Nonlinear Analysis: Theory, Methods & Applications
Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces
Introduction
We let be the distance (in some sense) between subsets and of a space . Recall that the best proximity point theory is concerned with the conditions under which the following optimization problems have solutions:
(OP1). For given and where does not have a fixed point in (i.e. ), we have a question of whether there exists (a point is called a best proximity point for ) such that .
(OP2). For given and where is cyclic (i.e. and ), we have a question of whether there exists (a point is called a best proximity point for ) such that and is a singleton (i.e. ).
(OP3). For given and where is noncyclic (i.e. and ), we have a question of whether there exists (a point is called a best proximity point for ) such that and and are endpoints of (i.e. and ).
The best proximity point theory, although initiated quite a long time ago (by the Fan best approximation theorem [1]), is still very interesting (in particular for mathematicians working in fixed point theory), has become a very popular topic in recent years and its range of application has been considerably extended (see e.g. [2], [3], [4], [1], [5], [6], [7], [8], [9] for single-valued maps and [10], [11], [12], [13] for set-valued maps). In a short review it is impossible to even glance over its main ideas and trends. The papers [2], [3], [4], [5], [6], [7], [8], [13], [9] contain some general overviews of the best proximity point theory which are informative and fun to read.
Note that in the existing literature all of the best approximation, best proximity and convergence theorems which solve these problems are obtained when is a locally convex space or a Banach space, when and are convex or compact and when the distance is defined using a pseudonorm or a norm; see e.g. Fan [1], Eldred, Kirk and Veeramani [3], Eldred and Veeramani [4, Theorem 3.10] and Di Bari, Suzuki and Vetro [2, Theorem 2].
In this paper, in cone uniform spaces , using the concept of the -family of cone pseudodistances, the distance between two sets and in is defined, the concepts of cyclic and noncyclic set-valued dynamic systems of -relatively quasi-asymptotic contractions are introduced and the best approximation and best proximity point theorems for such contractions are proved (see Theorem 2.2, Theorem 2.4, Theorem 2.5). Also conditions are given which guarantee that for each starting point each generalized sequence of iterations of these contractions (in particular, each dynamic process) converges and the limit is a best proximity point (see Theorem 2.4, Theorem 2.5). Moreover, -families are constructed, characterized and compared (see Theorem 2.6, Theorem 2.7 and Section 8). The results are new and of quite different natures for set-valued and single-valued dynamic systems in cone uniform, cone locally convex and cone metric spaces. In investigations we do not use the assumptions that the sets and are convex or compact. Various examples illustrating ideas, methods, definitions and results are constructed (see Section 8). This paper is a continuation of [14].
Section snippets
Definitions, notation and statement of results
We define a real normed space to be a pair , with the understanding that a vector space over carries the topology generated by the metric , .
Definition 2.1 Let be a real normed space. A nonempty closed convex set is called a cone in if it satisfies: ; ; and .
It is clear that each cone defines, by virtue of “ if and only if ”, an order of under which is an ordered normed space with cone . We shall write to indicate
Proof of Theorem 2.2
The proof will be broken into four steps.
Step I. The following holds:
First, we see that Indeed, the set-valued dynamic system is cyclic or noncyclic on , and accordingly, . Also . Hence, by () and (), . On the
Proof of Theorem 2.4
By Theorem 2.3, the map has a unique endpoint in and the map has a unique endpoint in , i.e. and each of the sequences and , where for and , satisfies and Moreover, Theorem 2.2(P1) implies since a -family is a -semifamily.
By (4.1), we have Additionally, since
Proof of Theorem 2.5
By Theorem 2.3, we conclude that has a unique endpoint in , has a unique endpoint in and for every two sequences and , where for and , we have , . Moreover, by Theorem 2.2(P2), we have . Now, using where , we can obtain , and accordingly, . □
Proof of Theorem 2.6
(a) The condition () is satisfied since, for , such that , the following three cases hold: Case 1. If or , then and . Case 2. If , then or or and, consequently, . Case 3. If and and , then we get .
To prove (), assume that there exist and , , such that . Then we have two cases: Case 1. If
Proof of Theorem 2.7
Suppose our assertions were not true. Thus there exists such that , and the assertions (a)–(d) of Theorem 2.4 hold for and . Then, by the definition of the -family and the assertion of (d), we get , a contradiction. □
Examples
In Example 8.1(A)–(C) and 8.2 we construct various -families and -semifamilies.
Example 8.1 Let be an ordered normed space with cone , let the family be a -family and let be a Hausdorff cone uniform space, with a normal solid cone , containing at least two different points. (A) For each and satisfying the family defined by: if or , if and if , , is a
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