The convergence of Iteration Scheme to Fixed Points in Modular Spaces

Abstract

The aim of this paper is to study the convergence of an iteration scheme for multi-valued mappings which defined on a subset of a complete convex real modular. There are two main results, in the first result, we show that the convergence with respect to a multi-valued contraction mapping to a fixed point. And, in the second result, we deal with two different schemes for two multivalued mappings (one of them is a contraction and other has a fixed point) and then we show that the limit point of these two schemes is the same. Moreover, this limit will be the common fixed point the two mappings.

A New Iterative Sequence of (λ, ρ)-Firmly Nonexpansive Multi-Valued Mappings in Modular Function Spaces with Applications

“…To prove 𝑙𝑖𝑚 𝑛⟶∞ 𝜌(𝑓 𝑛 − 𝑠) exists. By Definitions (11,12), convexity of ρ, and Lemmas (3,5), we get:…”

“…Let s is fixed point of 𝑇, and 𝜀 𝑛 = 𝜌(𝑓 𝑛+1 − 𝑚 𝑛 ), by ( 5), convexity of 𝜌, Lemma 3, Definitions (11,12) The iterative scheme in ( 5) is summably almost T-stable.…”

“…In order to achieve the best approximation in modular spaces, Abed and Sada [11] used fixed point theorems of compactnonexpansive multivalued mapping. For iteration schemes in multivalued mappings in modular function spaces, Abdul Jabbar and Abed [12] studied convergence. For shared fixed points and convergence, Morwal and Panwar [13] presented a three-step iterative technique in three multivalued -nonexpansive mappings and studded approximation.…”

افضل تقريب في فضاءات b - المعيارية

“…It possible to combine the set 𝑃 𝛺 (∑) into work in 11,12 and 16 to posing the following question: Could the limit of convergence iterative sequences in 11,12…”

“…It was then redefined with some modification by Musielak and Ortiz in 1959 In 2017, the best approximation modular spaces have been defined by Abed 8 and results about proximinal set, Chebysev set are proven also, see 9 . Various results in modular spaces and other related spaces about fixed point problem can be seen in Turkoglu and Nesrin 9 , Albundi 10 , Abdul Jabbar and Abed 11,12 , Ahmed 13 , Mohammed and Abed 14 , Pathak and Beg 15 , Abed and Abdul Jabbar 16,17 , Ege and, Alac 18 also Abed and Salman 19 .…”