Convergence of Implicit Iterative Processes for Semigroups of Nonlinear Operators Acting in Regular Modular Spaces

This paper focuses on one-parameter semigroups of ρ-nonexpansive mappings Tt:C→C, where C is a subset of a modular space Xρ, the parameter t ranges over [0,+∞), and ρ is a convex modular with the Fatou property. The common fixed points of such semigroups can be interpreted as stationary points of a dynamic system defined by the semigroup, meaning they remain unchanged during the transformation Tt at any given time t. We demonstrate that, under specific conditions, the sequence {xk} generated by the implicit iterative process xk+1=ckTtk+1(xk+1)+(1−ck)xk is ρ-convergent to a common fixed point of the semigroup. Our findings extend existing convergence results for semigroups of operators, from Banach spaces to a broader class of regular modular spaces.