕ2 (I) for a.e. s ∈ I, 3. There exist β, u0 > 0 such that R(u) ≤ β M(u) , u for u ≥ u0 , (K4) ϕ2 is an N-function satisfying ZZ ϕ2 (M(|K(t, s)|)) dtds < ∞ I2 and 2k1 · (2 + (b − a)(1 + ϕ2 (1))) · |λ| · G0 · kKkϕ2 ◦M · R(r0 ) < 1, where   ω 1 − kbkN . r0 = ϑ 2|λ| · k1 · G0 · (2 + (b − a)(1 + ϕ2 (1))) · kKkϕ2 ◦M Then there exist a number ρ > 0 and a number ̟ > 0 such that for all λ ∈ R with |λ| < ρ and for all g ∈ Eϕ (I) with kgkϕ < ̟ there exists a solution x ∈ Eϕ (I) of (5.9) which is a.e. nondecreasing on I. Proof. We will indicate only the points of the proof if they differ from the previous case. I. In this case the operator B can be considered as continuous when acting on the whole Eϕ (I). 74 By [84, Lemma 15.1 and Theorem 19.2] and the assumption (K4): kA(x)χT kϕ2 ≤ 2·(2+(b−a)(1+ϕ2 (1)))·kK·χT ×I kϕ2 ◦M (kbkN + kR(|x(·)|)kN ) (5.12) for arbitrary x ∈ Lϕ (I) and arbitrary measurable subset T of I. Let us note, that the assumption (C5) 3. implies that there exist constants ω, u0 > 0 and η > 1 such that N(ωR(u)) ≤ ηu for u ≥ u0 . Thus for x ∈ Lϕ (I)   Z 1 1 + N(ωR(|x(s)|) ds kR(|x(·)|)kN ≤ ω I   Z 1 ≤ 1 + ηu0 (b − a) + η |x(s)| ds . ω I The remaining estimations can be derived as in the first main theorem and then we obtain, that A : Eϕ (I) → Eϕ2 (I), so by the properties of G we get B : Eϕ (I) → Eϕ (I). II. Put ρ= 1  2 · k1 · G0 · (2 + (b − a)(1 + ϕ2 (1))) · kKkϕ2 ◦M · kbkN + 1 ω (1 + ηu0(b − a)) Fix λ with |λ| < ρ. Choose a positive number r in such a way that kgkϕ + . 2rk1 · (2 + (b − a)(1 + ϕ2 (1))) · G0 · |λ| · kKkϕ2 ◦M (kbkN  1 + (1 + ηu0(b − a) + ηϑr) ≤ r. (5.13) ω As a domain for the operator B we will consider the ball Br (Eϕ (I)). Let us remark, that the above inequality is of the form a + (b + vr)cr ≤ r with a, b, c, v > 0. Then vc > 0 and if we assume that bc−1 < 0 and that the discriminant is positive, then Viète’s formulas imply that the quadratic equation has two positive solutions r1 < r2 for sufficiently small λ. By the definition of ρ it is clear, that our assumptions guarantee the above requirements, so there exists a positive number r satisfying this inequality. Put C = (2 + (b − a)(1 + ϕ(1))). Let us note, in view of the above considerations, that the assumption about the discriminant which implies the existence of solutions for the above problem is of the form: 1 1 ]2 (1 + ηu0 (b − a)) − ω 2|λ| · k1 · G0 · C · kKkϕ2 ◦M 2kgkϕ ηϑ × |λ| · k1 · G0 · C · kKkϕ2 ◦M > ω [kbkN + 75 i.e. 2  1 1 ̟ = kbkN + (1 + ηu0(b − a)) − ω 2|λ| · k1 · G0 · C · kKkϕ2 ◦M |λ| · k1 · G0 · C · ωkKkϕ2 ◦M × . 2ηϑ For x ∈ Br (Eϕ (I)) we have the following estimation: kB(x)kϕ ≤ kgkϕ + kUxkϕ kgkϕ + kG(x) · A(x)kϕ = kgkϕ + k1 kG(x)kϕ1 · kA(x)kϕ2 Z b kgkϕ + k1 |λ|kG(x)kϕ1 · k K(·, s)f (s, x(s)) dskϕ2 ≤ = a ≤ + ≤ + ≤ + = + kgkϕ + 2k1 · C · G0 · |λ| · kxkϕ kKkϕ2 ◦M [kbkN  Z 1 (1 + N(ωR(u0 )) · (b − a) + η |x(s)| ds) ω I kgkϕ + 2k1 · C · G0 · |λ| · kxkϕ kKkϕ2 ◦M [kbkN  1 (1 + N(ωR(u0 )) · (b − a) + ηkxk1 ) ω kgkϕ + 2k1 · C · G0 · |λ| · kxkϕ kKkϕ2 ◦M [kbkN  1 (1 + N(ωR(u0 )) · (b − a) + ηϑkxkϕ ) ω kgkϕ + 2rk1 · C · G0 · |λ| · kKkϕ2 ◦M (kbkN  1 (1 + ηu0 (b − a) + ηϑr) ≤ r. ω Then B : Br (Eϕ (I)) → Br (Eϕ (I)). Note, that the parts III. and IV. of the previous proof are similar to those from the first theorem, so we omit the details. V. We will prove that B is a contraction