Best Proximity Results on Condensing Operators via Measure of Noncompactness with Application to Integral Equations

We prove the best proximity point results for condensing operators on C-class of functions, by using a concept of measure of noncompactness. The results are applied to show the existence of a solution for certain integral equations. We express also an illsutrative examples to indicate the validity of the observed results. MSC: 47H08; 54H25


Introduction and Preliminaries
In 1930 Kuratowski [1], introduced the measure of non-compactness α(S) where S is a bounded subset of a metric space X. This notion was used effectively in the definition of a Hausdorff measure of non-compactness, χ(S), see e.g. [2] and the references therein. One of the main aim of this paper is to derive best proximity point results for certain mappings, by using the concept of a measure of noncompactness.
We shall present some definitions, notations and results which will be needed in the sequel. Throughout this paper, the letter E represents an infinite dimensional Banach space. The symbols co(C) denotes the closure of convex hull of C ⊂ E, which is the smallest closed and convex set that contains C. Furthermore, the expressions M E and N E indicated the family of nonempty bounded subsets of E and the subfamily consisting of all relatively compact subsets of E, respectively.
A mapping F : [0, ∞) 2 → R is called C-class function [4] if it is continuous and satisfies the following axioms: (1) F (s, t) ≤ s ; (2) F (s, t) = s implies that either s = 0 or t = 0; for all s, t ∈ [0, ∞). We denote C-class functions as C, for short. (A5) If (X n ) is a sequence of closed sets in M E such that X n+1 ⊆ X n , for each positive integer n, and if lim n→∞ µ(X n ) = 0 then the intersection set X ∞ = ∞ n=1 X n is nonempty. The family Kerµ described in (A1) is said to be the kernel of the measure of noncompactness µ. Note that the intersection set lies in = Ker, that is, X ∞ ∈ Kerµ, since µ(X ∞ ) ≤ µ(X n ) for any n.
The following is one of the pioneer results in the direction of finding fixed point via the measure of non-compactness and it extend the well-known Schauder fixed point theorem. 5]). Let C be a nonempty, bounded, closed, and convex subset of a Banach space E and let T : C → C be a continuous mapping. Assume that there exists a constant k ∈ [0, 1) such that µ(T (X)) ≤ kµ(X), for any subset X of C, then T has a fixed point. Definition 1.3. Let X be a Banach space. We say that X is strictly convex if the following implication holds, for all x, y, p ∈ X and R > 0: Let A and B be two nonempty subsets of a normed linear space Y . The pair (A, B) satisfies a property if both A and B satisfy that property. So, we say that (A, B) is closed if and only if both A and B are closed; (A, B) ⊆ (C, D) ⇔ A ⊆ C, B ⊆ D. From now on, B(x; r) will mean the closed ball in the Banach space X centered at x ∈ X with radius r > 0. We shall also adopt the following notations We mention that if A is a nonempty and compact subset of a Banach space X, then co(A) is compact (see ). In addition, we set It is remarkable to note that if (A, B) is a nonempty, bounded, closed and convex pair in a reflexive Banach space X, then (A 0 , B 0 ) is also nonempty, closed and convex.
is compact (see [7]). Definition 1.5. Let (A, B) be a nonempty pair in a Banach space X and T : A ∪ B → A ∪ B be a mapping. If T is cyclic, then a point p ∈ A ∪ B is said to be a best proximity point for T provided that Also, if T is noncyclic, then the pair (p, q) ∈ A × B is called a best proximity pair for T provided that p = T p, q = T q, p − q = dist(A, B).
Existence of best proximity points (pairs) for cyclic (noncyclic) relatively nonexpansive mappings was first studied by Eldred-Kirk-Veeramani ( [8]), under a geometric concept of proximal normal structure. Here, we state the following existence results which play important roles in our coming discussions. Theorem 1.6 ( [7]). Let (A, B) be a nonempty, bounded, closed and convex pair in a reflexive Banach space X. Assume that T : A∪B → A∪B is a cyclic relatively nonexpansive mapping. If T is compact, then it admits a best proximity point. Theorem 1.7 ([9]). Let (A, B) be a nonempty, bounded, closed and convex pair in a reflexive and strictly convex Banach space X. Assume that T : A ∪ B → A ∪ B is a noncyclic relatively u-continuous mapping. If T is compact, then it admits a best proximity pair.
The cyclic (noncyclic) version of condensing mappings was introduced in [7] in order to study the existence of best proximity points (pairs) and to generalize Theorems 1.6 and 1.7 above. Definition 1.8. Let (A, B) be a nonempty and convex pair in a Banach space X and µ a measure of non-compactness on X. A cyclic (noncyclic) mapping T : A ∪ B → A ∪ B is said to be a condensing operator if there exists r ∈ (0, 1) such that for any nonempty, bounded, closed, convex, proximal and T -invariant pair (H 1 , Next results are real extensions of Theorem 1.2 due to Darbo. Theorem 1.9 ([7]). Let (A, B) be a nonempty, bounded, closed and convex pair in a reflexive Banach space X and µ an measure of non-compactness on X. If T : A ∪ B → A ∪ B is a cyclic relatively nonexpansive mapping which is condensing in the sense of Definition 1.8, then it admits a best proximity point.
The above theorem holds true for noncyclic relatively nonexpansive mapping whenever we add an additional condition "strict convexity": Theorem 1.10 ( [7]). Let (A, B) be a nonempty, bounded, closed and convex pair in a reflexive and strictly convex Banach space X and µ an measure of non-compactness on X. If T : A ∪ B → A ∪ B is a noncyclic relatively nonexpansive mapping which is condensing in the sense of Definition 1.8, then it admits a best proximity pair.
We also refer to Gabeleh-Vetro [10] for the generalizations of Theorems 1.9 and 1.10, by considering a class of cyclic (noncyclic) Meir-Keeler condensing operators.

