Czechoslovak Mathematical Journal, first online, pp. 1-21
Stieltjes differential problems with general boundary value conditions. Existence and bounds of solutions
Valeria Marraffa, Bianca Satco
Received March 21, 2023. Published online March 22, 2024.
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.
Keywords: boundary value differential inclusion; Stieltjes derivative; Kurzweil-Stieltjes integral; periodic problem
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Affiliations: Valeria Marraffa, Department of Mathematics and Computer Sciences, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy, e-mail: valeria.marraffa@unipa.it; Bianca Satco (corresponding author), Faculty of Electrical Engineering and Computer Science, Stefan cel Mare University of Suceava, Universitatii 13, 720225, Suceava, Romania, e-mail: bisatco@usm.ro
