Fixed Point Theorems for ws-Compact Mappings in Banach Spaces

1 Department of Mathematical Sciences, Florida Institute of Technology, 150 West University Boulevard, Melbourne, FL 32901, USA 2 Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia 3 Department of Mathematics, National University of Ireland, Galway, Ireland 4 Université Cadi Ayyad, Laboratoire de Mathématiques et de Dynamique de Populations, Marrakech, Morocco


Introduction
Let X be a Banach space, and let M be a subset of X. Following 1 , a map A : M → X is said to be ws-compact if it is continuous and for any weakly convergent sequence x n n∈N in M the sequence Ax n n∈N has a strongly convergent subsequence in X. This concept arises naturally in the study of both integral and partial differential equations see 1-5 . In this paper, we continue the study of ws-compact mappings, investigate the boundary conditions, and establish new fixed point theorems. Specifically, we prove several fixed point theorems for ws-compact mappings under Sadovskii, Leray-Schauder, Rothe, Altman, Petryshyn and Furi-Pera type conditions. Finally, we note that ws-compact mappings are not necessarily sequentially weakly continuous see Example 2.14 . This explains the usefulness of our fixed point results in many practical situations. For the remainder of this section, we gather some notations and preliminary facts. Let X be a Banach space, let B X denote the collection of 2 Fixed Point Theory and Applications all nonempty bounded subsets of X and W X the subset of B X consisting of all weakly compact subsets of X. Also, let B r denote the closed ball centered at 0 with radius r.
In our considerations, the following definition will play an important role.
Definition 1.1 see 6 . A function ψ : B X → R is said to be a measure of weak noncompactness if it satisfies the following conditions. 1 The family ker ψ {M ∈ B X : ψ M 0} is nonempty and ker ψ is contained in the set of relatively weakly compact sets of X.
3 ψ co M ψ M , where co M is the closed convex hull of M.
5 If M n n≥1 is a sequence of nonempty weakly closed subsets of X with M 1 bounded and M 1 ⊇ M 2 ⊇ · · · ⊇ M n ⊇ · · · such that lim n → ∞ ψ M n 0, then M ∞ : ∞ n 1 M n is nonempty.
The family ker ψ described in 1 is said to be the kernel of the measure of weak noncompactness ψ. Note that the intersection set M ∞ from 5 belongs to ker ψ since ψ M ∞ ≤ ψ M n for every n and lim n → ∞ ψ M n 0. Also, it can be easily verified that the measure ψ satisfies The first important example of a measure of weak noncompactness has been defined by De Blasi 7 as follows: Notice that w · is regular, homogeneous, subadditive, and set additive see 7 . In what follows, let X be a Banach space, C a nonempty closed convex subset of X, F : C → C a mapping and x 0 ∈ C. For any M ⊆ C, we set for n 2, 3, . . .

Definition 1.2.
Let X be a Banach space, C a nonempty closed convex subset of X, and ψ a measure of weak noncompactness on X. Let F : C → C be a bounded mapping that is it takes bounded sets into bounded ones and x 0 ∈ C. We say that F is a ψ-convex-power condensing operator about x 0 and n 0 if for any bounded set M ⊆ C with ψ M > 0, we have Obviously, F : C → C is ψ-condensing if and only if it is ψ-convex-power condensing operator about x 0 and 1.

Fixed Point Theorems
Theorem 2.1. Let X be a Banach space, and let ψ be a regular and set additive measure of weak noncompactness on X. Let C be a nonempty closed convex subset of X, x 0 ∈ C, and let n 0 be a positive integer. Suppose that F : C → C is ψ-convex-power condensing about x 0 and n 0 . If F is ws-compact and F C is bounded, then F has a fixed point in C.

