Abstract

In this paper we study some fixed point results in pseudocompact Tichnovo space using Edelstein type contractive conditions. The results presented in this paper include the generalization of some fixed point theorems established by Fisher and Pathak.

1. Introduction

Fixed point theory is a fascinating topic for research in modern mathematics. In this direction the Banach contraction mapping theorem of 1922 popularly known as Banach contraction mapping principle is a rewarding result in analysis and fixed point theory. It has widespread applications in both pure and applied mathematics. The well known Banach ² contraction mapping principle states that if "X is a complete metric space and is a contraction mapping of X into itself then T has unique fixed point in X". This celebrated principle has been generalized by several authors. In 1961, Edelstein ⁶ introduced the concept of contractive mapping defined on compact metric spaces which is generalizetion of Banach contraction mapping principle. According to Edelstein "if T is a continuous mapping of a compact metric space X into itself satisfying for all then T has unique fixed point in X". Edelstein’s contractive mapping theorem has been extensively generalized and improved by several mathematicians for fixed points in several different ways viz, Bailey ¹, Chatterjee ⁴, Ciric ⁵, Iseki ⁹, Kannan and Sharma ¹², Pachpatte ¹⁵, Popa ¹⁷, Sahu ¹⁸, Sharma and Sahu ¹⁹ and Soni ²⁰ on complete and compact metric spaces. The concept of fixed point results for contractive mappings in pseudocompact Tichonov spaces was introduced by Harinath ¹⁰. Later on, Jain and Dixit ¹¹ and Liu ¹³ also established fixed point results for several classes of contractive type mappings in pseudocompact Tichonov spaces. Inspired by the ideas of Fisher ⁷ and Pathak ¹⁶ the aim of the present paper is to prove the existence and uniqueness of fixed point results for self mapping in the setting of pseudocompact Tichonov spaces satisfying contractive type conditions.

The following fixed point theorems were proved in ⁷ and ¹⁶.

Theorem 1.1. ⁷ If T is a mapping of the complete metric space X into itself satisfying the inequality

for all in where and then has a fixed point.

Theorem 1.2. ¹⁶ Let P be a Pseudocompact Tichonov space and be a non-negative real valued continuous function over ( is Tichnovo but need not be pseudocompact). Suppose μ also satisfies

(i)

If S and T are two continuos self maps of P satisfying

(ii) ST = TS and

(iii)

for all distinct with where Then S and T have a unique common fixed point in P which is unique whenever

2. Main Results

Theorem 2.1. Let P be a pseudocompact Tichonov space and d be a nonnegative real valued continuous function over satisfying the conditions:

(i)

(ii)

If S and T are continuous self maps of P satisfying

(2.1)

(iii)

(2.2)

for all distinct with and then S and T have a common fixed point in X, which is unique whenever

Proof. Define a function by for all . Clearly is continuous being the composite of three continuous functions and Since is compact, every real valued continuous function over is bounded and attain its bounds. Thus there exists a point such that We now affirm that is a fixed point for If not, let us suppose that then using (2.2) we have

or

or

or

or

which is contradiction because Hence is a fixed point for that is Using (2.1), we have

(2.3)

Now we shall prove that If possible, let then by using (2.2) and (2.3), we have

which is a contradiction because Hence is a fixed point of i.e.

Uniqueness: To prove the uniqueness of v, if possible, let w be another fixed point for S and T i.e. and Then, using (2.2), we have

leading to a contradiction because which proves that is unique and this establishes the theorem.

Theorem (2.1) yields the following corollary.

Corollary 2.2 Let P be a pseudocompact Tichonov space and d be a non-negative real valued continuous function over satisfying:

(i)

(ii)

and is a continuous map satisfying the inequality

for all distinct with and then S has a fixed point in P which is unique.

Proof If we take T = S, then theorem (2.1) shows that S has a unique fixed point in

Theorem 2.3. Let P be a pseudocompact Tichonov space and d be a non-negative real valued continuous function over satisfy the conditions:

(i)

(ii)

If S and T are continuous self maps of P satisfying

(2.4)

(2.5)

for all and are non negative real numbers such that then S and T have a common fixed point in X, which is unique whenever,

Proof. Define a function by for all Clearly is continuous being the composite of three continuous functions S, T and d. Since X is compact, every real valued continuous function over X is bounded and attain its bounds. Thus there exists a point such that We now affirm that v is a fixed point for S. If not, let us suppose that then using (2.5) we have

Case(i)-If

Then

or

or

or

or

where

But which is a contradiction.

Case-(ii) If

Then

or

or

or

where

But which is a contradiction.

Case-(iii) If

Then

or

or

where

But which is a contradiction.

Here

Hence, is a fixed point of and so Using (2.4), we have

(2.6)

Now we shall prove that If possible, let then by using (2.5) and (2.6) we have

which is contradiction because Hence is a fixed point of that is

Uniqueness: Let, if possible be another fixed pioint of S and T that is and then using (2.5), we get

This is a contradiction because Hence is unique fixed point of and

Now we give an example to support our results.

Remark: If we in Theorem (2.3) then we get Theorem (2.1).

Example 2.1. Let and let is the discrete topology on Define as

and let d be a non-negative real valued continuous function over such that Then, it is clear that is a pseudocompact Tichonov space and and are continuous self maps of which satisfy all the conditions of Theorem (2.1) and Theorem (2.3) with 1 as the only common fixed point by setting for

References