New Approximating Results by Weak Convergence of Forked Sequences

Abstract

The modular function spaces are natural generalization of spaces like Lebesgue space, Orlicz space, Lorentz p-space, Orlicz–Lorentz space, Musielak–Orlicz space, et al. The function modulars lack basic and flexible properties that norm functions have, as they are functional lacks homogeneity and subadditivity and, therefore, it might be surprising to use techniques involving asymptotic centers, normal structure and uniform convexity to obtain fixed point theorems. The purpose of this paper is to give a new accelerated iterative algorithm for multi valued\ single valued mappings in modular function spaces and to prove some results about their convergence (strong or weak) to a fixed point (or a common fixed point). Through the work, the modular function satisfies (UUC1) property and -condition. Sometimes the work required the use of the Opial’s property or demi-closed condition.      The intent of this manuscript is proving the existence and uniqueness of fixed point inducing from weak convergence of a forked iterative scheme. This scheme is constructed by five-step iterative for (λ, ρ) -firmly nonexpansive (multi\ single) mappings in modular spaces with respect to modular ρ satisfies (UUC1) property and Δ2-condition. To obtain these results and other finding, the definitions of weak convergence, demi-closeness and Opial’s condition format for the case of double sequences. Note that the authors presented a previous study on the strong convergence of forked double sequences including important results, see references.

Received 05/06/2023

Revised 03/03/2024

Accepted 05/03/2024

Published Online First 20/08/2024