Continuous dependence on parameters of differential inclusion using new techniques of fixed point theory

Abstract

In this paper, we establish the global existence and the continuous dependence on parameters for a set solutions to a class of time-fractional partial differential equation in the form
\begin{align*}
\left\{ \begin{gathered}
\frac{\partial}{\partial t} u(t) +\mathcal{K}\mathcal{A}^{\sigma_1} \frac{\partial}{\partial t}u(t)+\mathcal{A}^{\sigma_2} u(t) \in F(t,u(t), \mu),\,\, \hfill t\in \mathcal I, \\
%u(t,x)=0, \,\, \hfill (t,x)\in (0,T)\times \partial \Omega, \\
u(T) = h,\,\, (\text{ resp. } u(0)=h) \,\, \hfill \text{ on } \Omega,
\end{gathered}\right. %\label{MainProblem0}
\end{align*}
where $\sigma_1, \sigma_2>0$ and $\mathcal I=[0,T)$ (resp. $\mathcal I=(0,T]$). Precisely, first results are about the global existence of mild solutions and the compactness of the mild solutions set. These result are mainly based on some necessary estimates derived by considering the solution representation in Hilbert spaces. The remaining result is the continuous dependence of the solutions set on some special parameters. The main technique used in this study include the fixed point theory and some certain conditions of multivalued operators.