𝐿-Fuzzy Fixed Points Theorems for 𝐿-Fuzzy Mappings via 𝛽_ℱ𝐿-Admissible Pair

Abstract

We define the concept of -admissible for a pair of -fuzzy mappings and establish the existence of common -fuzzy fixed point theorem. Our result generalizes some useful results in the literature. We provide an example to support our result.

1. Introduction

A large variety of the problems of analysis and applied mathematics relate to finding solutions of nonlinear functional equations which can be formulated in terms of finding the fixed points of nonlinear mappings. Heilpern [1] first introduced the concept of fuzzy mappings and established a fixed point theorem for fuzzy contraction mappings in complete metric linear spaces, which is a fuzzy extension of Banach contraction principle and Nadler’s [2] fixed point theorem. Subsequently several other authors [3–17] generalized this result and studied the existence of fixed points and common fixed points of fuzzy mappings satisfying a contractive type condition.

Zadeh published his important paper “Fuzzy sets” [18], after that Goguen published the paper “-Fuzzy sets” [19]. The concept of -fuzzy sets is a generalization of the concept of fuzzy sets. Fuzzy set is a special case of -fuzzy set when . There are basically two understandings of the meaning of , one is when is a complete lattice equipped with a multiplication operator satisfying certain conditions as shown in the initial paper [19] and the second understanding of the meaning of is that is a completely distributive complete lattice with an order-reversing involution (see, e.g., [20–22], etc.).

In 2012, Samet et al. [23] introduced the concept of -admissible mapping and established fixed point theorems via -admissible and also showed that these results can be utilized to derive fixed point theorems in partially ordered spaces and coupled fixed point theorems. Moreover, they applied the main results to ordinary differential equations. Afterwards, Asl et al. [24] extended the concept of -admissible for single valued mappings to multivalued mappings. Recently, Mohammadi et al. [25] introduced the concept of -admissible for multivalued mappings which is different from the notion of -admissible which has been provided in [24] and Azam and Beg [26] obtained a common -fuzzy fixed point of a pair of fuzzy mappings on a complete metric space under a generalized contractive condition for -level sets via Hausdorff metric for fuzzy sets.

In this paper we introduce the concept of -admissible for a pair of -fuzzy mappings and establish the existence of common -fuzzy fixed point theorem. We also have given an example to support our main theorem.

2. Preliminaries

Let be a metric space, and denote  is nonempty closed and bounded subset of ,  is nonempty compact subset of .

For and the sets define

Then the Hausdorff metric on induced by is defined as

Lemma 1 (see [2]). Let be a metric space and ; then for each

Lemma 2 (see [2]). Let be a metric space and ; then for each , , there exists an element such that

Definition 3 (see [19]). A partially ordered set is called (i)a lattice, if ,   for any ; (ii)a complete lattice, if ,   for any ; (iii)distributive if ,   for any .

Definition 4 (see [19]). Let be a lattice with top element and bottom element and let . Then is called a complement of , if , and . If has a complement element, then it is unique. It is denoted by .

Definition 5 (see [19]). An -fuzzy set on a nonempty set is a function , where is complete distributive lattice with and .

Remark 6. The class of -fuzzy sets is larger than the class of fuzzy sets as an -fuzzy set is a fuzzy set if .

The -level set of -fuzzy set is denoted by and is defined as follows:

Here denotes the closure of the set .

We denote and define the characteristic function of an -fuzzy set as follows:

Definition 7. Let be an arbitrary set and a metric space. A mapping is called -fuzzy mapping if is a mapping from into . An -fuzzy mapping is an -fuzzy subset on with membership function . The function is the grade of membership of in .

Definition 8. Let be a metric space and -fuzzy mappings from into . A point is called an -fuzzy fixed point of if , where . The point is called a common -fuzzy fixed point of and if .

