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

Convergent Disfocality and Nondisfocality Criteria for Second-Order Linear Differential Equations

Copyright © 2013 Pedro Almenar and Lucas Jódar. 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.This paper presents a method to determine whether the second-order linear differential equation y(n) + q(x)y = 0 is either disfocal or nondisfocal in a fixed interval. The method is based on the recursive application of a linear operator to certain functions and yields upper and lower bounds for the distances between a zero and its adjacent critical points, which will be shown to converge to the exact values of such distances as the recursivity index grows.This work has been supported by the Spanish Ministry of Science and Innovation Project DPI2010-C02-01.Almenar, P.; Jódar Sánchez, LA. (2013). Convergent Disfocality and Nondisfocality Criteria for Second-Order Linear Differential Equations. Abstract and Applied Analysis. 2013:1-11. doi:10.1155/2013/987976S1112013Kwong, M. K. (1981). On Lyapunov’s inequality for disfocality. Journal of Mathematical Analysis and Applications, 83(2), 486-494. doi:10.1016/0022-247x(81)90137-2Kwong, M. K. (1999). Integral Inequalities for Second-Order Linear Oscillation. Mathematical Inequalities & Applications, (1), 55-71. doi:10.7153/mia-02-06Harris, B. . (1990). On an inequality of Lyapunov for disfocality. Journal of Mathematical Analysis and Applications, 146(2), 495-500. doi:10.1016/0022-247x(90)90319-bBrown, R. C., & Hinton, D. B. (1997). Proceedings of the American Mathematical Society, 125(04), 1123-1130. doi:10.1090/s0002-9939-97-03907-5Tipler, F. J. (1978). General relativity and conjugate ordinary differential equations. Journal of Differential Equations, 30(2), 165-174. doi:10.1016/0022-0396(78)90012-8Došlý, O. (1993). Conjugacy Criteria for Second Order Differential Equations. Rocky Mountain Journal of Mathematics, 23(3), 849-861. doi:10.1216/rmjm/1181072527Moore, 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.1955.5.125Almenar, 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.2011.11.023Almenar, P., & Jódar, L. (2013). The Distribution of Zeroes and Critical Points of Solutions of a Second Order Half-Linear Differential Equation. Abstract and Applied Analysis, 2013, 1-6. doi:10.1155/2013/147192Bellman, R. (1943). The stability of solutions of linear differential equations. Duke Mathematical Journal, 10(4), 643-647. doi:10.1215/s0012-7094-43-01059-

An upper bound for the distance between a zero and a critical point of a solution of a second order linear differential equation

This paper presents an upper bound for the distance between a zero and a critical point of a solution of the second order linear differential equation (p(x)y¿)¿+q(x)y(x)=0(p(x)y¿)¿+q(x)y(x)=0, with p(x),q(x)>0p(x),q(x)>0. It also compares it with previous results.This work has been supported by the Spanish Ministry of Science and Innovation project DPI2010-20891-C02-01.Almenar, P.; Jódar Sánchez, LA. (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 and Mathematics with Applications. 63:310-317. https://doi.org/10.1016/j.camwa.2011.11.023S3103176