Cybernetics and Systems Analysis, Vol. 36, No. 5, 2000 PENALTY METHOD FOR VARIATIONAL INEQUALITIES WITH MULTIVALUED MAPPINGS. PART II 1 M. Z. Zgurovskii a and V. S. Mel’nik b UDC 517.91 By means of penalty operators, a method of regularization of variational inequalities is generalized to a class of variational inequalities with multivalued mappings. Keywords: variational inequalities, multivalued mappings, penalty method, coercitivity conditions. In the present paper, a further development of the penalty-operator method for variational inequalities of the form (5), (6) from [1] is used. In particular, for inequality (5), it turned out to be possible to relax significantly the coercitivity conditions and to prove the statements on resolvability, which generalize the results obtained for this class of inequalities using other methods [2, 3]. The method of combined penalty is proposed for inequality (6), apparently, for the first time. The structures and notation introduced in [1] are used in the present paper. The following statement is a generalization of Theorem 2 of [1]. THEOREM 1. Let X V W = ∩ , V and W be reflexive Banach spaces, K be closed and convex in X, AV V : * → 2 be a bounded demiclosed operator satisfying the condition α ( ) V ( ( )) α 2 V and 5 ∈ y K 0 such that 2200 > ε 0 || || [ ( ), ] ( ), y Ay y y y y y x W - - - + -       →+∞ 1 0 0 1 ε β as || || . y x →∞ () 1 Then 2200 > ε 0 and f X V W ∈ = + * * * , the operator inclusion A y Ay y f ε ε β () () () = + ⊃ co 1 () 2 is solvable and from the sequence of its solutions { } y ε , a subsequence { } y τ can be selected such that y y w τ → in X, y y τ → in V for τ → 0, and an element y K ∈ satisfies the inequality Ay y f y K X ( ), , ξ ξ ξ -       ≥ - 2200 ∈ + . (3) Proof. Let BX W : * → 2 be a multivalued operator with the following properties: a) B is a bounded radially semicontinuous (r.s.c) operator with semibounded variation (s.b.v.); 667 1060-0396/00/3605-0667$25.00 © 2000 Kluwer Academic/Plenum Publishers a National Technical University of Ukraine “Kiev Polytechnical Institute,” Kiev, Ukraine. b Scientific and Training Complex of the Institute of Applied Systems Analysis of the National Academy of Sciences of Ukraine and Ministry of Education and Science of Ukraine, Kiev, Ukraine. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 41-53, September-October, 2000. Original article submitted May 15, 2000. 1 The first part of the paper was published in No. 4, 2000.