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International audience; The aim of this paper is to introduce of notion of weak variational solution in an abstract setting, although we are mainly interested in almost periodic type solutions. We give two existence and uniqueness theorems. Even if assumptions are strong, we obtain two theorems with explict bounds.

International audience

Travaux dirigés d'algèbre linéaire, fonctions de plusieurs variables et intégrales multiples. Reprises et complétées à partir de celles fournies par mon collègue P. Gourdel.

International audience; This paper is concerned with weighted pseudo almost automorphic functions, which are more general and complicated than weighted pseudo almost periodic functions. Using these properties, we establish an existence and uniqueness theorem for pseudo almost automorphic mild solutions to semilinear differential equations in a Banach spaces which extends known results.

International audience; We are concerned in this paper with the pseudo almost automorphy of mild solutions for the semilinear evolution equation $x'(t)=Ax(t)+f(t,x)$ where $A$ is a sectorial operator not necessarily densely defined in $X$ generating an hyperbolic semigroup $(T(t))_{t\geq 0}$ in a Banach space $X$ and $f:\R\times X_\alpha \to X$, where $X_{\alpha}$ is an intermediate space. We prove the existence and uniqueness of a pseudo almost automorphic solution in $X_\alpha$, when the function $f:\R\times X_\alpha\longrightarrow X$ is pseudo almost automorphic.

International audience; In this paper, we study pseudo almost automorphic solutions to perturbations to Eq.\;(\ref{1}) consisting of the class of abstract partial evolution equations of the form \begin{eqnarray}\label{2} \displaystyle{\frac{d}{dt}} \left[u(t) + f(t, u(t))\right] = A u(t) \; \; t \in \R, \end{eqnarray} where $A$ is the infinitesimal generator of an exponentially stable $c_0$-semigroup acting on $\X$, $B, C$ are two densely defined closed linear operators on $\X$, and $f$ is continuous functions. Under some appropriate assumptions, we establish the existence and uniqueness of an almost automorphic (mild) solution to Eq.\,(\ref{2}) using the Banach fixed-point principle.

International audience; In this paper we consider an equation of type $q''(t)=\partial_2 V(t,q(t))$, where $V$ is quasi-periodic (q.p.) in $t$, uniformly w.r.t. to $q$, is convex w.r.t. $q$ for each $t$. We look for q.p. solutions $q:\R \to \IH$ (where $\IH$ is an Hilbert space) to this equation. By using the formalism introduced by the Physician Percival, it is possible to transform this problem in a elliptic degenerate Partial Differential Equation on the torus. By using a singular perturbation method, we obtain existence results under technical assumptions on $V$.

International audience; In this paper we prove the existence and uniqueness of $C^{(n)}$-almost periodic solutions to the nonautonomous ordinary differential equation $x'(t)=A(t)x(t)+f(t),\;\;t\in {\mathbb R}$, where $A(t)$ generates an exponentially stable family of operators $(U(t,s))_{t\geq s}$ and $f$ is a $C^{(n)}$-almost periodic function with values in a Banach space $\X$. We also study a Volterra-like equation with a $C^{(n)}$-almost periodic solution.

International audience; We build spaces of q.p. (quasi-periodic) functions and we establish some of their properties. They are motivated by the Percival approach to q.p. solutions of Hamiltonian systems. The periodic solutions of an adequat Partial Differential Equation are related to the q.p. solutions of an Ordinary Differential Equation. We use this approach to obtain some regularization theorems of weak q.p. solutions of differential equations. For a large class of differential equations, the first theorem gives a result of density: a particular form of perturbated equations have strong solutions. The second theorem gives a condition which insures that any essentially bounded weak solution is a strong one.

We will prove, besides other things like localization and (in)stability, that the differential equations $x'+x^3-\lambda x=\varepsilon r(t)$, $\lambda> 0$, and $x''+x^3-x=\varepsilon r(t)$, where $r\colon\mathbb{R}\to\mathbb{R}$ are uniformly limit-periodic functions, possess for sufficiently small values of $\varepsilon > 0$ uniformly limit-periodic solutions, provided $r$ in the first-order equation is strictly positive. As far as we know, these are the first nontrivial effective criteria, obtained for limit-periodic solutions of nonlinear differential equations, in the lack of global lipschitzianity restrictions. A simple illustrative example is also indicated for difference equations.