On the existence of fixed points that belong to the zero set of a certain function

Erdal Karapinar,Bessem Samet,Donal O’Regan

doi:10.1186/s13663-015-0401-7

Abstract

AbstractLet $T: X\to X$ T : X → X be a given operator and $F_{T}$ F T be the set of its fixed points. For a certain function $\varphi: X\to[0,\infty)$ φ : X → [ 0 , ∞ ) , we say that $F_{T}$ F T is φ-admissible if $F_{T}$ F T is nonempty and $F_{T}\subseteq Z_{\varphi}$ F T ⊆ Z φ , where $Z_{\varphi}$ Z φ is the zero set of φ. In this paper, we study the φ-admissibility of a new class of operators. As applications, we establish a new homotopy result and we obtain a partial metric version of the Boyd-Wong fixed point theorem.

Highlights

The set of fixed points of T is denoted by FT, that is, FT = {x ∈ X : Tx = x}
In order to prove the uniqueness of the fixed point, let us assume that w ∈ FT with d(z, w) >
The Boyd-Wong fixed point theorem cannot be applied in this case

Summary

Introduction

Let F be the set of functions F : [ , ∞) → [ , ∞) satisfying the following conditions: (F ) max{a, b} ≤ F(a, b, c), for all a, b, c ≥ ; (F ) F(a, , ) = a, for all a ≥ ; (F ) F is continuous. We obtain an homotopy result and a partial metric version of the Boyd-Wong fixed point theorem. Let (X, d) be a complete metric space and T : X → X be a given operator.

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Conclusion