Filomat 33:17 (2019), 5519–5530 https://doi.org/10.2298/FIL1917519D Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Solvability of Infinite Systems of Fractional Differential Equations in the Spaces of Tempered Sequences Anupam Dasa , Bipan Hazarikaa,b , Ravi P. Agarwalc , Hemant Kumar Nashined,∗ a Department of Mathematics, Rajiv Gandhi University,Rono Hills, Doimukh-791112, Arunachal Pradesh, India b Department of Mathematics, Gauhati University, Guwahati-781014, Assam, India c Department of Mathematics, Texas A & M University, Kingsville, Texas 78363-8202, USA d Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam Abstract. In this paper we discuss the existence of solution of infinite systems of fractional differential equations with the help of Hausdorff measure of noncompactness and Meir–Keeler fixed point theorem in the tempered sequence spaces. We provide examples to established the applicability of our results. 1. Introduction and Definitions The fractional differential equations describe many phenomena in the fields of engineering, physics, biophysics, chemistry, biology, economics, control theory, signal and image processing, aerodynamics, viscoelasticity, electromagnetic and rheology etc. The fractional differential equations have important tool for the description of hereditary properties of various materials and processes than the corresponding integer order differential equations. For different types of applications of fractional differential equations we refer [2, 3, 12] and references therein. The theory of infinite systems of ordinary differential equations is a very important branch of the theory of differential equations in Banach spaces. Infinite systems of ordinary differential equations describes many real life problems which can found in the theory of neural nets, the theory of branching processes and mechanics etc (see [9, 11, 19]). In functional analysis the measure of noncompactness play important role which was introduced by Kuratowski [13]. The idea of measure of noncompactness has been used by many authors in obtaining the existence of solutions of infinite systems of integral equations and differential equations (see [8]). Mursaleen and Mohiuddine [16] proved existence theorems for the infinite systems of differential equations in the space ℓp . On the other hand, existence theorems for the infinite systems of linear equations in ℓ1 and ℓp was discussed by Alotaibi et al. [5]. Mursaleen and Alotaibi [18] proved existence theorems for the infinite systems of differential equations in some BK-spaces. Mursaleen et al. [15] proved the existence of infinite systems of fractional differential equations in the spaces c0 and ℓp . Srivastava et al. [20] studied the existence 2010 Mathematics Subject Classification. Primary 45G05; Secondary, 34A34, 46B45, 47H10, 93C15 Keywords. hausdorff measure of noncompactness, infinite systems of fractional differential equations, Meir–Keeler fixed point theorem. Received: 27 October 2018; Revised: 01 February 2019; Accepted: 09 April 2019 Communicated by Jelena Manojlović ∗ Corresponding author: Hemant Kumar Nashine Email addresses: math.anupam@gmail.com (Anupam Das), bh_rgu@yahoo.co.in (Bipan Hazarika), agarwal@tamuk.edu (Ravi P. Agarwal), hemantkumarnashine@tdtu.edu.vn (Hemant Kumar Nashine) A. Das et al. / Filomat 33:17 (2019), 5519–5530 5520 of solutions of infinite systems of nth order differential equations in the spaces c0 and ℓ1 via the measure of noncompactness. Definition 1.1. [13] Let (X, d) be a metric space and Q a bounded subset of X. Then the Kuratowski measure of noncompactness (α-measure or set measure of non-compactness) of Q, denoted by α(Q), is the infimum of the set of all numbers ǫ > 0 such that Q can be covered by a finite number of sets with diameters ǫ > 0, that is, n o α(Q) = inf ǫ > 0 : Q ⊂ ∪ni=1 Si , Si ⊂ X, diam(Si ) < ǫ (i = 1, 2, ..., n) , n ∈ N The function α is called Kuratowski measure of noncompactness. It was introduced by Kuratowski [13]. Clearly α(Q) ≤ diam(Q) for each bounded subset Q of X. Suppose E is a real Banach space with the norm k . k . Let B(x0 , r) be a closed ball in E centered at x0 and with radius r. If X is a nonempty subset of E then by X̄ and Conv(X) we denote the closure and convex closure of X. Moreover let ME denote the family of all nonempty and bounded subsets of E and NE its subfamily consisting of all relatively compact sets. Definition 1.2. [8] A function µ : ME → [0, ∞) is called a measure of noncompactness if it satisfies the following conditions:  (i) the family ker µ = X ∈ ME : µ (X) = 0 is nonempty and ker µ ⊂ NE . (ii) X ⊂ Y =⇒ µ (X) ≤ µ (Y) .
