Global optimality results for multivalued non-self mappings in b-metric spaces

In this paper, we introduce a new class of multivalued contractions with respect to b-generalized pseudodistances and prove a best proximity point theorem for this class of non-self mappings. In this way, we improve and extend several existing results in the literature. Examples are given to support our main results. This work is a continuation of studies on the use of a new type of distances in the fixed point theory. The pioneering effort in direction of defining distance is inter alia paper of O. Kada, T. Suzuki and W. Takahashi.


Introduction
In 2008, Suzuki [1] presented a weaker notion of contractions in order to characterize the completeness of metric spaces and established the following interesting theorem.
Then for a metric space (X, d), the following are equivalent: (i) X is complete.
(ii) There exists r ∈ [0, 1) such that every mapping T on X satisfying the following: has a fixed point.
After that the multivalued version of Theorem 1.1, which is an extension of Nadler's fixed point theorem, was presented as below.
Let (X, d) be a complete metric space and let T : X → 2 X be a multivalued mapping such that T (x) is a nonempty, bounded and closed subset of X for each x ∈ X . Assume that there exists r ∈ [0, 1) such that

η(r )D(x, T x) ≤ d(x, y) implies H(T x, T y) ≤ rd(x, y),
for all x, y ∈ X, where H denotes the Hausdorff metric. Then there exists z ∈ X such that z ∈ T z. Now, let (A, B) be a nonempty pair of subsets of a metric space (X, d) and let T : A → 2 B be a multivalued non-self mapping. Then for each x ∈ A we have D(x, T x) ≥ dist (A, B), where dist (A, B) := inf{d(x, y) : (x, y) ∈ A × B} and D(x, T x) := dist ({x}, T x). So, it is quite natural to seek an approximate solution x ∈ A that is optimal in the sense that the distance D(x, T x) with respect to D is minimum. As the minimality of the value D(x, T x) connotes the highest closeness between the elements x and T x, one attempts to determine an element x for which D(x, T x) assumes the least possible value dist (A, B). Such an optimal solution x for which D(x, T x) = dist (A, B), is called a best proximity point of the multivalued non-self mapping T . Existence of best proximity points for multivalued nonself mappings was first studied in [3] for multivalued nonexpansive non-self mappings in hyperconvex metric spaces and in Hilbert spaces (see also [4][5][6][7][8][9][10] for different approaches to the same problem).
The aim of this article is to elicit a best proximity point theorem for a new class of multivalued non-self mappings with respect to b-generalized pseudodistances. Our results improve and extend some recent results in the previous works.

Preliminaries
Let A and B be two nonempty subsets of a metric space (X, d). When we say that a pair (A, B) satisfies a special property, we mean that both A and B satisfy the mentioned property.
We denote by CB(X ) the family of all nonempty closed bounded subsets of X . We will use the following notations: It is easy to see that if (A, B) is a nonempty weakly compact pair in a Banach space X then (A 0 , B 0 ) is a nonempty pair.
The following notion is weaker than the notion of P-property which was first introduced in [12].
where x 1 , x 2 ∈ A 0 and y 1 , y 2 ∈ B 0 . Here, we state the next best proximity point theorem which is a main result of [14].
for all x, y ∈ A. If T (x) ∈ CB(X ) for all x ∈ A, and T (x 0 ) ⊂ B 0 for each x 0 ∈ A 0 , then T has a best proximity point in A.
The notion of b-metric space was introduced by Czerwik [15] as below.
Definition 2.4 [15] Let X be a nonempty set and s ≥ 1 be a given real number. A function for any x, y, z ∈ X the following three conditions are satisfied: If d is a b-metric on X (with constant s ≥ 1), then the pair (X, d) is called a b-metric space. Note that every metric space is a b-metric space. Throughout this paper, we assume that the b-metric d : Here, we mention the following fixed point theorem which is the main result of [15].
Theorem 2.5 [15] Let (X, d) be a complete b-metric space and T : X → CB(X ) be a multivalued mapping. Suppose there exists r ∈ (0, 1 for all x, y ∈ X . Then T has a fixed point.
, is said to be a b-generalized pseudodistance on X if the following two conditions hold: is a b-generalized pseudodistance on X . However, there exists a b-generalized pseudodistance on X which is not a b-metric (for details see Example 4.1 of [16]).

