Approximate and Near Weak Invariance for Nonautonomous Differential Inclusions

Abstract

Let X be a real Banach space and I a nonempty interval. Let $K:I\rightsquigarrow X$ be a multi-function with the graph $\mathcal {K}$. We give here a characterization for $\mathcal {K}$ to be approximate/near weakly invariant with respect to the differential inclusion $x^{\prime }(t)\in F(t, x(t))$ 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 Cârjă et al. (Trans Amer Math Soc. [2009];361:343–90) (see also Cârjă 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 $x^{\prime }(t)\in F(t,x(t))$.