Abstract and Applied Analysis

Volume 2012 (2012), Article ID 750403, 9 pages

doi:10.1155/2012/750403

Research Article

A Unique Common Triple Fixed Point Theorem for Hybrid Pair of Maps

K. P. R. Rao,¹ G. N. V. Kishore,² and Kenan Tas³

¹Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar, Guntur 522510, India

²Department of Mathematics, Swarnandhra Institute of Engineering and Technology, Seetharampuram, Narspur 534 280, India

³Department of Mathematics and Computer Science, Cankaya University, 06810 Ankara, Turkey

Received 25 June 2012; Accepted 29 August 2012

Academic Editor: Nikolaos Papageorgiou

Copyright © 2012 K. P. R. Rao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We obtain a unique common triple fixed point theorem for hybrid pair of mappings in metric spaces. Our result extends the recent results of B. Samet and C. Vetro (2011). We also introduced a suitable example supporting our result.

1. Introduction

The study of fixed points for multivalued contraction mappings using the Hausdorff metric was initiated by Nadler [1].

Let be a metric space. We denote the family of all nonempty closed and bounded subsets of and the set of all nonempty closed subsets of . For and , we denote . Let be the Hausdorff metric induced by the metric on , that is,

for every .

It is clear that for and , we have .

Definition 1.1. An element is said to be a fixed point of a set-valued mapping if and only if .

In 1969, Nadler [1] extended the famous Banach contraction principle [2] from single-valued mapping to multivalued mapping and proved the following fixed point theorem for the multivalued contraction.

Theorem 1.2 (see, Nadler [1]). Let be a complete metric space and let be a mapping from into . Assume that there exists such that

for all . Then, has a fixed point.

Lemma 1.3   (see, Nadler [1]). Let and . Then for every , there exists such that .

Lemma 1.4   (see, Nadler [1]). Let . If with , then for each , there exists such that .

Lemma 1.5   (see, Nadler [1]). Let be a sequence in with , for . If and , then .

The existence of fixed points for various multivalued contractive mappings has been studied by many authors under different conditions. For details, we refer the reader to [1, 3–11] and the references therein.

The concept of coupled fixed point for multivalued mapping was introduced by Samet and Vetro [12], and later several authors, namely, Hussain and Alotaibi [13], Aydi et al. [14], and Abbas et al. [15], proved coupled coincidence point theorems in partially ordered metric spaces.

Definition 1.6   (see, Samet and Vetro [12]). Let be a given mapping. We say that is a coupled fixed point of if and only if

Definition 1.7   (see, Hussain and Alotaibi [13]). Let the mappings and be given. An element is called

(1)a coupled coincidence point of a pair if and ;(2)a coupled common fixed point of a pair if and .

Berinde and Borcut [16] introduced the concept of triple fixed points and obtained a tripled fixed point theorem for single valued map.

Now we give the following.

Definition 1.8. Let be a nonempty set, (collection of all nonempty subsets of ). .

(i)The point is called a tripled fixed point of if

(ii)The point is called a tripled coincident point of and if

(iii)The point is called a tripled common fixed point of and if

Definition 1.9. Let be a multivalued map and be a self map on . The Hybrid pair is called -compatible if whenever is a tripled coincidence point of and .

2. Main Results

Theorem 2.1. Let be a metric space and let and mappings satisfying

(2.1.1), for all and with , where is a fixed number, (2.1.2) and is a complete subspace of .

Then the maps and have a tripled coincidence point.

Further, and have a tripled common fixed point if one of the following conditions holds.

(2.1.3) (a) is -compatible, there exist such that , and , whenever is a tripled coincidence point of and is continuous at . (b) There exist such that ,   and whenever is a tripled coincidence point of and is continuous at , and .

Proof. Let . From , there exist sequences , , and in such that , and , .

For simplification, denote

From (2.1.1), we obtain

Let   denoted by  .

Clearly, .

Then,

It is clear that .

Now we prove by induction that

where

Equation (2.3) is true for .

Assume that (2.3) is true for some . Consider

We have

Similarly, we have

Thus (2.3) is true for all integer values of .

Now from (i)–(vi) and continuing this process, we get

for all . That is,

For , we have

Hence is a Cauchy. Similarly, we can show that and are Cauchy.

Suppose is complete, the sequences ,  , and are convergent to some in , respectively. There exist such that ,  , and .

Now, we have

Letting , we get so that . That is, . Similarly, we can show that and . Thus is a tripled coincidence point of and . Suppose (2.1.3) (a) holds.

Since is a tripled coincidence point of and , there exist such that , and .

Since is continuous at and , we have ,   and .

Since , we have .

Since , we have .

Since , we have .

Then is tripled coincidence point of and .

Similarly, we can show that is a tripled coincidence point of and .

Also it is clear that

From (2.1.1), we have

Letting , we obtain

which implies that

Thus . Similarly, we can show that and . Thus is a tripled common fixed point of and . Suppose (2.1.3) (b) holds.

Since is a tripled coincidence point of , there exist such that , and .

Since is continuous at and , we have ,   and . Thus , and . Hence is a tripled common fixed point of .

The following example illustrates Theorem 2.1.

Example 2.2. Let and defined as and . Then

It is clear that all conditions of Theorem 2.1 are satisfied and is the tripled common fixed point of and .

The following example shows that and have no tripled common fixed point if (a) or (b) is not satisfied.

Example 2.3. Let , and . Then is a tripled coincidence point of and . Clearly and have no tripled common fixed point.

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