International Journal of Mathematics and Mathematical Sciences

Volume 2011, Article ID 705943, 12 pages

http://dx.doi.org/10.1155/2011/705943

## Fixed Point and Common Fixed Point Theorems for Generalized Weak Contraction Mappings of Integral Type in Modular Spaces

Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Received 11 January 2011; Accepted 19 April 2011

Academic Editor: S. M. Gusein-Zade

Copyright © 2011 Chirasak Mongkolkeha and Poom Kumam. 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 prove new fixed point and common fixed point theorems for generalized weak contractive mappings of integral type in modular spaces. Our results extend and generalize the results of A. Razani and R. Moradi (2009) and M. Beygmohammadi and A. Razani (2010).

#### 1. Introduction

Let be a metric space. A mapping is a contraction if where . The Banach Contraction Mapping Principle appeared in explicit form in Banach's thesis in 1922 [1]. For its simplicity and usefulness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis. Banach contraction principle has been extended in many different directions; see [2–6]. In 1997 Alber and Guerre-Delabriere [7] introduced the concept of weak contraction in Hilbert spaces, and Rhoades [8] has showed that the result by Akber et al. is also valid in complete metric spaces A mapping is said to be weakly contractive if where is continuous and nondecreasing function such that if and only if . If one takes where , then (1.2) reduces to (1.1). In 2002, Branciari [9] gave a fixed point result for a single mapping an analogue of Banach's contraction principle for an integral-type inequality, which is stated as follow.

Theorem 1.1. *Let be a complete metric space, , a mapping such that for each ,
**
where is a Lebesgue integrable which is summable, nonnegative, and for all , . Then, f has a unique fixed point such that for each , .*

Afterward, many authors extended this work to more general contractive conditions. The works noted in [10–12] are some examples from this line of research.

The notion of modular spaces, as a generalize of metric spaces, was introduced by Nakano [13] and redefined by Musielak and Orlicz [14]. A lot of mathematicians are interested, fixed points of Modular spaces, for example [15–22]. In 2009, Razani and Moradi [23] studied fixed point theorems for -compatible maps of integral type in modular spaces.

Recently, Beygmohammadi and Razani [24] proved the existence for mapping defined on a complete modular space satisfying contractive inequality of integral type.

In this paper, we study the existence of fixed point and common fixed point theorems for -compatible mapping satisfying a generalize weak contraction of integral type in modular spaces.

First, we start with a brief recollection of basic concepts and facts in modular spaces.

*Definition 1.2. *Let be a vector space over (or ). A functional is called a modular if for arbitrary and , elements of satisfy the following conditions: (1) if and only if ; (2) for all scalar with ; (3), whenever and . If we replace (3) by(4), for , with an , then the modular is called s-convex modular, and if , is called convex modular.

If is modular in , then the set defined by is called a modular space. is a vector subspace of .

*Definition 1.3. *A modular is said to satisfy the - if as , whenever as .

*Definition 1.4. *Let be a modular space. Then,(1)the sequence in is said to be - to if , as , (2)the sequence in is said to be -if , as ,(3)a subset of is said to be - if the - of a -convergent sequence of always belong to , (4)a subset of is said to be - if any - sequence in is - sequence and its is in ,(5)a subset of is said to be - if = sup .

*Definition 1.5. *Let be a subset of and an arbitrary mapping. is called a - if for each there exists such that

*Definition 1.6. *Let be a modular space, where satisfies the -. Two self-mappings and of are called - if as , whenever is a sequence in such that and for some point .

#### 2. A Common Fixed Point Theorem for *ρ*-Compatible Generalized Weak Contraction Maps of Integral Type

Theorem 2.1. *Let be a - modular space, where satisfies the -. Let , and are two - mappings such that and
**
for all , where is a Lebesgue integrable which is summable, nonnegative, and for all , and is lower semicontinuous function with for all and if and only if . If one of or is continuous, then there exists a unique common fixed point of and . *

*Proof. *Let and generate inductively the sequence as follow: . First, we prove that the sequence converges to 0. Since,
This means that the sequence is decreasing and bounded below. Hence, there exists such that
If , then . Taking in the inequality (2.2) which is a contradiction, thus . This implies that

Next, we prove that the sequence is -. Suppose is not -, then there exists and sequence of integers with such that
We can assume that

