On some new fixed point results in fuzzy b-metric spaces

This paper consists of several fixed point theorems in the fuzzy b-metric spaces. As an important result, we give a sufficient condition for a sequence to be Cauchy in the fuzzy b-metric space. Thus we simplify the proofs of many fixed point theorems in the fuzzy b-metric spaces with the well-known contraction conditions.


Introduction and preliminaries
The notion of the fuzzy logic is introduced by Zadeh [31]. Unlike the theory of traditional logic, where some element does or does not belong to the set, in the fuzzy logic the affiliation of the element to the set is expressed as a number from the interval [0, 1]. Uncertainty, as an essential part of a real problem, has prompted Zadeh to study the theory of the fuzzy sets to face the problem of indeterminacy. Theory of a fixed point in the fuzzy metric spaces can be viewed in different ways, and one of them is using a fuzzy logic. After Zadeh's result, Heilpern [13] represents the concept of the fuzzy mapping and proves a theorem on a fixed point for fuzzy contraction mapping in linear metric spaces, which represents a fuzzy generalization of Banach's contraction principle, followed by interest of many authors to study various contractions conditions within the framework of the fuzzy mapping. If the distance between the elements is not an exact number, then the imprecision is included in the metric, as it is introduced in the definition of fuzzy metric spaces introduced by Kaleva and Seikkala [14]. After that, first by Kramosil and Michalek [17] and further by George and Veeramani [6], the notion of a fuzzy metric space was introduced. For more details about fuzzy metric spaces and fixed point theorems in these spaces, among a huge number of the papers regarding this topic, we suggest for reading the papers [5,8,19,21,22,[25][26][27]30].
In addition to fuzzy metric spaces, there are still many extensions of metric and metric space terms. Bakhtin [1] and Czerwik [3] introduced a space where, instead of the triangle inequality, a weaker condition was observed, with the aim of generalization of the Banach contraction principle [2]. They called these spaces b-metric spaces. For more information about these spaces, we refer the readers to the papers [15,18,20,28].
Relation between b-metric and fuzzy metric spaces is consider in [12]. On the other hand, in [23] the notion of a fuzzy b-metric space was introduced, where the triangle inequality is replaced by a weaker one. In this paper, we deal with this type of spaces. Using the notion of a countable extension of the t-norm, we prove a very useful lemma in the fuzzy b-metric space settings that ensure that a sequence {x n } is a Cauchy sequence. Using this lemma, we simplify the proofs of many well-known fixed point theorems. We present some of them in the main part of the paper.
We start with basic notions important for further work.
We say that a t-norm T is of H-type if the family {T n (x)} n∈N is equicontinuous at x = 1.
A trivial example of t-norm of H-type is T min ; for a nontrivial example, see [9]. Each t-norm T can be extended (see [16]) by associativity in a unique way to an n-ary operation taking for (x 1 , . . . , x n ) ∈ [0, 1] n the values Example 1.1 ([10]) n-ary extensions of the t-norms T min , T L , and T P , are the following: A t-norm T (see [16]) can be extended to a countable infinite operation taking for any The sequence (T n i=1 x i ) n∈N is nonincreasing and bounded from below, and hence the limit In the fixed point theory (see [10,11]), it is of interest to investigate the classes of t-norms T and sequences (x n ) from the interval [0, 1] such that lim n→∞ x n = 1 and In [10] the following proposition is obtained.

Proposition 1.3
Let (x n ) n∈N be a sequence of numbers from [0, 1] such that lim n→∞ x n = 1, and let T be a t-norm of H-type. Then is called a fuzzy metric space if X is an arbitrary (nonempty) set, T is a continuous t-norm, and M is a fuzzy set on X 2 × (0, ∞) satisfying the following conditions for all x, y, z ∈ X and t, s > 0: Remark 1.5 In this paper, we use X 2 := X × X.
satisfying the following conditions for all x, y, z ∈ X, t, s > 0 and a given real number b ≥ 1: The class of fuzzy b-metric spaces is effectively larger than that of fuzzy metric spaces, since a fuzzy b-metric is a fuzzy metric when b = 1.
The next example shows that a fuzzy b-metric on X need not be a fuzzy metric on X.
where p > 1 is a real number. Then M is a fuzzy b-metric with b = 2 p-1 .
Noted that in the preceding example, for p = 2, it is easy to see that (X, M, T) is not a fuzzy metric space. Before presentation of our main results, we give some definitions and proposition in a fuzzy b-metric space.  In a fuzzy b-metric space we have the following proposition.

Main results
We will further use a fuzzy b-metric space in the sense of Definition 1.6 with additional condition lim t→∞ M(x, y, t) = 1.
and there exist x 0 , x 1 ∈ X and ν ∈ (0, 1) such that Then {x n } is a Cauchy sequence.
By Proposition 1.3 the next corollary immediately follows.

Corollary 2.2 Let {x n } be a sequence in a fuzzy b-metric space (X, M, T), and let T be of H-type. If there exists
then {x n } is a Cauchy sequence.

Theorem 2.4
Let (X, M, T) be a complete fuzzy b-metric space, and let f : X → X. Suppose that there exists λ ∈ (0, 1 b ) such that M(fx, fy, t) ≥ M x, y, t λ , x, y ∈ X, t > 0, (2.5) and there exist x 0 ∈ X and ν ∈ (0, 1) such that Then f has a unique fixed point in X.
Proof Let x 0 ∈ X and x n+1 = fx n , n ∈ N. If we take x = x n and y = x n-1 in (2.5), then we have for all t > 0. By (2.7), as n → ∞, we get Suppose that x and y are fixed points for f . By x, y ∈ X, t > 0, for all x, y ∈ X, t > 0, and there exist x 0 ∈ X and ν ∈ (0, 1) such that for all t > 0. Then f has a unique fixed point in X.