A Counterexample to "An Extension of Gregus Fixed Point Theorem"

Abstract

In the paper by Olaleru and Akewe (2007), the authors tried to generalize Gregus fixed point theorem. In this paper we give a counterexample on their main statement.

1. Introduction

Let be a Banach space and be a closed convex subset of . In 1980 Greguš [1] proved the following results.

Theorem 1.1.

Let be a mapping satisfying the inequality

(1.1)

for all , where , and . Then has a unique fixed point.

Several papers have been written on the Gregus fixed point theorem. For example, see [2–6]. We can combine the Gregus condition by the following inequality, where is a mapping on metric space :

(1.2)

for all , where , and .

Definition 1.2.

Let be a topological vector space on . The mapping is said to be an such that for all

(i),

(ii),

(iii),

(iv) for all with ,

(v)if and , then .

In 2007, Olaleru and Akewe [7] considered the existence of fixed point of when is defined on a closed convex subset of a complete metrizable topological vector space and satisfies condition (1.2) and extended the Gregus fixed point.

Theorem 1.3.

Let be a closed convex subset of a complete metrizable topological vector space and a mapping that satisfies

(1.3)

for all , where is an on , , and . Then has a unique fixed point.

Here, we give an example to show that the above mentioned theorem is not correct.

2. Counterexample

Example 2.1.

Let endowed with the Euclidean metric and . Let defined by . Let and such that . Then for all such that , we have that

(2.1)

We have two cases, or .

If , then , and hence inequality (2.1) is true. If , then , and so , and hence inequality (2.1) is true. So condition (1.3) holds for and , but has not fixed point.