with respect to a measure of noncompactness. Assume that X is a nonempty subset of Qr and let the fixed constant ε > 0 be arbitrary. Then for an arbitrary x ∈ X and for a set D ⊂ I, meas(D) ≤ ε 76 we obtain kB(x) · χD kϕ ≤ = ≤ kgχD kϕ + kU(x) · χD kϕ kgχD kϕ + kG(x) · A(x)χD kϕ kgχD kϕ + k1 kG(x)χD kϕ1 · kA(x) · χD kϕ2 Z = kgχD kϕ + k1 · |λ| · G0 · kxχD kϕ · k K(·, s)f (s, x(s)) dskϕ2 D Z ≤ kgχD kϕ + k1 |λ|G0 kxχD kϕ k |K(·, s)|(b(s) + R(|x(s)|)) dskϕ2 D  Z ≤ kgχD kϕ + k1 · |λ| · G0 · kxχD kϕ · k |K(·, s)|b(s) dskϕ2 D  Z + k |K(·, s)|R(|x(s)|)) dskϕ2 D ≤ kgχD kϕ + 2 · C · k1 · G0 · |λ| · kxχD kϕ · ϑ · kKkϕ2 ◦M kbχD kN Z + 2 · C · k1 · G0 · kxχD kϕ · k |K(·, s)|R(|x(s)|) dskϕ2 D ≤ ≤ kgχD kϕ + 2Ck1 G0 |λ| · kxχD kϕ kKkϕ2 ◦M [kbχD kN + R(r)] kgχD kϕ + 2Ck1 G0 |λ| · kxχD kϕ kKkϕ2 ◦M [kbχD kN + R(r0 )], where   ω 1 − kbkN . r0 = ϑ 2|λ| · k1 · G0 · (2 + (b − a)(1 + ϕ2 )) · kKkϕ2 ◦M Let us note, that r0 is an upper bound for solutions of (5.13). Similarly as in the previous theorem we get βH (B(X)) ≤ 2 · k1 · C · G0 · |λ|kKkϕ2 ◦M · R(r0 ) · βH (X). The inequality obtained above together with the properties of the operator B and the set Qr established before and then the assumption (K4) allow us to apply the Theorem 1.6.5, which completes the proof. We need to stress on some aspects of our results. First of all we can observe, that our solutions are not necessarily continuous as in previously investigated cases. In particular, we need not to assume, that the Hammerstein operator transforms the space C(I) into itself. For the examples and conditions related to Hammerstein operators in Orlicz spaces we refer the readers to [106, Chapter VI.6.1., Corollary 6 and Example 7]. We have two more information about the set of solutions: it is included in Eϕ (I) and in view of Theorem 1.6.5 it can be proved, that this set is compact as a subset of Lϕ (I). 77 5.4.3 Remarks on classes of solutions. There is one more interesting question related to the case of continuous solutions. This is the question if we are able to put in our results the same Orlicz space (as in the case of C(I)). In other words, the case of Banach-Orlicz algebras. Note, that the presented case in this Chapter seems to be more general we need to mention, that as claimed by Kalton [77] this property is true for ϕ(x) = x(1 + x)−1 or ϕ(x) = log+ x. These spaces are not of big interest in the theory of integral equations, then let us present some remarks on operators satisfying our assumptions. Let X, Y be ideal spaces. A superposition operator F : X → Y is called improving (cf. remark 1.3.3). The applications of such operators are based on the observation that large classes of linear integral operators Z Hy(t) = λ k(t, s)y(s)ds, D although not being compact, map sets with equiabsolutely continuous norms into precompact sets. In contrast to the classical (non-quadratic) case, for quadratic integral equations even such a nice assumption is not sufficient for using the Schauder fixed point theorem. Moreover, we assume in our main theorem, that G maps sets with equiabsolutely continuous norms into the same family, but we don’t need to assume, that it is an improving operator. Conversely, any improving operator G can be considered in our results. Let us note, that for operators from Lebesgue spaces Lp (I) into Lr (I) (i.e. Orlicz spaces with p(x) = xp and r(x) = xr , respectively), the characterization of improving operators is known ([123]): a superposition operator F : Lp (I) → Lr (I) is improving if and only if there exists a continuous and even function M satisfying limu→∞ Mu(u) = ∞ and such that G(x)(t) = M(f (t, x(t))) is also an operator from Lp (I) into Lr (I) (for an appropriate growth condition of f see [123]). The aspect of applicability of our results deal also with the technique of Orlicz spaces for partial differential equations, so for an appropriate class of integral equations. In this context one can consider more singular equations than in a classical case. Motivated by previously considered equations (see [36, 37, 45, 120] or [100, 112]) we extend this method to the case of quadratic integral equations. It should be recalled that our method of the proof can be also adapted to classical equations considered in [83, 36, 45, 120]. For more information we refer the readers to the Chapter IX ”Nonlinear PDEs and Orlicz spaces” in [105]. Finally, let us remark, that our results can be applied also for Lebesgue spaces Lp (I) (p ≥ 1) (cf. [89, 90]). As mentioned above this class of spaces is also included into the class of Orlicz spaces. But even in this case we allow for f or K to be strongly 78 nonlinear. The simplest case is that when F : L1 (I) → LN (I), H : LN (I) → Lp (I), G : L1 (I) → Lq (I) ( 1p + 1q = 1). Thus we will have integrable solutions (in L1 (I)), but f or K can be strongly nonlinear. Of course, strong nonlinearity of one of them implies that we need to consider the weak one for another (cf. [83, Chapter IV.19]). Let us present an example of such spaces. By N1 , N2 we denote complementary functions for M1 , M2 , respectively. Put M1 (u) = exp |u| − |u| − 1 and M2 (u) = u2 = N2 (u). Note, that M1 satisfies the ∆3 -condition. In this case N1 (u) = (1 + 2 |u|) · ln (1 + |u|) − |u|. If we define an N-function either as Ψ(u) = M2 [N1 (u)] or Ψ(u) = N1 [M2 (u)], then by choosing arbitrary kernel K from the space LΨ (I) we are able to apply [83, Theorem 15.4]. Thus H : LM1 (I) → LM2 (I) is continuous and we may apply our result (Theorem 5.4.2) for operators G : L1 (I) → Lq (I) and F : L1 (I) → LM1 (I) (with natural growth conditions, see Lemma 1.3.3). Let us also to pay attention to the particular case of our problem with G(x) = a(t)x(t): Z 1 x(t) = g(t) + λa(t) · x(t) K(t, s)f (s, x(s))ds. 0 Since we are motivated by some study on quadratic integral equations, this is of our particular interest. Note, that a full description for acting and continuity conditions for G(x) = a(t)x(t) can be found in [83, Theorem 18.2]. 79 5.5 Conclusions and fixed point theorems. If we are looking for the proofs of our main results, in contrast to earlier theorems we stress on some properties of spaces rather on continuity of solutions and the properties of this particular space C(I). We will present a general approach for differential and integral problems by presenting a new fixed point theorem specialized to quadratic equations. For completness, we need to recall some necessary facts. In this section some properties of function spaces play a major role. We need to consider the triples of spaces with the following property: for a triple of spaces E, E1 , E2 there