Oscillation criteria for half-linear dynamic equations on time scales

Abstract

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.