Condensing Mappings and Best Proximity Point Results

Best proximity pair results are proved for noncyclic relatively u-continuous condensing mappings. In addition, best proximity points of upper semicontinuous mappings are obtained which are also fixed points of noncyclic relatively u-continuous condensing mappings. It is shown that relative u-continuity ofT is a necessary condition that cannot be omitted. Some examples are given to support our results.


Introduction
e concept of measure of noncompactness was first introduced by Kuratowski [1]. However, the interest in the concept was revived in 1955 when Darbo [2] proved a generalization of Schauder's fixed point theorem using this concept. Sadovskii [3], in 1967, defined condensing mappings and extended Darbo's theorem. Since then a lot of work has been done using this concept, and several interesting results have appeared, see, for instance, [4][5][6][7][8][9].
Let (W, Z) be a nonempty pair in a Banach space (that is, both W and Z are nonempty sets). A mapping T: W ∪ Z ⟶ W ∪ Z is called noncyclic provided T(W)⊆W and T(Z)⊆Z. If there exists (w, z) ∈ W × Z which satisfies w � T(w), z � T(z), and ‖w − z‖ � dist(W, Z), then we say that the noncyclic mapping T has a best proximity pair. For a multivalued nonself mapping S: W ⟶ 2 Z , a point w ∈ W is called a fixed point of S if w ∈ S(w). e necessary condition for the existence of a fixed point for such S is W ∩ Z ≠ ∅. If W ∩ Z � ∅, then dist(w, S(w)) > 0 for each w ∈ W. Best proximity point theorems provide sufficient conditions for the existence of at least one solution for the minimization problem, min w∈W dist(w, S(w)). If dist(w, S(w)) � dist(W, Z), the point w is called a best proximity point of S. e existence results of best proximity points for multivalued mappings were obtained in [10][11][12][13][14] and [15]. Best proximity point theorems for relatively nonexpansive and relatively u-continuous were established by Elderd et al. in [16,17] and by Markin and Shahzad in [18]. In recent years, the topics of best proximity points of single-valued and multivalued mappings have attracted the attention of many researchers, see, for example, the work in [6,7,19,20] and the references cited therein. In this paper, we prove best proximity pair theorems for noncyclic relatively u-continuous condensing mappings. In addition, we obtain best proximity points of upper semicontinuous mappings which are fixed points of noncyclic relatively u-continuous condensing mappings. Also, we give examples to support our results and show by giving an example that relative u-continuity of T is a necessary condition that cannot be omitted. Our results extend and complement results of [6,7,11].
(1) Theorem 1. Let X be a metric space. en, for any nonempty bounded pair (C 1 , C 2 ) in X (that is, both C 1 and C 2 are nonempty and bounded sets), the following hold: For a normed space X: , for any number λ (iv) α(con(C 1 )) � α(con(C 1 )) � α(C 1 ), where con(C 1 ) represents the convex hull of C 1 Theorem 2. Let F j be a decreasing sequence of nonempty closed subsets of a complete metric space For more details about the measure of noncompactness, see [4]. Definition 2. Let (W, Z) be a nonempty pair in Banach space X and T: W ∪ Z ⟶ W ∪ Z a mapping. en, T is said to be noncyclic relatively u-continuous. If T is noncyclic and for each ϵ > 0, there is c > 0 such that for each w ∈ W and z ∈ Z.
Definition 3. Let (W, Z) be a nonempty convex pair in Banach space X. A mapping T: W ∪ Z ⟶ W ∪ Z is said to be affine if for each α, β ∈ [0, 1] with α + β � 1 and denotes the collection of all nonempty, convex, and compact subsets of Z ), then by the commutativity of T and S, we mean that T(S(w))⊆S(T(w)) holds for each w ∈ W. Given (W, Z), a nonempty pair in Banach space, its proximal pair (W 0 , Z 0 ) is given by Moreover, if (W, Z) is a nonempty, convex, and compact pair in X, then (W 0 , Z 0 ) is also a nonempty, convex, and compact pair. Definition 6. Let X be a normed space. For a nonempty subset C of X, the metric projection operator P C : X ⟶ 2 C is given by For a nonempty, convex, and compact subset C of a strictly convex Banach space, P C is a single-valued mapping. Furthermore, for a nonempty, convex, and compact subset C of a Banach space X, P C is upper semicontinuous with nonempty, convex, and compact values.

Lemma 2.
(see [11]). Let (W, Z) be a nonempty, convex, and compact pair in a strictly convex Banach space X. Let T: W ∪ Z ⟶ W ∪ Z be a noncyclic relatively u-continuous and P: W ∪ Z ⟶ W ∪ Z be a mapping given by Theorem 3. (see [18]). Let (W, Z) be a nonempty, convex, and compact pair in a strictly convex Banach space X. If T: W ∪ Z ⟶ W ∪ Z is a noncyclic relatively u-continuous mapping. en, T has best proximity pair.
In [6], Gabeleh and Markin introduced the class of noncyclic condensing operators.
Recall that a nonempty pair Definition 7. Let (W, Z) be a nonempty convex pair in a strictly convex Banach space X. A mapping T: W ∪ Z ⟶ W ∪ Z is called noncyclic condensing operator provided that, for any nonempty, bounded, closed, convex, proximinal, and Lemma 3. (see [11]). Let (W, Z) be a nonempty, convex, and compact pair in a strictly convex Banach space X. If

Main Results
roughout this paper, we will assume that X is a strictly convex Banach space and α is the measure of noncompactness on X.
Remark 1. Let T: W ⟶ W be condensing in the sense of Definition 7 with k ∈ (0, 1). en, for any bounded subset H of W, T satisfies To see this, in (7), set W � Z and Theorem 4. Let (W, Z) be a nonempty, convex, and closed pair in X such that W is bounded and W 0 is nonempty.

