Philippe Cieutat

New approach for weighted pseudo-almost periodic functions under the light of measure theory, basic results and applications

The aim of this work is to present new approach to study weighted pseudo almost periodic functions using the measure theory. We present a new concept of weighted ergodic functions which is more general than the classical one. Then we establish many interesting results on the functional space of such functions like completeness and composition theorems. The theory of this work generalizes the classical results on weighted pseudo almost periodic functions. For illustration, we provide some applications for evolution equations which include reaction diffusion systems and partial functional differential equations.

Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations

We establish new theorems for the composition of pseudo almost periodic and pseudo almost automorphic functions in Banach spaces. Our results extend the recent ones [H. Li, F. Huang and J. Li, Composition of pseudo almost-periodic functions and semilinear differential equations, J. Math. Anal. Appl. 255 (2001), pp. 436–446; J. Liang, J. Zhang, T.J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. 340 (2001), pp. 1493–1499]. We also study some sufficient conditions for the continuity of the superposition operator. As an application to the abstract results, we give some existence theorems of pseudo almost periodic/automorphic solutions for some semilinear evolution equations and examples with the heat equation.

Bounded and almost periodic solutions of convex Lagrangian systems

Abstract We study some properties of bounded and almost periodic solutions of convex Lagrangian systems in the presence of almost periodic forcing d dt ∇ x′ L(t,x(t),x′(t))= ∇ x L(t,x(t),x′(t)), then we establish a result of existence and uniqueness of the almost periodic solution.

Local attractivity in nonautonomous semilinear evolution equations

Abstract We study the local attractivity of mild solutions of equations in the form u’(t) = A(t)u(t) + f (t, u(t)), where A(t) are (possible) unbounded linear operators in a Banach space and where f is a (possible) nonlinear mapping. Under conditions of exponential stability of the linear part, we establish the local attractivity of various kinds of mild solutions. To obtain these results we provide several results on the Nemytskii operators on the space of the functions which converge to zero at infinity

Gradient Flows, Second-Order Gradient Systems and Convexity

We disclose an interesting connection between the gradient flow of a

Several kinds of oscillations in forced Liénard equations

\mathcal{C}^2

Almost Periodic Solutions of Monotone Second-Order Differential Equations

-smooth function

Measure theory and pseudo almost automorphic functions: New developments and applications

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Superposition operators between various almost periodic function spaces and applications.

and evanescent orbits of the second order gradient system defined by the square-norm of

Almost-periodic oscillations of monotone second-order systems

\nabla\psi