Proof. Let
The set F is nonempty since C ∈ F. Set M A∈F A. Now, we show that for any positive integer n we have

Fixed Point Theory and Applications
To see this, we proceed by induction. Clearly M is a closed convex subset of C and F M ⊆ M. Thus, M ∈ F. This implies co This shows that P 1 holds. Let n be fixed, and suppose that P n holds. This implies F n 1, As a result Notice F C is bounded implies that M is bounded. Using the properties of the measure of weak noncompactness, we get which yields that M is weakly compact. Now, we show that F M is relatively compact. To see this, consider a sequence y n n∈N in F M . For each n ∈ N, there exists x n ∈ M with y n Fx n . Now, the Eberlein-Smulian theorem 9, page 549 guarantees that there exists a subsequence S of N so that x n n∈S is a weakly convergent sequence. Since F is ws-compact, then Fx n n∈S has a strongly convergent subsequence. Thus, F M is relatively compact. Keeping in mind that F M ⊆ M, the result follows from Schauder's fixed point theorem.
As an easy consequence of Theorem 2.1, we recapture 10, Theorem 3.1 .

Corollary 2.2.
Let X be a Banach space, and let ψ be a regular and set additive measure of weak noncompactness on X. Let C be a nonempty closed convex subset of X. Assume that F : C → C is ws-compact and F C is bounded. If F is ψ-condensing, that is, ψ F M < ψ M , whenever M is a bounded nonweakly compact subset of C, then F has a fixed point. Theorem 2.3. Let X be a Banach space, and let ψ a measure of weak noncompactness on X. Let C be a closed, convex subset of X, U an open subset of C, and p ∈ U. Assume that F : X → X is ws-compact and ψ-convex-power condensing about p and n 0 .
Proof. Suppose that ii does not hold and F has no fixed points on ∂U otherwise, we are finished . Then, u / λF u 1 − λ p for u ∈ ∂U and λ ∈ 0, 1 . Consider In addition, the continuity of F implies that A is closed. Notice that Fixed Point Theory and Applications 5 therefore, by Urysohn's lemma, there exists a continuous μ :

2.7
It is immediate that N : C → C is continuous. Now we show that N is ws-compact. To see this, let x n n∈N be a sequence in C which converges weakly to some x ∈ C. Without loss of generality, we may take x n n∈N in U. Notice that μ x n n∈N is a sequence in 0, 1 . Hence, by extracting a subsequence if necessary, we may assume that μ x n n∈N converges to some λ ∈ 0, 1 . On the other hand, since F is ws-compact, then there exists a subsequence S of N so that Fx n n∈S converges strongly to some y ∈ C. Consequently, the sequence Nx n n∈S converges strongly to λy 1 − λ p. This proves that N is ws-compact. Our next task is to show that N is ψ-convex-power condensing about p and n 0 . To see this, let S be a bounded subset of C. Clearly By induction, note for all positive integer n, we have Indeed, fix an integer n ≥ 1 and suppose that 2.9 holds. Then,

2.10
In particular, we have Thus, This proves that N is ψ-convex-power condensing about p and n 0 . Theorem 2.1 guarantees the existence of for all x ∈ X. Then, i μ is nonnegative and continuous on X. ii Lemma 2.6. Let X be a Banach space, ψ a set additive measure of weak noncompactness on X, and Q a closed convex subset of X with 0 ∈ int Q . Let μ be the Minkowski functional defined in Lemma 2.5, and, r be the map defined on X by ii For any subset A of X we have r A ⊆ co A ∪ {0} .
iii For any bounded subset A of X we have ψ r A ≤ ψ A .
Proof. i The continuity of r follows immediately from Lemma 2.5 i . Now, let x ∈ X. Using Lemma 2.5 ii , we get μ r x μ x max 1, μ x ≤ 1.

2.15
This implies that r x ∈ Q. The last statement follows easily from Lemma 2.5 v . Now, we prove ii . To this end, let A be a subset of X, and let x ∈ A. Then,

2.16
Thus, r A ⊆ co A ∪ {0} . Using the properties of a measure of weak noncompactness, we get This proves iii .
Fixed Point Theory and Applications 7 Theorem 2.7. Let X be a Banach space, and let ψ a regular set additive measure of weak noncompactness on X. Let Q be a closed convex subset of X with 0 ∈ Q, and let n 0 a positive integer. Assume that F : X → X is ws-compact and ψ-convex-power condensing about 0 and n 0 and F Q is bounded and if x j , λ j is a sequence in ∂Q × 0, 1 converging to x, λ with x λF x and 0 < λ < 1, then λ j F x j ∈ Q for j sufficiently large 2.18 holding. Also, suppose the following condition holds: there exists a continuous retraction r : X −→ Q with r z ∈ ∂Q for z ∈ X \ Q and r D ⊆ co D ∪ {0} for any bounded subset D of X.

