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Perturbations of the half-linear Euler differential equation. (English) Zbl 0958.34029

The authors investigate oscillation/nonoscillation properties of the perturbed half-linear Euler differential equation

(x ' n* ) ' +γ 0 t n+1 [n+2(n+1)δ(t)]x n* =0,(*)

where the function δ(t) is piecewise continuous on (t 0 ,), t 0 0, n>0 is a fixed real number and u n* =|u| n sgnu. The number γ 0 =n n (n+1) n+1 is a critical constant, which means that in case δ(t)0 the half-linear Euler differential equation is for γγ 0 nonoscillatory while for γ>γ 0 is oscillatory. Using a Riccati technique and a transformation of the independent variable, there are proved the following main results:

Suppose that there exists the finite limit

lim T t 0 T δ(t)dt t

such that t δ(s)ds s0 for t>t 0 .

(a) If n>1 and the linear equation

z '' +δ(e s )z=0(**)

is nonoscillatory then equation (*) is also nonoscillatory.

(b) If 0<n<1 and equation (*) is nonoscillatory then the linear equation (**) is also nonoscillatory.

In addition, the authors establish an asymptotic form of the solution to (*) provided that the solutions to (**) satisfy two integral inequalities.

MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
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