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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.

Keywords

Double sequence, Firmly nonexpansive, Fixed point, Strong convergence, Weak convergence

Subject Area

Mathematics

First Page

623

Last Page

631

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Receive Date

6-5-2023

Revise Date

3-3-2024

Accept Date

3-5-2024

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