Fixed Point Results for Generalized Fuzzy Contractive Mappings in Fuzzy Metric Spaces with Application in Integral Equations

Abstract

In this paper, we introduce generalized --fuzzy contractive mappings and generalized --fuzzy contractive mappings and prove existence of fixed point for such mappings. Our results generalize and improve the recent work of Gopal and Vetro (Iranian journal of fuzzy systems, 11 (2014), 95–107). Some equivalent conditions of our results are presented. Also, an example is given to support our new results.

1. Introduction

The notion of fuzzy sets was introduced initially by Zadeh [1] in 1965. Afterwards, Kramosil and Michalek [2] introduced the notion of fuzzy metric space (FMS). In 1994, George and Veeramani [3] modified the notion of FMS, introduced by Kramosil and Michalek [2], and presented a Hausdorff topology of a FMS. Recently, Samet et al. [4] introduced a new concept of -admissible and --contractive-type mappings which is described below.

Let be a metric space. A mapping is said to be an -contractive mapping if there exist two functions and so that is nondecreasing and , for all , such thatfor all . They established and proved some fixed point theorems for such mappings in complete metric spaces. Very recently, motivated by Samet et al. [4], the concept of -admissible and --fuzzy contractive mappings and --fuzzy contractive mappings in a FMS was discussed by Gopal and Vetro [5]. Gopal and Vetro introduced the concept of --fuzzy contractive mappings which is described below.

Let be a FMS. A mapping is called an --fuzzy contractive mapping if there exist two functions and such that is right continuous and , for all , andfor all and for all .

In [6], the authors studied the existence of common -fuzzy fixed points for fuzzy mappings via F-contractions on a metric space. They obtained some common fixed points of fuzzy (multivalued) mappings satisfying an F-contraction associated with the Hausdorff metric.

In this paper, we introduce generalized --fuzzy contractive and generalized --fuzzy contractive mappings, which are motivated by Gopal and Vetro [5] and show that the fixed point exists for the presented mappings. Our results generalize and improve the results of Gopal and Vetro [5]. To show the usefulness of our results, some equivalent conditions are presented.

2. Preliminaries

In this section, we recall some definitions and notions which will be used in this paper.

Definition 1. (see [7]). A binary operation is a continuous t-norm if it satisfies the following conditions: () is associative and commutative () is continuous () for all  () whenever and for all Three examples of a continuous t-norm are Lukasievicz t-norm, that is, , product t-norm, that is, , and minimum t-norm, that is, . The concept of FMS which was defined by George and Veeramani [3] which is as follows.

Definition 2 (see [3]). Let be an arbitrary nonempty set, be a continuous t-norm, and be a fuzzy set on . The triple is called a FMS if, for each and for all , (1)  (2) if and only if  (3)  (4)  (5) is continuousIf property () be replaced by , for each and , then the triple is called a non-Archimedean FMS.

Example 1. Let be a metric space. Define (or ) for all , and define as for all and . Then, is a FMS. We call this induced fuzzy metric by the metric the standard fuzzy metric.

Definition 3 (see [3, 8, 9]). Let be a FMS.(i)A sequence in is said to be convergent to a point whenever for all ; in this case, we write (ii)A sequence in is called -Cauchy if, for each and , there exists such that , for each (iii)A sequence in is called -Cauchy if , for each and (iv)A FMS in which every -Cauchy (-Cauchy) sequence is convergent is said to be -complete (-complete)

Definition 4 (see [10]). Let be a FMS. The fuzzy metric is said to be triangular if the following condition holds,for all and for all .

3. Main Results

In this section, we give our main results of this study. Firstly, we give some notions that will be needed in the sequel.

Denote by the collection of all nondecreasing right continuous functions such that for all and . It is well known that for all , where denotes the th iterate of .

Proposition 1 (see [11]). Let be a FMS. Then, is a continuous function on .

The notion of -admissibility for self-mappings in a FMS is given in [5] as follows.

Definition 5. Let be a FMS, , and . is called -admissible whenever, for all ,

Definition 6. Let be a FMS. is called a generalized --fuzzy contractive mapping whenever there exist two functions and satisfyingfor all and , whereNow, we are ready to state and prove our first main result.

Theorem 1. Let be a -complete FMS. Let be triangular and be a generalized --fuzzy contractive mapping so that(i) is -admissible,=(ii)There exists such that , for all (iii) is continuousThen, possesses a fixed point.

Proof. Define the sequence in by , for all . If for some , then is a fixed point of . Assume that for all . Since and is -admissible, so we get for all . Continuing this process we get for all and for all . By (5), for any and , we havewhereOn the contrary,From (8) and (9), we get . Substituting in (7), we obtainNow, if , by the property , thenwhich is a contradiction. So, we have and soHence, . So, the sequence is strictly increasing in the interval , for all . Let , for all . We claim that , for all . Suppose the contradiction, , for some . Taking limit in both sides of (12), we obtainwhich is a contradiction. Thus, we have , for all . Now, for a fixed , we haveas . So, is a -Cauchy sequence. Since is -complete, there exists such that , for all . Since is continuous, so we get , for all . Now, we have , for all , that is, . Uniqueness of the limit implies that .
In the above theorem, the fixed point is unique if for all , whenever and are fixed points of .
In the following theorem, we omit the continuity condition of the mapping .

Theorem 2. We have as a -complete FMS, a triangular fuzzy set, and as a generalized --fuzzy contractive mapping such that(i) is -admissible(ii)There exists such that for all (iii)For each sequence in with , for all and such that , we have , for all and Then, admits a fixed point.

