2002,22B(2): 213-221 MULTIVALUED DIFFERENTIAL EQUATIONS IN BANACH SPACES AND THEIR APPLICATIONS 1 Liu Zhenhai ( *')#.;';: ) Department of Mathematics, Changsha University of Electric Power, Hunan 410077, China School of Mathematics and Information Sciences, Wenzhou University, Wenzhou 325027, China Ivan Szanto Departamento de Matematica, Universidad T.F.Santa Maria, Casilia 110- V, Valparaiso, Chile Abstract This paper studies the existence of solutions to a class of multi valued differ- ential equations by using a surjectivity result for multivalued (5+) type mappings. The authors then apply their results to evolution hemivariational inequalities and parabolic equations with nonmonotone discontinuities, which generalize and extend previously known theorems. Key words Evolution equations, differential inclusions, mulitvalued mappings, existence results, hemivariational inequality 1991 MR Subject Classification 34G05,47H19 1 Introduction Consider the initial value problem { it + Au + Fu 3 l, u(O) = un, 0< t < T, (1.1) where A is of class (8+) and. F is a multivalued map. If F is single-valued, Hirano(1] got existence results for F being compact. Ahmed and Xiang[2] extended Hirano's results for F satisfying that < FUn, Un - U >-+ 0 if Un converges weakly to u, while Liu[3] extended the above mentioned results to corresponding class of variational inequalities'and introduced a more general condition that admits a broader class of perturbation, i.e., that Un converges weakly to U implies lim < Fu, Un - U >:2: 0. Our purpose in this paper is to extend the main results in Liu[3] to nonlinear and multi- valued systems, which is motivated by many engineering problems, mechanical problems and parabolic partial differential equations with discontinuous nonlinearities. Here we refer to the works of Naniewicz and Panagiotopoulosl'", Car[5], Miettinenl'f, Rauchl?], and Liu[8]. We in- vestigate the existence on the base of a surjectivity result for multivalued (8+) type mappings 1 Received September 8, 2000. This research is supported by the National Natural Science Foundation of China(10171008)