On the zeroes and the critical points of a solution of a second order half-linear differential equation

This paper presents two methods to obtain upper bounds for the distance between a zero and an adjacent critical point of a solution of the second-order half-linear di¿erential equation p x ¿ y q x ¿ y 0, with p x and q x piecewise continuous and p x > 0, ¿ t |t| r¿2 t and r being real such that r > 1. It also compares between them in several examples. Lower bounds i.e., Lyapunov inequalities for such a distance are also provided and compared with other methods.This work has been supported by the Spanish Ministry of Science and Innovation Project DPI2010-C02-01.Almenar, P.; Jódar Sánchez, LA. (2012). On the zeroes and the critical points of a solution of a second order half-linear differential equation. Abstract and Applied Analysis. 2012(ID 78792):1-18. doi:10.1155/2012/787920S1182012ID 78792Almenar, P., & Jódar, L. (2012). An upper bound for the distance between a zero and a critical point of a solution of a second order linear differential equation. Computers & Mathematics with Applications, 63(1), 310-317. doi:10.1016/j.camwa.2011.11.023Li, H. J., & Yeh, C. C. (1995). Sturmian comparison theorem for half-linear second-order differential equations. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 125(6), 1193-1204. doi:10.1017/s0308210500030468Yang, X. (2003). On inequalities of Lyapunov type. Applied Mathematics and Computation, 134(2-3), 293-300. doi:10.1016/s0096-3003(01)00283-1Lee, C.-F., Yeh, C.-C., Hong, C.-H., & Agarwal, R. P. (2004). Lyapunov and Wirtinger inequalities. Applied Mathematics Letters, 17(7), 847-853. doi:10.1016/j.aml.2004.06.016Pinasco, J. P. (2004). Lower bounds for eigenvalues of the one-dimensionalp-Laplacian. Abstract and Applied Analysis, 2004(2), 147-153. doi:10.1155/s108533750431002xPinasco, J. P. (2006). Comparison of eigenvalues for the p-Laplacian with integral inequalities. Applied Mathematics and Computation, 182(2), 1399-1404. doi:10.1016/j.amc.2006.05.027Almenar, P., & Jódar, L. (2009). Improving explicit bounds for the solutions of second order linear differential equations. Computers & Mathematics with Applications, 57(10), 1708-1721. doi:10.1016/j.camwa.2009.03.076Moore, R. (1955). The behavior of solutions of a linear differential equation of second order. Pacific Journal of Mathematics, 5(1), 125-145. doi:10.2140/pjm.1955.5.12