Proof. Following the proof of Theorem 1, we obtain a sequence in which is -Cauchy, , for all , and there exists such that . From condition (iii), we get , for all and . Now, we haveOn the contrary,whereThus, taking limit on both sides of (16) as , we get . Now, taking limit on both sides of (15) as and from right continuity of , we obtain . This implies . Therefore, and so .

Remark 1. Since implies , for all and , thus any --fuzzy contractive mapping in the sense of [5] which is a generalized --fuzzy contractive mapping. Therefore, Theorems 1 and 2 are generalizations of Theorems 3.5 and 3.6 in [5], respectively.
The following example shows that Theorem 2 is a real generalization of Theorem 3.6 in [5].

Example 2. Letand . Let , for all , and , for all and . Define byand byIf , thenIf , thenIf , then , and clearly (8) holds true.
If and , thenSo, if , thenand if , thenWe see that , for all . So, by definition of the mapping , we get for all . Thus, is a generalized --fuzzy contractive mapping with . Also, for , we have . It is easy to check that is -admissible and the condition (iii) in Theorem 2 holds. So, by Theorem 2, possesses a fixed point. Here, . Now, let . Then, we haveThus, is not an --fuzzy contractive mapping. Therefore, the main result of [5] is not applicable in this example.
Let be a poset. A mapping is called nondecreasing if implies , for all .

Definition 7. Let be a poset and be a FMS. We say that is a generalized fuzzy ordered -contractive mapping whenever there exists such thatfor all with and , where

Corollary 1. Let be a poset and be a -complete FMS. Let be triangular and be a generalized fuzzy ordered -contractive such that(i) is nondecreasing(ii)There exists such that (iii) is continuousThen, admits a fixed point.

Proof. Define the function by if and otherwise. Then, apply Theorem 1.

Corollary 2. Let be a poset and be a -complete FMS. Let be triangular and be a generalized fuzzy ordered -contractive mapping such that(i) is nondecreasing(ii)There exists such that (iii)For each nondecreasing sequence in which , we have , for all Then, possesses a fixed point.

Proof. Define the function by if and otherwise. Then, apply Theorem 2.
Using equivalent conditions from [12], we can find also some equivalent results for Theorems 1 and 2.

Proposition 2. Let be a -complete FMS. Let be triangular and be a mapping such that there exist a mapping , a lower semicontinuous function and a function with and such thatfor all and . Moreover, assume that(i) is -admissible(ii)There exists such that , for all (iii) is continuous or for each sequence in with , for all and , for which , and we have for all and Then, admits a fixed point.

Proposition 3. Let be a -complete FMS. Let be triangular and be a mapping such that there exist a mapping , a continuous from right function with for all and a function with for all such thatfor all and . Moreover, let(i) is -admissible(ii)There exists such that for all (iii) is continuous or for each sequence in with for all and for which , and we have for all and Then, possesses a fixed point.

4. Generalized --Fuzzy Contractive Mappings

In this section, we introduce generalized --fuzzy contractive mappings and prove existence of fixed point for such mappings. Denote by the collection of all nondecreasing left continuous functions such that , for all . It is well known that and , for all , where denotes the th iterate of .

Definition 8 (see [5]). Let be a FMS, , and be a mapping. It is called that is -admissible whenever, for all ,

Definition 9. Let be a FMS. We say that is a generalized --fuzzy contractive mapping whenever there exist two functions and satisfyingfor all and , where

Theorem 3. Let be a -complete non-Archimedean FMS and be a generalized --fuzzy contractive mapping so that(i) is -admissible(ii)There exists such that for all (iii)For each sequence in with , for all and , we have , for all , with and (iv)For each sequence in with , for all and , in which , and we have , for all and Then, admits a fixed point.

Proof. Define the sequence in by , for all . If for some , then is a fixed point of . Assume that , for all . Since and is -admissible, so we get for all . Continuing this process, we get for all and for all . By (32), for any and , we havewhere, as in the proof of Theorem 1, we haveSubstituting in (34), we obtainNow, if , thenwhich is a contradiction. So, we havesoFrom (39), we obtainas . So, we have , for all . Now, if is not an -Cauchy sequence, then there are , and such that, for each with , there exist with satisfyingLet, for each , be the least positive integer exceeding satisfying the above property, that is,Now, we haveThus, , for all . Also, we haveOn the contrary,Taking the limit in the above inequality, we obtainwhich impliesTaking limit in (44) and from left continuity of , we obtainThis contradiction shows that is a -Cauchy sequence. Let . Then, from assumption (iv), , for all and . Now, we haveOn the contrary, as in the proof of Theorem 2, we haveTaking limit in (49) and from left continuity of , we obtainwhich implies , so .

Remark 2. Since , for all and , thus any --fuzzy contractive mapping in the sense of [5] which is a generalized --fuzzy contractive mapping. Therefore, Theorem 3 is a generalization of Theorem 4.4 in [5].

5. Application

This section is devoted to the application for the uniqueness of the solution of an integral equation. Consider the integral equation,and Banach space of all continuous functions defined on equipped with supremum norm,with induced metric

Now, consider the FMS with product norm as

According to [3], standard FMS and the corresponding metric space have same topologies. So, FMS defined in (55) is complete.

Theorem 6. Consider the integral operator in aswhere satisfies the following conditions: there exists such that and for all , , we havewhere

Then, the integral equation (52) possesses a unique solution.

Proof. Let and considerConsequently,Using (55), we can writeTherefore,So, (5) holds for for all with , for all . As is complete, so all the remaining conditions of Theorem 1 hold. Hence, (52) admits a unique solution.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this manuscript.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.