No abstract

We develop a new concept of a solution, called the limit solution, to fully nonlinear differential inclusions in Banach spaces. That enables us to study such kind of inclusions under relatively weak conditions. Namely we prove the existence of this type of solutions and some qualitative properties, replacing the commonly used compact or Lipschitz conditions by a dissipative one, i.e., one-sided Perron condition. Under some natural assumptions we prove that the set of limit solutions is the closure of the set of integral solutions.

In this paper we give a criterion for a given set K in Banach space to be approximately weakly invariant with respect to the differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A generates a C 0-semigroup and F is a given multi-function, using the concept of a tangent set to another set. As an application, we establish the relation between approximate solutions to the considered differential inclusion and solutions to the relaxed one, i.e., x′(t) ∈ Ax(t) + $\overline {co} $ F(x(t)), without any Lipschitz conditions on the multi-function F.

In this paper we study different types of (generalized) solutions for semilinear evolution inclusions in general Banach spaces, called limit and weak solutions, which are extensions of the weak solutions studied by T. Donchev [Nonlinear Anal., 16 (1991), pp. 533-542] and the directional solutions studied by J. Tabor [Set-Valued Anal., 14 (2006), pp. 121-148]. Under appropriate assumptions, we show that the set of the limit solutions is compact R δ . When the right-hand side satisfies the one-sided Perron condition, a variant of the well-known lemma of Filippov-Plis, as well as a relaxation theorem, are proved.

We study evolution inclusions given by multivalued perturbations of m-dissipative operators with nonlocal initial conditions. We prove the existence of solutions. The commonly used Lipschitz hypothesis for the perturbations is weakened to one-sided Lipschitz ones. We prove an existence result for the multipoint problems that cover periodic and antiperiodic cases. We give examples to illustrate the applicability of our results.