FIXED POINT FREE MAPPINGS WHICH SATISFY A DARBO TYPE CONDITION ELISABETTA MALUTA, STANISLAW PRUS, AND JACEK WOŚKO Abstract. We give constructions of continuous mappings on normed spaces which satisfy a Darbo type condition with respect to a measure of noncompactness and do not have fixed points or even have positive minimal displacement. An estimate for the minimal displacement is given. Introduction In [13], Lin and Sternfeld proved that if a bounded closed convex subset C of a Banach space is not compact, then there exists a lipschitzian mapping F : C → C with positive minimal displacement. Additionally F can be chosen in such a way that its Lipschitz constant is arbitrarily close to 1. Moreover, various estimates of minimal displacement of lipschitzian mappings in terms of the Lipschitz constant are known (see [10]). Instead of the Lipschitz condition some authors consider an analogous condition in which measures of noncompactness of a set and its image are compared. The best constant in this condition is called µ-norm of a mapping, where µ is a given measure of noncompactness. In particular, Darbo [5] proved a counterpart of the well-known Banach contraction principle for mappings with µ-norm less than 1. Using the result of Lin and Sternfeld it is easy to show that if a measure µ is equivalent to the Kuratowski measure, then for each bounded closed convex subset C of a Banach space which is not compact there exist lipschitzian mappings F : C → C with positive minimal displacement whose µ-norms are arbitrarily close to 1 (see [4]). On the other hand, Darbo’s theorem implies that a continuous mapping with µ-norm one does not have positive minimal displacement. Other applications of measures of noncompactness to minimal displacement problems can be found in [12] and [16]. Measures of noncompactness appear also in the spectral theory (see for instance [8]). A coefficient introduced in this theory motivated some authors to study the infimum of µ-norms of fixed point free continuous mappings of the unit ball of an infinite dimensional Banach space (see [18]). Darbo’s theorem shows that it is not less than 1 and in [1] it was proved that for an arbitrary measure of noncompactness µ there exists a fixed point free continuous mapping of the unit ball whose µ-norm equals 1. The method used in [1] works actually not only for the unit ball but for any set with nonempty interior. In [19] and [1] estimates for the minimal displacement of continuous mappings satisfying a Darbo type condition are also given. Again the results concern only the unit ball or sets containing balls. In this paper we give some constructions of fixed point free continuous mappings for arbitrary convex subsets of a normed space. In the case when µ is the Kuratowski or 2000 Mathematics Subject Classification. 47H09, 47H10. Key words and phrases. Measures of noncompactness, Darbo type conditions, minimal displacement. 1 2 E. MALUTA, S. PRUS, AND J. WOŚKO separation measure of noncompactness we obtain mappings with µ-norm one without any additional assumption. If µ is the Hausdorff measure we need to consider weakly compact subsets of Banach spaces with the non-strict Opial property. These results can be seen as analogs of the Lin–Sternfeld theorem, but in this case the minimal displacement of the resulting mappings is zero. Actually our construction gives mappings which are asymptotically regular. Mappings with this property were also obtained in [1] but only on the unit ball of a Banach space with a special basis. In the last part of the paper we study the minimal displacement problem. We begin with the case of unbounded sets. The authors are indebted to T. Kuczumow for suggesting them this problem. We show that if A is a convex unbounded subset of a normed space X, then there exists a continuous mapping F : X → A with µ-norm