Continuous dependence on parameters of second order dierential inclusion and self-adjoint operator

By Vo Viet Tri
DOI: 10.37550/tdmu.EJS/2022.04.356

In this paper, we establish compactness and continuous dependence on parameters for solution-set of the second order differential inclusion including self-adjoint operator in the form
\begin{align*}
\left\{ \begin{gathered}
\frac{\partial^2}{\partial t^2} u(t,x) +2\mathcal{A} \frac{\partial}{\partial t}u(t,x)+\mathcal{A}^{2} u(t,x) \in F(t,u(t),\mu),\,\, \hfill (t,x)\in [0,T)\times\Omega \\
u(0,x)=\frac{\partial }{\partial t}u(0,x)=0, \,\, \hfill x \in \Omega, \\
%u(T,x) = h(x), \,\, \hfill x\in\Omega,
\end{gathered}\right. %\label{MainProblem}
\end{align*}
where $\mathcal A$ is a self-adjoint operator.
We use the spectral theory on Hilbert spaces to obtain formulation for mild solutions. Using the mild solution formula together with a measure of non-compactness with values in an ordered space, we construct useful bounds for solution operators. Then, we establish necessarily upper semi-continuous and condensing settings, which mainly help to obtain the global existence of mild solutions and the compactness of the mild solution set. Finally, we provide a brief discussion on the continuous dependence of the solution-set on parameter $\mu$.