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Got it We value your privacy We use cookies to offer you a better experience, personalize content, tailor advertising, provide social media features, and better understand the use of our services. To learn more or modifyprevent the use of cookies, see our Cookie Policy and Privacy Policy. Fead Optimizer Extract Tool Download Citation ShareAccept Cookies top See all 23 Citations See all 63 References Download citation Share Facebook Twitter LinkedIn Reddit Download full-text PDF Regularization of quasi-variational inequalities Article (PDF Available) in Optimization 64(8):1-22 April 2015 with 207 Reads How we measure reads A read is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more DOI: 10.108002331934.2015.1028935 Cite this publication Akhtar A. Khan 27.05 Rochester Institute of Technology Christiane Tammer 28.64 Martin Luther University Halle-Wittenberg Constantin Zalinescu 32.24 Octav Mayer Institute of Mathematics, Iasi Abstract An ill-posed quasi-variational inequality with contaminated data can be stabilized by employing the elliptic regularization. Under suitable conditions, a sequence of bounded regularized solutions converges strongly to a solution of the original quasi-variational inequality. Moreover, the conditions that ensure the boundedness of regularized solutions, become sufficient solvability conditions. It turns out that the regularization theory is quite strong for quasi-variational inequalities with set-valued monotone maps but restrictive for generalized pseudo-monotone maps. The results are quite general and are applicable to ill-posed variational inequalities, hemi-variational inequalities, inverse problems, and split feasibility problem, among others. Fead Optimizer Extract Tool Free Advertisement ContentDiscover the worlds research 17 million members 135 million publications 700k research projects Join for free Advertisement Content uploaded by Constantin Zalinescu Author content All content in this area was uploaded by Constantin Zalinescu on Nov 24, 2015 Content may be subject to copyright. Khan a, Christiane Tammer b and Constantin Zalinescu c a Center for Applied and Computational Mathematics, School of Mathematical Sciences, Rochester Institute of T echnology, Rochester, NY, USA; b Institute of Mathematics, Martin-Luther-University of Halle-Wittenber g, Halle-Saale, Germany; c Faculty of Mathematics, University Al. I. Cuza Iasi, Iasi, Romania ( Received 30 November 2014; accepted 5 March 2015 ) An ill-posed quasi-variational inequality with contaminated data can be stabilized by employing the elliptic regularization. Moreover, the conditions that ensure the bounded- ness of regularized solutions, become sufcient solvability conditions. Keywords: quasi-variational inequalities; variational inequalities; hemi-variational inequalities; regularization; ill-posed; monotone; pseudo-monotone; generalized pseudo-monotone AMS Subject Classications: 49J20; 90C51; 90C30 1. Introduction Throughout this paper, B is a reexive Banach space and B is its topological dual. W e assume that B has been renormed so that B and B are locally uniformly convex. W e denote the duality pairing between B and B by,, whereas stands for the norm in B as well as the associated norm in B. Let C B be non-empty, closed and convex, and let K: C C be a set-valued map such that for every v C, the set K (v) is a non-empty, closed and convex subset of C.L e t F: B B be a given set-valued map, let: B R: R be a given functional, and let f B. The domain and the graph of F are given by D ( F ): x B F ( x ) and G ( F ): ( x, y ) B B x D ( F ), y F ( x ), respectively. The strong convergence and the weak convergence in B as well as in B are specied by and, respectively. In this work, we study the following quasi-variational inequality: Find x C with x K ( x ) such that for some w F ( x ), we have w f, z x ( x ) ( z ), for every z K ( x ). Corresponding author.
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