Fixed point, approximate fixed point and Kantorovich-Rubinstein maximum principle in convex metric spaces

Abstract

We obtain necessary conditions for the existence of fixed point and approximate fixed point of nonexpansive and asymptotically nonexpansive maps defined on a closed bounded convex subset of a uniformly convex complete metric space and study the structure of the set of fixed points. We construct Mann type iterative sequences in convex metric space and study its convergence. As a consequence of fixed point results, we prove best approximation results. We also prove KantorovichRubinstein maximum principle in convex metric spaces.Keywords Fixed point · Convex metric space · Uniformly convex metric space · Best approximation · Kantorovich-Rubinstein maximum principle Mathematics Subject Classification (2000) 47H09 · 47H10 · 54H25 · 49J35 · 30C80 I. Beg ( ) · M. Abbas Centre for Advanced Studies in Mathematics,

Existence and Approximation of Fixed Points of Enriched Contractions and Enriched φ-Contractions

“…(5) There exist other important results regarding the solution of the fixed point problem in convex metric spaces (see [11,12,14,17,[19][20][21][22][23][24][25][27][28][29][30]) or in Banach spaces, metric spaces and generalized metric spaces (see [2,26,[31][32][33][34][35][36][37][38][39][40][41][42][43][47][48][49]56,65,66]) that could be developed by means of the approach considered in the present paper.…”

“…A convex metric space offers the minimal tools for constructing various fixed point iterative methods for approximating fixed points of nonlinear operators, such as Krasnoselskij, Mann and Ishikawa fixed point iterative schemes, which require the linearity and convexity of the ambient topological space. This is the main reason, after the pioneering work by Takahashi [4], several authors studied fixed point problems in the setting of a Takahashi convex metric space, e.g., Machado [5], Talman [6], Itoh [7], Naimpally, Singh and Whitfield [8,9], Ding [10], Ciric [11], Shimizu and Takahashi [12], Huang [13], Popa [14], Beg [15], Chang, Kim and Jin [16], Sharma and Deshpande [17], Tian [18], Beg and Abbas [19,20], Beg, Abbas and Kim [21], Aoyama, Eshita and Takahashi [22], Shimizu [23], Abbas [24], Agarwal, O'Regan and Sahu [25], Xue, Lv and Rhoades [26], Phuengrattana and Suantai [27,28], Khan and Abbas [29], and Siriyan and Kangtunyakarn [30], among others.…”