Almost periodic linear differential equations with non-separated solutions

Abstract

A celebrated result by Favard states that, for certain almost periodic linear differential systems, the existence of a bounded solution implies the existence of an almost periodic solution. A key assumption in this result is the separation among bounded solutions. Here we prove a theorem of anti-Favard type: if there are bounded solutions which are non-separated (in a strong sense) sometimes almost periodic solutions do not exist. Strongly non-separated solutions appear when the associated homogeneous system has homoclinic solutions. This point of view unifies two fascinating examples by Zhikov-Levitan and Johnson for the scalar case. Our construction uses the ideas of Zhikov-Levitan together with the theory of characters in topological groups. RésuméUn résultat célébré de Favard affirme que, pour certains systèmes différentiels linéaires presque pé-riodiques, l'existence d'une solution bornée implique l'existence d'une solution presque périodique. Une supposition essentielle dans ce résultat est la séparation des solutions bornées. Ici nous prouvons un théo-rème de type anti-Favard : s'il y a des solutions bornées qui sont non-séparées (dans un sens fort), parfois il n'existent pas des solutions presque périodiques. Des solutions fortement non-séparées apparaissent quand le système homogène associé a des solutions homoclines. Ce point de vue unifie deux exemples fascinants de Zhikov-Levitan et Johnson pour le cas scalaire. Notre construction utilise des idées de Zhikov-Levitan ainsi que la théorie de caractères dans groupes topologiques.

Favard theory for the adjoint equation and Fredholm Alternative

“…Many concrete examples of such strong failure of the Favard separation condition can be found in the literature, typically aimed to show that the almost periodic world behaves very differently from the periodic one: see for instance [3], [34], [10] and [18]. There is another way for (F a ) to fail that is relevant to the present paper and, in some sense, is transversal to condition (5.7).…”

“…Although necessity is an open problem, it is well known since longtime that (F A ) is optimal for the validity of Theorem 3.1, even if we restrict ourselves to the Θ's which are almost periodic: see [34], [10] and [18]. However, testing (F A ) in concrete situations is not always an easy task.…”

“…The actual need for (F ) is an open question but optimality is well known. That is, if we omit (F ), then the conclusion of the Favard result may be false: see [34], [10] and [18] for almost periodic examples where Favard condition fails, which admit bounded solutions but no almost periodic solutions. In Section 2 we show that the minimality of H(A, f) is also optimal for the Favard result.…”

Almost periodicity and almost automorphy for some evolution equations using Favard’s theory in uniformly convex Banach spaces

“…Recently, Almost periodic and almost automorphic solutions for Eq. (2) via Favard's approach has been largely investigated in the literature [9,10,20,26,29,32]. More results about almost periodic differential equations in finite dimensional spaces can be found in [17].…”

A Stepanov version for Favard theory

“…In [10] these counterexamples are merged into a general principle: when, for some B(t), the bounded solutions to (1.2) are all homoclinic to zero, and at least one of them in nontrivial, then there exist an almost periodic function f (t) for which the conclusions of Favard's theorem do not hold.…”