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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 $$ \bigl (r(t)\Phi (x')\bigr)'+c(t)\Phi (x)=0,\quad \Phi (x)=\vert x\vert ^{p-2}x. \qquad {(1)} $$ One of the main results of the paper reads as follows. Let $h$ be a positive solution of (1) such that $h'(t)\ne 0$ for large $t$. Suppose that $\int ^\infty R^{-1}(t)\,{\operatorname {d}}t=\infty $, where $R(t)=r(t)h^{2}(t)\vert h'(t)\vert ^{p-2}$, and $\liminf _{t\to \infty }r(t)h(t)\vert h'(t)\vert ^{p-1}>0.$ 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 {\it {J. Jaroš}}, {\it K. Takaŝi} and {\it {T. Tanigawa}} [Result. Math. 43, No. 1--2, 129--149 (2003; Zbl 1047.34034)], where (1) reduces to the half-linear Euler differential equation.
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34D05Asymptotic stability of ODE
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