On a class of bounded trajectories for some non‐autonomous systems

Abstract

We prove, by variational arguments, the existence of a solution to the boundary value problem in the half linewhere c ≥ 0 and a belongs to a certain class of positive functions. The existence of such a solution in the case c = 0 means that the system (0.1) behaves in significantly different way from its autonomous counterpart.

A non-autonomous second order boundary value problem on the half-line

“…The theorem is stronger than that in a recent paper by Gavioli and Sanchez [5]. In their result, V was assumed to be C 2 , inf a must be positive, a must be of bounded variation, a must be nondecreasing on [t 0 , ∞) for some t 0 > 0, and the coefficient of t in condition (A 2 ) is smaller.…”

“…In their result, V was assumed to be C 2 , inf a must be positive, a must be of bounded variation, a must be nondecreasing on [t 0 , ∞) for some t 0 > 0, and the coefficient of t in condition (A 2 ) is smaller. The arguments here are also more elementary than those of [5].…”

On the existence of heteroclinic trajectories for asymptotically autonomous equations

“…Then the equation to be studied takes the following form:ẍ = a(t)V (x), x ∈ R, where V is a double-well potential, whose behaviour outside the interval of the two equilibria is actually not involved. Situations of this kind do not appear very often in the literature: for instance, we refer to [3] ( §5, Example 1), where a result is given which includes the case in which the coefficient a(t) is definitively increasing with respect to |t|, and also to [2, Chapter 2, Theorem 2.2]), where the equality a(t) ≤ l is supposed to hold everywhere, and the required heteroclinic connection is found as a minimizer of the associated functional (2.3) (see also [5], [6,Corollary 1.8]) for the case in which the two equilibria are not consecutive, but a and V are both even). In this paper we begin to study the case a(t) ≥ l, again through a variational approach, even if our assumptions do not allow to find the required solution as an absolute minimum of the functional F : indeed, this minimum is attained only in the trivial case a(t) ≡ l almost everywhere.…”