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Asymptotic formulas for nonoscillatory solutions of conditionally oscillatory half-linear equations. (English) Zbl 1240.34180

Asymptotic properties of nonoscillatory solutions of a special half-linear differential equation are investigated. This equation is viewed as a perturbation of the general nonoscillatory equation

r (t) Φ (x ' ) ' +c(t)Φ(x)=0,Φ(x)=|x| p-2 x·(1)

One of the main results of the paper reads as follows. Let h be a positive solution of (1) such that h ' (t)0 for large t. Suppose that R -1 (t)dt=, where R(t)=r(t)h 2 (t)|h ' (t)| p-2 , and lim inf t r(t)h(t)|h ' (t)| p-1 >0· Then the equation

r (t) Φ (x ' ) ' +c(t)+μ h p (t)R(t) t R -1 (s) d s 2 Φ(x)=0,

possesses a pair of linearly independent solutions, which, depending on the value of the parameter μ, can be expressed by the asymptotic formula

x i (t)=h(t) t R -1 (s)ds λ i L i (t),

where λ i are roots of a certain quadratic equation and L i are normalized slowly varying functions. The results of the paper extend, among others, asymptotic formulas given in the paper J. Jaroš, K. Takaŝi and T. Tanigawa [Result. Math. 43, No. 1–2, 129–149 (2003; Zbl 1047.34034)], where (1) reduces to the half-linear Euler differential equation.

MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34D05Asymptotic stability of ODE