Approximate and Near Weak Invariance for Nonautonomous Differential Inclusions

Abstract

Let X be a real Banach space and I a nonempty interval. Let K:IźX$K:I\rightsquigarrow X$ be a multi-function with the graph K$\mathcal {K} $. We give here a characterization for K$\mathcal {K} $ to be approximate/near weakly invariant with respect to the differential inclusion xź(t)źF(t,x(t))$x^{\prime }(t)\in F(t, x(t))$ by means of an appropriate tangency concept and Lipschitz conditions on F. The tangency concept introduced in this paper extends in a natural way the quasi-tangency concept introduced by Cârjăź et al. (Trans Amer Math Soc. [2009];361:343---90) (see also Cârjăź et al. ([2007])). Viability, invariance and applications. Amsterdam: Elsevier Science B V) in the case when F is independent of t. As an application, we give some results concerning the set of solutions for the differential inclusion xź(t)źF(t,x(t))$x^{\prime }(t)\in F(t,x(t))$.