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

Abstract

We are concerned with first order set-valued problems with very general boundary value conditions $\begin{cases} u'_g(t)\in F(t,u(t)),\quad\mu_g \text{-a.e.} t\in[0,T] , \\ L(u(0), u(T))=0 \end{cases}$ involving the Stieltjes derivative with respect to a left-continuous nondecreasing function $g\colon[0,T]\to\mathbb{R}$, a Carathéodory multifunction $F\colon[0,T]\times\mathbb{R}\to\mathcal{P}(\mathbb{R})$ and a continuous $L\colon\mathbb{R}^2\to\mathbb{R}$. 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.