Convergence Theorems for Modified Inertial Viscosity Splitting Methods in Banach Spaces

Chanjuan Pan,Yuanheng Wang

doi:10.3390/math7020156

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.

Highlights

Throughout this paper, we let E be a real Banach space with norm k.k and E∗ be its dual space. ∗The normalized duality mapping J from E into 2E is defined by the following equation:J ( x ) = { f ∈ E∗ : h x, f i = k f kk x k = k x k2 } ∀ x ∈ E.we denote the generalized duality pairing between E and E∗ by h., .i and the single-valued duality mapping by j.The inclusion problem is to find x ∈ E such that 0 ∈ ( A + B) x where A : E → E is an operator and B : E → 2E is a set-valued operator
The strong convergence theorems are established, and the numerical experiments are presented to support that the inertial extrapolation greatly improves the efficiency of the algorithm
In Theorem 1 and Corollary 1, if f = u and A is an inverse strongly monotone operator in Hilbert space, it is the main results of Cholamjiak et al [20]

Summary

Introduction

Throughout this paper, we let E be a real Banach space with norm k.k and E∗ be its dual space. In 2012, Takashashi et al [8] studied a Halpern-type iterative method for an α-inverse strongly monotone mapping A and a maximal monotone operator B in a Hilbert space as follows: xn+1 = β n xn + (1 − β n )(αn u + (1 − αn ) JrBn ( xn − rn Axn )), under certain conditions, the algorithm was showed to converge strongly to a solution of A + B. In 2015, Lorenz et al [19] applied inertial extrapolation technique to forward-backward algorithm for monotone operators in Hilbert spaces They proved that the iterative process defined by y n = x n + θ n ( x n − x n −1 ), xn+1 = ( I + rn B)−1 (yn − rn Ayn ). The strong convergence theorems for such iterations are established and some applications including the numerical experiments are presented to support our main theorem

Preliminaries

Main Results

Applications and Numerical Experiments

Conclusions