On Regularization Parameter Choice and Convergence Rates in Regularization for Ill-Posed Mixed Variational Inequalities

Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 4, 181 - 198

On Regularization Parameter Choice

and Convergence Rates in Regularization

for Ill-Posed Mixed Variational Inequalities

Nguyen Buong

Vietnamse Academy of Science and Technology

Institute of Information Technology

18, Hoang Quoc Viet, q. Cau Giay, Ha Noi, Vietnam

[email protected]

Nguyen Thi Thu Thuy

Department of Mathematics

Thainguyen University, Vietnam

[email protected]

Abstract

In this paper, for ill-posed inverse-strongly monotone mixed varia￾tional inequalities in Banach spaces, the convergence rates of the reg￾ularized solution and regularized solution in connection with finite￾dimensional approximations of the spaces are estimated on the base

of a priori regularization parameter choice as well as a posteriori choice

by the generalized discrepancy principle.

Mathematics Subject Classification: 47H17

Keywords: Monotone operators, hemi-continuous, strictly convex Banach

space, Fr´echet differentiable, weakly lower semicontinuous and convex func￾tional and Tikhonov regularization

1. INTRODUCTION

Let X be a real reflexive Banach space having the property: the weak

convergence and convergence of norms of any sequence in X follow its strong