## 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 $\bigl (r(t)\Phi (x')\bigr)'+c(t)\Phi (x)=0,\quad \Phi (x)=| x| ^{p-2}x. \qquad {(1)}$ One of the main results of the paper reads as follows. Let $$h$$ be a positive solution of (1) such that $$h'(t)\neq 0$$ for large $$t$$. Suppose that $$\int ^\infty R^{-1}(t)\,{\operatorname {d}}t=\infty$$, where $$R(t)=r(t)h^{2}(t)| h'(t)| ^{p-2}$$, and $$\liminf _{t\to \infty }r(t)h(t)| h'(t)| ^{p-1}>0.$$ Then the equation $\bigl (r(t)\Phi (x')\bigr)'+\left [c(t)+ \frac {\mu }{h^p(t)R(t)\bigl (\smallint ^t R^{-1}(s)\,{\operatorname {d}}s\bigr)^2}\right ]\Phi (x)=0,$ possesses a pair of linearly independent solutions, which, depending on the value of the parameter $$\mu$$, can be expressed by the asymptotic formula $x_i(t)=h(t)\left (\int^t R^{-1}(s)\,{\operatorname {d}}s\right)^{\lambda _i}L_i(t),$ where $$\lambda _i$$ 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:

 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations

Zbl 1047.34034
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### References:

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