STUDIA UNIV. “BABES ¸–BOLYAI”, MATHEMATICA, Volume XLVII, Number 3, September 2002 TWO-VARIABLEVARIATIONAL-HEMIVARIATIONAL INEQUALITIES ENDRE BUZOG ´ ANY, ILDIK ´ O ILONA MEZEI, VIORICA VARGA Abstract. In this paper we guarantee the solution for two-variable variational-hemivariational inequalities and we give some applications. 1. Introduction The aim of this paper is to establish a two-variable result concerning the hemivariational inequalities. These inequalities appear as a generalisation of varia- tional inequalitis, but they are more general than these ones, having applications in several branches of mathematics, mechanics, economy engineering. The paper is organized as follows. In the Section 2 we formulate the prob- lem and give some notions and results which will be used later. In Section 3 we establish the main results of this paper, i.e. we guarantee solution for hemivaria- tional inequality. Finally in Section 4 we give some applications. More preciselly, we obtain a Brouwer’s type variational inequality, the Schauder fixed point theorem (and Brouwer fixed point theorem), a hemivariational inequality of Panagiotopoulos- Fundo-R˘adulescu type, and a result concerning the Nash equilibrium theory. 2. Preliminaries Let X be a Banach space, X ∗ its dual. We consider the following hypotheses: (H T ) T : X → L p (Ω, R k ) is a linear, continuous operator, where p ∈ [1, ∞), k ≥ 1 and Ω is a bounded open set in R N . (H j ) j :Ω×R k → R is a Carath´ eodory function which is locally Lipschitz with respect to the second variable and there exist h 1 ∈ L p p−1 (Ω, R) and h 2 ∈ L ∞ (Ω, R) Received by the editors: 10.04.2002 This paper was supported by Sapientia Foundation. 31