The Scientific World Journal

Volume 2014 (2014), Article ID 981578, 12 pages

http://dx.doi.org/10.1155/2014/981578

Research Article

Implicit Contractive Mappings in Modular Metric and Fuzzy Metric Spaces

N. Hussain¹ and P. Salimi²

¹Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²Young Researchers and Elite Club, Rasht Branch, Islamic Azad University, Rasht, Iran

Received 21 April 2014; Accepted 26 May 2014; Published 5 June 2014

Academic Editor: Abdullah Alotaibi

Copyright © 2014 N. Hussain and P. Salimi. 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.

The notion of modular metric spaces being a natural generalization of classical modulars over linear spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, and Calderon-Lozanovskii spaces was recently introduced. In this paper we investigate the existence of fixed points of generalized -admissible modular contractive mappings in modular metric spaces. As applications, we derive some new fixed point theorems in partially ordered modular metric spaces, Suzuki type fixed point theorems in modular metric spaces and new fixed point theorems for integral contractions. In last section, we develop an important relation between fuzzy metric and modular metric and deduce certain new fixed point results in triangular fuzzy metric spaces. Moreover, some examples are provided here to illustrate the usability of the obtained results.

1. Introduction and Basic Definitions

Chistyakov introduced the notion of modular metric spaces in [1, 2]. The main idea behind this new concept is the physical interpretation of the modular. Informally speaking, whereas a metric on a set represents nonnegative finite distances between any two points of the set, a modular on a set attributes a nonnegative (possibly, infinite valued) “field of (generalized) velocities”: to each “time” the absolute value of an average velocity is associated in such a way that in order to cover the “distance” between points it takes time to move from to with velocity . But the way we approached the concept of modular metric spaces is different. Indeed we look at these spaces as the nonlinear version of the classical modular spaces introduced by Nakano [3] on vector spaces and modular function spaces introduced by Musielak [4] and Orlicz [5, 6].

For the study of electrorheological fluids (for instance lithium polymethacrylate), modeling with sufficient accuracy using classical Lebesgue and Sobolev spaces, and , where is a fixed constant is not adequate, but rather the exponent should be able to vary [7]. One of the most interesting problems in this setting is the famous Dirichlet energy problem [8, 9]. The classical technique used so far in studying this problem is to convert the energy functional, naturally defined by a modular, to a convoluted and complicated problem which involves a norm (the Luxemburg norm). The modular metric approach is more natural and has not been used extensively. In recent years, there was a strong interest to study the fixed point property in modular function spaces after the first paper [10] was published in 1990. For more on metric fixed point theory, the reader may consult the book [11] and for modular function spaces [12, 13].

Let be a nonempty set. Throughout this paper for a function , we will write

for all and .

Definition 1 (see [1, 2]). A function is said to be modular metric on if it satisfies the following axioms:

(i) if and only if , for all ;(ii), for all , and ;(iii), for all and .

If instead of (i), we have only the condition (i′)

then is said to be a pseudomodular (metric) on . A modular metric on is said to be regular if the following weaker version of (i) is satisfied:

Finally, is said to be convex if for and , it satisfies the inequality

Note that for a metric pseudomodular on a set and any , the function is nonincreasing on . Indeed, if , then

Following example presented by Abdou and Khamsi [14] is an important motivation of the concept of modular metric spaces.

Example 2. Let be a nonempty set and a nontrivial -algebra of subsets of . Let be a -ring of subsets of , such that for any and . Let us assume that there exists an increasing sequence of sets such that . By we denote the linear space of all simple functions with supports from . By we will denote the space of all extended measurable functions; that is, all functions such that there exists a sequence , , and for all . By we denote the characteristic function of the set . Let be a nontrivial, convex, and even function. We say that is a regular convex function pseudomodular if

(i);(ii) is monotone; that is, for all implies , where ;(iii) is orthogonally subadditive; that is, for any such that ;(iv) has the Fatou property; that is, for all implies , where ;(v) is order continuous in ; that is, and implies .

