Asymptotic behavior of a class of nonlinear differential equations of 𝑛th order

Abstract

ABSTRACT. In this paper we obtain a result of the asymptotic behavior of the nth order equation r¿ ("' + f(t,u,u',... ,«("-1)) = 0 under some assumptions. For n = 2 and f{t,u,u') = f{t,u), it revises the result given by Jingcheng Tong, which is not true in general.Much work has been done on the asymptotic behavior of the second order equation (ii) for u > 0, giu) is positive and nondecreasing, (iii) |/(i, u)| < vit)(j>it)gi\u\/t) for t > 1, -oo < u < oo, then the equation (1) has solutions which are asymptotic to a + bt, where a,b are constants and 6^0.According to the proof it seems to be true that every solution w(i) of Equation (1) satisfies that u'(i) -► b £ 72 as t -► oo. But unfortunately, the Proposition does not hold in general because of two mistakes in its proof.The first error arises because Gic3)+f1 t;(s)0(s) ds may be outside the domain of G_1 (see (4) in [1]). Subsequently, the second error arises because one cannot hence conclude that f^° |/(s, u(s))| ds < oo. For example, the equation u" -(2/í4)u2 = 0 has a solution u = r2 which does not satisfy the property u'(£) -► b it -> oo) as depicted in the proof of the Proposition.For the same reason, the example in [1] cannot be true.In this paper we obtain a result of the asymptotic behavior of the nth order equation (2) u^A-fit,u,u',...,u^-^)=0.In a special case, for n = 2 and /(i,u,u') = fit,u), it revises the result in [1].

Ghizzetti′s theorem for piecewise continuous solutions

“…We wish to study the second order impulsive differential equation where and y'(t) f(t,y(t),y'(t)), ti, > a > 0 (P1) Ay(t) gl(t,y(t),y'(t)), (P2) Ay'(t) g2(t,y(t),y'(t)), [2] for ordinary differential equations (see [5] to [9]) to problem (P).…”