Relaxation Theorem for Stieltjes Differential Inclusions on Infinite Intervals

For a very general first-order differential problem on an infinite-time horizon involving the Stieltjes derivative with respect to a left-continuous non-decreasing function g:[0,infinity)-> R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g:[0,\infty )\rightarrow \mathbb {R}$$\end{document}xg '(t)is an element of F(t,x(t)),t is an element of[0,infinity)x(0)=x0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{l} x'_g(t) \in F(t,x(t)),\; t\in [0,\infty )\\ x(0)=x_0, \end{array} \right. \end{aligned}$$\end{document}we study the possibility to approximate the solutions of the convexified inclusion by the solutions of the non-convexified problem. Via a generalization to this framework of a classical result concerning continuous selection of trajectory, we thus present a relaxation theorem which states that, similarly to the setting of usual differential inclusions, the approximation can be achieved once we allow to the initial value to differ (but remaining close to) from the initial value of the considered solution of the relaxed inclusion.