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Oscillations of second-order nonlinear impulsive ordinary differential equations. (English) Zbl 1042.34063

The authors study the second-order impulsive ordinary differential equation

r(t)x ' (t) σ ' +f(t,x(t))=0,tt 0 ,tt k ,k=1,2,(1)

where rC(,(0,)), fC(×,) and f satisfies the sign condition xf(t,x)>0 for all x0 and the growth condition f(t,x) φ(x)q(t) for all x0 where qC(,[0,)), xφ(x)>0 for x0 and φ ' (x)0. The impulses

x(t k + )=g k (x(t k )),x ' (t k + )=h k (x ' (t k )),k=1,2,

are supposed to satisfy g k ,h k C(,) and

a ¯ k g k (x) xa k ,b ¯ k h k (x) xb k ,k=1,2,,

where a k , a ¯ k , b k and b ¯ k are positive real numbers. Sufficient conditions for the oscillation of equation (1) are established. One of the typical results is the following theorem:

Assume that

lim t t 0 t 1 r(s) 1/σ t o <t k <s b ¯ k a k ds=+

and there exists a positive integer k 0 such that a ¯ k 1 for kk 0 . If

ε du (φ(u)) 1/σ <+, -ε - du (φ(u)) 1/σ <+,

hold for some ε>0 and

k=0 t k + t k+1 1 r(s) 1/σ lim t+ s t s<t k <u 1 b k σ q(u)du 1/σ ds=+,

then every solution of equation (1) is oscillatory.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34A37Differential equations with impulses