Approximate Endpoints for Set-Valued Contractions in Metric Spaces

N Hussain,Yj Cho,A Amini-Harandi

doi:10.1155/2010/614867

Abstract

AbstractThe existence of approximate fixed points and approximate endpoints of the multivalued almost "Equation missing"-contractions is established. We also develop quantitative estimates of the sets of approximate fixed points and approximate endpoints for multivalued almost "Equation missing"-contractions. The proved results unify and improve recent results of Amini-Harandi (2010), M. Berinde and V. Berinde (2007), Ćirić (2009), M. Păcurar and R. V. Păcurar (2007) and many others.

Highlights

Introduction and PreliminariesIn fixed point theory, one of the main directions of investigation concerns the study of the fixed point property in topological spaces
One of the main directions of investigation concerns the study of the fixed point property in topological spaces
The interest in approximate fixed point results arise in the study of some problems in economics and game theory, including, for example, the Nash equilibrium approximation in games; see [1,2,3] and references therein

Summary

Introduction and Preliminaries

One of the main directions of investigation concerns the study of the fixed point property in topological spaces. Recall that a topological space X is said to have the fixed point property if every continuous mapping f : X → X has a fixed point. Every compact convex subset of a locally convex space has the fixed point property. Another important branch of fixed point theory is the study of the approximate fixed point property. We establish some existence results concerning approximate fixed points, endpoints, and approximate endpoints of multivalued contractions. Let X and Y be two Hausdorff topological spaces and T : X → P Y a multivalued mapping with nonempty values. It is important to note that any mapping satisfying Banach, Kannan, Chatterjea, Zamfirescu, or Ciricwith the constant k in 0, 1/2 type conditions is a single-valued almost contraction; see 5, 6, 8, 11

Approximate Fixed Points of Multivalued Contractions

Endpoints of Multivalued Nonlinear Contractions