Abstract and Applied Analysis
Volume 2011 (2011), Article ID 182827, 16 pages
doi:10.1155/2011/182827
Research Article

Two-Parametric Conditionally Oscillatory Half-Linear Differential Equations

1Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic
2Department of Mathematics, Mendel University in Brno, Zemĕdĕlská 1, 613 00 Brno, Czech Republic

Received 2 November 2010; Accepted 5 January 2011

Academic Editor: Miroslava Růžičková

Copyright © 2011 Ondřej Došlý and Simona Fišnarová. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study perturbations of the nonoscillatory half-linear differential equation ( 𝑟 ( 𝑡 ) Φ ( 𝑥 ) ) + 𝑐 ( 𝑡 ) Φ ( 𝑥 ) = 0 , Φ ( 𝑥 ) = | 𝑥 | 𝑝 2 𝑥 , 𝑝 > 1 . We find explicit formulas for the functions ̂ 𝑟 , ̂ 𝑐 such that the equation [ ( 𝑟 ( 𝑡 ) + 𝜆 ̂ 𝑟 ( 𝑡 ) ) Φ ( 𝑥 ) ] + [ 𝑐 ( 𝑡 ) + 𝜇 ̂ 𝑐 ( 𝑡 ) ] Φ ( 𝑥 ) = 0 is conditionally oscillatory, that is, there exists a constant 𝛾 such that the previous equation is oscillatory if 𝜇 𝜆 > 𝛾 and nonoscillatory if 𝜇 𝜆 < 𝛾 . The obtained results extend the previous results concerning two-parametric perturbations of the half-linear Euler differential equation.