This paper concerns the oscillation of solutions to the second-order dynamic equation ( r ( t ) x Δ ( t ) ) Δ + p ( t ) x Δ ( t ) + q ( t ) f ( x σ ( t ) ) = 0 , on a time scale T which is unbounded above. No sign conditions are imposed on r ( t ) , p ( t ) , and q ( t ) . The function f ∈ C ( R , R ) is assumed to satisfy x f ( x ) > 0 and f ′ ( x ) > 0 for x ≠ 0 . In addition, there is no need to assume certain restrictive conditions and also the both cases ∫ t 0 ∞ Δ t r ( t ) = ∞ and ∫ t 0 ∞ Δ t r ( t ) < ∞ are considered. Our results will improve and extend results in (Baoguo et al. in Can. Math. Bull. 54:580-592, 2011; Bohner et al. in J. Math. Anal. Appl. 301:491-507, 2005; Hassan et al. in Comput. Math. Anal. 59:550-558, 2010; Hassan et al. in J. Differ. Equ. Appl. 17:505-523, 2011) and many known results on nonlinear oscillation. These results have significant importance to the study of oscillation criteria on discrete time scales such as T = Z , T = h Z , h > 0 , or T = { t : t = q k , k ∈ N 0 , q > 1 } and the space of harmonic numbers T = H n . Some examples illustrating the importance of our results are also included. MSC: 34K11, 39A10, 39A99.
Hassan Advances in Difference Equations 2012, 2012 :171 http://www.advancesindifferenceequations.com/content/2012/1/171
R E S E A R C H Open Access Oscillation criteria for second-order nonlinear dynamic equations Taher S Hassan * * C pondence: orres tshassan@mans.edu.eg Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt
Abstract This paper concerns the oscillation of solutions to the second-order dynamic equation ( r ( t ) x ( t ) ) + p ( t ) x ( t ) + q ( t ) f ( x σ ( t ) ) = 0, on a time scale T which is unbounded above. No sign conditions are imposed on r ( t ), p ( t ), and q ( t ). The function f ∈ C ( R , R ) is assumed to satisfy xf ( x ) > 0 and f ( x ) > 0 for x = 0. In addition, there is no need to assume certain restrictive conditions and also the both cases ∞ t t 0 ∞ r ( t ) t = ∞ and t 0 r ( t ) < ∞ are considered. Our results will improve and extend results in (Baoguo et al. in Can. Math. Bull. 54:580-592, 2011; Bohner et al. in J. Math. Anal. Appl. 301:491-507, 2005; Hassan et al. in Comput. Math. Anal. 59:550-558, 2010; Hassan et al. in J. Differ. Equ. Appl. 17:505-523, 2011) and many known results on nonlinear oscillation. These results have significant importance to the study of oscillation criteria on discrete time scales such as T = Z , T = h Z , h > 0, or T = { t : t = q k , k ∈ N 0 , q > 1 } and the space of harmonic numbers T = H n . Some examples illustrating the importance of our results are also included. MSC: 34K11; 39A10; 39A99 Keywords: oscillation; second order; dynamic equations; time scales