Corollary 1. Let (W, Z) be a nonempty, convex, and closed
pair in X such that W is bounded and W 0 is nonempty. Suppose T: W ⟶ W is a continuous, affine, and condensing mapping. If S: W ⟶ KC(W) is an upper semicontinuous multivalued mapping, T and S commute, and then there is w ∈ W which satisfies w ∈ Fix(T) ∩ Fix(S). Theorem 5. Let (W, Z) be a nonempty, convex, and closed pair in X such that W is bounded and W 0 is nonempty. If T 1 , T 2 : W ∪ Z ⟶ W ∪ Z are commuting, noncyclic relatively u-continuous, affine, and condensing mappings, then there exists Proof. Since W 0 is nonempty and by relative u-continuity of T 1 , for w 0 ∈ W 0 , there exists z 0 ∈ Z such that us, α(Fix W (T 1 )) � 0, and thus, Fix W (T 1 ) is compact. Furthermore, is a continuous mapping on a compact convex set. By Schauder's fixed point theorem, there is Let v 0 in Z 0 be the unique closest point to u 0 . By relative u-continuity of T 1 and T 2 , we infer that, since en, the mappings in C have common fixed points u 0 ∈ W 0 and v 0 ∈ Z 0 .
Proof. For each T ∈ C, consider Fix(T), Fix W (T), and Fix Z (T) defined previously. en, Fix W (T) is nonempty, compact, and convex. Let T 1 , T 2 , . . . , T k be a finite sub- for n ∈ N. en, F n is a decreasing sequence of compact subsets of X. Furthermore, F n ≠ ∅ for each n ∈ N. Indeed, for w ∈ F and each m ∈ 1, 2, . . . , k { }, then T m (T k+1 (w)) � T k+1 (T m (w)) � T k+1 (w), and this implies that T k+1 (w) ∈ F. us, F is invariant under T k+1 . By Schauder's fixed point theorem, we get that F 1 ≠ ∅. Now, for each n ∈ N and m ∈ 1, 2, . . . , k + n { }, pick x ∈ F n : that is, T k+n+1 (x) ∈ F n . So, T k+n+1 : F n ⟶ F n is continuous on F n , and then there is y ∈ F n such that T k+n+1 (y) � y. erefore, y ∈ F n+1 ≠ ∅. By eorem 2, ∩ n∈N F n ≠ ∅. Hence, □ Theorem 6. Let (W, Z) be a nonempty, convex, and closed pair in X such that W is bounded and W 0 is nonempty. Let C be the collection of the commuting, noncyclic, relatively u-continuous, affine, and condensing mappings on W ∪ Z.
en, there is (u 0 , v 0 ) ∈ W × Z such that, for each T ∈ C: Proof. Based on the previous lemma, the mappings in C have a fixed point in common u 0 ∈ W, that is, T(u 0 ) � u 0 , for each T ∈ C. Let v 0 ∈ Z be the unique closest point to u 0 . By relative u-continuity of T, since ‖u 0 − v 0 ‖ � dist(W, Z), (24) Proof. By Lemma 4, ( ∩ T∈C Fix W (T), ∩ T∈C Fix Z (T)) is a nonempty compact convex pair. Also, in view to the proof of eorem 4, for T ∈ C and for each x ∈ Fix W (T), we have S(x) and Z 0 are invariant under T. So, Define f: . en, f is an upper semicontinuous multivalued mapping with nonempty, compact, and convex values. Moreover, P W : By relative u-continuity of T, one can conclude that ‖T(x) − T(y)‖ � dist(W, Z). us, T(y) � P Z (T(x)) and T(x) � P W (T(y)), and by (26), T(x) � T(P W (y)) � P W (T(y)) � P W (y). us, P W (y) ∈ ∩ T∈C Fix W (T). Note that P W°f : ∩ T∈C Fix W (T) ⟶ 2 Fix W (T) , and by Lemma 1, there is w ∈ ∩ T∈C Fix W (T) such that w ∈ (P W°f )(w), that is, for T ∈ C, we have T(w) � w and w ∈ (P W (f(w))). So, there We infer that ‖z − w‖ � dist(z, W). But z ∈ Z 0 , then there is w * ∈ W such that ‖w * − z‖ � dist(W, Z). en, Hence, dist(w, S(w)) � dist(W, Z). and Z � ζ 1 e 1 + ζ 2 e 2 : ζ 1 ∈ [0, 3], ζ 2 ∈ R .

Conclusion
We have proved some best proximity pair theorems for noncyclic relatively u-continuous and condensing mappings. We have also obtained best proximity points of upper semicontinuous mappings which are fixed points of noncyclic relatively u-continuous condensing mappings.

Journal of Mathematics
Moreover, we have given some examples to support our results. It has been shown that relative u-continuity of T is a necessary condition that cannot be omitted. We have extended recent results of [6,11].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.