Open Access

Some Common Fixed Point Theorems for Weakly Compatible Mappings in Metric Spaces

Fixed Point Theory and Applications20092009:804734

https://doi.org/10.1155/2009/804734

Received: 23 October 2008

Accepted: 18 January 2009

Published: 3 February 2009

Abstract

We establish a common fixed point theorem for weakly compatible mappings generalizing a result of Khan and Kubiaczyk (1988). Also, an example is given to support our generalization. We also prove common fixed point theorems for weakly compatible mappings in metric and compact metric spaces.

1. Introduction

In the last years, fixed point theorems have been applied to show the existence and uniqueness of the solutions of differential equations, integral equations and many other branches mathematics (see, e.g., [13]). Some common fixed point theorems for weakly commuting, compatible, -compatible and weakly compatible mappings under different contractive conditions in metric spaces have appeared in [415]. Throughout this paper, is a metric space.

Following [9, 16], we define,
(1.1)
For all , we define
(1.2)

where , for some and , for some .

If for some , we denote , and for , and , respectively. Also, if , then one can deduce that .

It follows immediately from the definition of that, for every ,
(1.3)

We need the following definitions and lemmas.

Definition 1.1 (see [16]).

A sequence of nonempty subsets of is said to be convergent to if:

(i)each point in is the limit of a convergent sequence , where is in for ( := the set of all positive integers),

(ii)for arbitrary , there exists an integer such that for , where denotes the set of all points in for which there exists a point in , depending on such that .

is then said to be the limit of the sequence .

Definition 1.2 (see [9]).

A set-valued function is said to be continuous if for any sequence in with , it yields .

Lemma 1.3 (see [16]).

If and are sequences in converging to and in , respectively, then the sequence converges to .

Lemma 1.4 (see [16]).

Let be a sequence in and let be a point in such that . Then the sequence converges to the set in .

Lemma 1.5 (see [9]).

For any , it yields that .

Lemma 1.6 (see [17]).

Let be a right continuous function such that for every . Then, for every , where denotes the -times repeated composition of with itself.

Definition 1.7 (see [15]).

The mappings and are weakly commuting on if and for all .

Definition 1.8 (see [13]).

The mappings and are said to be -compatible if whenever is a sequence in such that , and for some .

Definition 1.9 (see [13]).

The mappings and are weakly compatible if they commute at coincidence points, that is, for each point such that , then (note that the equation implies that is a singleton).

If is a single-valued mapping, then Definition 1.7 (resp., Definitions 1.8 and 1.9) reduces to the concept of weak commutativity (resp., compatibility and weak compatibility) for single-valued mappings due to Sessa [18] (resp., Jungck [11, 12]).

It can be seen that
(1.4)

but the converse of these implications may not be true (see, [13, 15]).

Throughtout this paper, we assume that is the set of all functions satisfying the following conditions:

(i) is upper semi-continuous continuous at a point from the right, and non-decreasing in each coodinate variable,

(ii)For each , .

Theorem 1.10 (see [19]).

Let be mappings of a complete metric space into and be a mapping of into itself such that and are continuous, , , , and for all ,
(1.5)

where satisfies (i) and for each , and , with . Then and have a unique common fixed point such that .

In the present paper, we are concerned with the following:

(1)replacing the commutativity of the mappings in Theorem 1.10 by the weak compatibility of a pair of mappings to obtain a common fixed point theorem metric spaces without the continuity assumption of the mappings,

(2)giving an example to support our generalization of Theorem 1.10,

(3)establishing another common fixed point theorem for two families of set-valued mappings and two single-valued mappings,

(4)proving a common fixed point theorem for weakly compatible mappings under a strict contractive condition on compact metric spaces.

2. Main Results

In this section, we establish a common fixed point theorem in metric spaces generalizing Theorems 1.10. Also, an example is introduced to support our generalization. We prove a common fixed point theorem for two families of set-valued mappings and two single-valued mappings. Finally, we establish a common fixed point theorem under a strict contractive condition on compact metric spaces.

