An admissible Hybrid contraction with an Ulam type stability

Abstract

In this manuscript, we introduce a new hybrid contraction that unify several nonlinear and linear contractions in the set-up of a complete metric space. We present an example to indicate the genuine of the proved result. In addition, we consider Ulam type stability and well-posedness for this new hybrid contraction.

A Note on Extended Z-Contraction

“…Further, all observed results can be derived in the framework of standard metric space, by let s = 1. 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.…”

“…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.…”

Admissible Hybrid Z-Contractions in b-Metric Spaces

“…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.…”

“…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.…”

Stability analysis for a class of implicit fractional differential equations involving Atangana–Baleanu fractional derivative

“…Also we derive various results for Ulam–Hyers type stability analysis of our problem as the said stability has been investigated in the last few years very well in respect of FDEs. Also some researchers have investigated various results of the concerned stability in nonlinear analysis (see [ 21 – 25 ]).…”