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Abstract
In this paper, we present a new type of extra-gradient method for generalized variational inequalities with multi-valued mapping in an infinite-dimensional Hilbert space. For this method, the generated sequence possesses an expansion property with respect to the initial point, and the existence of the solution to the problem can be verified through the behavior of the generated sequence. Furthermore, under mild conditions, we show that the generated sequence of the method strongly converges to the solution of the problem which is closest to the initial point.MSC:90C30, 15A06.
Highlights
Let F be a multi-valued mapping from H into H with nonempty values, where H is a realHilbert space
If the multi-valued mapping F is a single-valued mapping from H to H, the GVIP collapses to the classical variational inequality problem [, ]
In this paper, inspired by the work in [ ] for finding the zeros of maximal monotone operators in a real Hilbert space, we proposed a new type of extragradient solution method for the GVIP which has the following expansion property w.r.t. the initial point, i.e., xk – x ≤ xk+ – x, ∀k
Summary
In this paper, inspired by the work in [ ] for finding the zeros of maximal monotone operators in a real Hilbert space, we proposed a new type of extragradient solution method for the GVIP which has the following expansion property w.r.t. the initial point, i.e., xk – x ≤ xk+ – x , ∀k. Throughout this paper, we assume that the multi-valued mapping F : X → H is maximal monotone and continuous on X with nonempty compact convex values, where X ⊆ H is a nonempty, closed, and convex set. The projection residue r(x, ξ ) can verify the solution set of problem
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