An Inertial Iterative Algorithm with Strong Convergence for Solving Modified Split Feasibility Problem in Banach Spaces

Huijuan Jia,Shufen Liu,Yazheng Dang,Ching-Feng Wen

doi:10.1155/2021/9974351

Abstract

In this paper, we propose an iterative scheme for a special split feasibility problem with the maximal monotone operator and fixed-point problem in Banach spaces. The algorithm implements Halpern’s iteration with an inertial technique for the problem. Under some mild assumption of the monotonicity of the related mapping, we establish the strong convergence of the sequence generated by the algorithm which does not require the spectral radius of A T A. Finally, the numerical example is presented to demonstrate the efficiency of the algorithm.

Highlights

E CQ algorithm is one of the most popular solvers for SFP which was first proposed by Byrne [11], taking an initial point arbitrarily and defining the iterative step as xk+1 PC􏼐xk − μAT􏼐I − PQ􏼑Axk􏼑, ∀k ≥ 1, (2)
Motivated by the above results, in this paper, we present an inertial algorithm for solving (4) in p-uniformly convex and uniformly smooth Banach spaces which have strong convergence
We introduced an inertial iterative algorithm for approximating a common solution of the split feasibility problem, monotone inclusion problem, and fixed-point problem for the class of Bregman weak relative nonexpansive mapping in p-uniformly convex and uniformly smooth Banach spaces

Summary

Preliminaries

We recall some basic definitions and preliminaries’ results which will be useful for our convergence analysis in this paper. Let E be a smooth, strictly convex, and reflexive Banach space and A: E ⟶ 2E∗ be a maximal monotone operator. Let C be a nonempty, closed, and convex subset of a reflexive, strictly convex, and smooth Banach space E, x0 ∈ C and x ∈ E. en, the following assertions are equivalent:. Let E be a smooth and uniformly convex real Banach space. Let q ≥ 1 andr > 0 be two fixed real numbers, a Banach space E is uniformly convex if and only if there exists a continuous, strictly, increasing, and convex function g: R+ ⟶ R+, g(0) 0 such that for all x and y ∈ Br and0 ≤ α ≤ 1, ‖αx +(1 − α)y‖q ≤ α‖x‖q +(1 − α)‖y‖q − Wq(α)g(‖x − y‖),. Where Wq ≔ αq(1 − α) + α(1 − α)q and Br ≔ {x ∈ E: x ≤ r}

Inertial Iteration Algorithm and Its Strong Convergence

Numerical Example

Conclusion

Disclosure