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.


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


Zbl 1047.34034
Full Text: DOI


[1] DOŠLÝ, O.– PÁTÍKOVÁ, Z.: Hille-Wintner type comparison criteria for half-linear second order differential equations, Arch. Math. (Brno) 42 (2006), 185–194. · Zbl 1164.34386
[2] DOŠLÝ, O.– ŘEHÁK, P.: Half-Linear Differential Equations. North-Holland Math. Stud. 202, Elsevier, Amsterdam, 2005.
[3] DOŠLÝ, O.– UNAL, M.: Conditionally oscillatory half-linear differential equations, Acta Math. Hungar. (To appear). · Zbl 1199.34169
[4] HOWARD, H. C.– MARIĆ, V.: Regularity and nonoscillation of solutions of second order linear differential equations, Bull. Cl. Sci. Math. Nat. Sci. Math. 20 (1990), 85–98. · Zbl 0947.34015
[5] JAROŠ, J.– KUSANO, T.– TANIGAWA, T.: Nonoscillation theory for second order half-linear differential equations in the framework of regular variation, Results Math. 43 (2003), 129–149. · Zbl 1047.34034
[6] JAROŠ, J.– KUSANO, T.– TANIGAWA, T.: Nonoscillatory half-linear differential equations and generalized Karamata functions, Nonlinear Anal. 64 (2006), 762–787. · Zbl 1103.34017
[7] PÁTÍKOVÁ, Z.: Asymptotic formulas for non-oscillatory solutions of perturbed Euler equation, Nonlinear Anal. 69 (2008), 3281–3290. · Zbl 1158.34027
[8] PÁTÍKOVÁ, Z.: Asymptotic formulas for solutions of half-linear Euler-Weber equation, Electron. J. Qual. Theory Differ. Equ. 15 (2008), 1–11 (Proc. 8’th Coll. Qualitative Theory of Differential Equations).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.