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Generalized partially relaxed pseudomonotone variational inequalities and general auxiliary problem principle

Journal of Inequalities and Applications volume 2006, Article number: 90295 (2006) Cite this article

Abstract

Let be a nonlinear mapping from a nonempty closed invex subset of an infinite-dimensional Hilbert space into. Let be proper, invex, and lower semicontinuous on and let be continuously Fréchet-differentiable on with, the gradient of,-strongly monotone, and-Lipschitz continuous on. Suppose that there exist an, and numbers,, such that for all and for all, the set defined by is nonempty, where and is-Lipschitz continuous with the following assumptions. (i), and. (ii) For each fixed, map is sequentially continuous from the weak topology to the weak topology. If, in addition, is continuous from equipped with weak topology to equipped with strong topology, then the sequence generated by the general auxiliary problem principle converges to a solution of the variational inequality problem (VIP): for all.

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  1. Department of Mathematics, University of Toledo, Toledo, OH, 43606, USA

    Ram U. Verma

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  1. Ram U. Verma
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Correspondence to Ram U. Verma.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Verma, R.U. Generalized partially relaxed pseudomonotone variational inequalities and general auxiliary problem principle. J Inequal Appl 2006, 90295 (2006). https://doi.org/10.1155/JIA/2006/90295

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