| Abstract: |
| Let $g:[0,T]\to \mathbb{R}$ be a left-continuous nondecreasing function and $\;\mu_g\;$ the Lebesgue-Stieltjes measure generated by $\;g$ on $[0, T]$.
The aim of the talk is to present an existence result (\cite{ms cmj}) for first order set-valued problems with a very general boundary condition
\begin{equation*}
\left\{
\begin{array}{l}
u'_g(t) \in F(t,u(t)),\; \mu_g{\rm -a.e.\;} t\in [0,T]\
L(u(0),u(T))=0
\end{array}
\right.
\end{equation*}
involving the Stieltjes derivative with respect to $g$ (see \cite{pouso}). The multifunction $F : [0,T] \times \mathbb{R} \to \mathcal{P}(\mathbb{R}) $ is of Carath\'{e}odory type with
convex compact values and $L:\mathbb{R}^2\to \mathbb{R}$ is continuous and nonincreasing with respect to its second argument.
The applied method is that of lower and upper solutions inspired from \cite{mf}.
The announced result generalizes several already known outcomes since the theory of Stieltjes differential equations encompasses various classical problems: ordinary differential equations (when $g$ is the identity map), impulsive differential problems (if $g$ can be written as a sum of the identical function with a sum of Heaviside functions) and dynamic equations on time scales.
Moreover, for particular maps $L$ some existence results for Stieltjes differential inclusions with initial value conditions ($L(x,y)=x-u_0$) or with periodic conditions ($L(x,y)=x-y$) are covered. |
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