(iii) µ X̄ = µ (X) . (iv) µ (ConvX) = µ (X) . (v) µ (λX + (1 − λ) Y) ≤ λµ (X) + (1 − λ) µ (Y) for λ ∈ [0, 1] . T (vi) if Xn ∈ ME , Xn = X̄n , Xn+1 ⊂ Xn for n = 1, 2, 3, ... and lim µ (Xn ) = 0 then ∞ n=1 Xn , φ. n→∞ The family kerµ is said to be the kernel of measure µ. A measure µ is said to be the sublinear if it satisfies the following conditions: (1) µ (λX) = |λ| µ (X) for λ ∈ R. (2) µ (X + Y) ≤ µ (Y) + µ (Y) . A sublinear measure of noncompactness µ satisfying the condition:  µ (X ∪ Y) = max µ (X) , µ (Y) and kerµ = NE is said to be regular.  Definition 1.3. [8] Let (X, d) be a metric space, Q be a bounded subset of X and B(x, r) = y ∈ X : d(x, y) < r . Then the Hausdorff measure of noncompactness χ(Q) of Q is defined by   n   [     χ(Q) := inf  B(x , r ), x ∈ X, r < ǫ (i = 1, 2, . . . , n), n ∈ N ǫ > 0 : Q ⊂ .  i i i i     i=1 The definition of the Hausdorff measure of noncompactness of the set Q it is not supposed that centers of the balls that cover Q belong to Q. Hence it can equivalently be stated as follows: χ(Q) = inf {ǫ > 0 : Q has a finite ǫ − net in X} . Consider the following sequence spaces, which are Banach spaces with their respective norms ( ) c0 = x ∈ ω : lim xk = 0, k x kc0 = sup |xk | k→∞ k A. Das et al. / Filomat 33:17 (2019), 5519–5530 5521 the space of all null sequences and ( ) c = x ∈ ω : lim xk = l, l ∈ C , k x kc = sup |xk | k→∞ k the space of all convergent sequences.
In [8], the Hausdorff measure of noncompactness χ in the Banach space c0 , k . kc0 is defined by " # χ (B) = lim sup max | uk | , where B ∈ Mc0 . n→∞ u∈B  (1) k≥n In [17], the most convenient measure of noncompactness µ for the Banach space (c, k . kc ) is defined by       µ (B) = lim sup  sup | uk − lim um p→∞ u∈B  m→∞ k≥p     |  ,  (2) where B ∈ Mc . The measure µ is regular. Recently Banaś and Krajewska [7] have introduced tempering
∞ sequence and space of tempered sequences. Let us fix a positive non increasing real sequence β = βn n=1 , such a sequence is called the tempering sequence. Let the set X consisting of all real (or complex) sequences x = (xn )∞ such that βn xn → 0 as n → ∞. It is n=1 β obvious that X forms a linear space over the field of real (or complex ) numbers. We denote the space by c0 . β It is easy to see that c0 is a Banach space with the norm  k x kcβ = sup βn |xn | . 0 n∈N
Similarly, let the set X consisting of all real (or complex) sequences x = (xn )∞ such that βn xn converges n=1 to a finite limit. It is obvious that X forms a linear space over the field of real (or complex) numbers. We denote the space by cβ . It is easy to see that cβ is a Banach space with the norm  k x kcβ = sup βn |xn | . n∈N β Also there is a isometry between the spaces c0 and c0 and between the spaces cβ and c. In [7], the Hausdorff measure of noncompactness χ(B) for B ∈ Mcβ is defined by 0 ( " #)
χ(B) = lim sup sup βk |xk | . n→∞ x∈B k≥n Similarly the analogue of the measure of noncompactness µ on cβ defined by formula (2) has the form       
