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

Valeria Marraffa; Bianca Satco

Czechoslovak Mathematical Journal (2025)

  • Issue: 1, page 235-255
  • ISSN: 0011-4642

Abstract

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We are concerned with first order set-valued problems with very general boundary value conditions u g ' ( t ) ∈ F ( t , u ( t ) ) , μ g -a.e. ∈ [ 0 , T ] , L ( u ( 0 ) , T ) ) = 0 involving the Stieltjes derivative with respect to a left-continuous nondecreasing function g : [ 0 , T ] → ℝ , a Carathéodory multifunction F : [ 0 , T ] × ℝ → 𝒫 ( ℝ ) and a continuous L : ℝ 2 → 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.

How to cite

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Marraffa, Valeria, and Satco, Bianca. "Stieltjes differential problems with general boundary value conditions. Existence and bounds of solutions." Czechoslovak Mathematical Journal (2025): 235-255. <http://eudml.org/doc/299921>.

@article{Marraffa2025,
abstract = {We are concerned with first order set-valued problems with very general boundary value conditions \[ \{\left\lbrace \begin\{array\}\{ll\} u^\{\prime \}\_g(t)\in F(t,u(t)),\quad \mu \_g \text\{-a.e.\} \in [0,T] , \\ L(u(0),T))=0 \end\{array\}\right.\} \] involving the Stieltjes derivative with respect to a left-continuous nondecreasing function $g\colon [0,T]\rightarrow \mathbb \{R\}$, a Carathéodory multifunction $F\colon [0,T]\times \mathbb \{R\}\rightarrow \mathcal \{P\}(\mathbb \{R\})$ and a continuous $L\colon \mathbb \{R\}^2\rightarrow \mathbb \{R\}$. 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.},
author = {Marraffa, Valeria, Satco, Bianca},
journal = {Czechoslovak Mathematical Journal},
keywords = {boundary value differential inclusion; Stieltjes derivative; Kurzweil-Stieltjes integral; periodic problem},
language = {eng},
number = {1},
pages = {235-255},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Stieltjes differential problems with general boundary value conditions. Existence and bounds of solutions},
url = {http://eudml.org/doc/299921},
year = {2025},
}

TY - JOUR
AU - Marraffa, Valeria
AU - Satco, Bianca
TI - Stieltjes differential problems with general boundary value conditions. Existence and bounds of solutions
JO - Czechoslovak Mathematical Journal
PY - 2025
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 235
EP - 255
AB - We are concerned with first order set-valued problems with very general boundary value conditions \[ {\left\lbrace \begin{array}{ll} u^{\prime }_g(t)\in F(t,u(t)),\quad \mu _g \text{-a.e.} \in [0,T] , \\ L(u(0),T))=0 \end{array}\right.} \] involving the Stieltjes derivative with respect to a left-continuous nondecreasing function $g\colon [0,T]\rightarrow \mathbb {R}$, a Carathéodory multifunction $F\colon [0,T]\times \mathbb {R}\rightarrow \mathcal {P}(\mathbb {R})$ and a continuous $L\colon \mathbb {R}^2\rightarrow \mathbb {R}$. 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.
LA - eng
KW - boundary value differential inclusion; Stieltjes derivative; Kurzweil-Stieltjes integral; periodic problem
UR - http://eudml.org/doc/299921
ER -

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