Generalized partially relaxed pseudomonotone variational inequalities and general auxiliary problem principle

• Main Author: Verma Ram U
• Format: Article
• Language: English
• Published: SpringerOpen 2006-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2006/90295
• ISSN: 1025-5834; 1029-242X

<p/> <p>Let <inline-formula><graphic file="1029-242X-2006-90295-i1.gif"/></inline-formula> be a nonlinear mapping from a nonempty closed invex subset <inline-formula><graphic file="1029-242X-2006-90295-i2.gif"/></inline-formula> of an infinite-dimensional Hilbert space <inline-formula><graphic file="1029-242X-2006-90295-i3.gif"/></inline-formula> into <inline-formula><graphic file="1029-242X-2006-90295-i4.gif"/></inline-formula>. Let <inline-formula><graphic file="1029-242X-2006-90295-i5.gif"/></inline-formula> be proper, invex, and lower semicontinuous on <inline-formula><graphic file="1029-242X-2006-90295-i6.gif"/></inline-formula> and let <inline-formula><graphic file="1029-242X-2006-90295-i7.gif"/></inline-formula> be continuously Fr&#233;chet-differentiable on <inline-formula><graphic file="1029-242X-2006-90295-i8.gif"/></inline-formula> with <inline-formula><graphic file="1029-242X-2006-90295-i9.gif"/></inline-formula>, the gradient of <inline-formula><graphic file="1029-242X-2006-90295-i10.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2006-90295-i11.gif"/></inline-formula>-<it>strongly</it> monotone, and <inline-formula><graphic file="1029-242X-2006-90295-i12.gif"/></inline-formula>-<it>Lipschitz</it> continuous on <inline-formula><graphic file="1029-242X-2006-90295-i13.gif"/></inline-formula>. Suppose that there exist an <inline-formula><graphic file="1029-242X-2006-90295-i14.gif"/></inline-formula>, and numbers <inline-formula><graphic file="1029-242X-2006-90295-i15.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2006-90295-i16.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2006-90295-i17.gif"/></inline-formula> such that for all <inline-formula><graphic file="1029-242X-2006-90295-i18.gif"/></inline-formula> and for all <inline-formula><graphic file="1029-242X-2006-90295-i19.gif"/></inline-formula>, the set <inline-formula><graphic file="1029-242X-2006-90295-i20.gif"/></inline-formula> defined by <inline-formula><graphic file="1029-242X-2006-90295-i21.gif"/></inline-formula> is nonempty, where <inline-formula><graphic file="1029-242X-2006-90295-i22.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2006-90295-i23.gif"/></inline-formula> is <inline-formula><graphic file="1029-242X-2006-90295-i24.gif"/></inline-formula>-<it>Lipschitz</it> continuous with the following assumptions. (i) <inline-formula><graphic file="1029-242X-2006-90295-i25.gif"/></inline-formula>, and <inline-formula><graphic file="1029-242X-2006-90295-i26.gif"/></inline-formula>. (ii) For each fixed <inline-formula><graphic file="1029-242X-2006-90295-i27.gif"/></inline-formula>, map <inline-formula><graphic file="1029-242X-2006-90295-i28.gif"/></inline-formula> is sequentially continuous from the weak topology to the weak topology. If, in addition, <inline-formula><graphic file="1029-242X-2006-90295-i29.gif"/></inline-formula> is continuous from <inline-formula><graphic file="1029-242X-2006-90295-i30.gif"/></inline-formula> equipped with weak topology to <inline-formula><graphic file="1029-242X-2006-90295-i31.gif"/></inline-formula> equipped with strong topology, then the sequence <inline-formula><graphic file="1029-242X-2006-90295-i32.gif"/></inline-formula> generated by the general auxiliary problem principle converges to a solution <inline-formula><graphic file="1029-242X-2006-90295-i33.gif"/></inline-formula> of the variational inequality problem (VIP): <inline-formula><graphic file="1029-242X-2006-90295-i34.gif"/></inline-formula> for all <inline-formula><graphic file="1029-242X-2006-90295-i35.gif"/></inline-formula>.</p>