First we state and prove the following.

Theorem 2.1.

Let be two sefmaps of a metric space and let be two set-valued mappings with

(2.1)
Suppose that one of and is complete and the pairs and are weakly compatible. If there exists a function such that for all ,
(2.2)

then there is a point such that .

Proof.

Let be an arbitrary point in . By (2.1), we choose a point in such that and for this point there exists a point in such that . Continuing this manner we can define a sequence as follows:
(2.3)
for . For simplicity, we put for . By (2.2) and (2.3), we have that
(2.4)
If , then
(2.5)
This contradiction demands that
(2.6)
Similarly, one can deduce that
(2.7)
So, for each , we obtain that
(2.8)
where . By (2.8) and Lemma 1.6, we obtain that . Since
(2.9)

then . Therefore, is a Cauchy sequence.

Let be an arbitrary point in for . Then and is a Cauchy sequence. We assume without loss of generality that is complete. Let be the sequence defined by (2.3). But for all . Hence, we find that
(2.10)

as . So, is a Cauchy sequence. Hence, for some . But by (2.3), so that . Consequently, . Moreover, we have, for , that . Therefore, . So, we have by Lemma 1.4 that . In like manner it follows that and .

Since, for ,
(2.11)
and as , we get from Lemma 1.3 that
(2.12)
This is absurd. So, . But , so such that . If , , then we have
(2.13)

We must conclude that .

Since and the pair is weakly compatible, so . Using the inequality (2.2), we have
(2.14)

This contradiction demands that . Similarly, if the pair is weakly compatible, one can deduce that . Therefore, we get that .

The proof, assuming the completeness of , is similar to the above.

To see that is unique, suppose that . If , then
(2.15)

which is inadmissible. So, .

Now, we give an example to show the greater generality of Theorem 2.1 over Theorem 1.10.

Example 2.2.

Let endowed with the Euclidean metric . Assume that for every . Define and as follows:
(2.16)
We have that and . Moreover, if . If , then and . So, we obtain that
(2.17)

for all . It is clear that is a complete metric space. Since is a closed subset of , so is complete. We note that is a -compatible pair and therefore a weakly compatible pair. Also, and , that is, and are weakly compatible. On the other hand, if , so that even though , that is, is not a -compatible pair. We know that is the unique common fixed point of and . Hence the hypotheses of Theorem 2.1 are satisfied. Theorem 1.10 is not applicable because for all , and the maps I, J and G are not continuous at .

In Theorem 2.1, if the mappings and are replaced by and , where is an index set, we obtain the following.

Theorem 2.3.

Let be a metric space, and let be selfmaps of , and for , be set-valued mappings with and . Suppose that one of and is complete and for the pairs and are weakly compatible. If there exists a function such that, for all ,
(2.18)

then there is a point such that for each .

Proof.

Using Theorem 2.1, we obtain for any , there is a unique point such that and . For all
(2.19)

This yields that .

Inspired by the work of Chang [9], we state the following theorem on compact metric spaces.

Theorem 2.4.

Let be a compact metric space, selfmaps of set-valued functions with and Suppose that the pairs , are weakly compatible and the functions , are continuous. If there exists a function , and for all , the following inequality:
(2.20)

holds whenever the right-hand side of (2.20) is positive, then there is a unique point in such that .

Declarations

Acknowledgment

The author wishes to thank the refrees for their comments which improved the original manuscript.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Assiut University