Similarly, as in the case of measure spaces, we say that a set is -null if for every . We say that a property holds -almost everywhere if the exceptional set is -null. As usual we identify any pair of measurable sets whose symmetric difference is -null as well as any pair of measurable functions differing only on a -null set. With this in mind we define

where each is actually an equivalence class of functions equal to -a.e. rather than an individual function. Where no confusion exists we will write instead of . Let be a regular function pseudomodular.

(a)We say that is a regular function semimodular if for every implies -a.e.(b)We say that is a regular function modular if implies -a.e.

The class of all nonzero regular convex function modulars defined on will be denoted by . Let us denote for . It is easy to prove that is a function pseudomodular in the sense of Definition in [13] (see also [15, 16]). Let be a convex function modular.

(a)The associated modular function space is the vector space , or briefly , defined by

(b)The following formula defines a norm in (frequently called Luxemburg norm):

A modular function space furnishes a wonderful example of a modular metric space. Indeed, let be a modular function space. Define the function modular by

for all , and . Then is a modular metric on . Note that is convex if and only if is convex. Moreover we have

for any .

Other easy examples may be found in [1, 2].

Definition 3. Let be a modular metric space.

(1)The sequence in is said to be -convergent to if and only if for each , , as . will be called the -limit of .(2)The sequence in is said to be -Cauchy if for each , , as , .(3)A subset of is said to be -closed if the -limit of a -convergent sequence of always belongs to .(4)A subset of is said to be -complete if any -Cauchy sequence in is a -convergent sequence and its -limit is in .(5)A subset of is said to be -bounded if we have

In 2012, Samet et al. [17] introduced the concepts of --contractive and -admissible mappings and established various fixed point theorems for such mappings defined on complete metric spaces. Afterwards Salimi et al. [18] and Hussain et al. [19–21] modified the notions of --contractive and -admissible mappings and established certain fixed point theorems.

Definition 4 (see [17]). Let be self-mapping on and a function. One says that is an -admissible mapping if

Definition 5 (see [18]). Let be self-mapping on and two functions. One says that is an -admissible mapping with respect to if

Note that if we take then this definition reduces to Definition 4. Also, if we take, then we say that is an -subadmissible mapping.

Definition 6 (see [20]). Let be a metric space. Let and be functions. One says is an --continuous mapping on , if, for given and sequence with

Example 7 (see [20]). Let and be a metric on . Assume and are defined by

and . Clearly, is not continuous, but is --continuous on .

A mapping is called orbitally continuous at if implies that . The mapping is orbitally continuous on if is orbitally continuous for all .

Remark 8 (see [20]). Let be self-mapping on an orbitally -complete metric space . Define by

where is an orbit of a point . If is an orbitally continuous map on , then is --continuous on .

In this paper, we investigate existence and uniqueness of fixed points of generalized -admissible modular contractive mappings in modular metric spaces. As applications, we derive some new fixed point theorems in partially ordered modular metric spaces, Suzuki type fixed point theorems in modular metric spaces and new fixed point theorems for integral contractions. At the end, we develop an important relation between fuzzy metric and modular metric and deduce certain new fixed point results in triangular fuzzy metric spaces. Moreover, some examples are provided here to illustrate the usability of the obtained results.

2. Fixed Point Results for Implicit Contractions

Let us first start this section with a definition of a family of functions.

Definition 9. Assume that denotes the collection of all continuous functions satisfying the following.

is increasing in its 1th variable and nonincreasing in its 5th variable.if with and , then, there exists such that

Notice that here we denote with the family of nondecreasing functions such that for all , where is the th iterate of .

Example 10. Let , where ; then .

Example 11. Let , where ; then .

Theorem 12. Let be a complete modular metric space and self-mapping satisfying the following assertions:

(i) is an -admissible mapping with respect to ;(ii)there exists such that ;(iii) is an --continuous function;(iv)assume that there exists such that for all and with we have

 where .

