Research Article | Open Access

# Some Fixed Point Results for Generalized Weak Contraction Mappings in Modular Spaces

**Academic Editor:**Stefan Kunis

#### Abstract

We prove the existence theorem of fixed points for a generalized weak contractive mapping in modular spaces.

#### 1. Introduction

In 1997, Alber and Guerre-Delabriere [1] introduced the concept of weak contraction in Hilbert spaces. Later, Rhoades [2] proved that the result which Alber et al. is also valid in complete metric spaces, the result of Rhoades in the following: A mapping where is a metric space, is said to be *weakly contractive* if
where is continuous and nondecreasing function such that if and only if . In 2008, Dutta and Choudhury [3] introduced a new generalization of contraction in metric spaces and proved the following theorem.

Theorem 1. *Let be a complete metric space, and let be a self-mapping satisfying the following inequality:
**
where are both continuous and monotone nondecreasing function with if and only if . Then has a unique fixed point. *

We note that, if one takes , then (2) reduces to (1).

Recall that the theory of modular on linear spaces and the corresponding theory of modular linear spaces were founded by Nakano [4, 5] and redefined by Musielak and Orlicz [6]. Furthermore, the most complete development of these theories is due to Mazur, Luxemburg, and Turpin [7–9]. In the present time, the theory of modular and modular spaces is extensively applied, in particular, in the study of various Orlicz spaces which in their turn have broad applications [10–14]. In many cases, particularly in applications to integral operators, approximation and fixed point theory, modular-type conditions are much more natural as modular-type assumptions can be more easily verified than their metric or norm counterparts. Even though a metric is not defined, many problems in metric fixed point theory can be reformulated in modular spaces. For instance, fixed point theorems are proved in [15, 16] for nonexpansive mappings. The existences for contraction mapping in modular spaces has been studied in [3, 17–24] and the references therein.

From the above mentioned, we will study the existence of fixed point theorems for mappings satisfying generalized weak contraction mappings in modular spaces.

#### 2. Preliminaries

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

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

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

*Definition 3. *A modular is said to satisfy the -condition if , whenever as .

*Definition 4. *Let be a modular space. (1)The sequence in is said to be -*convergent* to if , as . (2)The sequence in is said to be -*Cauchy* if , as . (3)A subset of is said to be -*closed* if the -limit of a -*convergent* sequence of always belong to . (4)A subset of is said to be -*complete* if any -*Cauchy* sequence in is -*convergent* sequence and its -limit is in . (5)A subset of is said to be -*bounded* if any .

*Definition 5. *Let be a nonempty set and . A point is a fixed point of if and only if .

*Definition 6. *Let be a subset of a real numbers . A mapping is called monotone increasing (or monotone nondecreasing) if and only if , for all and are elements in . A mapping is called monotone decreasing (or monotone nonincreasing), if and only if for all and are elements in .

*Definition 7. *A sequence of a real number is said to be monotone increasing (or monotone nondecreasing), if it satisfies . It is also said to be monotone decreasing (or monotone nonincreasing) if it satisfies .

#### 3. A Generalized Weak Contraction in Modular Spaces

In this section, we prove fixed point theorems for mappings satisfying generalized weak contractions in modular spaces.

Proposition 8. *Let be a modular space on . If with , then .*

*Proof. *In case , we are done. Suppose , and then one has and

Proposition 9. *Let be a modular space in which satisfies the -condition and let be a sequence in . If as , then as , where with and . *

*Proof. *Since as , by the -condition, we get
for . Using (5), Proposition 8, and the sandwich theorem, we conclude that
for . From the fact that , we get , then there exist such that
By Proposition 8, we get
From (6) and (8), we obtain

Theorem 10. *Let be a - complete modular space, where satisfies the -condition. Let , , and be a mapping satisfying the inequality
*

*for all , where are both continuous and monotone nondecreasing functions with if and only if . Then, has a unique fixed point.*

*Proof. *Let , and we construct the sequence by , . First, we prove that the sequence converges to 0. Indeed
By monotone nondecreasing of and Proposition 8, we have

This means that the sequence is monotone decreasing and bounded below. Hence there exists such that
If , taking in the inequality (11), we get
which is a contradiction, thus . So we have
Next, we prove that the sequence is a -*Cauchy*. Suppose that is not -*Cauchy*, then there exist and subsequence , with such that
Now, let such that , then we get
which implies that
We have
By (16), (18), and (19), we get
Using (15) and Proposition 9, we have
From (20) and (21), we obtain

Letting in (17), by property of and (22), we get
which is a contradiction. Therefore, is -*Cauchy*. Since is -*complete* there exists a point such that as . Consequently, as . Next, we prove that is a unique fixed point of . Putting and in (10), we obtain
Taking in the inequality (24), we have
which implies that and . Suppose that there exists such that and , and then we have
which is a contradiction. Hence and the proof is complete.

Corollary 11. *Let be a - complete modular space, where satisfies the -condition. Let , , and be a mapping satisfying the inequality
*

*for all , where is continuous and monotone nondecreasing function with if and only if . Then, has a unique fixed point.*

*Proof. *Take , and then we obtain the Corollary 11.

Theorem 12. *Let be a - complete modular space, where satisfies the -condittion and let be a mapping satisfying the inequality
*

*for all ,where*

*and are both continuous and monotone nondecreasing functions with if and only if . Then, has a unique fixed point.*

*Proof. *First, we prove that the sequence converges to 0. Since,

By monotone nondecreasing of , we have
From the definition of , we get
If , then . Furthermore it is implied that
which is a contradiction, and, hence,
So, we have that the sequence is monotone decreasing and bounded below. Hence there exists such that
If , taking in the inequality (30), we get
which is a contradiction, and thus . So, we have
Next, we prove that the sequence is -*Cauchy*. Suppose is not -*Cauchy*, and there exist and sequence of integers , with such that
Since,
which implies that
On the other hand,
For the last term in , by Proposition 8, we have
It follow from (41) and (42) that
By (37), (38), (40), (43), and the -condition of , we have
Taking in (39), by (44) and the continuity of , we get
which is a contradiction. Hence, is -*Cauchy*. Since is -*complete*, there exists a point such that as . Next, we prove that is a unique fixed point of . Suppose that , then .

Since,
Taking in (46), by using (47), we get
which is a contradiction. Hence, and . If there exists point such that and , then using an argument similar to the above we get
which is a contradiction. Hence, and the proof is complete.

Corollary 13. *Let be a - complete modular space, where satisfies the -condition, and let be a mapping satisfying the inequality
*

*for all , where , , , and is continuous and monotone nondecreasing function with if and only if . Then, has a unique fixed point.*

*Proof. *Taking , we obtain the Corollary 13.

#### Acknowledgments

This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission. The authors would like to thank the referee for his comments and suggestion. C. Mongkolkeha was supported from the Thailand Research Fund through the the Royal Golden Jubilee Ph.D. Program (Grant no. PHD/0029/2553).

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#### Copyright

Copyright © 2013 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.