Some Fixed Point Results for Generalized Weak Contraction Mappings in Modular Spaces
We prove the existence theorem of fixed points for a generalized weak contractive mapping in modular spaces.
In 1997, Alber and Guerre-Delabriere  introduced the concept of weak contraction in Hilbert spaces. Later, Rhoades  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  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.
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 . 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.
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.
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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