Abstract
The purpose of this paper is to solve the hierarchical variational inequality with the constraint of a general system of variational inequalities in a uniformly convex and 2-uniformly smooth Banach space. We introduce implicit and explicit iterative algorithms which converge strongly to a unique solution of the hierarchical variational inequality problem. Our results improve and extend the corresponding results announced by some authors.
Highlights
Let X be a real Banach space with its topological dual X∗, and C be a nonempty closed convex subset of X
It is well known [ ] that, in a smooth Banach space, this problem is equivalent to a fixed-point equation, containing a sunny nonexpansive retraction from any point of the space onto the feasible set, which is usually assumed to be closed and convex
We introduce implicit and explicit iterative algorithms for finding a solution of the problem and derive the strong convergence of the proposed algorithms to a unique solution of the problem
Summary
Let X be a real Banach space with its topological dual X∗, and C be a nonempty closed convex subset of X. Aoyama et al [ ] proposed an iterative scheme to find the approximate solution of ( ) and proved the weak convergence of the sequences generated by the proposed scheme It is well known [ ] that, in a smooth Banach space, this problem is equivalent to a fixed-point equation, containing a sunny nonexpansive retraction from any point of the space onto the feasible set, which is usually assumed to be closed and convex. Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space X, and let T be a nonexpansive mapping of C into itself with the fixed point set Fix(T) = ∅. Lemma ([ ]) Let C be a nonempty closed convex subset of a real -uniformly smooth Banach space X. Let F : C → X be δ-strongly accretive and ζ -strictly pseudocontractive with δ
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