A Variational Approach For Almost Periodic Solutions in Retarded Differential Equations

Moez Ayachi (1) , Joël Blot (1)

1 Marin Mersenne - Pour les Mathématiques, l'Informatique et l'Interdisciplinarité

Abstract. To study the a.p. (almost periodic) solutions of retarded functional differential equations in the form $u''(t)=\int_{-r}^{0}D_{1}f(u(t),u(t+\theta))d\theta+\int_{-r}^{0}D_{2}f(u(t-\theta),u(t))d\theta $, we introduce variational formalisms to characterize the a.p. solutions as a critical points of functionals defined on Banach spaces of a.p. functions. We obtain an existence result of weak a.p. solutions and a result of density of the a.p. forcing termes e(.) for which the equation possesses usual a.p. solutions.

Mots clés: Retarded functional differential equations; Almost periodic functions; Variational method

Domaines: Analyse fonctionnelle [math.FA]; Optimisation et contrôle [math.OC]

Moez Ayachi, Joël Blot. A Variational Approach For Almost Periodic Solutions in Retarded Differential Equations. Differential Equations and Applications, 2009, Volume 1 (2009) (Number 1), pp.69-84. ⟨hal-00379284⟩