Nonexpansive Mappings on New Premodular Special Space of Sequences

Awad A Bakery,Om Kalthum S K Mohamed,Sun Young Cho

doi:10.1155/2022/2098840

Awad A Bakery, Om Kalthum S K Mohamed + Show 1 more

Open Access

https://doi.org/10.1155/2022/2098840

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Abstract

For different premodular, which is a generalization of modular, defined by weighted Orlicz sequence space and its prequasi operator ideal, we have examined the existence of a fixed point for both Kannan contraction and nonexpansive mappings acting on these spaces. Some numerous numerical experiments and practical applications are presented to support our results.

Highlights

Every Orlicz sequence space contains a subspace that is isomorphic to c0 or lr, for some 1 ≤ r < ∞ ([4], eorem 4.a.9). e space of all bounded linear operators from a Banach space X into a Banach space Y will be denoted by
Bakery and Mohamed [15] introduced the concept of the prequasi norm on l((ra)) with variable exponent in (0, 1]. ey looked at the Fatou property of different prequasi norms on l((ra)), as well as the sufficient requirements on l((ra)) with the definite prequasi norm to construct prequasi Banach and closed space. ey demonstrated the existence of a fixed point of Kannan prequasi norm contraction maps on l((ra)) and the prequasi Banach operator ideal constructed by l((ra)) and s-numbers
Reich and Zaslavski [16] showed the existence of a unique fixed point for nonlinear contractive self-mappings of a nonbounded closed subset of a Banach space. ey extended this conclusion to contractive mappings, which map into a Banach space a closed subset of the space

Summary

Definitions and Preliminaries

The space of all functions μ: Y ⟶ [0, ∞) is [0, ∞)Y, θ is the zero vector of Y, [x/2] is the integral part of x/2, F is the space of finite sequences, and B is the class of each bounded linear mapping between any two Banach spaces. E function μ(β) (􏽐x∈Z+ 􏽰5 |β x |) is a premodular (not a modular) on the vector space l1/5. E function μ ∈ [0, ∞)Y is said to be prequasi norm on Y, if it holds the settings (i), (ii), and (iii) of Definition 6. (vii) A prequasi norm μ on X satisfies the Fatou property, if for every sequence 􏼈ηx􏼉⊆(X)μ with limx⟶∞ μ(ηx − η) 0 and any β ∈ (X)μ, we have μ(β − η) ≤ supminf x≥m μ(β − ηx). [6] A function Υ ∈ [0, ∞)G is said to be a prequasi norm on the ideal G if the following conditions verify:. [15] e function Υ(H) μ(sx(H))∞ x 0 is a prequasi norm on SYμ, when Yμ is a premodular (sss). X 0 λx, βx, ηx ∈ [0, ∞), for all x ∈ Z+ and 􏽐∞ x 0 λx 1

Main Results

Kannan μ-Contraction Operator

Kannan Nonexpansive Operator

16. An operator

Applications on Summable Equations

Conclusion