We investigate the problem of the existence and continuous dependence of solution
to the following Dirichlet problem
$$
\left\{
\begin{array}
[c]{l}
f_{x_{1}}(t,x,D_{a+}^{\alpha}x,y,D_{a+}^{\beta}y,u)=D_{b-}^{\alpha}f_{x_{2}
}(t,x,D_{a+}^{\alpha}x,y,D_{a+}^{\beta}y,u)\\
f_{y_{1}}(t,x,D_{a+}^{\alpha}x,y,D_{a+}^{\beta}y,u)=D_{b-}^{\beta}f_{y_{2}
}(t,x,D_{a+}^{\alpha}x,y,D_{a+}^{\beta}y,u)
\end{array}
\right. \label{p1}
$$
$$\left\{
\begin{array}
[c]{l}
I_{a+}^{1-\alpha}x\left( a\right) =x\left( a\right) =x\left( b\right)
=0,\\
I_{a+}^{1-\beta}y\left( a\right) =y\left( a\right) =y\left( b\right) =0,
\end{array}
\right. \label{b1}
$$
Since the assumptions we made does not guarantee the uniqueness of solution to
(\ref{p1})-(\ref{b1}) we use the notion of Kuratowski--Painlev\'{e} limit to
describe the mentioned continuous dependence.
Applying continuous dependence we also prove theorem on
existence of optimal solution to the following Bolza problem:
\begin{enumerate}
[(B)] minimize
\begin{multline*}
\mathcal{B}\left( u,x_{u},y_{u}\right) :=
\int_{a}^{b}B_{1}( t,x_{u}\left( t\right) ,D_{a+}^{\alpha}x_{u}\left(
t\right) ,y_{u}\left( t\right) ,D_{a+}^{\beta}y_{u}\left( t\right)
,u\left( t\right) ) dt
+B_{2}\left( x_{u}\left( T\right) ,y_{u}\left( T\right) \right)
\end{multline*}
where \(\left( x_{u}\left( .\right) ,y_{u}\left( .\right) \right) \) is
any solution to (\ref{p1})-(\ref{b1}) corresponding to \(u\left( .\right)
\in\mathcal{U}_{L}\), \(T:=\frac{b-a}{2}\), \begin{multline*}
\mathcal{U}_{L}:=\{ u\left( .\right) \in L^\infty\left([a,b],M\right):\left\vert u\left(
t_{1}\right) -u\left( t_{2}\right) \right\vert \leq L\left\vert t_{1}
-t_{2}\right\vert
\text{ for a.e. }t_{1},t_{2}\in\left[ a,b\right] \}
\end{multline*}and
\(M\subset \mathbb R^m\) is a given convex and compact set.
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