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

Let C be subset of a vector space, and consider a semigroup of nonlinear mappings Tt:C→C, where t∈[0,+∞). The common fixed points of this semigroup 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. This paper focuses on semigroups of ρ-nonexpansive mappings in an abstract modular space Xρ, where ρ is a regular convex modular. By employing recent results on the existence of such stationary points, 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 and modular function spaces to a broader class of regular modular spaces.