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Journal of Inequalities and Applications

Erschienen in:

Open Access 01.12.2006 | Research Article

Generalized partially relaxed pseudomonotone variational inequalities and general auxiliary problem principle

verfasst von: Ram U. Verma

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2006

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Abstract

Let

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq1_HTML.gif

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

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq3_HTML.gif

into

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

. Let

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq5_HTML.gif

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

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq7_HTML.gif

be continuously Fréchet-differentiable on

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq8_HTML.gif

with

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq9_HTML.gif

, the gradient of

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq10_HTML.gif

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

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq14_HTML.gif

, and numbers

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq15_HTML.gif

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

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq18_HTML.gif

and for all

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq19_HTML.gif

, the set

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq20_HTML.gif

defined by

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq21_HTML.gif

is nonempty, where

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq22_HTML.gif

and

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq23_HTML.gif

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)

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq25_HTML.gif

, and

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq26_HTML.gif

. (ii) For each fixed

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq27_HTML.gif

, map

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq28_HTML.gif

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):

https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F90295/MediaObjects/13660_2004_Article_1653_IEq34_HTML.gif

for all

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

Vorheriger Artikel Approximate solutions of the generalized Gołąb-Schinzel equation

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Titel: Generalized partially relaxed pseudomonotone variational inequalities and general auxiliary problem principle
verfasst von: Ram U. Verma
Publikationsdatum: 01.12.2006
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2006
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1155/JIA/2006/90295

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