Stieltjes differential problems with general boundary value conditions. Existence and bounds of solutions

We are concerned with first order set-valued problems with very general boundary value conditions{ug '(t)F(t,u(t)),mu g-a.e.t[0,T],L(u(0),u(T))=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\matrix{{u_g<^>\prime (t) \in F(t,u(t)),} \hfill \;\;\;\; {{\mu _g} - {\rm{a}}{\rm{.e}}{\rm{.}}\,\,t \in [0,T],} \hfill \cr {L(u(0),\,\,\,u(T)) = 0} \hfill \;\;\; {} \hfill \cr } } \right.$$\end{document}involving the Stieltjes derivative with respect to a left-continuous nondecreasing function g: [0, T] -> Double-struck capital R, a Caratheodory multifunction F:[0,T]xDouble-struck capital R ->(Double-struck capital R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F:[0,T] \times \mathbb{R} \to {\cal P}(\mathbb{R})$$\end{document} and a continuous L: Double-struck capital R2 -> Double-struck capital R. Using appropriate notions of lower and upper solutions, we prove the existence of solutions via a fixed point result for condensing mappings. In the periodic single-valued case, combining an existence theory for the linear case with a recent result involving lower and upper solutions (which can be seen as a consequence of our existence theorem mentioned before), we get not only the existence of solutions, but also lower and upper bounds, respectively, by imposing an estimation for the right-hand side.