Simple inertial methods for solving split variational inclusions in Banach spaces

Abstract

In this paper, we introduce two simple inertial algorithms for solving the split variational inclusion problem in Banach spaces. Under mild and standard assumptions, we establish the weak and strong convergence of the proposed methods, respectively. As theoretical realization we study existence of solutions of the split common fixed point problem in Banach spaces. Several numerical examples in finite and infinite dimensional spaces compare and illustrate the performances of our schemes. Our work generalize and extend some recent relate results in the literature and also propose a simple and applicable method for solving split variational inclusions.

A modified Ishikawa iteration scheme for b‐enriched nonexpansive mapping to solve split variational inclusion problem and fixed point problem in Hilbert spaces

“…This shows that the proposed method is robust and efficient. Finally, we plan on looking into two future directions: the first direction is to present the ALG 1 combining with inertial extrapolation for solving a variational inequality problem and a convex minimization problem in real Hilbert spaces, and the second direction is to present the ALG 1 in the setting of other spaces, such as Banach space (see, e.g., Tang et al [37]) and Metric space (see, e.g., Agarwal et al [38] and Farajzadeh et al [39]).…”

“…where A ∶ H 1 → H 2 is a linear bounded operator. We shall denote by S 𝛤 the solution set of (37). As a result of its applications in signal processing, image reconstruction, approximation theory, control theory, biomedical engineering, communications, and geophysics, the split feasibility problem has gained a significant amount of attention in recent years.…”

“…To this end, we establish a strong convergence of the proposed algorithm under suitable conditions. Moreover, we also apply our main result to the split feasibility problem (37) and provide numerical experiments to compare the proposed algorithm with some existing algorithms in the literature.…”