Here we present a necessary and sufficient conditions for the exact and approximate observability of the following linear difference equation \[\left\{\begin{array}{ll} z(n+1)=A(n)z(n), \hspace{0.2cm} n\in \mathbb{N}^*, \hspace{0.2cm} z(0)=z_0 \in Z,\\ y(n)=Cz(n), \end{array}\right. \] where $A\in l^{\infty}(\mathbb{N},L(Z))$, $C\in L(Z,U)$, $Z$, $U$ are Hilbert spaces and $\mathbb{N}^*=\mathbb{N}\cup\{0\}$. We apply these results to a flowdiscretization of the wave equation and the heat equation.

In this paper, we introduce a concept of almost automorphy for random sequences. Using the Banach contraction principle, we establish the existence and uniqueness of an almost automorphic solution to some Volterra stochastic difference equation in a Banach space. Our main results extend some known ones in the sense of mean almost automorphy. As an application, almost automorphic solution to a concrete stochastic difference equation is analyzed to illustrate our abstract results.

In this paper we study the Massera problem for the existence of a periodic mild solution of a class of nondensely nonautonomous semilinear differential equations with delay. We assume that the linear part satisfies the conditions introduced by Tanaka. First, we prove that the existence of a periodic solution for nonautonomous inhomogeneous linear differential equations with delay is equivalent to the existence of a bounded solution on the right half real line. Next, we undertake the analysis of the existence of periodic solutions in the semilinear case. To this end, we use a fixed point Theorem concerning setvalued maps. To illustrate the obtained results, we consider a periodic reaction diffusion equation with delay.

The aim of this work is to study the existence of periodic solutions in the $\alpha$-norm for some partial differential equations with infinite delay. A linear part of equations is assumed to generate an analytic semigroup. The delayed part is assumed to be continuous with respect to the fractional norm of the linear part and $\sigma$-periodic with respect to the first argument. Using Massera's approach we prove the existence of periodic solutions in the linear case. In the nonlinear case, a fixed point theorem for multivalued mapping is used to prove the existence of periodic solutions. We use also Horn's fixed point theorem to get the existence of periodic solutions when solutions are ultimate bounded. For illustration an example is provided for some reactiondiffusion equation involving infinite delay.

In this paper we investigate the existence and uniqueness of solutions for the initial value problems (IVP for short), for some classes of functional hyperbolic differential equations with finite and infinite delay by using some fixed point theorems.