Fixed point results in b-metric spaces approach to the existence of a solution for nonlinear integral equations

Abstract

The purpose of this work is to introduce new nonlinear mappings in setup of b-metric spaces and prove fixed point theorems for such mappings. Examples are provided in order to distinguish these results from the known ones. At the end of paper, we apply our fixed point result to prove the existence of a solution for the following nonlinear integral equation:

$$\begin{aligned} x(c) = \Omega (\phi (c),c ) + K(c,c,\phi (c))+\int ^b_a K(c,r,x(r))dr, \end{aligned}$$

(0.1)

where $a,b\in \mathbb {R}$ with $a<b$, $x \in C[a,b]$ (the set of all continuous real functions defined on [a, b]), $\phi :[a,b]\rightarrow \mathbb {R}$, $\Omega :\mathbb {R} \times [a,b]\rightarrow \mathbb {R}$ and $K : [a,b] \times [a,b] \times \mathbb {R} \rightarrow \mathbb {R}$ are given mappings.