    µcβ (B) = lim sup  , where B ∈ Mcβ . sup | βk xk − lim βm xm |   p→∞ x∈B  m→∞  k≥p β Let us consider the function spaces C(I, c0 ) and C(I, cβ ) where I = [0, T], T > 0 the spaces of all continuous functions on I Then β C(I, c0 ) β with values in c0 β and the spaces of all continuous functions on I with values in cβ respectively. and C(I, c ) are Banach spaces with respect to the norm   β k u kC(I,cβ ) = max k u(t) kcβ : t ∈ I , u ∈ C(I, c0 ) 0 0 5522 A. Das et al. / Filomat 33:17 (2019), 5519–5530 and k u kC(I,cβ ) = max {k u(t) kcβ : t ∈ I} , u ∈ C(I, cβ ) respectively. β For any non-empty, closed, bounded and convex subset X of C(I, c0 ) or C(I, cβ ) and t ∈ I, let X(t) = {x(t) : x ∈ X} , χC(I,cβ ) (X) = sup {χ(X(t)) : t ∈ I} 0 and  µC(I,cβ ) (X) = sup µcβ (X(t)) : t ∈ I . It was proved in [6] that for a bounded closed and convex X ⊂ C(I, E) where E is a Banach space the measure of noncompactness is given by µC(I,E) (X) = sup µE {X(t)} . t∈I β Thus χC(I,cβ ) and µC(I,cβ ) will satisfy all the axioms of measure of noncompactness on C(I, c0 ) and C(I, cβ ) 0 respectively. Definition 1.4. [4] Let E1 and E2 be two Banach spaces and let µ1 and µ2
be arbitrary measure of noncompactness on E1 and E2
, respectively. An operator f from E1 to E2 is called a µ1 , µ2 -condensing operator if it is continuous and µ2 f (D) < µ1 (D) for every set D ⊂ E1 with compact closure. Remark 1.5. If E1 = E2 and µ1 = µ2 = µ, then f is called a µ-condensing operator. Theorem 1.6. [10] Let Ω be a nonempty, closed, bounded and convex subset of a Banach space E
and let f : Ω → Ω be a continuous mapping such that there exists a constant k ∈ [0, 1) with the property µ2 f (Ω) < kµ1 (Ω). Then f has a fixed point in Ω. Definition 1.7. [14] Let (X, d) be a metric space. Then a mapping T on X is said to be a Meir–Keeler contraction if for any ǫ > 0, there exists δ > 0 such that
ǫ ≤ d(x, y) < ǫ + δ =⇒ d Tx, Ty < ǫ, ∀x, y ∈ X. Theorem 1.8. [14] Let (X, d) be a complete metric space. If T : X → X is a Meir-Keeler contraction, then T has a unique fixed point. Definition 1.9. [1] Let C be a nonempty subset of a Banach space E and let µ be an arbitrary measure of noncompactness on E. We say that an operator T : C → C is a Meir–Keeler condensing operator if for any ǫ > 0, there exists δ > 0 such that ǫ ≤ µ (X) < ǫ + δ =⇒ µ (T (X)) < ǫ for any bounded subset X of C. Theorem 1.10. [1] Let C be a nonempty, bounded, closed and convex subset of a Banach space E and let µ be an arbitrary measure of noncompactness on E. If T : C → C is a continuous and Meir–Keeler condensing operator, then T has at least one fixed point and the set of all fixed points of T in C is compact. 5523 A. Das et al. / Filomat 33:17 (2019), 5519–5530 2. Main Results For a function f : (0, ∞) → R, the fractional integral of order α is defined as follows Iα f (t) = 1 Γ(α) Z t 0 (t − s)α−1 f (s)ds, where α > 0, provided the integral exists. Similarly the fractional derivative of order α for a function f is defined by !n Z t 1 d 1 Dα f (t) = f (s)ds, α−n+1 Γ(n − α) dt (t − s) 0 where n = [α] + 1 = N + 1. We mention the following properties of the operator I and D for α, β > 0 Iα+β f (t) = Iα Iβ f (t), Dα Iα f (t) = f (t). For α > 0, the general solution of the fractional differential equation Dα y(t) = 0 with y(t) ∈ C(0, T) ∩ L1loc (0, ∞) is given by y(t) = C1 tα−1 + C2 tα−2 + ... + CN tα−N , where Ci ∈ R, i = 1, 2, . . . , N. We discuss the infinite systems of fractional differential equations by transforming the system into an infinite systems of integral equations with the help of Green’s function. Consider the infinite systems of fractional differential equations
(3) Dα yi (t) + hi t, y(t) = 0, 0 < t < T, hi ∈ C[0, T]
∞ with yi (0) = yi (T) = 0, where y(t) = yi (t) i=1 ∈ R∞ and i = 1, 2, 3, . . . . If yi (t) ∈ C[0, T] and 1 < α < 2, the unique solution of (3) is given by yi (t) = Z T G(t, s)hi (s, y(s))ds, (4) 0 where i = 1, 2, 3, ... and t ∈ I and the Green’s function associated to (3) is given by  i h α−1 1  α−1  − Tα−1 (t − s)α−1 , 0 ≤ s ≤ t ≤ T,  Tα−1 Γ(α) t (T − s) G(t, s) =    Tα−11Γ(α) tα−1 (T − s)α−1 , 0 ≤ t ≤ s ≤ T. (5) β In this article we establish the existence of solution of the infinite systems (3) for the sequence spaces C(I, c0 ) and C(I, cβ ). β 3. Solvability of infinite systems of fractional differential equations in C(I, c0 ) Suppose that (i) The functions hi are defined on the set I × R∞ , where I = [0, T] and take real values. The operator h β β defined on the space I × c0 into c0 as


∞ t, y(t) → hy (t) = hi (t, y(t)) i=1

β is the class of all functions hy (t) t∈I which is equicontinuous at every point of the space c0 . 5524 A. Das et al. / Filomat 33:17 (2019), 5519–5530 β (ii) For every y(t) ∈ c0 , t ∈ I, i ∈ N we have
hi t, y(t) ≤ ai (t) + bi (t) yi (t) , where
for all i ∈ N and both ai (t), bi (t) are real continuous functions defined on I such that the sequence βi ai (t) converges uniformly to zero on I and the sequence (bi (t)) is equibounded on I. Let us assume b(t) = sup {bi (t)} i∈N B = sup {b(t)} t∈I  A = sup βi ai (t) i∈N,t∈I and 2BTα Γ(α) < 1. β Theorem 3.1. Under the hypothesis (i)-(ii), infinite systems (3) has at least one solution y(t) = (yi (t)) ∈ C(I, c0 ) for all t ∈ I.  β ∈ C(I, c0 ) and t ∈ I, where L is a finite positive real Proof. We have sup βi | yi (t) | ≤ L for all y(t) = (yi (t))∞ i=1 i∈N number. By using (4) and (ii), for arbitrary fixed t ∈ I, we have " Z k y(t) kcβ = sup βi 0 i≥1 " Z ≤ sup βi i≥1 T G(t, s)hi (s, y(s))ds 0 T 0 # # |G(t, s)| hi (s, y(s)) ds " Z T #  2T ≤ sup βi ai (s) + bi (s) | yi (s) | ds Γ(α) i≥1 0 # "Z T 2Tα−1 sup (A + BL)ds ≤ Γ(α) i≥1 0 α−1 2(A + BL)Tα = d(say) Γ(α) i.e. k y(t) kcβ ≤ d. = 0 Thus max k y(t) kcβ ≤ d =⇒ k y kC(I,cβ ) ≤ d. t∈I 0 0 ∞ Let y0 (t) = y0i (t) , where y0i (t) = 0 ∀ t ∈ I, i ∈ N. i=1  Consider B = B y0 (t), d the closed ball centered at y0 (t) and radius d, thus B is a non-empty, bounded,  β closed and convex subset of C(I, c0 ). β β For arbitrary fixed t ∈ I, define the operator S = (Si )∞ i=1 on from C(I, c0 ) to C(I, c0 ) defined as follows

Sy (t) = Si y (t) ∞ i=1 = (Z T 0 )∞ , G(t, s)hi (s, y(s))ds i=1 A. Das et al. / Filomat 33:17 (2019), 5519–5530 5525
∞ β where y(t) = yi (t) i=1 ∈ C(I, c0 ) and yi (t) ∈ C(I, R).

