Some fixed point theorems of \(\alpha\)-convex generalized nonexpansive mappings in hyperbolic spaces

Abstract

This study focuses on \(\alpha\)-convex generalized nonexpansive (\(\alpha\)-CGNE) mappings, a generalization of nonexpansive mappings defined by an additional parameter \(\alpha \in (0, 1)\). These mappings extend classical nonexpansive properties and are applicable to problems involving variational inequalities and monotone operators. We investigate the fixed point theory of \(\alpha\)-CGNE mappings in hyperbolic metric spaces, emphasizing strong convergence results using projection mappings. Additionally, \(\Delta\)-convergence theorems are established under specific conditions, highlighting the asymptotic behavior of iterative methods in nonlinear settings. These results expand fixed point theory beyond Banach spaces, offering new approaches for studying nonlinear phenomena.