exists a constant k such that for arbitrary x ∈ E1 and y ∈ E2 a product (pointwise multiplication) x · y ∈ E and kx · ykE ≤ k · kxkE1 · kykE2 . Let us recall some special cases. Most known is the case of Banach algebras i.e. the space of continuous functions. In this case E = E1 = E2 = C(I, X) (k = 1). Moreover, some subalgebras of this space can be interesting. If we try to consider ”bigger” spaces we need to go outside the class of Banach algebras. For discontinuous functions, let us recall the Hölder inequality for Lebesgue spaces: kx · ykL1 ≤ kxkLp · kykLq whenever 1p + 1q = 1. Thus, a triple (L1 , Lp , Lq ) is good enough. Third important example (and most important) is for Orlicz spaces (for definitions see [83, 92], for instance). Generally speaking, the product of two functions x, y ∈ LM (I) is not in LM (I). However, if x and y belongs to some particular Orlicz spaces, then the product x · y belong to a third Orlicz space. Let us note, that one can find two functions belonging to Orlicz spaces: u ∈ LU (I) and v ∈ LV (I) such that the product uv does not belong to any Orlicz space (this product is not integrable). Nevertheless, Lemma 1.3.8 give us an interesting characterization for such a triple of spaces. An interesting discussion about necessary and sufficient conditions for product operators can be found in [83, 92]. Note, that since Lp = LM p for M(t) = tp the case of Lebesgue spaces Lp is included in the mentioned Lemma. Finally we have a special case for E2 = L∞ and some function spaces for which (E = E1 ) kx·ykE ≤ kxkE ·kykL∞ . The class of spaces with this property is known as preideal∗ spaces (cf. [119, p. 66] or [118]) or Köthe function spaces. Although this case seems to be general it has one weakness from our point of view: the measure of noncompactness in L∞ seems to be inapplicable and we do not discuss it in this Thesis. It is possible to check this property for a given triple of spaces. An open question is if is possible to characterize all such spaces? 80 Recall, that for any ε > 0, let c be a measure of equiintegrability of the set X in a ideal space E (introduced in [9], cf. also [119, Definition 3.9], [67, 66]): c(X) = lim sup ε→0 sup sup kx · χD kE , mesD≤ε x∈X where χD denotes the characteristic function of D. To distinguish between measures of noncompactness µ (or : c) in different spaces we will indicate an appropriate space as an index i.e. µE , cE1 , µE2 etc. The following theorem clarify the connections between the two coefficients in E. Proposition 5.5.1. ([66, Theorem 1]) Let X be a nonempty, bounded and compact in measure subset of an ideal regular space E. Then βH (X) = c(X). As a consequence, we obtain that bounded sets which are additionally compact in measure are compact in E iff they are equiintegrable in this space (i.e. have equiabsolutely continuous norms, in particular when X is a subset for a regular part of E). In contrast to the case considered in [25] we will need the following property: Lemma 5.5.1. Assume, that for a triple of regular ideal function spaces E, E1 , E2 there exists a constant k such that for arbitrary x ∈ E1 , y ∈ E2 and t ∈ I a product x · y ∈ E and kx · ykE ≤ k · kxkE1 · kykE2 . Then for any set X ⊂ E1 , Y ⊂ E2 we have X · Y ⊂ E and cE (X · Y ) ≤ k · cE1 (X) · cE2 (Y ), where cV stands for