An application of fixed point theorems in best approximation theory

Abstract

ABSTRACT. In this paper, we give an application of Jungck's fixed point theorem to best approximation Let X be a normed linear space. A mapping T X X is said to be contractwe on X (resp., on a subset C of X) if IITx-Tyll <_ IIx Yll for all x, y in X (resp., C). The set of fixed points of T on X is denoted by F(T). If is a point of X, then for 0 < a _< 1, we define the set Da of best (C, a)-approximants to consists of the points y in C such that Let D denote the set of best C-approximants to . For a 1, our definition reduces to the set D of best C-approximants to . A subset C of X is said to be starshaped with, respect to a point q E C [4] have shown that the assumption T" C C can be weakened to the condition T" OC C if y C, i.e., y E D is not necessarily in the interior of C, where OC denotes the boundary of C.Recently, Sahab, Khan and Sessa [9] generalized Theorem as in the following:

Common Fixed Point Theorem for Occasionally Weakly Baised Mappings and Its Application to Best Approximation

“…We given an example to illustrate our theorem. Our theorem improve the results of Gregus [9], Jungck [12], Pathak, Cho and Kang [22], Sharma and Deshpande [28]. We also give an application of our main theorem to best approximation theory.…”

“…We also give an application of our main theorem to best approximation theory. Our application improves the results Pathak, Cho and Kang [22], Sharma and Deshpande [26]- [28].…”

“…Many authors have studied the applications of fixed point theorems to best approximation theory including [2], [3], [5], [6]- [8], [10], [16], [18], [19], [22], [23], [25]- [34].…”

An extension of a theorem of Sahab, Khan, and Sessa

“…(i) In the light of Example 2.4, the classes of weakly commuting and compatible maps are different and so the statement "weakly commuting maps are compatible" on page 977 in [6] is not valid. Hence Theorem 2.3 cannot be implied by Theorem 5 of Pathak et al [11] even in Banach space setting.…”

Common fixed-points for Banach operator pairs in best approximation

“…A remarkable application of fixed-point theorems is to prove the existence of fixed-points in best approximation (see [3][4][5]13,16,[22][23][24]26]), which has special significance for the spaces that are not strictly convex (see [5]). As generalization of fixed-points, common fixed-points of two maps f and g satisfying some contractive or nonexpansive type condition have been studied by many authors and applied to various problems, especially to those associated with best approximation (see [1,2,6,10,18,21]).…”