Approximate and Near Weak Invariance for Nonautonomous Differential Inclusions

Let X be a real Banach space and I a nonempty interval. Let (K:Irightsquigarrow 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 Cârj? et al. (Trans Amer Math Soc. [2009];361:343–90) (see also Cârj? 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))).