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Keywords:
boundary value differential inclusion; Stieltjes derivative; Kurzweil-Stieltjes integral; periodic problem
boundary value differential inclusion; Stieltjes derivative; Kurzweil-Stieltjes integral; periodic problem
Summary:
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.
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