Approximate Weak Invariance and Relaxation for Fully Nonlinear Differential Inclusions

In this paper we prove that a given set K is approximately weakly invariant with respect to the fully nonlinear differential inclusion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x^\prime (t) \in Ax (t) + F (x (t))}$$\end{document}, where A is an m-dissipative operator, and F is a given multi-function in a Banach space, if and only if the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F(\xi)}$$\end{document} is A-quasi-tangent to the set K, for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\xi \in K}}$$\end{document} . As an application, we establish that the approximate solutions of the given differential inclusion approximate the solutions of the relaxed (convexified) nonlinear differential inclusion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x^\prime (t) \in Ax (t) + \overline{co}F (x (t))}$$\end{document}, with no hypotheses of Lipschitz type for multi-function F.