Marek Izydorek

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Marek Izydorek

Gdansk University of Technology (Poland)

Generalized heteroclinic solutions for a class of the second order Hamiltonian systems

We shall be concerned with the existence of heteroclinic orbits for the second order Hamiltonian system

q¨+ Vq(t,q) = 0,

(HS)

where q Rⁿ and V C¹(R ×Rⁿ,R), V ≤ 0. We will assume that V and a certain subset M ⊂ Rⁿ satisfy the following conditions:

#M ≥ 2 and $γ := inf{|x - y|: x,y ∈ M, x ⁄= y} > 0$ ,
there exists 0 < ɛ₀ ≤ G such that for every 0 < ɛ ≤ ɛ₀ there is δ > 0 such that for all (t,x) R×Rⁿ if d(x,M) ≥ ɛ then -V (t,x) > δ,
for every x Rⁿ \ M, $lim - V (t,x) = + ∞ --t→ ±∞$ ,
for every x M, $∫ √---- -+∞∞ - V (t,x) < 2α0$ , where $α := inf{- V (t,x): d(x,M ) ≥ ɛ }. 0 0$

Our result states that each point of M is joined with a certain other element of M by a solution of (HS). Since we should not expect that (HS) possesses a stationary solution the notion of a heteroclinic orbit is used in a generalized sense. Namely, q: R → Rⁿ is a generalized heteroclinic solution of (HS) if there exist x,y Rⁿ, x≠y such that q joins x to y, (i.e. lim_t→-∞q(t) = x, lim_t→+∞q(t) = y).

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