Fixed Point Theory in α -Complete Metric Spaces with Applications

Abstract

The aim of this paper is to introduce new concepts of $\alpha$-$\eta$-complete metric space and $\alpha$-$\eta$-continuous function and establish fixed point results for modified $\alpha$-$\eta$-$\psi$-rational contraction mappings in $\alpha$-$\eta$-complete metric spaces. As an application, we derive some Suzuki type fixed point theorems and new fixed point theorems for $\psi$-graphic-rational contractions. Moreover, some examples and an application to integral equations are given here to illustrate the usability of the obtained results.