zero and arbitrarily large minimal displacement. This contrasts the result proved in [15]: each infinite dimensional Banach space contains a closed convex unbounded set A such that every nonexpansive self-mapping of A has minimal displacement zero. We next consider weakly compact convex subsets A of a separable Banach space X. We give the general construction of a continuous mapping F : X → A whose χ-norm and minimal displacement are estimated in terms of the measure χ(A). Similar results were obtained in [19], but they mainly concern the case when A is a ball. 1. Preliminaries In this paper we consider real normed spaces. Let X be such space. By BX we denote the closed unit ball of X. A general measure of noncompactness in X is defined in an axiomatic way (see [2]). However, in this paper we will consider only specific measures. Let A ⊂ X be a nonempty bounded set. The Kuratowski measure of noncompactness α(A) is defined as the infimum of all d > 0 such that A may be covered with finitely many sets with diameters at most d. Similarly, the Hausdorff measure of noncompactness χ(A) is defined as the infimum of all r > 0 such that A may be covered with finitely many balls with radii at most r. Let now (xn ) be a sequence in X. We put sep(xn ) = inf{kxi − xj k : i 6= j}. The third standard measure of noncompactness is called the separation measure. It is defined by the formula β(A) = sup{sep(xn ) : (xn ) ⊂ A}. These measures are equivalent. More precisely, the following inequalities hold χ(A) ≤ β(A) ≤ α(A) ≤ 2χ(A). A subset A of a normed space X is called totally bounded if for every ǫ > 0 there is a finite ǫ-net for A. If A is relatively compact, then A is totally bounded and for Banach spaces also the opposite implication holds. Let µ ∈ {α, β, χ}. Clearly, a bounded set A is totally bounded if and only if µ(A) = 0. Let D be a nonempty subset of a normed space X. A mapping F : D → X is nonexpansive if it has Lipschitz constant one. Given a measure of noncompactness µ in X, we say that F satisfies a Darbo type condition if there exists a constant L ≥ 0 such that (1) µ(F (C)) ≤ Lµ(C) for every bounded set C ⊂ D. By [F ]µ we denote the infimum of all such constants L. We call it µ-norm of F . Clearly, condition (1) holds also with L = [F ]µ . In the FIXED POINT FREE MAPPINGS WHICH SATISFY A DARBO TYPE CONDITION 3 terminology of [2], a continuous mapping F satisfying condition (1) is said to be L-µcontractive. In case µ = α the term L-set-contractive is also used. Darbo [5] proved that if D is a bounded closed convex subset of a Banach space and a continuous mapping F : D → D satisfies the condition [F ]α < 1, then F has a fixed point. This result holds also for abstract measures of noncompactness (see [2]). Given a mapping F : D → X, by the minimal displacement of F we mean the number d(F ) = inf{kx − F (x)k : x ∈ D}. A mapping F : D → X is said to be asymptotically regular if lim kF n+1 (x) − F n (x)k = 0 n→∞ for every x ∈ D. Of course in this case d(F ) = 0. Curves are our main tool for constructing mappings with appropriate properties. By a curve in a normed space X we mean a continuous mapping Γ : I → X where I is an interval of the real line R. In this paper we will consider only the case when I is of the form [a, +∞). A curve is simple if it is one-to-one. We will identify a curve Γ with its image Γ(I). 