Abstract

The aim of this paper is to obtain the common fuzz fixed points of -fuzzy mappings satisfying generalized almost -contraction in complete metric spaces. Our results are extensions and improvements of the several well-known recent and classical results in literature. We give an example for supporting these results. As an application, we apply our obtained results to study the existence of a solution for a second order nonlinear boundary value problem.

1. Introduction and Preliminaries

The fixed point result of Banach [1] is an interesting tool that ensures the existence and uniqueness of a fixed point of self-mappings defined on metric spaces. Later, various extensions and generalizations of Banach’s theorem appeared by defining a variety of contractive type conditions for self and non-self-mappings on different spaces (see [227]).

In [17, 18], Berinde discussed many contraction type mappings. He initiated the notion of almost contractions.

Definition 1. A self-mapping on a metric space (in short MS) is named as an almost contraction, if there are and so that for all , This almost contraction has been generalized as follows.

Definition 2. A self-mapping on a MS is called a generalized almost contraction, if there are and so that for all , Wardowski [28] presented the notion of -contractions and established a related fixed point theorem.

Definition 3 (see [28]). A self-mapping on a MS said to be an -contraction if there are and such that where F is the set of functions such that
(F1) For all , ,
(F2) For each sequence of positive numbers, (F3) There is so that ,
Later on, Altun et al. [22] modified Definition 3 by adding the condition (F4):
(F4) , for all with
Denote by the set of functions verifying the conditions -.

Theorem 4 (see [28]). Let be an -contraction on a complete MS. Then, admits a unique fixed point , and for each , the sequence is convergent to .

Definition 5 (see [4]). A self-mapping on a MS is said to be a -contraction, if there are and so that where is the set of functions so that
() is nondecreasing,
() For each sequence , () There are and such that
() is continuous
The following result was established by Jleli and Samet [4].

Theorem 6 (see [4]). Each -contraction mapping on a complete MS possesses a unique fixed point.
As in [22], Hancer et al. [12] took the following:
() , for all with .
Denote by the set of functions so that , and hold.
As in [9], we denote by the set of functions so that
() is nondecreasing,
()
() is continuous.

Theorem 7 (see [9]). Let be a self-mapping on a MS . We have the equivalence of the following: (i) is a -contraction with ,(ii) is a -contraction with .Very recently, Liu et al. [9] introduced the notion of ()-type contractions and established new fixed point theorems for such kind of mappings in complete MS.

Definition 8. A self-mapping on a MS is said to be a -type contraction, if there are a comparison function and such that for all , where and is the set of functions so that
() is nondecreasing,
() For each sequence , () is continuous
As in [2], a function is named as a comparison function if (1) is increasing, that is, ,(2) for all , where stands for the nth iterate of .Denote by the set of comparison functions.

Lemma 9 (see [9]). Let be a nondecreasing and continuous function with , and be a sequence in . Then,

Example 10 (see [2]). Consider the following comparison functions(1) with ,

Example 11 (see [9]). For all , consider Here, .

On the other hand, using the notion of a fuzzy set, Heilpern [29] initiated the family of fuzzy mappings, which generalized the concept of set-valued mappings, and presented a fixed point result for fuzzy contraction mappings in the context of metric linear spaces. Mention that the theorem of Heilpern [29] is considered as a fuzzy extension of the BCP. Later on, several researchers worked on fixed point results involving fuzzy mappings, see ([3046]).

Let be the set family of bounded and closed subsets of a MS . For and , set

Given as

is a metric on . It is named as the Pompeiu-Hausdorff metric induced by . A fuzzy set in is a function with domain and values in . Denote by the set fuzzy sets in . Let be a fuzzy set and , then the function values is named as the grade of membership of in . The -level set of is denoted by and is defined by

Here, denotes the closure of the set . Let be the collection of all fuzzy sets in . For , means for each . The fuzzy set is denoted by unless and until it is stated, where is the characteristic function of the crisp set . If there is so that , then consider

If for each , then take

We write instead of . A fuzzy set in a metric linear space is said to be an approximate quantity if and only if is compact and convex in for each and . The set of approximate quantities in is denoted by . Let be an arbitrary set and be a MS. A mapping is called fuzzy mapping (in short, FM) if is a mapping from into . A fuzzy mapping is a fuzzy subset on with the membership function . The function is the grade of membership of in .

Definition 12. Let be FM from into . An element in is said to be an -fuzzy fixed point of if there is so that . is named as a common -fuzzy fixed point of and if there is so that . is called a common fixed point of fuzzy mappings.

In this manuscript, we ensure the existence of some common -fuzzy fixed points for fuzzy mappings for almost -contractions in the class of complete MS. Our theorems generalize some known results in literature.

Now, we need the following.

Lemma 13 (see [16]). Let be a MS and , then for every ,

Lemma 14 (see [30]). Let be a metric linear space and be a fuzzy mapping and . Then, there is so that .

Lemma 15 (see [47]). Let be a metric space, and be FM such that is a nonempty compact set for each . Then, if and only if for each .

2. Main Results

From now on, is assumed to be a complete MS.

Theorem 16. Let be FM and for each , and there exist , such that and are nonempty, closed, and bounded subsets of . Assume that there exist , , and such that for all implies where If is continuous, then there exists some such that

Proof. Let . By hypotheses, there exists such that is a nonempty, closed, and bounded subset of . For convenience, we denote by . Let , then there exists such that is a nonempty, closed, and bounded subset of . Since is nondecreasing, we have used (19) and Lemma 13where By , we have Thus, there exists such that Then from (22), we have where If then from (27), we have which is a contradiction. Thus, By (27), we get that Next, there exists such that is a nonempty, closed, and bounded subset of . By Lemma 13, using (19) and the fact that is nondecreasing, we have where From , we have Thus, there exists such that Then from (32), we have where If then from (37), we have which is a contradiction. Thus, By (37), we get that By continuing this process, we construct a sequence in such that and Thus, from (42) and (43), we have This implies that Letting , we get It yields that This together with and Lemma 9 gives that Now, we will prove that is a Cauchy sequence. Arguing by contradiction, we assume that there are and sequences and of integers so that for all with Thus, Letting in we get Again, Taking in (51) and (52), we get From (48) and (51), we can choose an integer so that by (19), we get where Letting in the above inequality, since and are continuous and by using (48), (49), (51), and (54), we get It is a contradiction, so is Cauchy. Since is complete, converges to i.e., . We claim that . Assume that (that is, ) then there are and a subsequence of so that , for all Since , for all so by we have where Letting and using the continuity of and , we have which is a contradiction. Hence, and Similarly, one can easily prove that . Therefore,

Example 17. Let endowed with the metric . Consider as and . Here, and are continuous. For , given bysuch that For with we have where Hence, all the conditions of Theorem 16 are satisfied, and .

Corollary 18. Let be FM from into , and for each , there exist , such that and are nonempty, closed, and bounded subsets of . Assume that there are and such that for all where is defined by (20). If is continuous, then there is such that

Proof. Set in Theorem 16.

Corollary 19. Let be a FM from into , and for each , there exist such that is nonempty, closed, and bounded subsets of . Assume that there are , , and such that for all