Condensing Operators on C-Class of Functions
Motivated by the class of condensing operators in Definition 1.8, we introduce the following new classes of cyclic (noncyclic) mappings.
Definition 2.1. Let (A, B) be a nonempty and convex pair in a Banach space X and µ an measure of non-compactness on X. A cyclic (noncyclic) mapping T : A ∪ B → A ∪ B is said to be a condensing operator on C-class of functions if for any nonempty, bounded, closed, convex, proximal and T -invariant pair ( for all ψ ∈ Ψ, ϕ ∈ Φ and F ∈ C. Remark 2.2. If in the above definition ψ(t) = t and F (s, t) = rs for all s, t ∈ [0, ∞) and for some r ∈ (0, 1), then T is a condensing operator in the sense of Definition 1.8.
is a function such that β(t n ) → 1 ⇒ t n → 0, then T is a β-condensing operator which was recently introduced in [9].
We begin our main results with the next existence theorem.
Theorem 2.4. Let (A, B) be a nonempty, disjoint, bounded, closed and convex pair in a strictly convex Banach space X such that A 0 is nonempty and µ is an measure of noncompactness on X. Let T : A∪B → A∪B be a noncyclic relatively u-continuous mapping which is a condensing operator on C-class of functions. Then T has a best proximity pair.
Proof. Notice that (A 0 , B 0 ) is closed, convex and proximinal. Relatively u-continuity of the mapping T ensures that By a similar notations of the proof of [9, Theorem 6], we set A 0 = A 0 and D 0 = B 0 and for all n ∈ N define C n = co(T (C n−1 )), D n = co(T (D n−1 )). Thus , and iteratively we have C n−1 ⊇ C n for all n ∈ N. Analogously, we find that D n−1 ⊇ D n for all n ∈ N. On the other hand, we have Equivalently, we have T (D n ) ⊆ D n . Thus, we conclude, for all n ∈ N, that each pair (C n , D n ) is T invariant and moreover each mentioned pair is closed and convex. Moreover, by the fact that T relatively u-continuous, if (x, y) On the other hand, if u ∈ C 1 = co(T (C 0 )), Hence, the pair (C 1 , D 1 ) is proximinal. By a similar argument we conclude that the is a compact pair and the result follows from Theorem 1.7. Thus we suppose that max{µ(C n ), µ(D n )} > 0 for all n ∈ N. In view of the fact that T is a condensing operator on C-class of functions, we obtain Since {µ(C n ∪ D n )} is a decreasing sequence we may assume that lim n→∞ µ(C n ∪ D n ) = r for some r ≥ 0. Now from (2) and the continuity of the ψ, F we must have and so by the property of the function F we conclude that either ψ(r) = 0 or ϕ(r) = 0.
In both cases, we must have r = 0. Thereby, It now follows from the condition (A5) of Definition 1.1 that the pair (C ∞ , D ∞ ) is nonempty, closed and convex which is C n = co(T (C n−1 )), D n = co(T (D n−1 )), where, C 0 := A 0 and D 0 := B 0 , then we have and so, T (C 1 ) ⊆ T (D 0 ) which ensures that C 2 = co(T (C 1 )) ⊆ co(T (D 0 )) = D 1 . Iteratively, we obtain C n+1 ⊆ D n which is equivalent to say that D n ⊆ C n−1 for all n ∈ N. Therefore, This concludes that {(C 2n , D 2n )} n≥0 is a decreasing sequence consisting of closed and convex pairs in A 0 × B 0 . Besides, Thus (C 2n , D 2n ) is T -invariant. We also can see that by a similar approach of the proof of Theorem 2.4, and that (C 2n , D 2n ) is also proximinal for all n ∈ N. Notice that if max{{µ(C 2k ), µ(D 2k )} = 0 for some k ∈ N, then the result follows from Theorem 1.9. Let max{{µ(C 2n ), µ(D 2n )} > 0 for all n ∈ N. By the fact that T is a condensing operator on C-class of functions, It is worth noticing that if in Theorem 2.4 A = B, then the existence of fixed points will be concluded as follows.
Consider 11]). Let C be a nonempty, bounded, closed, and convex subset of a Banach space E and let T : C → C be continuous mapping, such that ψ(µ(T (M ))) ≤ F (ψ(µ(M )), ϕ(µ(M ))), (3.9) for any subset M of C and where ψ ∈ Ψ, ϕ ∈ Φ and F ∈ C. Then T has a fixed point. Proof. By the above conditions, we shall prove the measure of noncompactness ω 0 (X) is satisfying the contraction (3.9). To do this we have some claims: Claim 2. Operator T : B r0 → B r0 is continuous.
Claim 3. Operator T satisfies (3.9) with respect to measure of noncompactness ω 0 in B r0 .
To prove Claim 1, we have  . This result shows that T x ∈ B r0 . To prove Claim 2; we prove that operator T : B r0 → B r0 is continuous. To do this, consider ε > 0 and any x, y ∈ B r0 such that |x i − y i | ≤ ε. Then we obtain the following inequalities by using conditions of Theorem To prove Claim 3; we show that operator T satisfies (3.9) with respect to measure of noncompactness ω 0 in B r0 .