Nonconvex Differential Inclusions with Nonlinear Monotone Boundary Conditions

Existence results for problems with monotone nonlinear boundary conditions obtained in the previous publications by the author for functional differential equations are transferred to the case of nonconvex differential inclusions with the help of the selection theorem due to A. Bressan and G. Colombo. The existence of solutions of boundary value problems for differential inclusions with possibly nonconvex right-hand sides was studied in [1–6]. The technique of continuous selections of multifunctions with decomposable values is helpful in these investigations. In particular, this technique allows us to establish a connection between the considered differential inclusion and a functional differential equation. Below we use this connection to transfer some assertions, previously proved by the author for functional differential equations [7–9], to the case of differential inclusions. Thus we obtain solvability results for problems for lower semicontinuous differential inclusions with nonlinear monotone boundary conditions. We essentially employ the selection theorem for multifunctions with decomposable values due to Bressan and Colombo [10]. A systematic account of different aspects of the theory of differential inclusions and the corresponding bibliography can be found in [11–13]. For results on boundary value problems, see e.g., [14, 15] in the case of equations and [16] in the case of convex-valued inclusions, and the references therein. The following notation is used below. We fix a norm in the n-dimensional space Rn and denote it by | · |n. Let Ck with an integer k ≥ 0 denote the space of k times continuously differentiable functions (C0 is the class of all continuous functions). The space L1 consists of all measurable integrable functions. Here and everywhere below we use the Lebesgue measure. CLm 1 1991 Mathematics Subject Classification. 34B15, 34A60, 34K10.