Mean Value-Mann Iterative Process
Mohd Tariq, Mayank Sharma
TL;DR
This work extends fixed-point theory for mean nonexpansive mappings to uniformly convex hyperbolic spaces by introducing a Mean Value-Mann hybrid iterative process. The update combines a mean nonexpansive mapping $\daleth$ with a hyperbolic convex combination parameterized by $\alpha_n$ and $r_n$, unifying and generalizing Picard, Mann, and Ishikawa iterations in this non-Euclidean setting. The authors establish monotonicity and convergence results: the distance to the fixed-point set is nonincreasing, the sequence is bounded with $\lim_{n\to\infty} d(\varkappa_n, \daleth\varkappa_n)=0$, and the sequence $\Delta$-converges to a fixed point of $\daleth$, with strong convergence under appropriate liminf conditions. These results extend fixed-point convergence theory from Banach and CAT(0) spaces to uniformly convex hyperbolic spaces, providing robust tools for iterative fixed-point computation in non-Euclidean geometries.
Abstract
In this paper, the Mean value iterative process is modified with the Mann iterative process for mean nonexpansive mapping in a hyperbolic metric space that satisfy the symmetry criteria and in uniformly convex hyperbolic spaces to validate the iterative process, we present strong and $Delta$-convergence theorems.