Algorithm method for solving the split general system of variational inequalities problem and fixed point problem of nonexpansive mapping with application

Abstract

The purpose of this research is to introduce the split general system of variational inequalities problem, by using the concept of the general system of variational inequalities problem in Ceng et al. Then, we establish and prove a strong convergence theorem for finding a common element of the set of solutions of this problem and the set of fixed points of a nonexpansive mapping for the first time in this research. By using our main theorem, strong convergence theorems of the split feasibility problem, the split variational inequality problem, and the minimization problem are obtained, respectively. In order to show the potential of our main result, we give three numerical examples of these classical problems introduced by previous studies, with the numerical example of our main theorem to compare the rate of convergence.

Iterative Scheme for Split Variational Inclusion and a Fixed-Point Problem of a Finite Collection of Nonexpansive Mappings

“…Haitao and Li [19] investigated the split variational inclusion problem and fixed-point problem of nonexpansive semigroup without prior calculation of operator norm. Later, many authors studied the common solution of split variational inequality/inclusion problem and fixedpoint problem of nonexpansive mappings in the framework of Hilbert/Banach spaces, see for example [18][19][20][21][22][23][24] and references therein.…”

An inertial forward-backward splitting method for approximating solutions of certain optimization problems

“…holds for every y ∈ H with x = y. Recently, several iterative methods have been devised for solving equilibrium problems, variational inequality problems, monotone inclusion problems, minimization problems and their related optimization problems; see, e.g., [6,7,8,9,10,11,12,13,14,15,16]) and the references therein.…”

“…where A, B : C → H are two mappings and λ , µ > 0 are two positive constants. Reently, Siriyan and Kangtunyakarn [15] introduced a split general system of variational inequalities, which consists of finding (x * , y * ) ∈ C ×C such that…”

“…Siriyan and Kangtunyakarn [15] proved the following theorem in the framework of real Hilbert spaces. Theorem 1.2.…”

“…Motivated by the ongoing interest on inertial-type algorithms and the results of Siriyan and Kangtunyakarn [15], Hieu [16], Cholamjiak, Pholasa and Suantai [25], Khan, Cholamjiak and Kazmi [26], Cheawchan and Kangtunyakarn [27], we introduce an inertial forward-backward splitting iterative algorithm for finding a common solution of a split general system of variational inequalities, the minimization problem and the fixed point problem of a finite family of quasi-nonexpansive mappings. We prove a strong convergence result in the framework of real Hilbert spaces and give some numerical examples for our main result.…”