Convergence Theorems for Modified Inertial Viscosity Splitting Methods in Banach Spaces

Abstract

In this article, we study a modified viscosity splitting method combined with inertial extrapolation for accretive operators in Banach spaces and then establish a strong convergence theorem for such iterations under some suitable assumptions on the sequences of parameters. As an application, we extend our main results to solve the convex minimization problem. Moreover, the numerical experiments are presented to support the feasibility and efficiency of the proposed method.

Approximation method for monotone inclusion problems in real Banach spaces with applications

“…Since ∂f is maximal monotone, imposing the condition that ∇g is α-inverse strongly monotone, then the FBA and its modifications can be used to approximate solutions of (21), which are minimizers of (20). Just as in the case of arbitrary monotone operators, the acceleration process has been an active topic of nonsmooth convex minimization.…”

“…x n+1 = (I + λ∂f ) -1 (y n -λ∇g(y n )), (22) where t 0 = 1, λ = 1 L , x 0 = x 1 ∈ H and, f and g are as defined in problem (20) in the setting of real Hilbert spaces. Beck and Teboulle [50] proved weak convergence of the sequence generated by (22) to a solution of the inclusion problem (21) in real Hilbert spaces.…”

“…An algorithm of inertial type is an iterative procedure in which subsequent terms are obtained using the preceding two terms. Many authors have shown numerically that adding the inertial extrapolation term in an existing algorithm improves its performance (see, e.g., [19][20][21][22][23][24]). The inertial technique has successfully been employed as an acceleration process for the FBA and its modifications.…”