Generalized Ulam-Hyers Stability, Well-Posedness, and Limit Shadowing of Fixed Point Problems forα-β-Contraction Mapping in Metric Spaces

Abstract

We study the generalized Ulam-Hyers stability, the well-posedness, and the limit shadowing of the fixed point problem for new type of generalized contraction mapping, the so-called α-β-contraction mapping. Our results in this paper are generalized and unify several results in the literature as the result of Geraghty (1973) and the Banach contraction principle.

Existence of a common solution for a system of nonlinear integral equations via fixed point methods inb-metric spaces

“…In 2014, Sintunavarat [17] (see also [6]) introduced the useful concept of transitivity for mappings as follows: Definition 2.15. Let X be a nonempty set.…”

A New Class of Contraction inb-Metric Spaces and Applications

“…In this interesting paper, Czerwik [1] generalized the Banach contraction principle in the context of complete b-metric spaces. After that many researchers reported the existence and uniqueness of fixed points of various operators in the setting of b-metric spaces (see, e.g., [2][3][4][5][6][7][8][9][10][11][12][13] and some references therein).…”

“…This function is known as Geraghty type function or mapping. Later on, many authors [11,[15][16][17] In 1973, Geraghty generalized the Banach contraction principle in the following form.…”

“…They established some fixed point theorems for such mappings in complete metric spaces and showed some examples and applications to ordinary differential equations. Since then, many researchers extended the idea and generalized fixed point results for single-valued and multivalued -admissible mappings in various abstract spaces (see, e.g., [2,3,11,19,20] and more references in the literature). The following definitions and examples reveal the basics of -admissible mappings.…”

“…In 2014, Sintunavarat [11] proved the fixed point theorem for generalized --contraction mapping in metric space and established Ulam-Hyers stability and well-posedness via fixed point results. The purpose of this paper is to define --contraction mapping for a pair of mappings and to prove coincidence point and common fixed point theorems in the context of b-metric space.…”