Some fixed point results in ordered $b$-metric space with auxiliary functions

Abstract

The purpose of this paper is to establish some fixed point results for a class of generalized $(\phi, \psi)$-weak contraction mapping in a partially ordered $b$-metric space. This mapping necessarily have a unique fixed point under ordered relation in a space. Also, the results for common and coincidence fixed points of the self mappings are presented. These results generalize and extend an existing results in the literature. Some illustrations are given at the end to support the results.