Unique fixed point theorems for α–ψ-contractive type mappings in fuzzy metric space

Abstract Fixed point theory is one of the most powerful tools in nonlinear analysis. The Banach contraction principle is the simplest and most versatile elementary result in fixed point theory. The principle has many applications and was extended by several authors. In this paper, we introduce a concept of α–ψ-contractive type mappings and establish fixed point theorems for such mappings in complete fuzzy metric spaces. Starting from the Banach contraction principle, the presented theorems are the extension, generalization, and improvement of many existing results in the literature. Some example and application to ordinary differential equations are given to illustrate the usability of obtained results.


Introduction
introduced and studied the concept of a fuzzy set in his seminal paper. The study of fuzzy sets initiated an extensive fuzzification of several mathematical concepts and has applications to various branches of applied sciences. The concept of fuzzy metric spaces was introduced initially by Kramosil and Michalek (1975). Later on, George and Veeramani (1994) modified the concept of fuzzy metric spaces due to Kramosil and Michalek (1975). The Banach contraction principle is certainly a classical result of modern analysis. This principle has been extended and generalized in different directions in metric spaces (Xu, He, & Man, 2012). Grabiec (1988) initiated the study of the fixed point theory in fuzzy metric space. Recently, Gregori and Sapena (2002) introduced new kind of contractive mappings in modified fuzzy metric spaces and proved a fuzzy version of Banach contraction principle (see Mihet, 2004Mihet, , 2007Phiangsungnoen, Sintunavarat, & Kumam, 2014). In particular, Miheţ (2008) introduced the concepts of fuzzy ψ-contractive mappings which enlarge the class of fuzzy

PUBLIC INTEREST STATEMENT
Fixed point theory is one of the most powerful tools and it is an active area of research with wide range of applications in various directions. This concept has been established in different directions by several authors. Fuzzy set is very interesting and constructive for applications in real world. The α-ψ-contractive type mapping has generalization of Banach contraction principle. In this paper, we introduced the concept of α-ψ-contractive mapping in fuzzy metric space and established various fixed point theorems for such mappings in complete fuzzy metric spaces. Different examples are considered to illustrate the usability of our results. We hope these results will further assist to comprehend the concept of fixed point theory on 2-fuzzy metric space.
contractions in Gregori and Sapena (2002) and many authors Abbas, Imded, andGopal (2011), Hong (2014) have used the result of Mihet (2008). Samet, Vetro, and Vetro (2012) introduced the concept of α-ψ-contractive mapping and utilized the same concept to prove several interesting fixed point theorems in setting of metric spaces (see Gopal, Imded, Vetro, & Hasan, 2012;Gopal & Vetro, 2014;Mursaleen, Mohiuddine, & Aggarwal, 2012;Mursaleen, Srivastava, & Sharma, 2016;Xu et al., 2012). Based on the same concept, we give some generalizations of the previous concepts of fuzzy contractive mappings in the setting of fuzzy metric spaces. We extend the results of Samet et al. (2012). The presented theorems extend, generalize, and improve many results in the literature specially the Banach contraction principle, and different examples and applications to ordinary differential equations are considered to illustrate the usability of our obtained results.
The main purpose of this paper is to obtain fixed point theorems for α-ψ-contractive type mappings and initiate individual in complete fuzzy metric spaces.

[B.1] * is commutative and associative
Definition 2.2. (George & Veeramani, 1994) The 3 − tuple (X, M, * ) is called a fuzzy metric space if X is an arbitrary non-empty set, * is a continuous t-norm and M is a fuzzy metric in X 2 × 0, ∞ → [0, 1], satisfying the following conditions: for all x, y, z ∈ X, and t, s > 0.
A fuzzy metric space in which every Cauchy sequence is convergent is called complete. It is called compact if every sequence contains a convergent subsequence.
Definition 2.6. (George & Veeramani, 1994) A self mapping T:X → X is called fuzzy contractive mapping if M(Tx, Ty, t) > M(x, y, t) for each x ≠ y ∈ X and t > 0.

Main results
Throughout in this paper, the standard notations and terminologies in nonlinear analysis are used. We start the main section by presenting the new notion of α-ψ-contractive and α-admissible mappings in fuzzy metric space.
Definition 3.2. Let (X, M, *) be a fuzzy metric space and T:X → X be a given mapping. We say that T is an α-ψ-contractive mapping if there exists two functions α:X × X × [0, ∞) → [0, 1] and ψ ∈ Ψ such that for all x, y ∈ X.
Then, T has a unique fixed point, that is, there exists x * ∈ X such that Tx * = x * .
Proof Let x 0 ∈ X such that (x 0 , Tx 0 , t) ≤ 1. Define the sequence {x n } ∈ X by x n+1 = Tx n , for all n ∈ N.
If x n = x n+1 for some n ∈ N, then x* = x n is a fixed point for T. Assume that x n ≠ x n+1 for all n ∈ N. Since T is α-admissible, we have

By induction, we get
Applying the inequality (1) with x = x n−1 and y = x n and using (2), we obtain By induction, we get Let n ∈ N and t > 0. Thus for any positive integer P we have i.e. {x n } is a Cauchy sequence in fuzzy metric space(X, M, *), hence convergent. Since (X, M, *) is complete, there exists x* ∈ X such that x n → x * as n → ∞. From the continuity of T, it follows that x n+1 = Tx n → Tx * as n → ∞. By the uniqueness of the limit, we get x* = Tx*, i.e. x* is a unique fixed point of T.
Theorem 3.2. Let (X, M, *) be a complete fuzzy metric space and T:X → X be a α-ψ-contractive mapping satisfying the following conditions: (ii) There exists x 0 ∈ X such that (x 0 , Tx 0 , t) ≤ 1; (iii) If {x n } is a sequence in X such that α(x n , x n+1 , t) ≤ 1 for all n and x n → x ∈ X as n → ∞, then α(x n , x, t) ≤ 1 for all n. Then, T has a unique fixed point.
Proof Following the proof of Theorem 3.1, we know that {x n } is a Cauchy sequence in complete fuzzy metric space(X, M, *). Then, there exists x * ∈ X such that x n → x * as n → ∞. On the other hand, form (2) and the hypothesis (iii) we have M(x n , x n+1 , t) ≥ n (M(x 0 , x 1 , t)), for all n ∈ N.
Moreover, there exists x 0 ∈ X such that α(x 0 , Tx 0 , t) ≤ 1. In fact, for x 0 = 1, we have Obviously T is continuous and so it remains to show that T is α-admissible. Let x, y ∈ X such that α(x, y, t) ≤ 1. This implies that x, y ∈ [0, 1] and by the definition of T and α, we have Then T is α-admissible.
Finally, let {x n } be a sequence in X such that α(x n , x n+1 , t) ≤ 1 for all n ∈ N and x n → x ∈ X as n → ∞. By the definition of α, we have x n ∈ [0, 1] for all n and x ∈ [0, 1], then the sequence {x n } is convergent i.e. {x n } is Cauchy sequence and lim n→∞ (x n , x n+1 , t) = 1. Now, all the hypotheses of Theorem 3.1 are satisfied. Consequently, T has a unique fixed point. In this example 0 is a unique fixed point.