β As hi t, y(t) ∈ c0 for each t ∈ I, we have " Z T #
  lim βi Si y (t) = lim βi G(t, s)hi (s, y(s))ds i→∞ i→∞ = Z 0 T   G(t, s) lim βi hi (s, y(s)) = 0. i→∞ 0
β Hence Sy
(t) ∈ C(I, c0 ). Also Si y (t) satisfies boundary conditions i.e. Z T Z T
Si y (0) = 0. fi (s, y(s))ds = 0, G(0, s)hi (s, y(s))ds = 0 0
Si y (T) = Z T G(T, s)hi (s, y(s))ds = 0 Z T 0. fi (s, y(s))ds = 0. 0

For fixed t ∈ I and y(t) ∈ B we have k Sy (t) − y0 (t) kcβ ≤ d gives maxt∈I k Sy (t) − y0 (t) kcβ ≤ d =⇒ k 0 0
Sy (t) − y0 (t) kC(I,cβ ) ≤ d thus S is self mapping on B. 0 ∈ B and there exists ǫ > 0 for each δ > 0 such that By assumption (i) we can assume z(t) = (zi (t))∞ i=1 ǫΓ(α) k (hy)(t) − (hz)(t) kcβ < 2Tα for each y(t), z(t) ∈ B, whenever k y(t) − z(t) k≤ δ, where t ∈ I. 0 For arbitrary fixed t ∈ I n o k (Sy)(t) − (Sz)(t) kcβ = sup βi (Si y)(t) − (Si z)(t) 0 i≥1 ( Z ≤ sup βi i≥1 T 0 2Tα−1 ≤ sup Γ(α) i≥1 |G(t, s)| hi (s, y(s)) − hi (s, z(s)) ds (Z T 0 ) ) βi hi (s, y(s)) − hi (s, z(s)) ds α−1 < ǫΓ(α) 2T .T < ǫ. . Γ(α) 2Tα β Thus S is continuous on B ⊂ C(I, c0 ). Since t is arbitrarily fixed therefore S is continuous on B for all t ∈ I. We have  ) ( Z T    G(t, s)hk (s, y(s))ds  χ (SB) = lim  sup sup βk i→∞ y(t)∈B k≥i 0  (Z T ) α−1     2T  ≤ βk ak (s) + βk bk (s) yk (s)  lim  sup sup Γ(α) i→∞  y(t)∈B k≥i 0 ≤ 2BTα χ(B) Γ(α) α i.e. χ (SB) ≤ 2BT Γ(α) χ(B). α β (SB) ≤ Thus supt∈I χ (SB) ≤ 2BT Γ(α) supt∈I χ(B) =⇒ χC(I,c ) 2BTα β Γ(α) χC(I,c0 ) (B). ǫΓ(α) χC(I,cβ ) (B) < 2BTα . 0 0 α β Hence χC(I,cβ ) (SB) ≤ 2BT Γ(α) χC(I,c0 ) (B) < ǫ =⇒ 0 ǫ α Taking δ = 2BTα (Γ(α) − 2BT ) we get ǫ ≤ χC(I,cβ ) (B) < ǫ + δ. Therefore S is a Meir–Keeler condensing operator β 0 defined on the set B ⊂ C(I, c0 ). Since t is arbitrarily fixed, thus for all t ∈ I, S satisfies all the conditions of β Theorem 3 which implies S has a fixed point in B. Thus the systems (3) has a solution in C(I, c0 ). 5526 A. Das et al. / Filomat 33:17 (2019), 5519–5530 4. Examples Let us consider the following systems of differential equations ∞ 2 X yi (t) e−it D yi (t) = − 2 − i 4 j2 j=i 3 2
with yi (0) = yi (1) = 0, where hi t, y(t) = Here T = 1, α = β 3 2. Let βi = 1 i2 (6) 2 e−it i2 for all i ∈ N. + yi (t) j=i 4j2 , P∞ ∀ i ∈ N, t ∈ (0, 1) . If y(t) ∈ C(I, c0 ) then for any t ∈ [0, 1] we have   ∞ X   e−it2 y (t) 1 i   lim βi hi t, y(t) = lim  4 + 2  = 0. i→∞ i→∞  i i j=i 4 j2 



