Asymptotic Behavior of a Class of Nonlinear Differential Equations of nth Order

Qingkai Kong
1988 Proceedings of the American Mathematical Society  
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 (1) u" + fit,u)=0. Some results of it are based on the integral inequalities of Gronwall-Bihari type. Here we quote the Theorem B in [1] as a proposition,
more » ... oposition, according to the author, which includes the Theorem in [2] as its special case. PROPOSITION. Let fit,u) be continuous on D: t > 0, -oo < u < oo. If there are two nonnegative and continuous functions vit), 0(t) for t > 0, and a continuous function giu) for u>0, such that (i) /j°° vit)it) dt < co, (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].
doi:10.2307/2046862 fatcat:ar4tietszvfkpeq7bbimtffuxm