Then has a fixed point. Moreover, if for all we have and for all , then has a unique fixed point.

Proof. Let such that . For , we define the sequence by . Now since is an -admissible mapping with respect to then . By continuing this process we have

for all . Also, let there exists such that . Then is fixed point of and we have nothing to prove. Hence, we assume for all . Now by taking and in (iv) we get

which implies

On the other hand,

Now since is nonincreasing in its 5th variable, so by (21) and (22) we obtain

From we deduce that

Inductively, we obtain

Taking limit as in the above inequality we get

Suppose with and be given. Then there exists such that

for all . Therefore we get

for all . This shows that is a Cauchy sequence. Since is complete, so there exists such that . Now since is an --continuous mapping, so as . Therefore,

Thus has a fixed point. Let all we have and for all . Then by (iv)

Now if , then

which is a contradiction. Hence, . That is, . Thus has a unique fixed point.

Corollary 13. Let be a complete modular metric space and a self-mapping satisfying the following assertions:

is an -admissible mapping;there exists such that ; is an -continuous function;assume that there exists such that for all and with we have

 where .

Then has a fixed point. Moreover, if for all we have and for all , then has a unique fixed point.

Theorem 14. Let be a complete modular metric space and self-mapping satisfying the following assertions:

(i) is an -admissible mapping with respect to ;(ii)there exists such that ;(iii)if is a sequence in such that with as , then either

 holds for all ;(iv)condition (iv) of Theorem 12 holds.

Then has a fixed point. Moreover, if for all we have and for all , then has a unique fixed point.

Proof. As in proof of Theorem 12, we can deduce a sequence such that with and there exists such that as . From (iii) either

holds for all . Let . Then by taking and in (iv) we have

which implies

Taking limit as in the above inequality we obtain

Now since, , , is increasing in its 1th variable and nonincreasing in its 5th variable, so we obtain

which is a contradiction. Now by taking and , from we have

Hence, ; that is, . Similarly we can deduce that when .

By using Example 10 and Theorem 14 we can obtain the following corollary.

Corollary 15. Let be a complete modular metric space. Let be self-mapping satisfying the following assertions:

(i) is an -admissible mapping with respect to ;(ii)there exists such that ;(iii)if is a sequence in such that with as , then either

 holds for all ;(iv)for all and with we have

 where and .

Then has a fixed point. Moreover, if for all we have , then has a unique fixed point.

Corollary 16. Let be a complete modular metric space. Let be self-mapping satisfying the following assertions:

(i) is an -admissible mapping with respect to ;(ii)there exists such that ;(iii)if is a sequence in such that with as , then either

 holds for all ;(iv)for all and with we have

 where and .

Then has a fixed point. Moreover, if for all we have , then has a unique fixed point.

Corollary 17. Let be a complete modular metric space. Let be self-mapping satisfying the following assertions:

(i) is an -admissible mapping with respect to ;(ii)there exists such that ;(iii)if is a sequence in such that with as , then either

 holds for all ;(iv)for all and with we have

 where and .

Then has a fixed point. Moreover, if for all we have , then has a unique fixed point.

Example 18. Let and . Define ,  , and by

Let ; then . On the other hand, for all . Then, . That is, is an -admissible mapping with respect to . If is a sequence in such that with as , then for all . That is,

hold for all . Clearly, . Let . Now, if or , then , which is a contradiction. So, . Therefore,

Hence all conditions of Corollary 16 hold and has a unique fixed point.

Let be a partially ordered modular metric space. Recall that is nondecreasing if for all , . Fixed point theorems for monotone operators in ordered metric spaces are widely investigated and have found various applications in differential and integral equations (see [20, 22–25] and references therein). From results proved above, we derive the following new results in partially ordered modular metric spaces.

Theorem 19. Let be a complete partially ordered modular metric space and self-mapping satisfying the following assertions:

(i) is nondecreasing;(ii)there exists such that ;(iii) is continuous function;(iv)assume that there exists such that for all and with we have

 where .