Definition 9 (see [23]). Let be a nonempty set, , and . We say that is -admissible if for all we have

Definition 10 (see [24]). Let be a nonempty set, , where is a collection of subset of , and . We say that is -admissible if for all we have

Definition 11 (see [25]). Let be a nonempty set, , where is a collection of subset of and . We say that is -admissible whenever for each and with we have for all .

3. Main Result

In this section, we introduce a new concept of -admissible for a pair of -fuzzy mappings and establish the existence of common -fuzzy fixed point theorem.

Definition 12. Let be a metric space, , and -fuzzy mappings from into . The order pair is said to be -admissible if it satisfies the following conditions: (i)for each and , where , with , we have for all , where ; (ii)for each and , where , with , we have for all , where .
If then is called -admissible.

Remark 13. It is easy to see that if is -admissible, then is also -admissible.

Next, we give a common -fuzzy fixed point theorem for -admissible pair.

Theorem 14. Let be a complete metric space, , and -fuzzy mappings from into satisfying the following conditions.(a) For each , there exists such that , are nonempty closed bounded subsets of and for , there exists with .(b) For all , we have  where , , , , and are nonnegative real numbers and and either or .(c) is -admissible pair.(d) If , such that and , then .
Then there exists such that .

Proof. We will prove the above result by considering the following three cases:(1),(2),(3) and .
Case 1. For in condition (a), there exist and such that and also there exists such that and are nonempty closed bounded subsets of . From Lemma 1, we obtain that
Now, inequality (9) implies that
Using together with the fact that , we get
It follows that , which further implies that
By condition (c), for and such that , we have for all . Since , therefore and hence
Again, inequality (9) implies that
Since and , we get which implies that and hence
Case 2. For in condition (a), there exist and such that and also there exists such that and are nonempty closed bounded subsets of . By condition (c), we have for all . From Lemma 1, we obtain that
Using together with the fact that , we get
It follows that , which further implies that
By condition (c), we have , and hence
Again, inequality (9) implies that
Since and , we get which implies that and hence
Case 3. Let and . Next, we show that if or , then .
If , then and so . Now if , then By condition (a), for , there exists such that is a nonempty closed bounded subset of . Since , by Lemma 2, there exists such that This implies that By the same argument, for , there exists such that is a nonempty closed bounded subset of . Since , by Lemma 2, there exists such that By condition (c), for and such that , we have for . So we have This implies that By repeating the above process, for , there exists such that is a nonempty closed bounded subset of . From Lemma 2, there exists such that By condition (c), for and such that , we have for . So we have This implies that By induction, we produce a sequence in such that Now, we have This implies that Similarly, This implies that From (36) and (38), it follows that, for each , Then for , we have Similarly, we obtain that Since , so by Cauchy’s root test, we get and are convergent series. Therefore, is a Cauchy sequence in . Now, from the completeness of , there exists such that as . By condition (d), we have for all . Now, we have Since so we get This implies that Letting , we have . It implies that . Similarly, by using we can show that . Therefore, . This completes the proof.

Next, we give an example to support the validity of our result.

Example 15. Let , , whenever ; then is a complete metric space. Let with , , and are not comparable; then is a complete distributive lattice. Define a pair of mappings as follows: Define as follows: For all , there exists , such that and all conditions of the above theorem are satisfied. Hence, there exists , such that .

Corollary 16. Let be a complete metric space, , and , fuzzy mappings from into satisfying the following conditions.(a) For each , there exists such that , are nonempty closed bounded subsets of and for , there exists with .(b) For all , we have  where , , , , and are nonnegative real numbers and and either or .(c) is -admissible pair.(d) If , such that and then .
Then there exists such that .

Proof. Consider an -fuzzy mapping defined by Then for , we have Hence by Theorem 14, we follow the result.
If we set for all in Corollary 16, we get the following result.

Corollary 17 (see [26]). Let be a complete metric space and , fuzzy mappings from into satisfying the following conditions:(a) for each , there exists such that , are nonempty closed bounded subsets of ;(b) for all , we have  where , , , , and are nonnegative real numbers and and either or .
Then there exists such that .

Conflict of Interests

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