Distributional, differential and integral problems: equivalence and existence results

• Main Authors: Giselle Monteiro, Bianca Satco
• Format: Article
• Language: English
• Published: University of Szeged 2017-01-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: distribution; derivative with respect to functions; regulated primitive integral; kurzweil-stieltjes integral; regulated function; variational measure; solution
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=4753
• ISSN: 1417-3875; 1417-3875

We are interested in studying the matter of equivalence of the following problems: \[ \begin{aligned} Dx&=f(t,x)Dg\nonumber \\ x(0)&=x_0 \end{aligned}\tag{1} \label{1a} \] where $Dx$ and $Dg$ stand for the distributional derivatives of $x$ and $g$, respectively; \[ \begin{aligned} x'_g(t)&=f(t,x(t)), \quad m_g\textrm{-a.e.}\\ x(0)&=x_0 \end{aligned}\tag{2}\label{2} \] where $x'_g$ denotes the $g$-derivative of $x$ (in a sense to be specified in Section 2) and $m_g$ is the variational measure induced by $g$; and \begin{equation}\tag{3}\label{3} x(t)=x_0+\int_0^t f(s,x(s))dg(s), \end{equation} where the integral is understood in the Kurzweil--Stieltjes sense. We prove that, for regulated functions $g$, \eqref{1a} and \eqref{3} are equivalent if $f$ satisfies a bounded variation assumption. The relation between problems \eqref{2} and \eqref{3} is described for very general $f$, though, more restrictive assumptions over the function $g$ are required. We provide then two existence results for the integral problem \eqref{3} and, using the correspondences established with the other problems, we deduce the existence of solutions for \eqref{1a} and \eqref{2}.