2. Mappings without fixed points To prove the main theorem of this section we need the following lemma. Lemma 2.1. Let X be a normed space and µ ∈ {α, β}. If a bounded convex set A ⊂ X is not totally bounded, then there exists a simple curve Γ in A such that µ(G) = µ(Γ) for every set G ⊂ Γ which is not totally bounded. Proof. We can assume that 0 ∈ A and diam A ≤ 1. There exist r > 0 and a sequence (xn )∞ n=0 in A such that (2) dist (xn , span{x0 , . . . , xn−1 }) ≥ r for every n ∈ N (see [11, Lemma 19.4]). Passing to a subsequence, we can assume that for any numbers a1 , a2 , a3 , a4 ∈ R the following limit exists lim n1 <···<n4 n1 →∞ 4 X ai xni i=1 (see [3] or [7, Proposition 19.17]). Given t ∈ [0, 1], we now put ϕ(t) = lim k(1 − t)xn−1 + txn − (1 − t)xm − txm+1 k. n<m n→∞ From (2) we see that ϕ(t) ≥ rt. Next, r(1 − t) ≤ k(1 − t)xm − ((1 − t)xn−1 + txn )k ≤ ktxm+1 + (1 − t)xm − ((1 − t)xn−1 + txn )k + tkxm+1 k   1 ≤ 1+ ktxm+1 + (1 − t)xm − ((1 − t)xn−1 + txn )k r whenever n < m. Consequently, c ϕ(t) ≥ c max{t, 1 − t} ≥ 2 4 E. MALUTA, S. PRUS, AND J. WOŚKO where c = r2 /(1 + r). We set c ((1 − t)xn + txn+1 ) 2ϕ(t) where t ∈ [0, 1] and n = 0, 1, 2, . . . . Obviously, Γn (t) ∈ A and c (3) lim kΓn−1 (t) − Γm (t)k = n<m 2 n→∞ Γn (t) = for every t ∈ [0, 1]. Next, ϕ(t) ϕ(t) kΓn (t) − Γn (s)k ≤ Γn (t) − Γn (t) + Γn (t) − Γn (s) ϕ(s) ϕ(s)   1 1 1 c − |t − s| kxn+1 − xn k kΓn (t)k + ≤ 2 ϕ(t) ϕ(s) ϕ(s) 2 ≤ |ϕ(t) − ϕ(s)| + |t − s| c for every n = 0, 1, 2, . . . and all t, s ∈ [0, 1]. Moreover, |ϕ(t) − ϕ(t)| ≤ 2|t − s|, so we finally obtain the inequality kΓn (t) − Γn (s)k ≤ M |t − s| (4) where M = 4/c + 1. We define the desired curve in A by the formula Γ(s) = Γ⌊s⌋ ({s}) where s ∈ [0, +∞), ⌊s⌋ = max{n ∈ Z : n ≤ s} and {s} = s − ⌊s⌋. We shall show that if (yn ) is a sequence on Γ without a convergent subsequence, then there exists a subsequence (ynk ) for which c (5) lim kyni − ynk k = . i<k 2 i→∞ Observe that if yn = Γ(sn ), then limn→∞ sn = +∞. Consequently, limn→∞ ⌊sn ⌋ = +∞, so there exists a subsequence (snk ) such that ⌊snk ⌋ + 1 < ⌊snk+1 ⌋ for every k ∈ N and ({snk }) converges to some t ∈ [0, 1]. From (4) we obtain kΓ(snk ) − Γ(⌊snk ⌋ + t)k ≤ M |{snk } − t| for every k ∈ N which shows that lim kΓ(snk ) − Γ(⌊snk ⌋ + t)k = 0. k→∞ This and (3) give lim kyni − ynk k = lim kΓ(⌊sni ⌋ + t) − Γ(⌊snk ⌋ + t)k i<k i→∞ i<k i→∞ = lim kΓ⌊sni ⌋ (t) − Γ⌊snk ⌋ (t)k = i<k i→∞ c . 2 Consider now a set G ⊂ Γ with µ(G) > 0. There exists a sequence (yn ) in G which does not have a convergent subsequence. We can therefore find a subsequence (ynk ) for FIXED POINT FREE MAPPINGS WHICH SATISFY A DARBO TYPE CONDITION 5 which formula (5) holds. For every ǫ > 0 there is m ∈ N such that sep(ynk )∞ k=m ≥ c/2−ǫ. This shows that β(G) ≥ c/2. Since β(G) ≤ α(G) ≤ α(Γ), it suffices to check that α(Γ) ≤ c/2. For this purpose we take an arbitrary ǫ > 0 and choose 0 < t1 < · · · < tN < 1 such that for every t ∈ [0, 1] there exists k with |t − tk | < ǫ/(3M ). By (3), there exists m0 ∈ N such that if m0 < n < m, then ǫ c kΓ(n − 1 + tk ) − Γ(m + tk )k − < 2 3 for any k = 1, . . . , N . We set n ǫ o Ak = Γ(m0 + 2p + t) : p ∈ N, t ∈ [0, 1], |t − tk | < 3M n ǫ o AN +k = Γ(m0 + 2p − 1 + t) : p ∈ N, t ∈ [0, 1], |t − tk | < 3M where k = 1, . . . , N . Let s1 , s2 ∈ [0, 1] be such that there is k ∈ {1, . . . , N } for which |si − tk | < ǫ/(3M ) for i = 1, 2. Then from (4) we obtain ǫ kΓ(n + si ) − Γ(n + tk )k < 3 for every n ∈ N and i = 1, 2. This shows that 2ǫ kΓ(n + s1 ) − Γ(n + s2 )k < . 3 Moreover, if n, m ∈ N, n, m > m0 and |n − m| > 1, then kΓ(n + s1 ) − Γ(m + s2 )k < 2ǫ c + kΓ(n + tk ) − Γ(m + tk )k < + ǫ. 3 2 It follows that diam Ak ≤ c +ǫ 2 for every k = 1, . . . , 2N . Clearly, {Γ(s) : s > m0 } ⊂ 2N [ Ak k=1 and the set {Γ(s) : s ≤ m0 } is compact. This shows that α(Γ) ≤ c/2 which completes the proof.  