| Contents |
| In this talk, we consider the following equation with infinite delay
\begin{equation}\label{eq1.1}
\displaystyle D^\alpha u(t) = Au(t)+\int_{-\infty}^t a(t-s)Au(s)ds +f(t,u(t)), \qquad t\in\mathbb{R},
\end{equation}
where $A$ is a closed linear
operator defined on a Banach space $X,$ $a\in L^1(\mathbb{R}_+)$ is a scalar-valued kernel, $f$ belongs to a closed
subspace of the space of continuous and bounded functions, and for $\alpha>0,$ the fractional derivative is understood in the Weyl's sense.
We study the existence and uniqueness of mild solutions for (\ref{eq1.1}) where the input data $f,$ is for example, almost periodic (resp. almost automorphic) and satisfies some Lipschitz type conditions. The unique mild solution $u$ of (\ref{eq1.1}) which is almost periodic (resp. almost automorphic) and is given by
\begin{equation}\label{eq1.2}
u(t)=\int_{-\infty}^t S_\alpha(t-s)f(s,u(s))ds, \quad t\in\mathbb{R},
\end{equation}
where $\{S_\alpha(t)\}_{t\geq 0}$ is the $\alpha$-{\it resolvent family} generated by $A$. It is remarkable that, in the scalar case, that is $A=-\rho I,$ with $\rho>0,$ some concrete examples of integrable $\alpha$-resolvent families are showed. |
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