Let be the smallest number exceeding for which (2.5) holds, and
Since and clearly , by well ordering principle, the minimum element of is denoted by and obviously (2.6) holds. Now, let be such that , then we get

Using the - and (2.4), we obtain
It follows that
From (2.8) and (2.11), we also have
which is a contradiction. Hence, is - and by the -, is -. Since is -, there exists a point such that as . If is continuous, then and as . Since as , by -, as . Next, we prove that is a unique fixed point of . Indeed,
Taking in the inequality (2.13), we have
which implies that and . Since , there exists such that . The inequality,
as , yields
and, thus,
which implies that, and also (see [25]). Hence, . Suppose that there exists such that and , we have and
which is a contradiction. Hence, and the proof is complete.

In fact, if take where and take , respectively, where is a nondecreasing and right continuous function with for all , we obtain following corollaries.

Corollary 2.2 (see [23]). *Let be a modular space, where satisfies the . Suppose , and are two mappings such that and
**
for some , where is a Lebesgue integrable which is summable, nonnegative, and for all , . If one of or is continuous, then there exists a unique common fixed point of and .*

Corollary 2.3 (see [23]). *Let be a modular space, where satisfies the . Suppose , and are two mappings such that and
**
where is a Lebesgue integrable which is summable, nonnegative, and for all , and is a nondecreasing and right continuous function with for all . If one of or is continuous, then there exists a unique common fixed point of and .*

#### 3. A Fixed Point Theorem for Generalized Weak Contraction Mapping of Integral Type

Theorem 3.1. *Let be a modular space, where satisfies the . Let , and be a mapping such that for each ,
**
where is a Lebesgue integrable which is summable, nonnegative, and for all , and is lower semicontinuous function with for all and if and only if . Then, has a unique fixed point. *

*Proof. *First, we prove that the sequence converges to 0. Since,
it follows that the sequence is decreasing and bounded below. Hence, there exists such that
If , then , taking in the inequality (3.2) which is a contradiction, thus . So, we have
Next, we prove that the sequence is -. Suppose is not -, there exists and sequence of integers with such that
We can assume that
Let be the smallest number exceeding for which (3.5) holds, and
Since and clearly , by well ordering principle, the minimum element of is denoted by and obviously (3.6) holds. Now, let be such that , then we get

Using the - and (3.4), we obtain
From (3.8) and (3.11), we have
which is a contradiction. Hence, is - and again by the -, is -. Since is -, there exists a point such that as . Next, we prove that is a unique fixed point of . Indeed,
Since as , we obtain
which implies that
So, we have
Thus and . Suppose that there exists such that and , we have and
which is a contradiction. Hence, and the proof is complete.

Corollary 3.2. *Let be a modular space, where satisfies the . Let be a mapping such that there exists an and where and for each ,
**
where is a Lebesgue integrable which is summable, nonnegative, and for all , . Then, has a unique fixed point in . *

Corollary 3.3 (see [24]). *Let be a modular space where satisfies the . Assume that is an increasing and upper semicontinuous function satisfying for all . Let be a Lebesgue integrable which is summable, nonnegative, and for all , and let be a mapping such that there are where ,
**
for each . Then, has a unique fixed point in .*

#### Acknowledgments

The authors would like to thank the National Research University Project of Thailand's Office of the Higher Education Commission for financial support under NRU-CSEC Project no. 54000267. Mr. Chirasak Mongkolkeha was supported by the Royal Golden Jubilee Grant for Ph.D. program at KMUTT, Thailand. Furthermore, this work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission.

