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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 \[ \bigl (r(t)\Phi (x')\bigr)'+c(t)\Phi (x)=0,\quad \Phi (x)=| x| ^{p-2}x. \qquad {(1)} \] 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

Citations:

Zbl 1047.34034
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References:

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