Some New Types of \(\alpha_\beta - 1\)-admissible Mappings and Fixed Point Results

Abstract

In this paper fixed point and common fixed point theorems are proved for \(\alpha - \phi \) contractions under the \(\alpha_\beta - 1\)-admissible and \(μ - \alpha_\beta - 1\) admissible conditions. These results generalize the \(\alpha_\beta - 1\)–admissible condition of self-mappings as explained by many researchers in the literature herein. In general, many research reports introduced the –admissible condition on the basis of \((p,q) > 1\). However, this condition is not applicable for finding the fixed points of self-mappings in many cases. We propose \(\alpha\)–admissible on the basis of \(\alpha (p,q)>\beta^−1\) which generalize the case we mentioned in the literature. To support our new concepts, we present two examples at the end of this paper.