A hybrid viscosity algorithm via modify the hybrid steepest descent method for solving the split variational inclusion in image reconstruction and fixed point problems

Ergodic-type method for a system of split variational inclusion and fixed point problems in Hilbert spaces

“…Our purpose is not only to extend the iterative methods (1.2), (1.3) and (1.4) to the case of the system of split variational inclusion and fixed point problems of a family of nonexpansive mappings in the framework of real Hilbert spaces, but also to establish a strong convergence theorem of the system of split variational inclusion and fixed point problems with variable coefficients instead of mean value, form which the minimum norm solution is deduced as a special case. Moreover, a numerical example is given to illustrate the operational reliability and convergence of our method and results which improve and extend the previously known results of [9,12,13,15,16] and many others.…”

“…In 2015, Wen and Chen [16] and Sitthithakerngkiet et al [13] extended scheme (1.2) to a general iterative method and a hybrid viscosity algorithm for solving the split variational inclusion in image reconstruction with fixed point problems, respectively. On the other hand, Shimizu and Takahashi [12] established an ergodic theorem of a family of nonexpansive mappings based on the Cesàro mean.…”

A viscosity iterative algorithm for the optimization problem system

“…The concept of viscosity solution for Hamilton-Jacobi equations, which plays a crucial role in control theory, game theory and partial differential equations has been introduced by Crandall and Lions [3]. Recently, the viscosity iterative algorithm have received rapid development, see, for example, [6], [9], [11], [21], [20], [24] and [25]. In this section, we introduce a viscosity iterative algorithma for finding a common element of the set of solution for an equilibrium problem (involving a bi-function defined on a closed convex subset) and the set of fixed points of a nonexpansive semigroup.…”

Parallel Extragradient-Proximal Methods for Split Equilibrium Problems

“…Recently, SEP (1.1) and its special cases have been recieved a lot of attention by many authors and some methods for solving them can be found, for instance, in [8,11,12,13,14,17,19,20,22,24,25,26,30,32]. Almost proposed methods for SEPs based on the proximal method [21] which consists of solving a regularized equilibrium problem, i.e., at current iteration, given x n , the next iterate x n+1 solves the following problem; …”