β β Thus if y(t) = yi (t) ∈ C(I, c0 ) i.e. hi t, y(t) ∈ c0 . β Let t ∈ [0, 1] and z(t) ∈ C(I, c0 ) be arbitrary, where z(t) = (zi (t))∞ i=1 . For ǫ > 0, we have n o k (hy)(t) − (hz)(t) kcβ = sup βi hi (t, y(t)) − hi (t, z(t)) 0 i≥1   )  ∞ (     1 X yi (t) zi (t)   − = sup     2 2 2   4j 4j  i≥1  i j=i     ∞  X   1   1 ≤ sup  yi (t) − zi (t)   2 2   j   i≥1  4i j=i ≤ π2 k y(t) − z(t) kcβ < ǫ 0 24
β whenever k y(t) − z(t) kcβ < δ = 24ǫ , which implies the equicontinuity of (hy)(t) t∈I on c0 , where I = [0, 1]. π2 0 Moreover for all i ∈ N and t ∈ I, we have 2 2 ∞ X e−it π2 e−it 1 yi (t) , ≤ + hi (t, y(t)) ≤ 2 + yi (t) 24 i 4 j2 i2 j=i where ai (t) = 2 e−it i2 , bi (t) = π2 24 are real continuous functions on I and B = π2 24 .
We observe that βi ai (t) = α 2 π √2 converges uniformly to zero on I and the sequence {bi (t)} is equibounded on I. Also 2BT Γ(α) = 2. 24 . π ≈ Thus by Theorem 3.1 the systems (6) has unique solution in  9.86 10.63 e−it i4 2  < 1. β C(I, c0 ). 5. Solvability of infinite systems of fractional differential equations in C(I, cβ ) Suppose that (i) The functions hi are defined on the set I × R∞ , where I = [0, T] and take real values. The operator h defined on the space I × cβ into cβ as


∞ t, y(t) → hy (t) = hi (t, y(t)) i=1

is the class of all functions hy (t) t∈I which is equicontinuous at every point of the space cβ . A. Das et al. / Filomat 33:17 (2019), 5519–5530 5527 (ii) For every y(t) ∈ cβ , t ∈ I, i ∈ N we have
hi t, y(t) = âi (t) + b̂i (t)yi (t), where for all i ∈ N and both âi (t), b̂i (t) are nonnegative continuous   functions defined on I, the sequence
βi âi (t) converges uniformly to zero on I and the sequence b̂i (t) is convergent on I. Let us consider n o b̂(t) = sup b̂i (t) , i∈N n o B̂ = sup b̂(t) , t∈I  Â = sup βi âi (t) i∈N,t∈I and 2B̂Tα Γ(α) < 1. ∈ C(I, cβ ) Theorem 5.1. Under the hypothesis (i)-(ii), infinite systems (3) has at least one solution y(t) = (yi (t))∞ i=1 for all t ∈ I.  Proof. We have sup βi | yi (t) | ≤ L1 for all y(t) = (yi (t))∞ ∈ C(I, cβ ) and t ∈ I, where L1 is a finite positive real i=1 i∈N number. By using (4) and (ii), we have for arbitrary fixed t ∈ I, " k y(t) kcβ = sup βi i≥1 Z " Z ≤ sup βi i≥1 T G(t, s)hi (s, y(s))ds 0 T 0 # |G(t, s)| hi (s, y(s)) ds # # " Z T o n 2T sup βi âi (s) + b̂i (s) | yi (s) | ds Γ(α) i≥1 0 # "Z T 2Tα−1 ≤ (Â + B̂L1 )ds sup Γ(α) i≥1 0 α−1 ≤ 2(Â + B̂L1 )Tα = d1 (say) Γ(α) i.e. k y(t) kcβ ≤ d1 . = Thus max k