#### References

- S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux
équations intégrales,”
*Fundamenta Mathematicae*, vol. 3, pp. 133–181, 1922. View at Google Scholar - 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 · View at Google Scholar - 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 · View at Zentralblatt MATH - G. Jungck and B. E. Rhoades, “Fixed points for set valued functions without continuity,”
*Indian Journal of Pure and Applied Mathematics*, vol. 29, no. 3, pp. 227–238, 1998. View at Google Scholar · View at Zentralblatt MATH - W. Sintunavarat and P. Kumam, “Coincidence and common fixed points for hybrid strict contractions without the weakly commuting condition,”
*Applied Mathematics Letters*, vol. 22, no. 12, pp. 1877–1881, 2009. View at Publisher · View at Google Scholar - W. Sintunavarat and P. Kumam, “Weak condition for generalized multi-valued $(f,\alpha ,\beta )$-weak contraction mappings,”
*Applied Mathematics Letters*, vol. 24, no. 4, pp. 460–465, 2011. View at Publisher · View at Google Scholar - Y. I. Alber and S. Guerre-Delabriere, “Principle of weakly contractive maps in Hilbert spaces,” in
*New Results in Operator Theory and Its Applications*, vol. 98 of*Operator Theory: Advances and Applications*, pp. 7–22, Birkhäuser, Basel, Switzerland, 1997. View at Google Scholar · View at Zentralblatt MATH - 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 · View at Google Scholar · View at Zentralblatt MATH - 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 · View at Google Scholar · View at Zentralblatt MATH - 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 · View at Google Scholar · View at Zentralblatt MATH - W. Sintunavart and P. Kumam, “Gregus-type common fixed point theorems for tangential multivalued mappings of integral type in metric spaces,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2011, Article ID 923458, 12 pages, 2011. View at Publisher · View at Google Scholar - W. Sintunavarat and P. Kumam, “Gregus type fixed points for a tangential multi-valued mappings satisfying contractive conditions of integral type,”
*Journal of Inequalities and Applications*. In Press. - H. Nakano,
*Modulared Semi-Ordered Linear Spaces*, Tokyo Mathematical Book Series, Maruzen Co. Ltd, Tokyo, Japan, 1950. - J. Musielak and W. Orlicz, “On modular spaces,”
*Studia Mathematica*, vol. 18, pp. 49–65, 1959. View at Google Scholar · View at Zentralblatt MATH - T. Dominguez Benavides, M. A. Khamsi, and S. Samadi, “Uniformly Lipschitzian mappings in modular function spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 46, no. 2, pp. 267–278, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. A. Khamsi, “Quasicontraction mappings in modular spaces without ${\mathrm{\Delta}}_{2}$-condition,”
*Fixed Point Theory and Applications*, vol. 2008, Article ID 916187, 6 pages, 2008. View at Google Scholar - M. A. Khamsi, W. M. Kozłowski, and S. Reich, “Fixed point theory in modular function spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 14, no. 11, pp. 935–953, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Kumam, “On nonsquare and Jordan-von Neumann constants of modular spaces,”
*Southeast Asian Bulletin of Mathematics*, vol. 30, no. 1, pp. 69–77, 2006. View at Google Scholar · View at Zentralblatt MATH - P. Kumam, “On uniform Opial condition, uniform Kadec-Klee property in modular spaces and application to fixed point theory,”
*Journal of Interdisciplinary Mathematics*, vol. 8, no. 3, pp. 377–385, 2005. View at Google Scholar · View at Zentralblatt MATH - P. Kumam, “Fixed point theorems for nonexpansive mappings in modular spaces,”
*Archivum Mathematicum*, vol. 40, no. 4, pp. 345–353, 2004. View at Google Scholar · View at Zentralblatt MATH - P. Kumam, “Some geometrical properties and fixed point theorems in modular spaces,” in
*International Conference on Fixed Point Theory and Applications*, pp. 173–188, Yokohama Publishers, Yokohama, Japan, 2004. View at Google Scholar · View at Zentralblatt MATH - A. Razani, E. Nabizadeh, M. B. Mohamadi, and S. H. Pour, “Fixed points of nonlinear and asymptotic contractions in the modular space,”
*Abstract and Applied Analysis*, vol. 2007, Article ID 40575, 10 pages, 2007. View at Google Scholar · View at Zentralblatt MATH - A. Razani and R. Moradi, “Common fixed point theorems of integral type in modular spaces,”
*Bulletin of the Iranian Mathematical Society*, vol. 35, no. 2, pp. 11–24, 2009. View at Google Scholar - M. Beygmohammadi and A. Razani, “Two fixed-point theorems for mappings satisfying a general contractive condition of integral type in the modular space,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2010, Article ID 317107, 10 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - H. Kaneko and S. Sessa, “Fixed point theorems for compatible multi-valued and single-valued mappings,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 12, no. 2, pp. 257–262, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH