In this paper we prove the interior approximate controllability of the following Semilinear Heat Equation $$ \left\{ \begin{array}{lr} z_{t}(t,x) = \Delta z(t,x) + 1_{\omega}u(t,x)+f(t,z,u(t,x)) & \mbox{in} \quad (0, \tau] \times \Omega,\\ z = 0, & \quad \mbox{on} \quad (0, \tau) \times \partial \Omega, \\ z(0,x) = z_{0}(x), & x \in\Omega, \end{array} \right. $$ where $\Omega$ is a bounded domain in $\mathbb{R}^{N}(N\geq1)$, $z_0 \in L^{2}(\Omega)$, $\omega$ is an open nonempty subset of $\Omega$, $1_{\omega}$ denotes the characteristic function of the set $\omega$,the distributed control $u$ belong to $\in L^{2}([0,\tau]; L^{2}(\Omega;))$ and the nonlinear function $f:[0, \tau] \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ is smooth enough and there are $a,b, c \in \mathbb{R}$, with $c \neq -1$, such that $$ \sup_{(t,z,u) \in Q_{\tau}} |f(t,z,u) -az-cu-b | < \infty, $$ where $Q_{\tau}= [0, \tau] \times \mathbb{R} \times \mathbb{R}$. Under this condition we prove the following statement: For all open nonempty subset $\omega$ of $\Omega$ the system is approximately controllable on $[0, \tau]$. Moreover, we could exhibit a sequence of controls steering the nonlinear system (1.1) from an initial state $z_0$ to an $\epsilon$ neighborhood of the final state $z_1$ at time $\tau \gt 0$, which is very important from a practical and numerical point of view.

We are concerned with the existence of solutions for fractional integro-differential equations with state-dependent delay on an infinite interval. Our results are based on Schauder's fixed point theorem combined with the diagonalization process.

In this paper some random fixed point theorems with PPF dependence are proved for random operators in separable Banach spaces with different domain and range spaces. The obtained abstract results are applied to prove PPF dependence existence results for first order periodic boundary value problems for functional random differential equations.

This note is dedicated to the existence of almost periodic solutions of a certain class of functional equations, of the form (1) in the text, in spaces like $AP_r(R, {\cal C}^n)$, $1\leq r\leq 2$. Frequency domain conditions are involved in this study.

This paper is concerned with positive almost automorphic solutions to a class of nonlinear infinite delay integral equation. By using a fixed point theorem in partially ordered Banach spaces, we establish an existence theorem about positive almost automorphic solutions to the addressed integral equation. Our theorem extend some earlier results to a more general class of integral equations.

This paper is concerned with the boundary-value problem for the steady incompressible Navier-Stokes equations with damping. Two cases are considered here: 1) the Dirichlet's boundary condition; 2) the nonhomogeneous boundary condition. we obtain the existence and uniqueness of the weak solutions for the steady incompressible Navier-Stokes equations with damping using different methods for the above cases.

In this paper, the Adomian decomposition method for solving nonlinear partial differential equations (NPDEs) is revisited. Then we show how this method can be extended and used to solve under-determined systems of NPDEs. The examples of Kompaneets, Novikov and Ginzburg-Landau equations are considered as illustration.

This paper is concerned with a class of nonautonomous semilinear evolution equation in a Banach space. We establish an existence theorem about asymptotically almost automorphic mild solution to the addressed evolution equation. An example is given to illustrate our abstract result.

In this paper, we first study further properties of S-asymptotically $\omega$-periodic functions taking values in Banach spaces including a theorem of composition. Then we apply the results obtained to study the existence and uniqueness of S-asymptotically $\omega$-periodic mild solutions to a nonautonomous semilinear differential equation.

We establish the existence of a Besicovitch almost periodic solution of a second-order differential equation, $u''(t)+ D_1V(u(t),t) = 0$, in a Hilbert space, when the potential $V(.,t)$ possesses a bump surrounded with a hollow. We use a variational method on a Hilbert space of Besicovitch almost periodic functions.

This is a survey paper on optimization problems via the technique of first and second order tangent cones to a nonempty subset of a Banach space X. Such a technique is also used in the study of the flow invariance of a closed set with respect to a second order differential equation (motion on a given orbit in a force field). Many of the known results in these areas are included here.