Fixed point theory for alpha-phi convex orbital contractions in geodesic spaces

Abstract

In this paper, we explore the concept of $\alpha$-$\phi$ convex orbital contractions in geodesic spaces, extending the classical framework of fixed point theory to a more general setting. We introduce a new class of mappings, termed $\alpha$-$\phi$ convex orbital contractions. By leveraging the geometric properties of geodesic spaces, we establish the existence and uniqueness of fixed points for these mappings under mild assumptions on the control function $\phi$. Furthermore, we analyze the convergence of Krasnosel'ski\u\i~iterative schemes for approximating fixed points. To reinforce our results, we present supporting examples.