a measure of equiintegrability in the space V for V = E, E1 or E2 , respectively. Proof. By the properties of spaces we obtain, that X · Y ⊂ E. Take arbitrary x ∈ X and y ∈ Y and arbitrary measurable subset D of I. Then k(x · y) · χD kE ≤ k · kx · χD kE1 · ky · χD kE2 . Then by the properties of supremum sup sup k(x · y) · χD kE ≤ k · sup sup kx · χD kE1 · ky · χD kE2 x∈X y∈Y x∈X y∈Y and by the property of lim supmeas(D)→0 lim sup sup sup k(x·y)·χD kE ≤ k · lim sup sup kx·χD kE1 · lim sup sup ky ·χD kE2 , meas(D)→0 x∈X y∈Y meas(D)→0 x∈X meas(D)→0 y∈Y which ends the proof. The last property is obvious but useful: Lemma 5.5.2. Let E be a regular ideal space. For any bounded subset X of E we have c(X) ≤ kXkE . 81 5.5.1 A fixed point theorem. Since we are interested on a fixed points of some product operators we will assume, that our operators have values in some intermediate spaces and then the product will again turn the values into the original space. First, let us apply our approach for the most applicable theorem of this type. Consider an arbitrary (in the sense of Definition 1.5.1) measure of noncompactness µ in C(I, E). An interesting fixed point theorem in Banach algebras was proved by Banaś and Lecko (cf. [25]). Let us consider different spaces of continuous functions with a suitable choice of measures of noncompactness µE on E = C(I), µE1 on E1 and µE2 on E2 . By using our approach we are able to present the following extension of the mentioned theorem: Theorem 5.5.1. Let E, E1 , E2 be regular ideal function spaces. Assume that T is nonempty, bounded, closed, and convex subset of the Banach space E, and the operators A : E → E1 and B : E → E2 . Moreover, assume: 1. (A1) A transform continuously the set T into T1 ⊂ E1 and A(T ) is bounded in E1 , 2. (A2) there exists a constant k1 > 0 such that A satisfies an inequality: µE (A(U)) ≤ k1 · µE1 (U) for arbitrary bounded subset U of E, 3. (B1) B transform continuously the set T into T2 ⊂ E2 and B(T ) is bounded in E2 , 4. (B2) there exists a constant k2 > 0 such that B satisfies an inequality: µE (B(U)) ≤ k2 · µE2 (U) for arbitrary bounded subset U of E, 5. (E1) for a triple of spaces E, E1 , E2 there exists a constant k such that for arbitrary x ∈ E1 , y ∈ E2 and t ∈ I a product x · y ∈ E and kx · ykE ≤ k · kxkE1 · kykE2 , 6. (E2) for every x ∈ T1 and y ∈ T2 one has x · y ∈ T , 7. (C) kA(T )kE1 · k2 + kB(T )kE2 · k1 < 1. Then there exists at least one fixed point for the operator K = A · B in the set T and that the set of all fixed points of K belongs to the kernel kerµE . 82 This theorem was proved by Banaś in a special case of Banach algebras E = E1 = E2 = C(I, R) (k = 1) (cf. also Dhage and Kumpulainen [53], for instance). We do not require, that the values of all operators are from the same space, but by using a property of considered spaces we are able to repeat the proof, so we omit the details. We will present a proof for a more general case. In the first theorem the Ascoli criterion of compactness in spaces of continuous functions simplify the proof, because the convergence of sequences is directly related with pointwise convergence. Now, we will consider the case of functions spaces without such a nice property. We will consider some subspaces of a space of L0 (I) of measurable functions, bigger