Best proximity points: global optimal approximate solutions

被引:106
作者
Sadiq Basha, S. [1 ]
机构
[1] Anna Univ, Dept Math, Madras 600025, Tamil Nadu, India
关键词
Global optimal approximate solution; Fixed point; Best proximity point; Contractive mapping; PAIR THEOREMS; EQUILIBRIUM PAIRS; EXISTENCE; MULTIFUNCTIONS; CONTRACTIONS; CONVERGENCE; EXTENSIONS; MAPPINGS;
D O I
10.1007/s10898-009-9521-0
中图分类号
C93 [管理学]; O22 [运筹学];
学科分类号
070105 ; 12 ; 1201 ; 1202 ; 120202 ;
摘要
Let A and B be non-empty subsets of a metric space. As a non-self mapping T : A -> B does not necessarily have a fixed point, it is of considerable interest to find an element x in A that is as close to Tx in B as possible. In other words, if the fixed point equation Tx = x has no exact solution, then it is contemplated to find an approximate solution x in A such that the error d(x, Tx) is minimum, where d is the distance function. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, to the fixed point equation Tx = x when there is no exact solution. As the distance between any element x in A and its image Tx in B is at least the distance between the sets A and B, a best proximity pair theorem achieves global minimum of d(x, Tx) by stipulating an approximate solution x of the fixed point equation Tx = x to satisfy the condition that d(x, Tx) = d(A, B). The purpose of this article is to establish best proximity point theorems for contractive non-self mappings, yielding global optimal approximate solutions of certain fixed point equations. Besides establishing the existence of best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.
引用
收藏
页码:15 / 21
页数:7
相关论文
共 21 条
[1]   Convergence and existence results for best proximity points [J].
Al-Thagafi, M. A. ;
Shahzad, Naseer .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2009, 70 (10) :3665-3671
[2]   Best Proximity Sets and Equilibrium Pairs for a Finite Family of Multimaps [J].
Al-Thagafi, M. A. ;
Shahzad, Naseer .
FIXED POINT THEORY AND APPLICATIONS, 2008, 2008 (1)
[3]   Best proximity pairs and equilibrium pairs for Kakutani multimaps [J].
Al-Thagafi, M. A. ;
Shahzad, Naseer .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2009, 70 (03) :1209-1216
[4]  
Basha S.S., 1977, Acta Sci. Math. (Szeged), V63, P289
[5]  
Basha SS, 2000, J APPROX THEORY, V103, P119
[6]  
Basha SS, 2001, INDIAN J PURE AP MAT, V32, P1237
[7]   Best proximity points for cyclic Meir-Keeler contractions [J].
Di Bari, Cristina ;
Suzuki, Tomonari ;
Vetro, Calogero .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2008, 69 (11) :3790-3794
[8]  
Edelstein M., 1962, J. London Math. Soc, V37, P74, DOI [10.1112/jlms/s1-37.1.74, DOI 10.1112/JLMS/S1-37.1.74]
[9]   Existence and convergence of best proximity points [J].
Eldred, A. Anthony ;
Veeramani, P. .
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2006, 323 (02) :1001-1006
[10]   Proximal normal structure and relatively nonexpansive mappings [J].
Eldred, AA ;
Kirk, WA ;
Veeramani, P .
STUDIA MATHEMATICA, 2005, 171 (03) :283-293