Hybrid Inertial Accelerated Algorithms for Solving Split Equilibrium and Fixed Point Problems

Yanlai Song

doi:10.3390/math9212680

Abstract

In this paper, we introduce a new hybrid inertial accelerated algorithm with a line search technique for solving fixed point problems for demimetric mapping and split equilibrium problems in Hilbert spaces. The algorithm is inspired by Tseng’s extragradient method and the viscosity method. Then, we establish and prove the strong convergence theorem under proper conditions. Furthermore, we also give a numerical example to support the main results. The main results are new and the proofs are relatively simple and different from those in early and recent literature.

Highlights

In order to find a common solution for the fixed point problem for nonexpansive mapping and the SEP, Kazmi and Rizvi [12] suggested the following projection algorithm:
Motivated and inspired by Kazmi and Rizvi [12], Jolaoso and Karahan [13], Alvarez and Attouch [14], and Cai et al [17], we suggest and analyze a hybrid inertial accelerated method for finding a common element of the set of fixed points of a demimetric mapping and the set of solutions of the split equilibrium problem in a real Hilbert space
We prove a strong convergence of the proposed method under some mild conditions

Summary

Introduction

Let C be a nonempty, closed, and convex subset of a real Hilbert space H with scalar product h·, ·i and generated norm k·k. In order to find a common solution for the fixed point problem for nonexpansive mapping and the SEP, Kazmi and Rizvi [12] suggested the following projection algorithm: xn+1 = αn u + β n xn + γn Syn , yn = PC (un − λn Tun ), F un = Trn ( xn − γA∗ ( I − TrFn2 ) Axn ), n ≥ 1, where S : C → C is nonexpansive and T : C → H1 is inverse strongly monotone. Generated by Algorithm 2 converges to a solution of the pseudo-monotone variational inequality problem under some suitable conditions. Motivated and inspired by Kazmi and Rizvi [12], Jolaoso and Karahan [13], Alvarez and Attouch [14], and Cai et al [17], we suggest and analyze a hybrid inertial accelerated method for finding a common element of the set of fixed points of a demimetric mapping and the set of solutions of the split equilibrium problem in a real Hilbert space. We provide a numerical experiment to demonstrate the efficiency of the proposed method over some existing ones

Preliminaries

Main Results

Application to Split Minimization Problems

Numerical Examples

Conclusions