than C(I). This allows us to apply the fixed point theorem for the problems with discontinuous solutions. This proof will be based on different compactness criterion (the Dunford-Pettis theorem and the Erzakova theorem). Theorem 5.5.2. Assume that T is nonempty, bounded, closed, convex and compact in measure subset of a regular ideal function space E, and the operators A : E → E1 and B : E → E2 . Put the following set of assumptions: 1. (A1) A transform continuously the set T into T1 ⊂ E1 and A(T ) is bounded in E1 , 2. (A2) there exists a constant k1 > 0 such that A satisfies an inequality: c(A(U)) ≤ k1 · cE1 (U) for arbitrary bounded subset U of E, 3. (B1) B transform continuously the set T into T2 ⊂ E2 and B(T ) is bounded in E2 , 4. (B2) there exists a constant k2 > 0 such that B satisfies an inequality: c(B(U)) ≤ k2 · cE2 (U) for arbitrary bounded subset U of E, 5. (E1) for a triple of regular ideal spaces E, E1 , E2 there exists a constant k such that for arbitrary x ∈ E1 and y ∈ E2 a product x · y ∈ E and kx · ykE ≤ k · kxkE1 · kykE2 , 6. (E2) for every x ∈ T1 and y ∈ T2 one has x · y ∈ T , 7. (C1) k · k1 · kB(T )kE2 < 1, 8. (C2) k · kA(T )kE1 · k2 < 1. 83 Assume, that (A1), (B1), (E1), (E2) and either (A2) and (C1) or (B2) and (C2) are satisfied. Then there exists at least one fixed point for the operator K = A · B in the set T and the set of all fixed points F ixK is relatively compact in E. Proof. Let us present the proof, when (A2) and (C1) are satisfied. The second case is similar. It is obvious that the operator K is well-defined on T and by (E2) it acts between T into itself. Denote M1 = supt∈T kA(t)kE1 and M2 = supt∈T kB(t)kE2 . Let (xn ) be an arbitrary sequence in T tending to x ∈ T . Then kK(xn ) − K(x)kE = ≤ ≤ ≤ kA(xn ) · B(xn ) − A(x) · B(x)kE kA(xn ) · B(xn ) − A(x) · B(x) − A(x) · B(xn ) + A(x) · B(xn )kE k(A(xn ) − A(x)) · B(xn )kE + k(B(xn ) − B(x)) · A(x)kE k · kA(xn ) − A(x)kE1 · M2 + k · kB(xn ) − B(x)kE2 · M1 . From our assumptions it follows that K is continuous from T into E. Now, we will investigate the contraction property for a measure c(X). Assume that X is a nonempty subset of T and let the fixed constant ε > 0 be arbitrary. Then for an arbitrary x ∈ X and for a set D ⊂ I, meas(D) ≤ ε we obtain kK(x) · χD kE ≤ kkA(x) · χD kE1 · kB(x)kE2 . Since for any non-negative real-valued functions f and g we have supI (f · g) ≤ supI f · supI g, by definition of c(x) and by taking the supremum over all x ∈ X and all measurable subsets D with meas(D) ≤ ε we get c(K(X)) ≤ k · k1 · kB(T )kE2 · c(X). Because X ⊂ T is a nonempty, bounded and compact in measure subset of an ideal regular space E, we can use Proposition 5.5.1 and get βH (K(X)) ≤ k · k1 · kB(T )kE2 · βH (X). The inequality obtained above together with the properties of the operator K and the set T established before, allow us to apply the classical Darbo fixed point theorem for βH . If we suppose, that βH (F ixK) 6= 0, then K = F ixK implies βH (F ixK) = βH (K) < βH (K), a contradiction, which completes the proof. Remark 5.5.1. We need to remark, that one of our assumptions can be easily relaxed. We assume, that the space is regular. Denote by 0 a regular part of E. It is 84 sufficient to assume, that K : T ∩ E0 → T ∩ E0 . 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