ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

9. Ordinary Differential Equations and Dynamical Systems

1991 MS Classification: 34B15 Kiguradze, Ivan, A. Razmadze Mathematical Institute, Georgia: On explosive kneser solutions of nonlinear ordinary differential equations. The differential equation

u^{(n)} = f(t, u, …, u^{(n−1)})  (1)

is considered, where n ≥ 2 and f : R_{+} × R^{n} → R is a continuous function. The sufficient conditions are found under which this equation has a solution satisfying the conditions

lim_{t→+∞} u(t) = +∞, (−1)^{i}u^{(i)}(t) > 0 (i = 0, 1, …, n − 1) for t > 0 (2)

and global two-sided estimates for such solutions are obtained. In particular, it is shown that if

p_{1}t^{μ}|x_{1}|^{λ} ≤ (−1)^{n}f(t, x_{1}, …, x_{n}) ≤ p_{2}t^{μ}|x_{1}|^{λ},

where μ > −n, λ > 1 and p_{2} ≥ p_{1} > 0, then problem (1), (2) is solvable and every its solution admits the estimates

r_{2}p2^{\frac{1}{1−λ}} t^{\frac{n+μ}{1−λ}} ≤ u(t) ≤ r_{1}p1^{\frac{1}{1−λ}} t^{\frac{n+μ}{1−λ}} for t > 0,

where r_{1} and r_{2} are constants depending only on λ, μ and n.