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International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 131068, 9 pages
http://dx.doi.org/10.1155/2009/131068
Research Article

Common Fixed Point Theorem of Two Mappings Satisfying a Generalized Weak Contractive Condition

Department of Mathematics, Lahore University of Management Sciences, Lahore 54792, Pakistan

Received 6 August 2009; Accepted 11 November 2009

Academic Editor: Evgeny Korotyaev

Copyright © 2009 M. Abbas and M. Ali Khan. 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

Existence of common fixed point for two mappings which satisfy a generalized weak contractive condition is established. As a consequence, a common fixed point result for mappings satisfying a contractive condition of integral type is obtained. Our results generalize, extend, and unify several well-known comparable results in literature.

1. Introduction and Preliminaries

Let be a metric space and a mapping. Recall that is contraction if for all where A point is a fixed point of provided If a map satisfies for each where denotes the set of all fixed points of then it is said to have property Banach contraction principle which gives an answer on existence and uniqueness of a solution of an operator equation is the most widely used fixed point theorem in all of analysis. Branciari [1] obtained a fixed point theorem for a mapping satisfying an analogue of Banach's contraction principle for an integral type inequality. Akgun and Rhoades [2] have shown that a map satisfying a Meir- Keeler type contractive condition of integral type has a property Rhoades and Abbas [3] extended [4, Theorem] for mappings satisfying contractive condition of integral type. They also studied several results for maps which have property defined on a metric space satisfying generalized contractive conditions of integral type. Rhoades [5] proved two fixed point theorems involving more general contractive condition of integral type (see, also [6, 7]). If maps and satisfy for each then they are said to have property Jeong and Rhoades [8] studied the property for pairs of maps satisfying a number of contractive conditions.

Recently Dutta and Choudhury [9] gave a generalization of Banach contraction principle, which in turn generalize [4, Theorem ] and corresponding result of [10]. Sessa [11] defined the concept of weakly commuting to obtain common fixed point for pairs of maps. Jungck generalized this idea, first to compatible mappings [12] and then to weakly compatible mappings [13]. There are examples that show that each of these generalizations of commutativity is a proper extension of the previous definition. The aim of this paper is to present a common fixed point theorem for weakly compatible maps satisfying a generalized weak contractive condition which is more general than the corresponding contractive condition of integral type. Our results substantially extend, improve, and generalize comparable results in literature [3, 14, 15].

The following definitions and results will be needed in the sequel.

Definition 1.1. Let be a set, and selfmaps of . A point in is called a coincidence point of and if and only if . We will call a point of coincidence of and

Definition 1.2. Two maps and are said to be weakly compatible if they commute at their coincidence points.

Lemma 1.3 (see [16]). Let and be weakly compatible self maps of a set . If and have a unique point of coincidence (say), then is the unique common fixed point of and .

2. A Common Fixed Point Theorem

Set is a Lebesgue integrable mapping which is summable and nonnegative and satisfies for each and is continuous and nondecreasing mapping with if and only if

The following is the main result of this paper.

Theorem 2.1. Let be two self maps of a metric space satisfying for all, where If range of contains the range of and is a complete subspace of , then and have a unique point of coincidence in Moreover if and are weakly compatible, and have a unique common fixed point.

Proof. Let be an arbitrary point of . Choose a point in such that This can be done, since the range of contains the range of Continuing this process, having chosen in , we obtain in such that Suppose for any since, otherwise, and have a point of coincidence. From (2.1), we have that is, , and hence It follows that is monotone decreasing sequence of numbers and consequently there exists such that as Suppose that , then which on taking limit as yields which is a contradiction. Therefore Now we prove that is a Cauchy sequence. If not, then there exist some and subsequences and of with such that for each As as , for large enough , we have and . Thus we obtain We may assume that are even and are odd and that for all . Put Now, implies that as . Furthermore gives , as . Therefore Taking limit as yields which is a contradiction. Hence is a Cauchy sequence. From completeness of , there exists a point in such that as Consequently, we can find in such that Now on taking limit as implies , and Hence is the point of coincidence of and Assume that there is another point of coincident in such that Then there exists in such that Using (2.1), we have which is a contradiction which proves the uniqueness of point of coincidence; the result now follows from Lemma 1.3

Corollary 2.2. Let be two self maps of a metric space satisfying for all , where and If range of contains the range of and is a complete subspace of , then and have a unique point of coincidence in Moreover if and are weakly compatible, and have a unique common fixed point.

Proof. Define by then and (2.15) becomes which further can be written as where and Clearly Hence by Theorem 2.1 and have unique common fixed point.

Now we present two examples in the support of Theorem 2.1.

Example 2.3. Let , Then is a complete metric space [17]. Consider , and as given in [9]: Let be defined as Assume that and discuss the following cases.
When then Taking in and in we obtain Hence Now, when and , then Obviously (2.31) holds. Finally when we have , and so that then Thus all conditions of Theorem 2.1 are satisfied. Moreover have a unique common fixed point.

Example 2.4. Let and   be given as Consider as and Then we have Note that is the unique coincidence point of and and and are commuting at Hence all conditions of Theorem 2.1 are satisfied. Moreover, is the unique common fixed point of and .

Following theorem can be viewed as generalization and extension of [3, Theorem 3].

Theorem 2.5. Let be a self map of a complete metric space satisfying for all , where and Then has a unique fixed point. Moreover has property .

Proof. Existence and uniqueness of fixed point of follows from Corollary 2.2. Now we prove that has property Let . We shall always assume that , since the statement for is trivial. We claim that If not, then, by (2.31), Continuing this process we arrive at which is a contradiction. Hence the result follows.

Remark 2.6 s. Existence and uniqueness of fixed point of in above theorem also follows from [9, Theorem  1].

Remark 2.7 s. (a) It is noted that if maps and involved in Theorem 2.1 are commuting, then they have property
(b) Suzuki [18] observed that Branciari [1, Theorem  1] is a particular case of Meir-Keeler fixed point theorem [19]. We pose an open problem to see if a link exists between the contractive conditions (2.15) and the Meir-Keeler condition.

Acknowledgment

The authors are thankful to referees for their precise remarks to improve the presentation of the paper.

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