<p style='text-indent:20px;'>In this paper, we investigate a source identification problem for a class of abstract nonlocal differential equations in separable Hilbert spaces. The existence of mild solutions and strong solutions for the problem of identifying parameter are obtained. Furthermore, we study the continuous dependence on the data and the regularity of the mild solutions and strong solutions of nonlocal differential equations. Examples given in anomalous diffusion equations illustrate the existence and regularity results.</p>

In this article, we study the existence of the integral solution to the neutral functional differential inclusion $${\frac{d}{dt}\mathcal{D}y_t-A\mathcal{D}y_t-Ly_t \in F(t,y_t), \quad\text{for a.e. }t \in J:=[0,\infty),\\  y_0=\phi \in C_E=C([-r,0];E),\quad r>0,}$$ and the controllability of the corresponding neutral inclusion $${\frac{d}{dt}\mathcal{D}y_t-A\mathcal{D}y_t-Ly_t \in F(t,y_t)+Bu(t),\quad  \text{for a.e. } t \in J,\\ y_0=\phi \in C_E,}$$ on a half-line via the nonlinear alternative of Leray-Schauder type for contractive multivalued mappings given by Frigon. We illustrate our results with  applications to a neutral partial differential inclusion with diffusion, and to a  neutral functional partial differential equation with obstacle constrains.

<p style='text-indent:20px;'>In this paper, we are interested in the existence of solutions to the anomalous diffusion equations with delay subjected to nonlocal initial condition:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \label{01} \begin{cases} \partial _t(k*(u-u_0)) +(- \Delta)^\sigma u = f(t,u,u_\rho) \; {\rm {in }}\ \mathbb R^+\times \Omega,\\ u\bigr |_{\partial \Omega} = 0\; {\rm {in }}\ \mathbb R^+\times \partial \Omega,\\ u(s)+g(u)(s) = \phi(s) \;{\rm {in }}\ \Omega, s\in [-h,0]. \end{cases} \notag \tag{1} \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded domain of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, the constant <inline-formula><tex-math id="M3">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula> is in <inline-formula><tex-math id="M4">\begin{document}$ (0,1] $\end{document}</tex-math></inline-formula>. Under appropriate assumptions on <inline-formula><tex-math id="M5">\begin{document}$ k $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ f,g $\end{document}</tex-math></inline-formula>, we obtain the existence of global solutions and decay mild solutions for (1). The tools used include theory of completely positive functions, resolvent operators, the technique of measures of noncompactness and some fixed point arguments in suitable function spaces. Two application examples with respect to the specific cases of the term <inline-formula><tex-math id="M7">\begin{document}$ k $\end{document}</tex-math></inline-formula> in (1) are presented.</p>