A Generalization of Kannan's Fixed Point Theorem

Yusuke Enjouji,Tomonari Suzuki,Masato Nakanishi

doi:10.1155/2009/192872

Yusuke Enjouji, Tomonari Suzuki + Show 1 more

Open Access

https://doi.org/10.1155/2009/192872

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Abstract

AbstractIn order to observe the condition of Kannan mappings, we prove a generalization of Kannan's fixed point theorem. Our theorem involves constants and we obtain the best constants to ensure a fixed point.

Highlights

A mapping T on a metric space X, d is called Kannan if there exists α ∈ 0, 1/2 such that d T x, T y ≤ αd x, T x αd y, T y1.1 for all x, y ∈ X
Throughout this paper we denote by N the set of all positive integers and by R the set of all real numbers
We show d z, T x ≤ βd x, T x ∀x ∈ X \ {z}

Summary

Introduction

A mapping T on a metric space X, d is called Kannan if there exists α ∈ 0, 1/2 such that d T x, T y ≤ αd x, T x αd y, T y. Kannan 1 proved that if X is complete, every Kannan mapping has a fixed point. A metric space X is complete if and only if every Kannan mapping on X has a fixed point. Kikkawa and Suzuki proved a generalization of Kannan’s fixed point theorem. We can consider “αd x, T x βd y, T y ” instead of “αd x, T x αd y, T y .” It is a quite natural question of what is the best constant for each pair α, β. We give the complete answer to this question

Preliminaries

Fixed Point Theorem

The Best Constants