In this paper, we introduce a multiple hybrid implicit iteration method for finding a solution for a monotone variational inequality with a variational inequality constraint over the common solution set of a general system of variational inequalities, and a common fixed point problem of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping in Hilbert spaces. Strong convergence of the proposed method to the unique solution of the problem is established under some suitable assumptions.
This paper discusses a monotone variational inequality problem with a variational inequality constraint over the common solution set of a general system of variational inequalities (GSVI) and a common fixed point (CFP) of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping in Hilbert spaces, which is called the triple hierarchical constrained variational inequality (THCVI), and introduces some Mann-type implicit iteration methods for solving it. Norm convergence of the proposed methods of the iteration methods is guaranteed under some suitable assumptions.
In this paper, we study a general system of variational inequalities with a
hierarchical variational inequality constraint for an infinite family of
nonexpansive mappings. We introduce general implicit and explicit iterative
algorithms. We prove the strong convergence of the sequences generated by
the proposed iterative algorithms to a solution of the studied problems.
Some new forward–backward multi-choice iterative algorithms with superposition perturbations are presented in a real Hilbert space for approximating common solution of monotone inclusions and variational inequalities. Some new ideas of constructing iterative elements can be found and strong convergence theorems are proved under mild restrictions, which extend and complement some already existing work.
In this paper, using a sunny generalized non-expansive retraction which is different from the metric projection and generalized metric projection in Banach spaces, we present a retractive iterative algorithm of Krasnosel’skii-type, whose sequence approximates a common solution of a mono-variational inequality of a finite family of η-strongly-pseudo-monotone-type maps and fixed points of a countable family of generalized non-expansive-type maps. Furthermore, some new results relevant to the study are also presented. Finally, the theorem proved complements, improves and extends some important related recent results in the literature.
The purpose of this paper is to introduce a new modified relaxed extragradient method and study for finding some common solutions for a general system of variational inequalities with inversestrongly monotone mappings and nonexpansive mappings in the framework of real Banach spaces. By using the demiclosedness principle, it is proved that the iterative sequence defined by the relaxed extragradient method converges strongly to a common solution for the system of variational inequalities and nonexpansive mappings under quite mild conditions.
We consider a general variational inequality and fixed point problem, which is to find a pointx*with the property that (GVF):x*∈GVI(C,A)andg(x*)∈Fix(S)whereGVI(C,A)is the solution set of some variational inequalityFix(S)is the fixed points set of nonexpansive mappingS, andgis a nonlinear operator. Assume the solution setΩof (GVF) is nonempty. For solving (GVF), we suggest the following methodg(xn+1)=βg(xn)+(1-β)SPC[αnF(xn)+(1-αn)(g(xn)-λAxn)],n≥0. It is shown that the sequence{xn}converges strongly tox*∈Ωwhich is the unique solution of the variational inequality〈F(x*)-g(x*),g(x)-g(x*)〉≤0, for allx∈Ω.
Hybrid viscosity CQ method for finding a common solution of a variational inequality, a general system of variational inequalities, and a fixed point problem