Now we can prove our theorem. Theorem 2.2. Let X be a normed space and µ ∈ {α, β}. If a bounded convex set A ⊂ X is not totally bounded, then there exists a continuous mapping F : X → A such that [F ]µ = 1, F is asymptotically regular and F does not have a fixed point. Proof. Consider the curve Γ in A constructed in the proof of Lemma 2.1. Observe that dist(xn+2 , Γn ) > 0 for every n = 0, 1, 2, . . . . We can therefore find a sequence (ǫn )∞ n=0 converging to 0 with 0 < ǫn < dist(xn+2 , Γn ) for every n. We use a modification of a construction given in [18]. Namely, we set Tn = {x ∈ X : dist(x, Γn ) ≤ ǫn }, T = ∞ [ n=0 Tn , Dn = n [ k=0 Tk ∪ (X \ Int T ) 6 E. MALUTA, S. PRUS, AND J. WOŚKO where n = 0, 1, 2, . . . . By induction we shall construct a sequence of continuous functions fn : Dn → [0, n + 2] such that fn (Γ(t)) = t if Γ(t) ∈ Dn , fn (x) = 0 if x ∈ X \ Int T and fn+1 is an extension of fn . As the first step, we set ψ0 (Γ(t)) = t if Γ(t) ∈ D0 , and ψ0 (x) = 0 if x ∈ X \ Int T . Using the Urysohn lemma, we extend ψ0 to a continuous function f0 : D0 → [0, 2]. Assuming that the desirable functions f0 , . . . , fn have been found, we put ψn+1 (Γ(t)) = t if Γ(t) ∈ Dn+1 \ Dn and ψn+1 (x) = fn (x) if x ∈ Dn . A continuous extension fn+1 : Dn+1 → [0, n + 3] of ψn+1 has the required properties. Now we define f : X → [0, +∞) in the following way. If x ∈ X, then x ∈ Dn for some n ∈ N∪{0} and we set f (x) = fn (x). The function f is continuous and f (Dn ) ⊂ [0, n+2]
1 for each n ∈ N ∪ {0}. Next we define g : [0, +∞) → Γ by the formula g(t) = Γ t + 1+t . We claim that F = g ◦ f has the required properties. Suppose that F (x) =
x. Then 1 x = Γ(s) for some s ∈ [0, +∞), so F (x) = g(f (x)) = g(s) = Γ s + 1+s . Hence
1 Γ s + 1+s = Γ(s) which contradicts the fact that Γ does not have a double point. Therefore F has no fixed point. Moreover, F (Γ) = {Γ(s) : s ≥ 1}, so µ(F (Γ)) = µ(Γ). This shows that [F ]µ ≥ 1. To prove the opposite inequality consider a bounded set C ⊂ X and put C0 = C ∩ T . Then F (C) \ F (C0 ) is empty or a singleton. Assume that µ(F (C)) > 0. It follows that C0 \ Dn 6= ∅ for every n ∈ N ∪ {0}. We can therefore find an increasing sequence of positive integers (mk ) and a sequence (vk ) in C0 such that vk ∈ Tmk for every k. Given k ∈ N we choose yk ∈ Γmk with kvk − yk k ≤ ǫk . Then (yk ) does not have a convergent subsequence, so there exists a subsequence (ynk ) for which formula (5) holds. Consequently, limi<k kvni − vnk k = c/2. Hence for every ǫ > 0 there exists m ∈ N such that sep(vnk )∞ k=m ≥ c/2 − ǫ. Therefore c/2 ≤ β(C0 ) ≤ µ(C0 ). This and the proof of Lemma 2.1 show that c µ(F (C)) = ≤ µ(C0 ) ≤ µ(C). 2 Hence [F ]µ ≤ 1 and we finally see that [F ]µ = 1. To show that F is asymptotically regular we take x ∈ X and put F n (x) = Γ(tn ). Then tn+1 = tn + 1/(1 + tn ) for every n ∈ N, so the sequence (tn ) tends to infinity and consequently, limn→∞ (tn+1 − tn ) = 0. In particular tn+1 − tn < 1 for n large enough. For such n either ⌊tn+1 ⌋ = ⌊tn ⌋ or ⌊tn+1 ⌋ − 1 ≤ tn < ⌊tn+1 ⌋. Applying (4), in the first case we see that kF n+1 (x) − F n (x)k ≤ M (tn+1 − tn ) and in the second case kF n+1 (x) − F n (x)k ≤ M (tn+1 − ⌊tn+1 ⌋) + M (⌊tn+1 ⌋ − tn ) = M (tn+1 − tn ). This shows that limn→∞ kF n+1 (x) − F n (x)k = 0.  Let us recall that χ(A) = α(A)/2 for any bounded set A in the space l∞ (see [2, p. 24]). Consequently, Theorem 2.2 holds for l∞ also with µ = χ. In general when considering Hausdorff measure, we obtain similar results under the additional hypothesis that X has the non-strict (or weak) Opial property, i.e. if for every weakly null sequence (xn ) in X and every x ∈ X we have lim inf kxn k ≤ lim inf kx + xn k. n→∞ n→∞ FIXED POINT FREE MAPPINGS WHICH SATISFY A DARBO TYPE CONDITION 7 Theorem 2.3. Let X be a Banach space which has the non-strict Opial property. If a convex weakly compact subset A of X is not compact, then there exists a continuous mapping F : X → A such that [F ]χ = 1, F is asymptotically regular and F does not have a fixed point. Proof. We can assume that A contains a sequence (xn )∞ n=0 which is weakly null but inf{kxn k : n = 0, 1, . . . } > 0. Passing to a subsequence if necessary, we may assume that it is a basic sequence (see [14]). Then cn = inf {k(1 − t)xm + txm+1 k : t ∈ [0, 1], m ≥ n} > 0 for every n. We set Γn (t) = cn ((1 − t)xn + txn+1 ) k(1 − t)xn + txn+1 k where