Existence of solutions for quasilinear second order differential inclusions with nonlinear boundary conditions

AbstractIn this paper we consider a quasilinear second-order differential inclusion with a convex-valued multivalued term and nonlinear, multivalued boundary conditions. Using the Leray–Schauder fixed-point theorem and techniques from multivalued analysis and from nonlinear analysis, we prove the existence of a solution. Our formulation of the problem is general and includes as special cases the Dirichlet, the Neumann, the periodic problems, as well as certain Sturm–Liouville-type problems

Existence and Relaxation Results for Nonlinear Second-Order Multivalued Boundary Value Problems in RN

AbstractIn this paper we study second order differential inclusions with nonlinear boundary conditions. Our formulation is general and incorporates as special cases well-known problems such as the Dirichlet (Picard), Neumann, and periodic problems. We prove existence theorems under various sets of hypotheses for both the convex and nonconvex problems. Also we show the existence of extremal solutions and that the extremal solutions are dense in the solutions of the convex problem for theW1,2(T,RN)-norm (strong relaxation theorem). Finally we examine the Dirichlet problem when the multivalued right-hand side does not depend on the derivative of x and satisfies a general growth hypothesis and a sign-type condition. For this problem we prove existence results and a relaxation theorem