Research Article | Open Access

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

**Academic Editor:**Evgeny Korotyaev

#### 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.

#### References

- A. Branciari, “A fixed point theorem for mappings satisfying a general contractive condition of integral type,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 29, no. 9, pp. 531–536, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - F. A. Akgun and B. E. Rhoades, “Maps satisfying a Meir-Keeler type contractive conditions of integral type for which $F(T)=F({T}^{n})$,” In press. View at: Google Scholar
- B. E. Rhoades and M. Abbas, “Maps satisfying generalized contractive conditions of integral type for which $F(T)=F({T}^{n})$,”
*International Journal of Pure and Applied Mathematics*, vol. 45, no. 2, pp. 225–231, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet - B. E. Rhoades, “Some theorems on weakly contractive maps,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 47, no. 4, pp. 2683–2693, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. E. Rhoades, “Two fixed-point theorems for mappings satisfying a general contractive condition of integral type,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2003, no. 63, pp. 4007–4013, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Aliouche, “A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type,”
*Journal of Mathematical Analysis and Applications*, vol. 322, no. 2, pp. 796–802, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. Vijayaraju, B. E. Rhoades, and R. Mohanraj, “A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2005, no. 15, pp. 2359–2364, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. S. Jeong and B. E. Rhoades, “Maps for which $F(T)=F({T}^{n})$,”
*Fixed Point Theory*, vol. 6, pp. 87–131, 2005. View at: Google Scholar - P. N. Dutta and B. S. Choudhury, “A generalisation of contraction principle in metric spaces,”
*Fixed Point Theory and Applications*, vol. 2008, Article ID 406368, 8 pages, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Ya. I. Alber and S. Guerre-Delabriere, “Principle of weakly contractive maps in Hilbert spaces,” in
*New Results in Operator Theory and Its Applications*, I. Gohberg and Y. Lyubich, Eds., vol. 98 of*Operator Theory: Advances and Applications*, pp. 7–22, Birkhäuser, Basel, Switzerland, 1997. View at: Google Scholar | Zentralblatt MATH | MathSciNet - S. Sessa, “On a weak commutativity condition of mappings in fixed point considerations,”
*Publications de l'Institut Mathématique*, vol. 32, pp. 149–153, 1982. View at: Google Scholar | Zentralblatt MATH | MathSciNet - G. Jungck, “Compatible mappings and common fixed points,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 9, no. 4, pp. 771–779, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Jungck, “Common fixed points for noncontinuous nonself maps on nonmetric spaces,”
*Far East Journal of Mathematical Sciences*, vol. 4, no. 2, pp. 199–215, 1996. View at: Google Scholar | Zentralblatt MATH | MathSciNet - I. Beg and M. Abbas, “Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition,”
*Fixed Point Theory and Applications*, vol. 2006, Article ID 74503, 7 pages, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. S. Khan, M. Swaleh, and S. Sessa, “Fixed point theorems by altering distances between the points,”
*Bulletin of the Australian Mathematical Society*, vol. 30, no. 1, pp. 1–9, 1984. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Abbas and G. Jungck, “Common fixed point results for noncommuting mappings without continuity in cone metric spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 341, no. 1, pp. 416–420, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. W. Boyd and J. S. Wong, “On nonlinear contractions,”
*Proceedings of the American Mathematical Society*, vol. 20, pp. 458–464, 1969. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Suzuki, “Meir-Keeler contractions of integral type are still Meir-Keeler contractions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2007, Article ID 39281, 6 pages, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Meir and E. Keeler, “A theorem on contraction mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 28, no. 2, pp. 326–329, 1969. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

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.