Weak convergence theorem for attractive point of finitely many families generalized nonexpansive mappings

Abstract

In this paper, we study iterative approximation of attractive points for finitely many families of generalized nonexpansive mappings in a uniformly convex Banach space. We introduce a new algorithm combining a viscosity step with an inertial extrapolation. Under suitable control of the inertial and viscosity parameters and standard conditions on the mappings, we prove that the generated sequence is bounded. We then show that every weak cluster point of the sequence is an attractive point common to all mapping families. The main result establishes that the sequence converges weakly to a unique such attractive point. This extends earlier results confined to two mappings by considering finite family. These findings confirm that the proposed viscosity-inertial iteration successfully approximates the common attractive point under the stated hypotheses. Overall, the work broadens convergence theory in Banach spaces by enabling new classes of algorithms for approximating solution points of generalized nonlinear problems.