Convergent disfocality and nondisfocality criteria for second-order linear differential equations. (English) Zbl 1470.34096

Almenar, Pedro; Jódar, Lucas

Abstr. Appl. Anal. 2013, Article ID 987976, 11 p. (2013).

Summary: This paper presents a method to determine whether the second-order linear differential equation \(y'' + q(x) y = 0\) 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.

MSC:

34C10

Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

References:

[1]	Kwong, M. K., On Lyapunov’s inequality for disfocality, Journal of Mathematical Analysis and Applications, 83, 2, 486-494 (1981) · Zbl 0504.34020 · doi:10.1016/0022-247X(81)90137-2
[2]	Kwong, M. K., Integral inequalities for second-order linear oscillation, Mathematical Inequalities & Applications, 2, 1, 55-71 (1999) · Zbl 0921.34035 · doi:10.7153/mia-02-06
[3]	Harris, B. J., On an inequality of Lyapunov for disfocality, Journal of Mathematical Analysis and Applications, 146, 2, 495-500 (1990) · Zbl 0702.34031 · doi:10.1016/0022-247X(90)90319-B
[4]	Brown, R. C.; Hinton, D. B., Opial’s inequality and oscillation of 2nd order equations, Proceedings of the American Mathematical Society, 125, 4, 1123-1129 (1997) · Zbl 0866.34026 · doi:10.1090/S0002-9939-97-03907-5
[5]	Pachpatte, B. G., Mathematical Inequalities (2005), New York, NY, USA: Elsevier, New York, NY, USA · Zbl 1091.26008
[6]	Tipler, F. J., General relativity and conjugate ordinary differential equations, Journal of Differential Equations, 30, 2, 165-174 (1978) · Zbl 0362.34023 · doi:10.1016/0022-0396(78)90012-8
[7]	Došlý, O., Conjugacy criteria for second order differential equations, The Rocky Mountain Journal of Mathematics, 23, 3, 849-861 (1993) · Zbl 0794.34025 · doi:10.1216/rmjm/1181072527
[8]	Moore, R. A., The behavior of solutions of a linear differential equation of second order, Pacific Journal of Mathematics, 5, 125-145 (1955) · Zbl 0064.08401 · doi:10.2140/pjm.1955.5.125
[9]	Almenar, P.; Jódar, L., 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 (2012) · Zbl 1238.34056 · doi:10.1016/j.camwa.2011.11.023
[10]	Almenar, P.; Jódar, L., The distribution of zeroes and critical points of solutions of a second order half-linear differential equation, Abstract and Applied Analysis, 2013 (2013) · Zbl 1276.34024 · doi:10.1155/2013/147192
[11]	Hutson, V.; Pym, J. S.; Cloud, M. J., Applications of Functional Analysis and Operator Theory. Applications of Functional Analysis and Operator Theory, Mathematics in Science and Engineering, 200 (2005), New York, NY, USA: Elsevier, New York, NY, USA · Zbl 1066.47001
[12]	Sagan, H., Boundary and Eigenvalue Problems in Mathematical Physics (1989), New York, NY, USA: Dover, New York, NY, USA · Zbl 1227.35003
[13]	Bellman, R., The stability of solutions of linear differential equations, Duke Mathematical Journal, 10, 643-647 (1943) · Zbl 0061.18502 · doi:10.1215/S0012-7094-43-01059-2