New Contributions to Fixed Point Techniques with Applications for Solving Fractional and Differential Equations

Abstract

In this article, we present two novel ideas of f-contractions, named dual \(f^{*}\)-weak rational contractions and triple \(f^{*}\)-weak rational contractions, generalizing and expanding many of the solid results in this direction. The endeavor to apply the generalized Banach contraction principle to the set of f-contraction type mappings by applying numerous f-type functions gave rise to these novel generalizations. Also, under appropriate conditions, related unique fixed-point theorems are established. Moreover, some illustrative examples are given to support and strengthen the theoretical results. Furthermore, the obtained results are applied to discuss the existence of solutions to a fractional integral equation and a second-order differential equation. Finally, the significance of the new results and some future work are presented.