t ∈ [0, 1] and n = 0, 1, 2, . . . . Clearly, Γn (t) ∈ A and the formula Γ(s) = Γ⌊s⌋ ({s}), where s ≥ 0, defines a simple curve in A. We have kΓ(s)k ≤ supn≥0 cn = limn→∞ cn for every s ≥ 0, which in particular shows that χ(Γ) ≤ limn→∞ cn . Now we proceed as in the proof of Theorem 2.2 and obtain a continuous mapping F : X → A which is asymptotically regular but does not have a fixed point. We need only to show that [F ]χ ≤ 1. To this end we consider a set C0 ⊂ T such that χ(F (C0 )) > 0, where T is the set constructed in the proof of Theorem 2.2. Then there exist an increasing sequence of positive integers (mk ) and points vk ∈ C0 , yk ∈ Γmk with kvk − yk k ≤ ǫk . Clearly, the sequence (yn ) is weakly null, so lim inf kx − vn k = lim inf kx − yn k ≥ lim inf kyn k = lim cn n→∞ n→∞ n→∞ n→∞ for every x ∈ X. Moreover, n o χ(C0 ) ≥ χ({vn : n ∈ N}) ≥ inf lim inf kx − vn k : x ∈ X . n→∞ It follows that χ(F (C0 )) ≤ χ(Γ) ≤ χ(C0 ).  3. Mappings with positive minimal displacement In our next theorem we deal with unbounded sets. In [15], it was proved that each infinite dimensional Banach space X contains an unbounded convex closed set A such that d(F ) = 0 for every nonexpansive mapping F : A → A. In the case when X is reflexive a simple characterization of such sets A was given in [17]. Namely, they are linearly bounded, i.e., the intersections of A with all lines are bounded. The case of mappings satisfying a Darbo type condition is completely different. Indeed, we will show that for every unbounded convex set there exists a continuous mapping with µ-norm zero and positive minimal displacement. Theorem 3.1. Let A be a convex unbounded subset of a Banach space X. For every ǫ > 0 there exists a continuous mapping F : X → A such that d(F ) ≥ ǫ and if a set C ⊂ X is bounded, then F (C) is relatively compact. 8 E. MALUTA, S. PRUS, AND J. WOŚKO Proof. The set A is not bounded, so there exists a norm-one linear functional x∗ on X for which sup{x∗ (x) : x ∈ A} = +∞. We can therefore find a sequence (xn ) in A such that x∗ (x1 ) > 2ǫ and x∗ (xn+1 ) ≥ 2x∗ (xn ) for every n ∈ N. Clearly, k(1 − t)xn + txn+1 − ((1 − s)xm + sxm+1 ) k ≥ x∗ (xn ) − x∗ (xm+1 ) > 2ǫ whenever t, s ∈ [0, 1] and n > m+1. Consequently, the segments Γn (t) = (1−t)xn +txn+1 where n ∈ N, t ∈ [0, 1], give us a simple curve Γ in A. We now follow the idea used in the proof of Theorem 1 in [19]. Namely, we put Gn = {x ∈ X : dist(x, Γn ) ≤ ǫ}, Tn = {x ∈ X : dist(x, Γn ) ≤ 2ǫ} S S∞ where n ∈ N, G = ∞ G , and T = n n=1 n=1 Tn . Using the Urysohn lemma we find a continuous function g : G → [1, +∞) such that g(Gn ) ⊂ [n, n + 1] and g(x) = n + 1 for every x ∈ Gn ∩ Gn+1 and every n ∈ N. Next, a construction similar to that in the proof of Theorem 2.2 gives us a continuous extension f : X → [1, +∞) of g for which f (x) = 1 for every x ∈ X \ T and f (Tn ) ⊂ [1, n + 3] for every n ∈ N. We finally put F (x) = Γ(f (x) + 2) for each x ∈ X. If x ∈ X \ G, then dist(x, Γ) > ǫ and in particular kx − F (x)k > ǫ. If in turn x ∈ G, then x ∈ Gn for some n ∈ N. Consequently, there exists y ∈ Γn such that kx − yk ≤ ǫ. But F (Gn ) ⊂ Γn+2 , so ky − F (x)k > 2ǫ which shows that kx − F (x)k > ǫ. We therefore see that d(F ) ≥ ǫ. Given r S > 0, we can find m ∈ N so that Tn ⊂ X \ rBX for every n ≥ m. Then F (rBX ) ⊂ m+1  k=3 Γk which shows that the set F (rBX ) is relatively compact. In [9], an upper estimate for the minimal displacement in terms of the α-norm was obtained. Namely, given a bounded subset A of a normed space X, by χ0 (A) we denote the inner Hausdorff measure of noncompactness of A, i.e. χ0 (A) is the infimum of all r > 0 such that A may be covered with finitely many balls with centers in A and radii at most r. Then   1 d(F ) ≤ 1 − χ0 (C) [F ]α for every continuous mapping F : C → C, where C is a bounded convex subset of X. In the last part of our paper we