y(t) kcβ ≤ d1 =⇒ k y kC(I,cβ ) ≤ d1 . t∈I  ∞ Let y0 (t) = y0i (t) , where y0i (t) = 0 ∀ t ∈ I, i ∈ N.  i=1  Consider B1 = B1 y0 (t), d1 , the closed ball centered at y0 (t) and radius d1 , thus B1 is a non-empty, bounded, closed and convex subset of C(I, cβ ). β β For arbitrarily fixed t ∈ I, define the operator S = (Si )∞ i=1 from C(I, c ) to C(I, c ) as follows

Sy (t) = Si y (t) ∞ i=1 = (Z T 0 )∞ , G(t, s)hi (s, y(s))ds i=1 A. Das et al. / Filomat 33:17 (2019), 5519–5530 5528
∞ where y(t) = yi (t) i=1 ∈ C(I, cβ ) and yi (t) ∈ C(I, R). Now let j ∈ N and βi (Si y)(t) − β j (Si y)(t) = βi Z ≤ βi Z ≤ Z T 0 T 0 T 0 G(t, s)hi (s, y(s))ds − β j Z T G(t, s)h j (s, y(s))ds 0 Z   G(t, s) âi (s) + b̂i (s)yi (s) ds − β j |G(t, s)| βi âi (s) − β j â j (s) ds + Z T 0 T 0   G(t, s) â j (s) + b̂ j (s)y j (s) ds |G(t, s)| βi b̂i (s)yi (s) − β j b̂ j (s)y j (s) ds. Also βi b̂i (s)yi (s) − β j b̂ j (s)y j (s) ≤ βi yi (s) b̂i (s) − b̂ j (s) + b̂ j (s) βi (s)yi (s) − β j (s)y j (s) . As i, j → ∞ we get b̂i (s) − b̂ j (s) → 0, βi (s)yi (s) − β j (s)y j (s) → 0 and βi âi (s) − β j â j (s) → 0 because 
 βi âi (t) , b̂i (t) are convergent on I and y(t) ∈ C(I, cβ ) for all t ∈ I. Thus as i, j → ∞ we get βi (Si y)(t) − β j (Si y)(t) → 0. Hence (Sy)(t) ∈ C(I, cβ ).

t ∈ I and y(t) ∈ B1 we have k Sy (t) − y0 (t) kcβ ≤ d1
gives maxt∈I k Sy (t) − y0 (t) kcβ ≤ d1 =⇒ k
For fixed 0 Sy (t) − y (t) kC(I,cβ ) ≤ d1 thus S is self mapping on B1 . Also Si y (t) satisfies boundary conditions i.e.
Si y (0) =
Si y (T) = Z T G(0, s)hi (s, y(s))ds = 0 Z T G(T, s)hi (s, y(s))ds = 0 Z T 0. fi (s, y(s))ds = 0, 0 Z T 0. fi (s, y(s))ds = 0. 0 By assumption (i) we can assume z̄(t) = (z̄i (t))∞ ∈ B1 and there exists ǫ > 0 for each δ > 0 such that i=1 ǫΓ(α) β k (hy)(t) − (hz̄)(t) kc < 2Tα for each y(t), z(t) ∈ B1 , whenever k y(t) − z̄(t) kcβ ≤ δ, where t ∈ I. For arbitrarily fixed t ∈ I, o n k (Sy)(t) − (Sz̄)(t) kcβ = sup βi (Si y)(t) − (Si z̄)(t) i≥1 ( Z ≤ sup βi i≥1 T 0 2Tα−1 sup ≤ Γ(α) i≥1 |G(t, s)| hi (s, y(s)) − hi (s, z̄(s)) ds (Z T 0 ) ) βi hi (s, y(s)) − hi (s, z̄(s)) ds α−1 < ǫΓ(α) 2T .T < ǫ. . Γ(α) 2Tα Thus S is continuous on B1 ⊂ C(I, cβ ). Since t is arbitrarily fixed therefore S is continuous on B1 for all t ∈ I. A. Das et al. / Filomat 33:17 (2019), 5519–5530 5529 We have for arbitrarily fixed t ∈ I, µcβ (SB1 )  ! ) ( Z T Z T    G(t, s)hk (s, y(s))ds − lim βm G(t, s)hm (s, y(s))ds  = lim  sup sup βk m→∞ i→∞ y(t)∈B k≥i 0 0 1  (Z T   )    = lim  sup sup G(t, s) βk b̂k (s)yk (s) − lim βm b̂m (s)ym (s) ds  m→∞ i→∞ y(t)∈B k≥i 0 1  ) (Z T   ≤ lim  sup sup |G(t, s)| βk b̂k (s)yk (s) − lim βm b̂m (s)ym (s) ds  m→∞ i→∞ y(t)∈B k≥i 0 1  ( ) Z T   2Tα−1  ≤ lim  sup sup ≤ βk b̂k (s)yk (s) − lim βm b̂m (s)ym (s) ds  m→∞ i→∞ y(t)∈B k≥i Γ(α) 0 1  ) (Z T  α−1     2T  lim  sup sup b̂k (s) βk (s)yk (s) − lim βm ym (s) + lim βm ym (s) b̂k (s) − b̂m (s) ds  ≤ m→∞ m→∞ Γ(α) i→∞ y(t)∈B1 k≥i 0 ≤ 2Tα B̂ µ β (B1 ) Γ(α) c 2B̂Tα 2B̂Tα β (SB1 ) ≤ β β Γ(α) µc (B1 ) =⇒ supt∈I µc Γ(α) supt∈I µc (B1 ). ǫΓ(α) 2B̂Tα Hence µC(I,cβ ) (SB1 ) ≤ Γ(α) µC(I,cβ ) (B1 ) < ǫ =⇒ µC(I,cβ ) (B1 ) < 2B̂Tα .   ǫ α we get ǫ ≤ µC(I,cβ ) (B1 ) < ǫ+δ. Therefore S is a Meir–Keeler condensing operator Taking δ = 2B̂T α Γ(α) − 2B̂T β defined on the set B1 ⊂ C(I, c ). Since t is arbitrarily fixed , thus for all t ∈ I, S satisfies all the conditions of Theorem 3 which implies S has a fixed point in B1 . Thus the systems (3) has a solution in C(I, cβ ). i.e. µcβ (SB1 ) ≤ 6. Examples Let us consider the following systems of differential equations   ∞ X t  1  D yi (t) = − 2 − 1 +  yi (t)  i 4 j2  j=i 3 2
with yi (0) = yi ( 14 ) = 0, where hi t, y(t) = (7) t i2  P + 1+ ∞ j=i 1 4 j2    yi (t), ∀ i ∈ N, t ∈ 0, 14 . Let βi = i12 for all i ∈ N. Here T = 14 , α = 23 . i h P π2 1 1 Also ai (t) = it2 , bi (t) = 1 + ∞ j=i 4j2 are real continuous functions on I = 0, 4 and B̂ = 1 + 24 . We observe that  
βi ai (t) = it4 converges uniformly to zero on I and the sequence {bi (t)} is convergent on I.   2 B̂Tα Also 2Γ(α) = 2. 1 + π24 . 18 . Γ(13 ) ≈ 1.41 3.55 < 1. 2 If y(t) ∈ C(I, cβ ) then for any t ∈ [0, 41 ] we have    ∞ X    t  y (t) 1 i    lim βi hi t, y(t) = lim  4 + 2 1 +  2  i→∞  i i→∞ i  4 j j=i


is unique and finite science y(t) ∈ C(I, cβ ) i.e. hi t, y(t) ∈ cβ . 5530 A. Das et al. / Filomat 33:17 (2019), 5519–5530 Let t ∈ I and z(t) ∈ C(I, cβ ) be arbitrary, where z(t) = (zi (t))∞ i=1 . For ǫ > 0, we have o n k (hy)(t) − (hz)(t) kcβ = sup βi hi (t, y(t)) − hi (t, z(t)) i≥1     ∞   X     1 1     ≤ sup  y (t) − z (t) 1 +   i i   i2  2   4 j   i≥1  j=i ! π2 k y(t) − z(t) kcβ < ǫ ≤ 1+ 24 whenever k y(t) − z(t) kcβ < δ = 0 ǫ 2 1+ π24 , which implies the equicontinuity of (hy)(t) Thus by Theorem 5.1, the systems (7) has unique solution in C(I, cβ ).
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