We are concerned with first order set-valued problems with very general boundary value conditions \(\left\{ {\matrix{{u_g^\prime (t) \in F(t,u(t)),} \hfill \;\;\;\; {{\mu _g} - {\rm{a}}{\rm{.e}}{\rm{.}}\,\,t \in [0,T],} \hfill \cr {L(u(0),\,\,\,u(T)) = 0} \hfill \;\;\; {} \hfill \cr } } \right.\) involving the Stieltjes derivative with respect to a left-continuous nondecreasing function g: [0, T] → ℝ, a Carathéodory multifunction \(F:[0,T] \times \mathbb{R} \to {\cal P}(\mathbb{R})\) and a continuous L: ℝ2 → 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.
