A boundedness Theorem for a Certain Third-Order Differential Equation
1967; Wiley; Volume: s3-17; Issue: 2 Linguagem: Inglês
10.1112/plms/s3-17.2.382-s
ISSN1460-244X
Autores Tópico(s)Advanced Mathematical Modeling in Engineering
ResumoLet ψ(Σ) denote the frontier of the square Σ (see § 3.1). A major part of the proof of the theorem rests on a tacit assumption, in §4.1, that if the condition (1) |ξ(t|⩽D3 and |η(t)|⩽ D3 for all t⩾t0 is not fulfilled then the corresponding curve γnecessarily intersects ψ(Σ)in such a way that, from a certain t onwards,all the intersection-points can be enumerated one after another thus giving rise to the sequence {E8}. This assumption is incorrect, as has been pointed out to me by Professor H. Pachale. Indeed, if (1) does not hold, γ may well behave as described in §4.1, but there is also a possibility that, as t → ∞, the representative point (ξ,η) on γ may move on the frontier ψ(Σ) itself for brief periods or that γ may intersect ψ(Σ) in a set of points with limit-points on ψ(Σ), and in either case the special enumeration, referred to above, of all the points of intersection of γ with ψ(Σ) would be impossible. With the additional condition (2) | p(t)| ⩽ A1 < ∞ for all t considered, on p(t), it is however possible to validate once again the entire content of §4.1, though with a somewhat different Σ; so that the boundedness theorem in the paper is valid subject to this further restriction (2) on p. Indeed let d1 = max{D3,(A1+1)δ0−1}, d2 = max{D3,[A1+1+1+δ2d1]δ1−1}; and letΣ*denote the rectangle |x| ⩽d1, | y| ⩽ d2 in the plane π. Here δ0, δ1 are the constants in the hypotheses of the theorem, and D3 is the constant given in §2.2. Our claim about the validity of the work in §4.1 under the present conditions rests upon the following LEMMA. Assume that (1) holds and also that all the previous conditions on a, f, g, p are satisfied. Then every curve γ in the plane π necessarily satisfies one or other of the following: (I) |ξ(tverbar; ⩽ d1 and |η(t)| for all t ⩾ t0, (II) 4as t increases to +∞ the curveγ intersects the frontier of Σ* repeatedly in such a way that the intersection-points can be enumerated one after another exactly as in Fig. 1 (p. 108) but with the vertices Q1, Q2, Q3, Q4 replaced by the points (−d1,d2), {d1,d2), {d1,−d2), (−d1, −d2)respectively. Proof of Lemma. Since d1 ⩾D3 and d2 ⩾ D3 it is clear from an argument in §3.2 that (ξ,η) cannot stay outside Σ* indefinitely. Therefore to prove the lemma it suffices to show that, under our present conditions, it is impossible to find an infinite sequence {tn} (n = 1,2,…) of values of t,with a finite limit-point, t* say, such that any one of the following is satisfied: (3) |ξ(tn)| ⩽ d1 and η(tn) =d2 (n=l,2,…), (4) |ξ(tn)| ⩽ d1 and η(tn) =−d2 (n=l,2,…), (5) ξ(tn)= d1 (n=l,2,…), (6) ξ(tn)= −d1 (n=l,2,…), Our proof of this will be by contradiction. Suppose, on the contrary, that there is an infinite sequence {tn} with a finite limit-point t*, satisfying (3). Then, because of the continuity of ξ(t),η(t), |ξ(t*)| ⩽ d1, ξ(t*)≡ η(t*)=d2. Also, from the fact that η(t1)=η(t2)=…=η(tn)=…= d2 there would be, by Rolle's theorem, an infinite sequence {τn}, with t*as a limit-point, such that (7) ξ ¨ (τn)≡ η ˙ τ(n)= 0 (n = 1,2,…); and from this, by virtue of the continuity of ξ ¨ (t), we derive the result ξ ¨ (t*)= 0. In view of (7) it is also clear, by applying Rolle's theorem to the sequence { ξ ¨ (τn)} and then using the continuity of ξ ˙¨ (t), that ξ ˙¨ (t* = 0. The substitution of these values, ξ ˙ (t*)=d2, ξ ¨ (t*)= 0 = ξ ˙¨ (t*), in the differential equation itself gives (8) f(ξ(t*))d2 = −g(ξ(t*))+p(t*). Since f⩾δ1, |p|⩽A1, |ξ(t*)|⩽d1, and since our hypotheses on g imply that |g(ξ)|⩽δ2|ξ|, it follows at once from (8) that (9) d2δ1⩽δ2d1+Aa. However, d2 has been fixed to satisfy d2⩾δ1−1(A1+1 +δ2d1), so that d2δ1−δ2d1⩾A1+1, and thus (9) cannot hold. Hence (3) is impossible. In the same way it can be shown that (4) cannot hold for any sequence {tn} with a finite limit point. It remains now to dispose of (5) and (6). Suppose, on the contrary, that there is an infinite sequence {tn}, with a finite limit-point t*, such that (5) holds. Then, in exactly the same way as before, we shall have ξ(t*)=d1, ξ ˙ (t*)=0= ξ ¨ (t*)= ξ ˙¨ (t*); and then, by substituting these values in the differential equation, one finds that g(d1)=p(t*) or, since |g(d1)|⩾δ0d1 and |p|⩽A1, that δ0d1⩽A1 But this is impossible since dx has been fixed so that δ0d1⩾A1+1. Similarly it can be shown that (6) cannot hold; and the lemma is thereby proved.
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