Then has a fixed point. Moreover, if for all we have and for all , then has a unique fixed point.

Theorem 20. Let be a complete partially ordered modular metric space and self-mapping satisfying the following assertions:

(i) is nondecreasing;(ii)there exists such that ;(iii)if is a sequence in such that with as , then either

 holds for all ;(iv)assume that there exists such that for all and with we have

 where .

Then has a fixed point. Moreover, if for all we have and for all , then has a unique fixed point.

Corollary 21. Let be a complete partially ordered modular metric space and self-mapping satisfying the following assertions:

(i) is nondecreasing;(ii)there exists such that ;(iii)if is a sequence in such that with as , then either

 holds for all ;(iv)assume that for all and with we have

 where and .

Then has a fixed point. Moreover, if for all we have , then has a unique fixed point.

Corollary 22. Let be a complete partially ordered modular metric space and self-mapping satisfying the following assertions:

(i) is nondecreasing;(ii)there exists such that ;(iii)if is a sequence in such that with as , then either

 holds for all ;(iv)assume that for all and with we have

 where and .

Then has a fixed point. Moreover, if for all we have , then has a unique fixed point.

Corollary 23. Let be a complete partially ordered modular metric space and self-mapping satisfying the following assertions:

(i) is nondecreasing;(ii)there exists such that ;(iii)if is a sequence in such that with as , then either

 holds for all ;(iv)assume that for all and with we have

 where and .

Then has a fixed point. Moreover, if for all we have , then has a unique fixed point.

3. Suzuki Type Fixed Point Results in Modular Metric Spaces

In 2008, Suzuki proved a remarkable fixed point theorem, that is, a generalization of the Banach contraction principle and characterizes the metric completeness. Consequently, a number of extensions and generalizations of this result appeared in the literature (see [26–30] and references therein). As an application of our results proved above we deduce Suzuki type fixed point theorems in the setting of modular metric spaces.

Theorem 24. Let be a complete modular metric space and continuous self-mapping on . Assume that

for all and , where . Then has a fixed point. Moreover, if for all , then has a unique fixed point.

Proof. Define by

Clearly, for all and . Since is continuous, is --continuous. Thus conditions (i)–(iii) of Theorem 12 hold. Let . Then . So from (58) we obtain

Therefore all conditions of Theorem 12 hold and has a unique fixed point.

Theorem 25. Let be a complete modular metric space and self-mapping on . Assume that

for all and , where and . Then has a unique fixed point.

Proof. Define by

where . Clearly, for all and . Then conditions (i)-(ii) of Corollary 17 hold. Let be a sequence such that as . Since for all , from (61) we obtain

which implies

Suppose there exists , such that

then

Therefore,

which is a contradiction. Hence, (iii) of Corollary 17 holds. Thus all conditions of Corollary 17 hold and has a unique fixed point.

Corollary 26. Let be a complete modular metric space and self-mapping on . Assume that

for all and , where and . Then has a unique fixed point.

4. Fixed Point Results for Integral Type Contractions

Recently, Azadifar et al. [31] and Razani and Moradi [32] proved common fixed point theorems of integral type in modular metric spaces. In this section we present more general fixed point theorems for integral type contractions.

Theorem 27. Let be a complete modular metric space and self-mapping satisfying the following assertions:

(i) is an -admissible mapping with respect to ;(ii)there exists such that ;(iii) is an --continuous function;(iv)assume that there exists such that for all and with we have

 where , is a Lebesgue-integrable mapping satisfying for .

Then has a fixed point. Moreover, if for all we have and for all , then has a unique fixed point.

Theorem 28. Let be a complete modular metric space and self-mapping satisfying the following assertions:

(i) is an -admissible mapping with respect to ;(ii)there exists such that ;(iii)if is a sequence in such that with as , then either

 holds for all ;(iv)condition (iv) of Theorem 27 holds.

Then has a fixed point. Moreover, if for all we have and for all , then has a unique fixed point.

Theorem 29. Let be a complete modular metric space and continuous self-mapping on . Assume that

for all and , where , is a Lebesgue-integrable mapping satisfying for . Then has a fixed point. Moreover, if for all , then has a unique fixed point.

Theorem 30. Let be a complete modular metric space and self-mapping on . Assume that

for all and , where , and is a Lebesgue-integrable mapping satisfying for . Then has a unique fixed point.

5. Modular Metric Spaces to Fuzzy Metric Spaces

In 1988, Grabiec [33] defined contractive mappings on a fuzzy metric space and extended fixed point theorems of Banach and Edelstein in such spaces. Successively, George and Veeramani [34] slightly modified the notion of a fuzzy metric space introduced by Kramosil and Michálek and then defined a Hausdorff and first countable topology on it. Since then, the notion of a complete fuzzy metric space presented by George and Veeramani has emerged as another characterization of completeness, and many fixed point theorems have also been proved (see for more details [35–39] and the references therein). In this section we develop an important relation between modular metric and fuzzy metric and deduce in fixed point results in a triangular fuzzy metric space.

Definition 31. A 3-tuple is said to be a fuzzy metric space if is an arbitrary set, is a continuous -norm and is fuzzy set on satisfying the following conditions, for all and :

(i);(ii) for all if and only if ;(iii);(iv);(v) is continuous.

The function denotes the degree of nearness between and with respect to .

Definition 32 (see [36]). Let be a fuzzy metric space. The fuzzy metric is called triangular whenever

for all and all .

Lemma 33 (see [33, 35]). For all , is nondecreasing on .

As an application of Lemma 33, we establish the following important fact that each triangular fuzzy metric on induces a modular metric.

Lemma 34. Let be a triangular fuzzy metric space. Define

for all and all . Then is a modular metric on .

Proof. Let . Then we get

for all and . Now, since is triangular, then we get

As an application of Lemma 34 and the results proved above we deduce following new fixed point theorems in triangular fuzzy metric spaces.

Theorem 35. Let be a complete triangular fuzzy metric space and self-mapping satisfying the following assertions:

(i) is an -admissible mapping with respect to ;(ii)there exists such that ;(iii) is an --continuous function;(iv)assume that there exists such that for all and with we have

 where .

Then has a fixed point. Moreover, if for all we have and for all , then has a unique fixed point.

Theorem 36. Let be a complete triangular fuzzy metric space and self-mapping satisfying the following assertions:

(i) is an -admissible mapping with respect to ;(ii)there exists such that ;(iii)if is a sequence in such that with as , then either

 holds for all ;(iv)condition (iv) of Theorem 35 holds.

Then has a fixed point. Moreover, if for all we have and for all , then has a unique fixed point.

Theorem 37. Let be a partially ordered complete triangular fuzzy metric space and self-mapping satisfying the following assertions:

(i) is nondecreasing;(ii)there exists such that ;(iii) is continuous function;(iv)assume that there exists such that for all and with we have

 where .

Then has a fixed point. Moreover, if for all we have and for all , then has a unique fixed point.

Theorem 38. Let be a partially ordered complete triangular fuzzy metric space and self-mapping satisfying the following assertions:

(i) is nondecreasing;(ii)there exists such that ;(iii)if is a sequence in such that with as , then either

 holds for all ;(iv)assume that there exists such that for all and with we have

 where .

Then has a fixed point. Moreover, if for all we have and for all , then has a unique fixed point.

Theorem 39. Let be a complete triangular fuzzy metric space and continuous self-mapping on . Assume that

for all and , where . Then has a unique fixed point.

Theorem 40. Let be a complete triangular fuzzy metric space and continuous self-mapping on . Assume that

for all and , where and . Then has a unique fixed point.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first author acknowledges with thanks DSR, KAU for financial support.

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