Convergence Theorem of Two Sequences for Solving the Modified Generalized System of Variational Inequalities and Numerical Analysis

Abstract

The purpose of this paper is to introduce an iterative algorithm of two sequences which depend on each other by using the intermixed method. Then, we prove a strong convergence theorem for solving fixed-point problems of nonlinear mappings and we treat two variational inequality problems which form an approximate modified generalized system of variational inequalities (MGSV). By using our main theorem, we obtain the additional results involving the split feasibility problem and the constrained convex minimization problem. In support of our main result, a numerical example is also presented.

Convergence theorem for an intermixed iteration in $p$-uniformly convex metric space

“…Then he proved the sequence {x n } converges strongly to z = P F (T ) f (z) under some suitable condition α n . After that, many researchers have modified the viscosity algorithm in which the sequence {x n } is involved in the sequence {y n } and the definition of the sequence {y n } is also involved in the sequence {x n }, see, for instance [31,27,28].…”

An Iterative Method for Solving Split Monotone Variational Inclusion Problems and Finite Family of Variational Inequality Problems in Hilbert Spaces

“…Under some control conditions, they proved that the sequence x n 􏼈 􏼉 converges strongly to P F(T) f(y * ) and y n 􏼈 􏼉 converges strongly to P F(S) f(x * ), respectively, where x * ∈ F(T), y * ∈ F(S), and P F(T) and P F(S) are the metric projection of H onto F(T) and F(S), respectively. After that, many authors have developed and used this algorithm to solve the fixed-point problems of many nonlinear operators in real Hilbert spaces (see for example [21][22][23][24][25][26][27]). Question: can we prove the strong convergence theorem of two sequences of split monotone variational inclusion problems and fixed-point problems of nonlinear mappings in real Hilbert spaces?…”