Admissible Hybrid Z-Contractions in b-Metric Spaces

Abstract

In this manuscript, we introduce a new notion, admissible hybrid Z-contraction that unifies several nonlinear and linear contractions in the set-up of a b-metric space. In our main theorem, we discuss the existence and uniqueness result of such mappings in the context of complete b-metric space. The given result not only unifies the several existing results in the literature, but also extends and improves them. We express some consequences of our main theorem by using variant examples of simulation functions. As applications, the well-posedness and the Ulam-Hyers stability of the fixed point problem are also studied.Axioms 2020, 9, 2 2 of 17As it was mentioned before, the theory has been advanced by reporting several new fixed point results that are obtained by changing the conditions on the given mappings. As a result, in the literature, there are so many different types of metric fixed point results that cause a disturbance, conflict, and disorder. For overcoming this problem, it needs to consider new theorems that cover several different results. One of the successful results in directions was given in [4] where admissible mappings were introduced to combine different structures. Other interesting results were given in [5] in which the notion of the simulation function was defined to combine many distinct contractions. The notion of the hybrid contractions can also be considered as a result of this trend: in two recent papers [6,7], the authors introduce a new type of contraction, namely admissible hybrid contraction, in order to unify several linear, nonlinear and interpolative contractions in the set-up of a complete metric and b-metric spaces.One of the main aims of this paper is to unify the several existing results in the literature by combining the interesting notions: admissible mappings, simulation functions, and hybrid contractions. Besides unifying the results, we express our results in the most generalized form: in the setting of a complete b-metric space. Next, we shall consider applications for our obtained results. In particular, we shall consider the well-posedness and the Ulam-Hyers stability of the fixed point problem. We shall give some consequences and we shall indicate how one can get more consequences from the main theorem of the paper. In the next section, we shall give some basic notions and results to provide a self-contained, easily readable paper. PreliminariesIn this section, we shall collect the necessary notations, notions, and results for the sake of the completeness of the paper. We first express the definition of the b-metric, as follows.Definition 1 (See, e.g., Bourbaki [1], Bakhtin [2], and Czerwik [3]). Let X be a nonempty set and let s ≥ 1 be a given real number. A functional d : X × X → [0, ∞) is said to be a b-metric with constant s, if

A Note on Extended Z-Contraction

“…Notice also that our results covers several existing results, such as the results in [5,7,18]. Regarding the richness of the simulations function, as it is done in [19], one can derive several well-known fixed point results from our main theorem. Indeed, explicitly writing the consequences/corollaries of our main result cannot easily fit on several pages.…”

“…Under these circumstances, the best thing to do is to evaluate the current results as widely as possible and to combine these existing results as much as possible. Recently, for that purpose, some interesting papers started to appear, such as, [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Among all these approaches, in this article, we focus on the notion of simulation function.…”

Multiparametric Contractions and Related Hardy-Roger Type Fixed Point Theorems

“…Notice that we assume that ψ (t) > 0 and β (t) < 1 for all t > 0, so φ (t) = β (t) ψ (t) < ψ (t) for all t > 0. Furthermore, condition (16) implies (c 2 ), so Theorem 4 is applicable.…”

“…Remark 2. Notice that condition (16) does not guarantee that β (t) < 1 for all t > 0. For instance, let consider β : (0, ∞) → {0.5, 2} defined by β (1) = 2 and β (t) = 0.5 if t ∈ (0, ∞) {1}.…”

Results in wt-Distance over b-Metric Spaces

“…It is known that b-metric space itself is a b-metric and, furthermore, b-metric turns out to be a standard metric for s = 1. As it is seen in [14,15,17,20], for different choices of σ ∈ Σ, we shall get more different consequences.…”