Shrinking projection methods for solving split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in Hilbert spaces

Abstract

In this paper, we propose a new iterative sequence for solving common problems which consist of split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in the framework of Hilbert spaces and prove some strong convergence theorems of the generated sequence {x n } by the shrinking projection method. Our results improve and extend the previous results given in the literature. MSC: 54E70; 47H25

Inertial viscosity with alternative regularization for certain optimization and fixed point problems

“…Inspired by the works of Cholamjiak et al [14], Yang and He [24], and Luo et al [25], in this paper, we propose two inertial viscosity type algorithms for finding the common element of the set of solutions of split equilibrium problem (1.1), variational inclusion problem 1.2, and fixed point problems of nonexpansive mappings in real Hilbert spaces. We prove two strong convergence theorems of the sequences generated by the algorithms without imposing upper semi-continuity on the equilibrium bifunction as in some existing works on SEP (see for example [14,26,27]). We also present some consequences of our strong convergence theorems and give an application.…”

An inertial based forward–backward algorithm for monotone inclusion problems and split mixed equilibrium problems in Hilbert spaces

“…This formalism is also at the core of modeling of many inverse problems arising for phase retrieval and other real-world problems; for instance, in sensor networks in computerized tomography and data compression; see, for example, [18,21]. Some methods have been proposed and analyzed to solve split equilib-rium problem and mixed split equilibrium problem in Hilbert spaces; see, for example, [24,25,28,29,36,37,51,54,59,60] and the references cited therein. Inspired and motivated by the above-mentioned results and the ongoing research in this direction, we aim to employ the modified inertial forward-backward algorithm to find a common solution of the monotone inclusion problem and the SEP in Hilbert spaces.…”

Algorithms for split common fixed point problems in Hilbert spaces

“…Such problems arise in the field of intensity-modulated radiation therapy when one attempts to describe physical dose constraints and equivalent uniform dose constraints within a single model (see [6]). Since the split feasibility problem in finite dimensional Hilbert spaces was first introduced by Censor and Elfving [5], many algorithms have been introduced to solve the SFP; see [2,7,18,13] and references therein. Note that, if (1.2) is consistent (i.e., (ref1.2) has a solution), it is no hard to see that x * solves (1.2) if and only if it solves the fixed point equation:…”