will obtain a lower estimate for the minimal displacement. For this purpose we need two preliminary results. Lemma 3.2. Let X be a separable Banach space and let A ⊂ X be convex and weakly compact but not compact. Then there exists a simple curve Γ in A such that 1 lim inf kx − Γ(t)k ≥ χ(A) t→+∞ 2 for every x ∈ X. Proof. S∞ Let (Xn ) be an ascending sequence of finite dimensional subspaces of X such that n=1 Xn is dense in X. We can assume for simplicity that χ(A) = 1. Let c ≥ 1 be such that kxk ≤ c for every x ∈ A. By induction we will choose a sequence (xn ) in A so that 2 n+1 for every x ∈ span (Xn ∪ {x1 , . . . , xn }). As x1 we take an arbitrary element of A. Next, having x1 , . . . , xn , we find a finite 1/(n + 1)-net {z1 , . . . , zm } for the set Y ∩ 2cBX where kx − xn+1 k ≥ 1 − FIXED POINT FREE MAPPINGS WHICH SATISFY A DARBO TYPE CONDITION 9 Y = span (Xn ∪ {x1 , . . . , xn }). Since χ(A) = 1, there exists xn+1 ∈ A such that kzi − xn+1 k ≥ 1 − 1/(n + 1) for every i = 1, . . . , m. Let now x ∈ Y . If kxk > 2c, then kx − xn+1 k ≥ kxk − kxn+1 k > c ≥ 1. If in turn kxk ≤ 2c, then there is i ∈ {1, . . . , m} such that kx − zi k < 1/(n + 1) and we see that 2 1 ≥1− . kx − xn+1 k > kzi − xn+1 k − n+1 n+1 We find a subsequence (ym ) of (xn ) which converges weakly to some y ∈ A and such that (ym − y) is a basic sequence (see [14]). Next, by induction we will choose an increasing sequence (mk ) of positive integers so that (6) kx − ymk+1 k ≥ kx − yk − 3 k+1 for every x ∈ span (Xk ∪ {ym1 , . . . , ymk }) ∩ 2cBX . To this end we set m1 = 1 and having ym1 , . . . , ymk , we take a finite 1/(k + 1)-net {v1 , . . . , vs } for the set Z ∩ 3cBX where Z = span (Xk ∪ {ym1 , . . . , ymk }). Given i ∈ {1, . . . , s}, we find a norm-one functional vi∗ ∈ X ∗ for which vi∗ (vi ) = kvi k. There exists mk+1 > mk such that vi∗ ymk+1 − y
< 1 k+1 for every i = 1, . . . , s. Let now x ∈ Z ∩ 2cBX . Then there exists i ∈ {1, . . . , s} such that kx − y − vi k < 1/(k + 1). Therefore, kx − ymk+1 k > vi − ymk+1 − y
− ≥ vi∗ vi − ymk+1 − y > kvi k − 2 k+1 ≥ kx − yk − 1 k+1

− 1 k+1 3 . k+1 Since (ymk ) is a subsequence of (xn ), (7) kx − ymk+1 k ≥ 1 − 2 k+1 for every x ∈ span (Xk ∪ {ym1 , . . . , ymk }). This easily implies that (8) lim inf kx − ymk+1 k ≥ 1 k→∞ for every x ∈ X. Since (ymk − y) is a basic sequence, the segments Γk (t) = (1 − t)ymk+1 + tymk , where t ∈ [0, 1], give us a simple curve Γ. Let x ∈ X and t ∈ [0, 1]. If kxk ≥ 2c, then kx − Γk (t)k ≥ c > 1/2. Assume that kxk < 2c. We take ǫ ∈ (0, 2c − kxk) and find i ∈ N 10 E. MALUTA, S. PRUS, AND J. WOŚKO and an element z ∈ Xi for which kx − zk < ǫ. Observe that kzk < 2c. If 1 − t ≥ 1/2, then (7) yields 1 t lim inf kz − Γk (t)k = (1 − t) lim inf z− ym − ymk+1 k→∞ k→∞ 1−t 1−t k   1 2 1 ≥ lim 1 − = . 2 k→∞ k+1 2 In the case when t > 1/2 from (6) and (8) we see that   t 3 1 z− ym − y − lim inf kz − Γk (t)k ≥ (1 − t) lim inf k→∞ k→∞ 1−t 1−t k k+1 1 1 1−t = t lim inf z − y − ymk ≥ t > . k→∞ t t 2 Hence lim inf kx − Γk (t)k ≥ lim inf kz − Γk (t)k − ǫ ≥ k→∞ k→∞ and passing to the limit with ǫ → 0, we finally obtain 1 1 lim inf kx − Γk (t)k ≥ = χ(A). k→∞ 2 2 This clearly implies the desired conclusion. 1 −ǫ 2  Let us point out that in general the coefficient 1/2 in Lemma 3.2 cannot be replaced by any bigger constant. To see this we consider the closed convex hull A of the standard basis (en ) of the unit vectors in c0 . Given x ∈ c0 we write x = (x(n)), where x(n) ∈ R for all n ∈ N. Then ( ) ∞ X A = x ∈ c0 : x(n) ≤ 1, x(n) ≥ 0, n = 1, 2, . . . n=1 and it is easy to see that χ(A) = 1. Let Γ be a curve in A. We will show that there exists x ∈ c0 such that 1 lim inf kx − Γ(t)k ≤ . t→+∞ 2 To this end assume that lim inf t→+∞ kΓ(t)k > 1/2. Then we can find s ≥ 0 such that inf{kΓ(t)k : t ≥ s} > 1/2. Consequently, for every t ≥ s there exists a unique positive integer n = n(t) for which (Γ(t))(n) > 1/2. Since n(t) is a continuous function of t we see that it is constant, i.e. there is m ∈ N such that (Γ(t))(m) > 1/2 for every t ≥ s. It follows that lim inf t→+∞ kem − Γ(t)k ≤ 1/2. Lemma 3.3. Let X be a separable Banach space S and (Xn ) be an ascending sequence of finite dimensional subspaces of X such that ∞ n=1 Xn is dense in X. Given δ > 0, we put U1 = {x ∈ X : dist(x, X1 ) < 2δ/3} and      1 1 Un = x ∈ X : dist(x, Xn ) < δ 1 − , dist(x, Xn−1 ) > δ 1 − n+2 n S∞ if n ≥ 2. Then X = k=1 Uk and a nonempty bounded subset A of X satisfies the condition χ(A) < δ if and only if the set {n ∈ N : Un ∩ A 6= ∅} is finite. FIXED POINT FREE MAPPINGS WHICH SATISFY A DARBO TYPE CONDITION 11 Proof. Let x ∈ X. If x ∈ / U1 , then dist(x, X1 ) > δ/2, so the set    1 M = n ∈ N : n ≥ 2, dist(x, Xn−1 ) > δ 1 − n is nonempty. It is also finite and we put k = max M . Then S dist(x, Xk ) ≤ δ(1 − 1/(k + 1)) which shows that x ∈ Uk . We therefore see that X = ∞ k=1 Uk . Let A be a nonempty bounded subset of X. If χ(A) < δ, then A can be covered with finitely many balls with radii less than δ. We can assume that their centers are contained in some subspace Xn . Thus there exists m such that dist(x, Xm ) < δ (1 − 1/(m S + 1)) for every x ∈ A. This shows that A ∩ Uk = ∅ for all k > m. Conversely, if A ⊂ nk=1 Uk for some n, then dist(x, Xn ) < δ (1 − 1/(n + 2)) for every x ∈ A. Using the assumption that A is bounded and the subspace Xn is finite dimensional, we now easily conclude that χ(A) < δ.  Using Lemma 3.2 and Theorem 1 in [19], one can show that if a weakly compact convex subset A of a separable Banach space X is not compact, then for every ǫ > 0 there exists a continuous mapping F : X → A such that [F ]χ ≤ 1/4 and d(F ) > χ(A)/4 − ǫ. Our next theorem generalizes this result. Theorem 3.4. Let X be a separable Banach space and let A be a convex weakly compact subset of X which is not compact. Then for every positive δ and ǫ there exists a continuous mapping F : X → A such that [F ]χ ≤ χ(A)/δ and 1 d(F ) ≥ χ(A) − δ − ǫ. 2 Proof. Let Γ be a simple curve in A constructed in the proof of Lemma 3.2. S∞We take an ascending sequence (Xn ) of finite dimensional subspaces of X such that n=1 Xn is dense in X and δ > 0. Let (Un ) be the corresponding sequence of sets constructed in Lemma 3.3. We choose an increasing sequence (tn ) of positive numbers such that for every n ∈ N 1 (9) kx − Γ(t)k ≥ χ(A) − ǫ 2 whenever x ∈ Xn and t ≥ tn . Given n ∈ N and x ∈ X, we set µn (x) φn (x) = P∞ k=1 µk (x) where µk (x) = dist(x, X \ Uk ). Observe that each x ∈ X belongs to Uk for at most two values of k. Only for those values µk (x) 6= 0. Moreover, there exist a ball B centered at x and a set S ⊂ N consisting of at most three numbers such that µk (y) = 0 for all y ∈ B and k ∈ N \ S. It follows that the formula ! ∞ X F (x) = Γ φn (x)tn n=1 gives a continuous function F : X → A. Consider a nonempty bounded set C ⊂ X. If χ(C) < δ, then by Lemma 3.3 there is m ∈ N such that µn (x) = 0 for every x ∈ C and every n > m. This shows that the set F (C) is compact. If in turn χ(C) ≥ δ, then χ(F (C)) ≤ χ(A) ≤ χ(A)χ(C)/δ. This shows that [F ]χ ≤ χ(A)/δ. 12 E. MALUTA, S. PRUS, AND J. WOŚKO If x ∈ X, then x ∈ Un ∪ Un+1 for some n. Consequently, F (x) = Γ (φn (x)tn + φn+1 (x)tn+1 ) . There exists y ∈ Xn such that kx − yk < δ (1 − 1/(n + 2)). From (9) we obtain 1 kx − F (x)k ≥ ky − F (x)k − kx − yk > χ(A) − ǫ − δ. 2  The separability assumption can be relaxed in Theorem 3.4. Namely, there is a large class of Banach spaces X such that for each separable subspace Y of X there exists a norm-one linear projection P on X for which P (X) is separable and Y ⊂ P (X). This class contains all reflexive spaces and more generally all weakly compactly determined spaces (see [6]). Given such space X, we can proceed as in the proof of