Oscillation criteria for half-linear dynamic equations on time scales

• Published in: Journal of mathematical analysis and applications 2008-09, Vol.345 (1), p.176-185
• Main Author: Hassan, Taher S.
• Format: Article
• Language: English
• Subjects: Difference and functional equations, recurrence relations; Exact sciences and technology; Finite differences and functional equations; Half-linear dynamic equations; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Oscillation; Real functions; Sciences and techniques of general use; Second-order; Time scales
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2008.04.019

This paper is concerned with oscillation of the second-order half-linear dynamic equation ( r ( t ) ( x Δ ) γ ) Δ + p ( t ) x γ ( t ) = 0 , on a time scale T where γ is the quotient of odd positive integers, r ( t ) and p ( t ) are positive rd-continuous functions on T . Our results solve a problem posed by [R.P. Agarwal, D. O'Regan, S.H. Saker, Philos-type oscillation criteria for second-order half linear dynamic equations, Rocky Mountain J. Math. 37 (2007) 1085–1104; S.H. Saker, Oscillation criteria of second order half-linear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2005) 375–387] and our results in the special cases when T = R and T = Z involve and improve some oscillation results for second-order differential and difference equations; and when T = h Z , T = q N 0 and T = N 0 2 , etc., our oscillation results are essentially new. Some examples illustrating the importance of our results are also included.