Second Order Tangency Conditions and Differential Inclusions: a Counterexample and a Remedy

In this paper we show that second order tangency conditions are superfluous not to say useless while discussing the existence condition for certain second order differential inclusions. In this regard, a counterexample is provided even in the simpler setting of second order differential equations, where a substitute condition is propound. In the setting of differential inclusions, the corresponding substitute condition allows for us to prove existence of sufficiently many approximate solutions without the use of any convexity, measurability, or upper semicontinuity assumption. Accordingly, some proofs in the related literature are greatly simplified. 1. A second order differential equation: the theory Consider the second order differential equation X ′′(t) = g(t,X(t), X ′(t)) (1.1) where g : [a, b) ×D → R is a function, D ⊆ R is a nonempty set, and [a, b) ⊆ R is a nonempty, possibly unbounded interval. The existence condition for the equation (1.1) states that for every (x, y) ∈ D and for every τ ∈ [a, b) there exist a subinterval [τ, υ) of [τ, b) and a solution X : [τ, υ) → R to the differential equation (1.1) such that X(τ) = x and X ′(τ) = y. (1.2) By a solution to the equation (1.1) we mean a Carathéodory solution, that is, a locally absolutely continuous function X : [τ, υ)→ R such that also X ′ : [τ, υ)→ R is locally absolutely continuous, such that (X(t), X ′(t)) ∈ D for all t ∈ [τ, υ), and such that (t,X(t), X ′(t), X ′′(t)) renders true the equality (1.1) for almost all t ∈ [τ, υ). Throughout this paper, by a solution to a differential equation, inclusion, and so on we mean a Carathéodory solution. A characterization of the existence condition (1.1) can be given by using a tangency concept which springs from two papers published, in 1931, in the same issue of the journal “Annales de la Société Polonaise de Mathématique.” The authors of these papers are Bouligand (see [6]) and Severi (see [14]). 2000 Mathematics Subject Classification. 34A60.