Theorem 3.4. We just consider a separable subspace Y containing the curve Γ and find a corresponding projection P . Then the proof of Theorem 3.4 gives us the mapping F : P (X) → Γ and we put F1 = F ◦ P . It is easy to see that if A ⊂ P (X), then the Hausdorff measures of noncompactness of A relative to P (X) and X coincide. Moreover, d(F1 ) = d(F ). We therefore obtain the following corollary. Corollary 3.5. The conclusion of Theorem 3.4 holds for any convex weakly compact, but not compact subset A of a weakly compactly determined Banach space X. We use results of Banach space theory in the proofs of Theorems 2.3, 3.1, and 3.4. However, the assumption of completeness of X is not essential in these theorems. Indeed, if X is not complete, then considering its completion X, we can obtain an appropriate mapping F defined on X and take its restriction to X. In Theorems 2.3 and 3.4 we deal with the Hausdorff measure of noncompactness related to the whole space, but it is easy to see that such measures evaluated with respect to X and X are equal. References [1] J. Appel, N. A. Erzakova, S. Falcon Santana, M. Väth, On some Banach space constants arising in nonlinear fixed point and eigenvalue theory, Fixed Point Theory and Appl. 2004 (2004), 317–336. [2] J. M. Ayerbe, T. Domı́nguez Benavides, G. López Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Birkhäuser-Verlag, Basel, Boston, Berlin, 1997. [3] A. Brunel, L. Sucheston, On B-convex Banach spaces, Math. System Theory 7 (1974), 294–299. [4] D. Caponetti, G. Trombetta, A note on the measure of solvability, Bull. Polish Acad. Sci. Math. 52 (2004), 179–183. [5] G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84–92. [6] R. Deville, G. Godefroy, V. Zizler Smoothness and Renormings in Banach Spaces, Longman Sci. & Tech., Harlow, 1993. [7] D. van Dulst, Reflexive and Superreflexive Banach Spaces, Mathematische Centrum, Amsterdam, 1978. [8] W. Feng, A new spectral theory for nonlinear operators and its applications, Abstr. Appl. Anal. 2 (1997), 163–183. [9] M. Furi, M. Martelli, On the minimal displacement of points under α-Lipschitz maps in normed spaces, Boll. Un. Mat. Ital. (4) 9 (1974), 791–799. [10] K. Goebel, Concise Course on Fixed Point Theorems, Yokohama Publishers, Yokohama, 2002. [11] K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990. FIXED POINT FREE MAPPINGS WHICH SATISFY A DARBO TYPE CONDITION 13 [12] T. Kuczumow, S. Reich, S. Stachura, Minimal displacement of points under holomorphic mappings and fixed point properties for unions of convex sets, Trans. Amer. Math. Soc. 343 (1994), 575–586. [13] P. K. Lin, Y. Sternfeld, Convex sets with the Lipschitz fixed point property are compact, Proc. Amer. Math. Soc. 93 (1985), 633–639. [14] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I. Sequence Spaces, Springer-Verlag, Berlin, New York, 1977. [15] E. Matoušková, S. Reich, Reflexivity and approximate fixed points, Studia Math. 159 (2003), 403– 415. [16] S. Reich, A minimal displacement problem, Comment. Math. Univ. St. Pauli 26 (1977), 131–135. [17] S. Reich, The almost fixed point property for nonexpansive mappings, Proc. Amer. Math. Soc. 88 (1983), 44–46. [18] M. Väth, Fixed point free maps of a closed ball with small measures of noncompactness, Collect. Math. 52 (2001), 101–116. [19] M. Väth, On the minimal displacement problem of γ-Lipschitz maps and γ-Lipschitz retractions onto the sphere, Z. Anal. Anwendungen 21 (2002), 901–914. Elisabetta Maluta, Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano MI, Italy E-mail address: elimal@mate.polimi.it Stanislaw Prus, Institute of Mathematics, M. Curie-Sklodowska University, 20-031 Lublin, Poland E-mail address: bsprus@golem.umcs.lublin.pl Jacek Wośko, Institute of Mathematics, M. Curie-Sklodowska University, 20-031 Lublin, Poland E-mail address: jwosko@golem.umcs.lublin.pl