MULTIVALUED p-LIENARD SYSTEMS

In the last decade, there have been many papers dealing with second-order multivalued boundary value problems. We mention the works of Erbe and Krawcewicz [5, 6], Frigon [7, 8], Halidias and Papageorgiou [9], Kandilakis and Papageorgiou [11], Kyritsi et al. [12], Palmucci and Papalini [17], and Pruszko [19]. In all the above works, with the exception of Kyritsi et al. [12], p = 2 (linear differential operator), G= 0, and g = 0. Moreover, in Frigon [7, 8] and Palmucci and Papalini [17], the inclusions are scalar (i.e., N = 1). Finally we should mention that recently single-valued p-Lienard systems were studied by Mawhin [14] and Manásevich and Mawhin [13]. In this work, for problem (1.1), we prove an existence theorem under conditions of nonuniform nonresonance with respect to the first weighted eigenvalue of the negative vector ordinary p-Laplacian with Dirichlet boundary conditions [15, 20]. Our approach is based on the multivalued version of the Leray-Schauder alternative principle due to Bader [1] (see Section 2).