References

  1. Pathak HK, Fisher B: Common fixed point theorems with applications in dynamic programming. Glasnik Matematički 1996,31(51):321–328.MathSciNetMATHGoogle Scholar
  2. Pathak HK, Khan MS, Tiwari R: A common fixed point theorem and its application to nonlinear integral equations. Computers & Mathematics with Applications 2007,53(6):961–971. 10.1016/j.camwa.2006.08.046MathSciNetView ArticleMATHGoogle Scholar
  3. Pathak HK, Mishra SN, Kalinde AK: Common fixed point theorems with applications to nonlinear integral equations. Demonstratio Mathematica 1999,32(3):547–564.MathSciNetMATHGoogle Scholar
  4. Ahmed MA: Common fixed point theorems for weakly compatible mappings. The Rocky Mountain Journal of Mathematics 2003,33(4):1189–1203. 10.1216/rmjm/1181075457MathSciNetView ArticleMATHGoogle Scholar
  5. Ahmed MA: Common fixed points for four mappings under a contractive condition of Kiventidis type. Proceedings of the Mathematical and Physical Society of Egypt 2005, (83):83–93.Google Scholar
  6. Ahmed MA, Rhoades BE: Some common fixed point theorems for compatible mappings. Indian Journal of Pure and Applied Mathematics 2001,32(8):1247–1254.MathSciNetMATHGoogle Scholar
  7. Banerjee A, Singh TB: A fixed point theorem for set-valued mappings. Applied Mathematics and Mechanics 2001,22(12):1397–1403.MathSciNetMATHGoogle Scholar
  8. Banerjee A, Thakur BS: A note on a theorem of Tas, Telci and Fisher. Applied Mathematics and Mechanics 1998,19(4):333–334. 10.1007/BF02457537MathSciNetView ArticleMATHGoogle Scholar
  9. Chang T-H: Fixed point theorems for contractive type set-valued mappings. Mathematica Japonica 1993,38(4):675–690.MathSciNetMATHGoogle Scholar
  10. Ćirić LjB, Nikolić NT, Ume JS: Common fixed point theorems for weakly compatible quasi contraction mappings. Acta Mathematica Hungarica 2006,113(4):257–267. 10.1007/s10474-006-0103-zMathSciNetView ArticleMATHGoogle Scholar
  11. Jungck G: Compatible mappings and common fixed points. International Journal of Mathematics and Mathematical Sciences 1986,9(4):771–779. 10.1155/S0161171286000935MathSciNetView ArticleMATHGoogle Scholar
  12. Jungck G: Common fixed points for commuting and compatible maps on compacta. Proceedings of the American Mathematical Society 1988,103(3):977–983. 10.1090/S0002-9939-1988-0947693-2MathSciNetView ArticleMATHGoogle Scholar
  13. Jungck G, Rhoades BE: Fixed points for set valued functions without continuity. Indian Journal of Pure and Applied Mathematics 1998,29(3):227–238.MathSciNetMATHGoogle Scholar
  14. Khan AR, Domlo AA, Hussain N: Coincidences of Lipschitz-type hybrid maps and invariant approximation. Numerical Functional Analysis and Optimization 2007,28(9–10):1165–1177. 10.1080/01630560701563859MathSciNetView ArticleMATHGoogle Scholar
  15. Sessa S, Khan MS, Imdad M: A common fixed point theorem with a weak commutativity condition. Glasnik Matematički. Serija III 1986,21(41):225–235.MathSciNetMATHGoogle Scholar
  16. Fisher B: Common fixed points of mappings and set-valued mappings. Rostocker Mathematisches Kolloquium 1981, (18):69–77.Google Scholar
  17. Matkowski J: Fixed point theorems for mappings with a contractive iterate at a point. Proceedings of the American Mathematical Society 1977,62(2):344–348. 10.1090/S0002-9939-1977-0436113-5MathSciNetView ArticleMATHGoogle Scholar
  18. Sessa S: On a weak commutativity condition of mappings in fixed point considerations. Publications de l'Institut Mathématique. Nouvelle Série 1982,32(46):149–153.MathSciNetMATHGoogle Scholar
  19. Khan MS, Kubiaczyk I: Fixed point theorems for point to set maps. Mathematica Japonica 1988,33(3):409–415.MathSciNetMATHGoogle Scholar

Copyright

© M. A. Ahmed. 2009

This article is published under license to BioMed Central Ltd. 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.