This paper is a continuation of the investigations done in the literature regarding the so called Bohr-Neugebauer property for almost periodic differential equations in Hilbert spaces. The aim of this work is to extend the investigation of this property to almost automorphic functional partial differential equations in Banach spaces. We use a compactness assumption which turns out to relax assumptions made in some earlier works for differential equations in Hilbert spaces. Two new integration theorems for almost automorphic functions are proven in the process. To illustrate our main results, we propose an application to a reaction-diffusion equation with continuous delay.

Let $\mathbb {X}$ be a Banach space. Let $A$ be the generator of an immediately norm continuous $C_0$-semigroup defined on $\mathbb {X}$. We study the uniform exponential stability of solutions of the Volterra equation

$u'(t) = Au(t)+\displaystyle \int _0^t a(t-s)Au(s)\,ds,\quad t\geq 0,\ u(0)=x$,

\noindent where $a$ is a suitable kernel and $x\in \mathbb {X}$. Using a matrix operator, we obtain some spectral conditions on $A$ that ensure the existence of constants $C,\omega >0$ such that $\|u(t)\|\leq Ce^{-\omega t}\|x\|$, for each $x\in D(A)$ and all $t\geq 0$. With these results, we prove the existence of a uniformly exponential stable resolvent family to an integro-differential equation with infinite delay. Finally, sufficient conditions are established for the existence and uniqueness of bounded mild solutions to this equation.

This note is intended to clarify some important points about the well-posedness of terminal value problems for fractional differential equations. It follows the recent publication of a paper by Cong and Tuan in this journal, in which a counter-example calls into question the earlier results in a paper by this note's authors. Here, we show in the light of these new insights, that a wide class of terminal value problems of fractional differential equations is well posed, and we identify those cases where the well-posedness question must be regarded as open.

In this work, we are concerned with uniform stabilization for an initial-boundary value problem associated with the Kirchhoff type wave equation with feedback terms and memory condition at the boundary. We prove that the energy decays exponentially when the boundary damping term has a linear growth near zero and polynomially when the boundary damping term has a polynomial growth near zero. Furthermore, we study the decay rate of the energy without imposing any restrictive growth assumption on the damping term near zero.

For $\nu \in (0,1)$, the nonautonomous integro-differential equation \[ \mathbf {D}_{t}^{\nu }u-\mathcal {L}_{1}u-\int _{0}^{t}\mathcal {K}_{1}(t-s)\mathcal {L}_{2}u(\cdot ,s)\,ds =f(x,t) \] is considered here, where $\mathbf {D}_{t}^{\nu }$ is the Caputo fractional derivative and $\mathcal {L}_{1}$ and $\mathcal {L}_{2}$ are uniformly elliptic operators with smooth coefficients dependent on time. The global classical solvability of the associated initial-boundary value problems is addressed.

In this paper, we present case studies to illustrate the dependence of the rate of convergence of numerical schemes for singular neutral equations (SNFDEs) on the particular mesh employed in the computation. In Ito and Turi, a semigroup theoretical framework was used to show convergence of semi- and fully- discrete methods for a class of SNFDEs with weakly singular kernels. On the other hand, numerical experiments in Ito and Turi demonstrated a ``degradation" of the expected rate of convergence when uniform meshes were considered. In particular, it was numerically observed that the degradation of the rate of convergence was related to the strength of the singularity in the kernel of the SNFDE. Following the idea used for Volterra equations with weakly singular kernels, see, e.g., Brunner, we investigate graded meshes associated with the kernel of the SNFDE in attempting to restore convergence rates.