##
**Conditionally oscillatory half-linear differential equations.**
*(English)*
Zbl 1199.34169

The authors assume that a nonoscillatory solution to the half-linear equation
\[
(r(t)\Phi(x'))+c(t)\Phi(x)=0,\;\Phi(x)=| x| ^{p-2}x,\;p>1,
\]
is known. Then they are able to construct a function \(d\) such that the (perturbed) equation
\[
(r(t)\Phi(x'))+(c(t)+\lambda d(t))\Phi(x)=0
\]
is conditionally oscillatory. They also establish an asymptotic formula for a solution of the perturbed equation in the critical case, i.e., when \(\lambda\) equals the oscillation constant. These results are then used to obtain new (non)oscillation criteria, which extend previous results for perturbed half-linear Euler type and Euler-Weber type equations. The concepts of generalized Riccati equation and of principal solution, and the Schauder-Tychonoff fixed point theorem play an important role in the proofs.

Reviewer: Pavel Rehák (Brno)

### MSC:

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

47N20 | Applications of operator theory to differential and integral equations |

### Keywords:

half-linear oscillation theory; conditionally oscillatory equation; oscillation and nonoscillation criteria; Riccati type equation
PDF
BibTeX
XML
Cite

\textit{O. Došlý} and \textit{M. Ünal}, Acta Math. Hung. 120, No. 1--2, 147--163 (2008; Zbl 1199.34169)

Full Text:
DOI

### References:

[1] | R. P. Agarwal, S. R. Grace and D. O’Regan, Oscillation Theory of Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations, Kluwer Academic (Dordrecht, 2002). · Zbl 1091.34518 |

[2] | O. Došlý, Perturbations of the half-linear Euler-Weber type differential equation, J. Math. Anal. Appl., 323 (2006), 426–440. · Zbl 1107.34030 |

[3] | O. Došlý and Á. Elbert, Integral characterization of the principal solution to half-linear second-order differential equations, Stud. Sci. Math. Hung., 36 (2000), 455–469. · Zbl 1012.34029 |

[4] | O. Došlý and A. Lomtatidze, Oscillation and nonoscillation criteria for half-linear second order differential equations, Hiroshima Math. J., 36 (2006), 203–219. · Zbl 1123.34028 |

[5] | O. Došlý and M. Ünal, Half-linear differential equations: Linearization technique and its application, J. Math. Anal. Appl., 335 (2007), 450–460. · Zbl 1128.34017 |

[6] | O. Došlý and P. Řehák, Half-Linear Differential Equations, North-Holland Mathematics Studies 202, Elsevier (Amsterdam, 2005). |

[7] | Á. Elbert and T. Kusano, Principal solutions of nonoscillatory half-linear differential equations, Adv. Math. Sci. Appl., 18 (1998), 745–759. · Zbl 0914.34031 |

[8] | Á. Elbert and A. Schneider, Perturbations of the half-linear Euler differential equation, Result. Math., 37 (2000), 56–83. · Zbl 0958.34029 |

[9] | J. Jaroš, T. Kusano and T. Tanigawa, Nonoscillation theory for second order half-linear differential equations in the framework of regular variation, Result. Math., 43 (2003), 129–149. · Zbl 1047.34034 |

[10] | J. Jaroš, T. Kusano and T. Tanigawa, Nonoscillatory half-linear differential equations and generalized Karamata functions, Nonlinear Anal., 64 (2006), 762–787. · Zbl 1103.34017 |

[11] | J. D. Mirzov, Principal and nonprincipal solutions of a nonoscillatory system, Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy, 31 (1988), 100–117. · Zbl 0735.34029 |

[12] | Z. Pátíková, Asymptotic formulas for nonoscillatory solutions of perturbed half-linear Euler equation, submitted. |

[13] | J. Sugie and N. Yamaoka, Comparison theorems for oscillation of second-order half-linear differential equations, Acta Math. Hungar., 111 (2006), 165–179. · Zbl 1116.34030 |

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.