We derive a differential inclusion governing the evolution of optimal trajectories to the Mayer problem. The value function is allowed to be discontinuous. This inclusion has convex compact right-hand sides.

In this article we consider the differential inclusion − div(|∇u|p(x)−2∇u) ∈ ∂F (x, u) in Ω, u = 0 on ∂Ω which involves the p(x)-Laplacian. By applying the nonsmooth Mountain Pass Theorem, we obtain at least one nontrivial solution; and by applying the symmetric Mountain Pass Theorem, we obtain k-pairs of nontrivial solutions in W 1,p(x) 0 (Ω).

Path-dependent impulse differential inclusions, and in particular, path-dependent hybrid control systems, are defined by a path-dependent differential inclusion (or path-dependent control system, or differential inclusion and control systems with memory) and a pathdependent reset map. In this paper, we characterize the viability property of a closed subset of paths under an impulse path-depende...

In this paper, we establish some existence results for higher-order nonlinear fractional differential inclusions with multi-strip conditions, when the right-hand side is convex-compact as well nonconvexcompact values. First, use alternative of Leray-Schauder type multivalued maps. We investigate next result by using well-known Covitz and Nadler?s fixed point theorem contractions. The are illust...