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International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 891375, 8 pages
http://dx.doi.org/10.1155/2008/891375
Research Article

Common Fixed Point Theorems for Weakly Compatible Maps Satisfying a General Contractive Condition

Dipartimento di Matematica ed Applicazioni, Università degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy

Received 21 May 2008; Accepted 3 September 2008

Academic Editor: Wolfgang Kuehnel

Copyright © 2008 Cristina Di Bari and Calogero Vetro. 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 introduce a new generalized contractive condition for four mappings in the framework of metric space. We give some common fixed point results for these mappings and we deduce a fixed point result for weakly compatible mappings satisfying a contractive condition of integral type.

1. Introduction and Preliminaries

The study of common fixed point of mappings satisfying contractive type conditions has been a very active field of research during recent years. The most general of the common fixed point theorems pertaining to four mappings, , and of a metric space , uses either a Banach-type contractive condition of the formwhereor a Meir-Keeler-type -contractive condition, that is, given , there exists a such thator a -contractive condition of the forminvolving a contractive gauge function such that for each . Clearly, Banach-type contractive condition is a special case of both conditions Meir-Keeler-type -contractive and -contractive. A -contractive condition does not guarantee the existence of a fixed point unless some additional condition is assumed. Moreover, a -contractive condition, in general, does not imply the Meir-Keeler-type -contractive condition [1, Example 1.1].

Recently, some fixed point results for mappings satisfying an integral-type contractive condition are obtained in [25]. Suzuki [6] showed that Meir-Keeler contractions of integral type are still Meir-Keeler contractions. Zhang [7] introduced a generalized contractive-type condition for a pair of mappings in metric space and proved common fixed point theorems that extend results in [35]. In this paper, we give a new generalized contractive-type condition for four mappings in metric space and prove some common fixed point results for these mappings. The results obtained extend well-known comparable results in [25, 7].

Lemma 1.1 (see [8]). For every function , let be the th iterate of . Then the following hold: (i)if is nondecreasing, then for each , implies ;(ii)if is right continuous with for , then .

2. Common Fixed Points

In this section, we give our main result. Two self-mappings and of a metric space are called weakly compatible if they commute at their coincidence points. Let and be self mappings of a metric space . In the sequel, we set

Lemma 2.1. Let and be self-mappings of a metric space such that , . Assume that there exist such that (i) is nondecreasing, continuous, and for every ;(ii) is nondecreasing, right continuous, and for every . If for all ,then for each , the sequence of points of defined by the ruleis a Cauchy sequence.

Proof. We have
Similarly
If for some we have either or , then by condition (2.2) we obtain that the sequence is definitely constant and thus is a Cauchy sequence. Suppose for each .
Fromwe deducefor all . Now, fromand (ii) of Lemma 1.1, we obtain , which implies
We prove that is a Cauchy sequence. Suppose not, then there exists such that for infinite values of and with . This assures that there exist two sequences of natural numbers, with , such thatIt is not restrictive to suppose that is the least positive integer exceeding and satisfying (2.10). We haveThen . We noteand thus as . We havewhere as and for all . Then fromas , being continuous and right continuous, we getThis is a contradiction. Therefore is a Cauchy sequence.

Lemma 2.2. Let be a metric space and let , and be as in Lemma 2.1. If one of and is a complete subspace of , then the following hold: (i) and have a coincidence point;(ii) and have a coincidence point.

Proof. Fix and let be the sequence defined in Lemma 2.1. If for some , then , and and have a coincidence point. If for some , then , and and have a coincidence point. Assume that for every and is complete. By Lemma 2.1, the sequence is Cauchy; as , there exists such that . Let be such that . To prove that . We haveIf , then definitely and consequently for large , being continuous, as , we obtainThis is a contradiction, therefore and is a coincidence point for and . From , which gives , we deduce that there exists such that . To prove that . We haveand hencewhich gives .
The same result holds if we suppose that one of is complete.

Theorem 2.3. Let , and be self-mappings of a metric space such that , . Assume that there exist such that (i) is nondecreasing, continuous, and for every ;(ii) is nondecreasing, right continuous, and for every ;(iii) for all . If one of and is a complete subspace of , then the following hold: (iv) and have a coincidence point;(v) and have a coincidence point.
Further, if and as well as and are weakly compatible, then , and have a unique common fixed point.

Proof. Fix and let be the sequence defined in Lemma 2.1. Assume that is complete and let and be as in Lemma 2.2. If and are weakly compatible, thentherefore is a coincidence point of and . To prove that . Suppose that . We haveThis is a contradiction, and thus . Since , we obtain that is a common fixed point for and .
Similarly, if and are weakly compatible, we deduce that is a common fixed point for and . Now if and as well as and are weakly compatible, then is a common fixed point for , and . If is also a common fixed point for , and with , thenwhich gives .

Let be a Lebesgue integrable function which is nonnegative and such thatThe function , with satisfies condition (i) of Lemma 2.1 and from Theorem 2.3 we deduce the following theorem.

Theorem 2.4 (see [2, Theorem 2.4 2.1]). Let , and be self-mappings of a metric space such that , . Assume that there exists a nondecreasing right continuous function , with for all , such thatwhere is a Lebesgue integrable function which is nonnegative and such thatIf one of , and is a complete subspace of , then the following hold: (i) and have a coincidence point;(ii) and have a coincidence point.
Further, if and as well as and are weakly compatible, then , and have a unique common fixed point.

Remark 2.5. Theorem 2.4 is a generalization of the main theorem in [3], of [4, Theorem 2], and of [5, Theorem 2].

If in Theorem 2.3, we assume , where is the identity map on , we obtain the following theorem.

Theorem 2.6. Let and be self-mappings of a metric space . Assume that there exist such that (i) is nondecreasing, continuous, and for every ;(ii) is nondecreasing, right continuous, and for every ;(iii) for all , whereIf one of and is a complete subspace of , then and have a unique common fixed point. Moreover, for each , the iterated sequence with and converges to the common fixed point of and .

Theorem 2.6 includes [7, Theorem 1].

Acknowledgment

The authors are supported by the University of Palermo (R.S. ex 60%).

References

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