Remark 2.8 From (J 1) and (J 2) it follows that for any
By using the notion of b-generalized pseudodistance on a b-metric space X , we can define the H J Hausdorff distance as below.
Similarly, the following definitions and notations can be constructed in b -metric spaces equipped with a b-generalized pseudodistance.
Let (X, d) be a b-metric space (with s ≥ 1) and let (A, B) be a nonempty pair of subsets of X and let the map J : X × X → [0, ∞) be a b-generalized pseudodistance on X . We set where x 1 , x 2 ∈ A 0 and y 1 , y 2 ∈ B 0 . (II) We say that the b-generalized pseudodistance J is associated with the pair (A, B) if for any sequences (x m : m ∈ N) and (y m : m ∈ N) in X such that lim m→∞ x m = x; lim m→∞ y m = y; and The following lemma will be used in the sequel. Then (x m : m ∈ {0} ∪ N) is a Cauchy sequence on X .

Main results
We begin our main result of this section with the following notion.
for all x, y ∈ A.
It is clear that the class of multivalued non-self mappings which are contraction of Suzuki type with respect to b-generalized pseudodistances contains the class of multivalued non-self mappings considered in Theorem 2.3. This can be seen by taking s = 1 and J = d.
We now prove the main result of this article. Thereby,

Now, for each m > n we have
Thus, as n → ∞ in above relation, we deduce that Hence

Examples illustrating Theorem 3.2 and some comparisons
In this section, we will present some examples illustrating the concepts having been introduced so far. We will show a fundamental difference between Theorems 3.2 and 2. 3. The examples will show that Theorem 3.2 is an essential generalization of Theorem 2.3. First, we present an example of generalized pseudodistance in metric spaces and b-metric spaces, respectively.
Example 4.1 Let X be a metric space (b-metric space respectively) where the metric d : Let the closed set E ⊂ X , containing at least two different points, be arbitrary and fixed. Let c > 0 such that c > δ(E), where δ(E) = sup{d(x, y) : x, y ∈ X } be arbitrary and fixed. Define the map J : X × X → [0, ∞) as follows: Next, we present an example which illustrate Theorem 3.2. To compare our results with some well-known best proximity point theorems in the literature, we start by giving an example where X is a metric space.  III. We see that T is contraction of Banach type (i.e. J (T x, T y) ≤ r J(x, y) for some r ∈ [0, 1) and for all x, y ∈ A). Indeed, let r = 1/2 and let x, y ∈ A be arbitrary and fixed. We see that by (4.2) We consider the following cases: Case 1 If x, y ∈ [0, 1], then by (4.2), T x = T y = 3/2. Now, by (4.1) we have: y). In consequence (4.4) and (4.5) implies that T is contraction of Banach type. IV. We see that T is contraction of Suzuki type (when s = 1, and T is single valued), i.e.
for some r ∈ [0, 1) and for all x, y ∈ A. It is consequence of Step III. V. We see that there exists a best proximity point of T .
Now, we will compare our results with two important results which existing in the literature. In 2013, Zhang et al. [18] proved the following theorem. for some r ∈ [0, 1) and for all x, y ∈ A. Assume that the following hold:

Then there exists a unique z ∈ A such that d(z, T z) = d(A, B).
In this same year, Suzuki [19] established the following interesting result. Theorem 4.2 [19] Let (A, B) be a pair of subsets of a metric space (X, d). Let T be a mapping from A into B. Assume that (i)-(iii) in Theorem 4.1 and the following hold: for all x, y ∈ A. Then there exists a unique z ∈ A so that d(z, T z) = d (A, B). I. We see that the map T is not contraction in the sense of Theorem 4.1.
II. We see that the mapping T is not contraction in the sense of Theorem 4.2. In this order, suppose the following condition holds for some α ∈ [0, 1 2 ) and for all x, y ∈ A. In particular, for x 0 = 1 2 and y 0 = 7 2 we have 2 ) = 1. Hence, and by (4.7) we get which is impossible.   I. We show that (A, B) has the W P J -property. Indeed, we observe that dist (A, B) = 1 and