2.19
Then, F has a fixed point.
Proof. Let r : X → Q be as described in 2.19 . Consider B {x ∈ X : x Fr x }.

2.20
We first show that B / ∅. To see this, consider rF : Q → Q. First, notice that rF Q is bounded since F Q is bounded and r F Q ⊆ co F Q ∪ {0} . Clearly, rF is continuous, since F and r are continuous. Now, we show that rF is ws-compact. To see this, let x n n∈N be a sequence in Q which converges weakly to some x ∈ Q. Since F is ws-compact, then there exists a subsequence S of N so that Fx n n∈S converges strongly to some y ∈ X. The continuity of r guarantees that the sequence rFx n n∈S converges strongly to ry. This proves that rF is ws-compact. Our next task is to show that rF is ψ-convex-power condensing about 0 and n 0 . To do so, let A be a subset of Q. In view of 2.19 , we have Hence,

2.23
Taking into account the fact that F is ψ-convex-power condensing about 0 and n 0 and using 2.19 , we get compact. To see this, let y n n∈N be a sequence in Fr B . For each n ∈ N, there exists x n ∈ r B with y n Fx n . Since r B is relatively weakly compact, then, by extracting a subsequence if necessary, we may assume that x n n∈N is a weakly convergent sequence. Now, F is ws-compact implies that y n n∈N has a strongly convergent subsequence. This proves that Fr B is relatively compact. From 2.25 , it readily follows that B is relatively compact. Consequently, B B is compact. We now show that B ∩ Q / ∅. To do this, we argue by contradiction. Suppose that B ∩ Q ∅. Then, since B is compact and Q is closed, there exists δ > 0 with dist B, Q > δ. Choose N ∈ {1, 2, . . .} such that Nδ > 1. Define To see this, let x n n∈N be a weakly convergent sequence in U i . Then, the set S : {x n : n ∈ N} is relatively weakly compact and so ψ S 0. In view of 2.19 , we infer that ψ r S 0 and so r S is relatively weakly compact. By extracting a subsequence if necessary, we may assume that rx n n∈N is weakly convergent. Now, F is ws-compact implies that Frx n n∈N has a strongly convergent subsequence. This proves that Fr is ws-compact. Our next task is to show that Fr is ψ-convex-power condensing about 0 and n 0 . To see this, let A be a bounded subset of U i and set A co A ∪ {0} . Then, keeping in mind 2.19 , we obtain

2.30
and by induction,

2.31
Thus, Fixed Point Theory and Applications whenever ψ A / 0. Applying Theorem 2.3 to Fr : U i → X, we may deduce that there exists y i , λ i ∈ ∂U i × 0, 1 with y i λ i Fr y i . Notice in particular since y i ∈ ∂U i × 0, 1 that {N, N 1, . . .}.

