Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces

https://doi.org/10.1016/j.na.2009.07.024Get rights and content

Abstract

In cone uniform spaces X, using the concept of the D-family of cone pseudodistances, the distance between two not necessarily convex or compact sets A and B in X is defined, the concepts of cyclic and noncyclic set-valued dynamic systems of D-relatively quasi-asymptotic contractions T:AB2AB are introduced and the best approximation and best proximity point theorems for such contractions are proved. 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. Moreover, D-families are constructed, characterized and compared. The results are new for set-valued and single-valued dynamic systems in cone uniform, cone locally convex and cone metric spaces. Various examples illustrating ideas, methods, definitions and results are constructed.

Introduction

We let D(A,B) be the distance (in some sense) between subsets A and B of a space X. Recall that the best proximity point theory is concerned with the conditions under which the following optimization problems have solutions:

(OP1). For given AX and T:A2X where T does not have a fixed point in A (i.e. xA{xT(x)}), we have a question of whether there exists wA (a point w is called a best proximity point for T) such that D({w},T(w))=D(A,T(w)).

(OP2). For given A,BX and T:AB2AB where T is cyclic (i.e. T:A2B and T:B2A), we have a question of whether there exists wAB (a point w is called a best proximity point for T) such that D({w},T(w))=D(A,B) and T(w) is a singleton (i.e. T(w)={T(w)}).

(OP3). For given A,BX and T:AB2AB where T is noncyclic (i.e. T:A2A and T:B2B), we have a question of whether there exists (u,v)A×B (a point (u,v) is called a best proximity point for T) such that D({u},{v})=D(A,B) and u and v are endpoints of T (i.e. T(u)={u} and T(v)={v}).

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 X is a locally convex space or a Banach space, when A and B are convex or compact and when the distance D(A,B) 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 X, using the concept of the D-family of cone pseudodistances, the distance between two sets A and B in X is defined, the concepts of cyclic and noncyclic set-valued dynamic systems of D-relatively quasi-asymptotic contractions T:AB2AB 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, D-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 A and B 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 (L,), with the understanding that a vector space L over R carries the topology generated by the metric (a,b)ab, a,bL.

Definition 2.1

Let L be a real normed space. A nonempty closed convex set HL is called a cone in L if it satisfies: (H1)λ(0,){λHH}; (H2)H(H)={0}; and (H3)H{0}.

It is clear that each cone HL defines, by virtue of “ab if and only if baH”, an order of L under which L is an ordered normed space with cone H. We shall write ab to indicate

Proof of Theorem 2.2

The proof will be broken into four steps.

Step I. The following holds:αA{limmDα(T[m](AB))=DD;α(A,B)}.

First, we see that αAγαL,0γα{limmDα(T[m](AB))DD;α(A,B)γα=0}. Indeed, the set-valued dynamic system (AB,T)is cyclic or noncyclic on AB, and accordingly, m{0}N{T[m](AB)AT[m](AB)B}. Also m{0}N{T[m+1](AB)T[m](AB)=T[m](A)T[m](B)AB}. Hence, by (D1) and (D3), αAm{0}N(x,y)T[m](A)×T[m](B){0DD;α(A,B)=inf({Dα({s,t}):(s,t)A×B})Dα({x,y})Dα(T[m](AB))}. On the

Proof of Theorem 2.4

By Theorem 2.3, the map T[2]:A2A has a unique endpoint w in A and the map T[2]:B2B has a unique endpoint v in B, i.e. T[2](w)={w}T[2](v)={v}, and each of the sequences {wm} and {vm}, where (wm,vm)T[m](w0)×T[m](v0) for mN and (w0,v0)A×B, satisfies limmw2m=limmv2m+1=w and limmw2m+1=limmv2m=v. Moreover, Theorem 2.2(P1) implies αA{limmDα({wm,wm+1})=limmDα({vm,vm+1})=DD;α(A,B)} since a D-family is a D-semifamily.

By (4.1), we have mN{T[2m](w)={w}T[2m](v)={v}}. Additionally, since w2mT[2m](

Proof of Theorem 2.5

By Theorem 2.3, we conclude that T:A2A has a unique endpoint u in A, T:B2B has a unique endpoint v in B and for every two sequences {um} and {vm}, where (um,vm)T[m](u0)×T[m](v0) for mN and (u0,v0)A×B, we have limmum=u, limmvm=v. Moreover, by Theorem 2.2(P2), we have αA{limmDα({um,vm})=DD;α(A,B)}. Now, using (sm,tm)=T[m](u)×T[m](v)=(u,v) where m{0}N, we can obtain αA{limmDα({sm,tm})=Dα({u,v})=DD;α(A,B)}, and accordingly, αA{limmDα({um,vm})=Dα({u,v})=DD;α(A,B)}.  

Proof of Theorem 2.6

(a) The condition (D3) is satisfied since, for E1, E22X such that E1E2, the following three cases hold: Case 1. If E2={a} or E2={b}, then E1=E2 and αA{Dα(E1)=Dα(E2)=0}. Case 2. If E2={a,b}, then E1={a} or E1={b} or E1={a,b} and, consequently, αA{Dα(E1)Dα(E2)=pα(a,b)}. Case 3. If E2{a,b} and E2{a} and E2{b}, then we get αA{Dα(E1)Dα(E2)=cα}.

To prove (D4), assume that there exist α0A and x0, y0, z0X such that Dα0({x0,y0})+Dα0({y0,z0})Dα0({x0,z0}). Then we have two cases: Case 1. If

Proof of Theorem 2.7

Suppose our assertions were not true. Thus there exists (w,T(w))A×B such that wT(w), {w,T(w)}{a,b} and the assertions (a)–(d) of Theorem 2.4 hold for w and T(w). Then, by the definition of the D-family and the assertion of (d), we get cα=Dα({w,T(w)})=DD;α(A,B)=pα(a,b)cα, a contradiction. 

Examples

In Example 8.1(A)–(C) and 8.2 we construct various D-families and D-semifamilies.

Example 8.1

Let L be an ordered normed space with cone HL, let the family P={pα:X×XL,αA} be a P-family and let (X,P) be a Hausdorff cone uniform space, with a normal solid cone H, containing at least two different points.

(A) For each a,bX and {cα}αAH satisfying αA{0pα(a,b)cα} the family D={Dα:2XL,αA} defined by: Dα(E)=0 if E={a} or E={b}, Dα(E)=pα(a,b) if E=E0 and Dα(E)=cα if EE0E, E2X,αA, E0={a,b} is a D

References (25)

  • W.A. Kirk et al.

    Fixed points for mappings satisfying cyclical contractive conditions

    Fixed Point Theory

    (2003)
  • P.S. Srinivasan et al.

    On best proximity pair theorems and fixed-point theorems

    Abstr. Appl. Anal.

    (2003)
  • View full text