We present some results of Eberlein-weakly almost periodic functions with values in a Banach space. Then, we apply these results to investigate the Eberlein-weakly almost periodic solutions of some differential equations in a Banach space.

We state sufficient conditions for the existence of positive pseudo almost automorphic solutions of the following nonlinear infinite delay integral equation:

$$ x(t)= \int^t_{-\infty}a(t,t-s)f(s,x(s))\: ds . $$

We deduce some corollaries on a finite delay integral equation and on a delay differential equation.

In this work, we study the local existence of mild solutions for some partial functional integrodifferential equations. We suppose that the linear part has a resolvent operator in the sense of Grimmer [13]. The nonlinear part is just assumed to be continuous. We have also that the solution may blow up in finite time. An application is provided to illustration.

In this paper we prove the interior controllability of the Thermoelastic Plate Equation

$$ \left\{ \begin{array}{ll} w_{tt}+\Delta^2w+\alpha\Delta w=1_{\omega}u_{1}(t,x),& \mbox{in} \quad (0, \tau) \times \Omega,\\ \theta_t-\beta\Delta\theta-\alpha\Delta w_t=1_{\omega}u_{2}(t,x), & \mbox{in} \quad (0, \tau) \times \Omega,\\ \theta=w=\Delta w=0, & \mbox{on} \quad (0, \tau) \times \partial \Omega, \end{array} \right.$$

where $\alpha\neq 0$, $\beta>0$, $\Omega$ is a sufficiently regular bounded domain in $\R^{N}$ $(N\geq 1)$, $\omega$ is an open nonempty subset of $\Omega$, $1_{\omega}$ denotes the characteristic function of the set $\omega$ and the distributed control $u_{i}\in L^{2}([0,\tau]; L^{2}(\Omega)), i=1,2.$ Specifically, we prove the following statement: For all $\tau >0$ the system is approximately controllable on $[0, \tau]$. Moreover, we exhibit a sequence of controls steering the system from an initial state to a final state in a prefixed time $\tau >0$ .

In this paper we first introduce and study the concepts of $p$-th mean pseudo almost automorphy and that of $p$-th mean pseudo almost periodicity for $p \geq 2$. Next, we make extensive use of the well-known Schauder fixed point principle to obtain the existence of $p$-th mean pseudo almost automorphic mild solutions to some nonautonomous stochastic differential equations.

An operator version of the Young's result [Proc. Amer. Math. Soc., 94 (1985), 636-640] is obtained, via non-differential techniques.

In this paper, we study Ostrowski--Grüss and Ostrowski-like inequalities on time scales and thus unify and extend corresponding continuous and discrete versions from the literature. We present corresponding inequalities by using the time scales $L^\infty$-norm and also by using the time scales $L^p$-norm. Several interesting inequalities representing special cases of our general results are supplied.

In this paper, we study the Gevrey smoothing property for the non-negative solution of the linearized spatially homogeneous Boltzmann equation. Using pseudo-differential calculus and some techniques of mathematical analysis, we show that in the non-cutoff and non-Maxwellian case with the inverse power law potential, if the solution is Lipschitz continuous on the velocity variable, then it has the local Gevrey regularity.

In this paper, consensus problem of the leader-following multi-agent system with a varying-velocity leader is analyzed. The system is considered with both time-varying input-delay and directed dynamic topologies, where the system delay is unknown and time-varying with a pre-specified upper bounded derivative. The stability analysis is performed with a proposed Lyapunov-Krosovskii functional. Sufficient delay-dependent condition in the form of Linear Matrix Inequalities (LMIs) is given to guarantee system consensus. Finally, numerical simulation verifies the theoretical results.

We introduce and study a new concept called doubly-weighted pseudo-almost periodicity, which generalizes the notion of weighted pseudo-almost periodicity due to Diagana. Properties of such a new concept such as the stability of the convolution, translation-invariance, existence of a doubly-weighted mean for almost periodic functions, and a composition result will be studied.