Nonlinear integral equations with new admissibility types in b-metric spaces

Abstract

In this paper, we aim to introduce new types of α-admissibility in the framework of b-metric spaces. Some examples to show the independently of each type of α-admissibility are given. Using these concepts, fixed point theorems satisfying generalized weak contractive condition in the setting of b-metric spaces are established. We furnish an illustrative example to demonstrate the validity of the hypotheses and the degree of utility of our results. As an application, we discuss the existence of a solution for the following nonlinear integral equation: $$x(c) = \phi (c) + {\int _{a}^{b}} K(c, r, x(r)) dr,$$ where \({a, b \in {\mathbb{R}}}\) such that \({a < b, x \in C[a, b]}\) (the set of all continuous functions from [a, b] into \({{\mathbb{R}}}\)), \({\phi : [a, b] \rightarrow {\mathbb{R}}}\) and \({K : [a, b] \times [a, b] \times {\mathbb{R}} \rightarrow {\mathbb{R}}}\) are given mappings.