2.34
Clearly, D is closed since F and r are continuous. Now, we claim that D is compact. To see this, first notice Thus,

2.38
Since Fr is ψ-convex-power condensing about 0 and n 0 , then ψ D 0, and so D is relatively weakly compact. Now, 2.19 guarantees that r D is relatively weakly compact. Now, we show that Fr D is relatively compact. To see this, let y n n∈N be a sequence in F D . For each n ∈ N, there exists x n ∈ r D with y n Fx n . Since r D is relatively weakly compact then, by extracting a subsequence if necessary, we may assume that x n n∈N is a weakly convergent sequence. Now, F is ws-compact implies that y n n∈N has a strongly convergent subsequence. This proves that Fr D is relatively compact. From 2.35 , it readily follows that D is relatively compact. Consequently, D D is compact. Then, up to a subsequence, we may assume that λ i → λ * ∈ 0, 1 and y i → y * ∈ ∂U i . Hence, λ i Fr y i → λ * Fr y * , and therefore y * λ * Fr y * . Notice λ * Fr y * / ∈ Q since y * ∈ ∂U i . Thus, λ * / 1 since B ∩ Q ∅. From assumption 2.18 , it follows that λ i Fr y i ∈ Q for j sufficiently large, which is a contradiction. Thus, B ∩ Q / ∅, so there exists x ∈ Q with x Fr x , that is, x Fx. Remark 2.8. If 0 ∈ int Q then we can choose r : X → Q in the statement of Theorem 2.7 as in Lemma 2.6. Clearly r z ∈ ∂Q for z ∈ X \ Q and r D ⊆ co D ∪ {0} for any bounded subset D of X.
Fixed Point Theory and Applications 11 Corollary 2.9. Let X be a Banach space, ψ a regular set additive measure of weak noncompactness on X, and Q a closed convex subset of X with 0 ∈ Q. Assume that F : X → X is ws-compact and ψ-convex-power condensing about 0 and n 0 , and assume that 2.19 holds. If F Q is bounded and F ∂Q ⊆ Q (the condition of Rothe type), then F has a fixed point in Q.
In the light of Remark 2.8, we have the following result. Corollary 2.10. Let X be a Banach space, ψ a regular set additive measure of weak noncompactness on X and Q a closed convex subset of X with 0 ∈ int Q . Assume F : X → X is ws-compact and ψ-convex-power condensing about 0 and n 0 . If F Q is bounded and F ∂Q ⊆ Q, then F has a fixed point in Q.
Theorem 2.11. Let Q be a closed convex set in a Banach space X, 0 ∈ int Q . Assume F : X → X is ws-compact and ψ-convex-power condensing about 0 and n 0 . If F Q is bounded and As a result λ − 1 2 ≥ λ 2 − 1 . This contradicts the fact that λ > 1. Therefore, F has a fixed point. Corollary 2.12. Let Q be a closed convex set in a Banach space X, 0 ∈ int Q . Assume that F : X → X is ws-compact and ψ-convex-power condensing about 0 and n 0 . If F Q is bounded and one of the following conditions are satisfied: Then, F has a fixed point in Q.
Remark 2.13. In Theorem 2.7 we need F : X → Xψ-convex-power condens-ing about 0 and n 0 : However, In Theorem 2.7 the condition F : X → Xws-compact can be replaced by F : Q → X ws-compact. This comment also applies to Corollaries 2.9, 2.10, Theorem 2.11, and Corollary 2.12.
In the following example, we give a broad class of ws-compact mappings which are not sequentially weakly continuous.

Fixed Point Theory and Applications
Example 2.14. Let g : 0, 1 × R → R be a function satisfying Carathéodory conditions, that is, g is Lebesgue measurable in x for each y ∈ R and continuous in y for each x ∈ 0, 1 . Additionally, we assume that for all x, y ∈ 0, 1 × R, where a x is a nonnegative function Lebesgue integrable on the interval 0, 1 and b ≥ 0. Let us consider the so-called superposition operator N g , generated by the function g, which to every function u defined on the interval 0, 1 assigns the function N g u given by the formula N g u x g x, u x , x ∈ 0, 1 .

2.42
Let L 1 L 1 0, 1 denote the space of functions u : 0, 1 → R which are Lebesgue integrable, equipped with the standard norm. It was shown 12 that under the above-quoted assumptions the superposition operator N g maps continuously the space L 1 into itself. Define the functional φ u

2.44
Clearly, K is continuous with norm K ≤ 1. Thus, φ is continuous. Now, we show that φ is ws-compact. To see this, let u n be a weakly convergent sequence of L 1 . Using 2.41 , we have for any for any subset D of 0, 1 that D N g u n x dx ≤ D a x dt b D |u n x |dx.

2.45
Taking into account the fact the sequence u n is weakly convergent and that any set consisting of one element is weakly compact and using Corollary 11 in 13, page 294 , we get uniformly in n. Applying Corollary 11 in 13, page 294 once again, we infer that N g u n has a weakly convergent subsequence, say N g u n k . Let u be the weak limit of N g u n k . Hence, Consequently, the sequence φu n k is convergent. This proves that φ is ws-compact. However, φ is not weakly sequentially continuous unless φ is linear with respect to the second variable see 14, 15 .