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Journal of Inequalities and Applications

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Open Access 01-12-2006 | Research Article

Generalized partially relaxed pseudomonotone variational inequalities and general auxiliary problem principle

Author: Ram U. Verma

Published in: Journal of Inequalities and Applications | Issue 1/2006

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Abstract

Let

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be a nonlinear mapping from a nonempty closed invex subset

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq2_HTML.gif

of an infinite-dimensional Hilbert space

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into

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq4_HTML.gif

. Let

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be proper, invex, and lower semicontinuous on

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq6_HTML.gif

and let

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be continuously Fréchet-differentiable on

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with

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, the gradient of

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https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq11_HTML.gif

-strongly monotone, and

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq12_HTML.gif

-Lipschitz continuous on

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq13_HTML.gif

. Suppose that there exist an

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, and numbers

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https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq16_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq17_HTML.gif

such that for all

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and for all

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, the set

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defined by

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is nonempty, where

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and

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https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq24_HTML.gif

-Lipschitz continuous with the following assumptions. (i)

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, and

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. (ii) For each fixed

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, map

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is sequentially continuous from the weak topology to the weak topology. If, in addition,

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq29_HTML.gif

is continuous from

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq30_HTML.gif

equipped with weak topology to

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq31_HTML.gif

equipped with strong topology, then the sequence

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq32_HTML.gif

generated by the general auxiliary problem principle converges to a solution

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq33_HTML.gif

of the variational inequality problem (VIP):

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for all

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq35_HTML.gif

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Title: Generalized partially relaxed pseudomonotone variational inequalities and general auxiliary problem principle
Author: Ram U. Verma
Publication date: 01-12-2006
Publisher: Springer International Publishing
Published in: Journal of Inequalities and Applications / Issue 1/2006
Electronic ISSN: 1029-242X
DOI: https://doi.org/10.1155/JIA/2006/90295

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