Harris, B. J., Kong, Q., On the oscillation of differential
equations with an oscillatory coefficient, Trans. Amer. Math. Soc. 347 (5),
1831-1839, 1995.
Hartman, P., Ordinary Differential Equations, Wiley, New
York, 1964 and Birkhuser, Boston, 1982.
Hartman, P., Wintner, A., On an oscillation criterion of de la
Vallée Poussin, The Quarterly J. Math. 13, 330--332, 1955.
Hilger, S., Analysis on measure chain--a unified approach to
continuous and discrete calculus, Results Math. 18, 18--56, 1990.
Karpuz, B., Kaymakçalan, B., Öcalan, Ö., A
generalization of Opial's inequality and applications to second order
dynamic equations, Diff. Eqns. Dyn. Syst. 18 (1--2), 11--18, 2010.
Kwong, M. K., On Lyapunov's inequality for disfocality, J. Math.
Anal. Appl. 83, 486-494, 1981.
Lee, C. F., Yeh, C. C., Hong, C. H., Agarwal, R. P., Lyapunov
and Wirtinger inequalities, Appl. Math. Lett. 17, 847--853, 2004.
Olech, Z., Asimple proof of a certain result of Z. Opial,
Ann. Polon. Math. 8, 61--63, 1960.
Opial, Z., Sur uné inegalité, Ann. Polon. Math.
8, 29--32, 1960.
Saker, S. H., Oscillation Theory of Dynamic Equations on
Time Scales: Second and Third Orders, Lambert Academic Publishing, Germany, 2010.
Saker, S. H., Lyapunov inequalities for half-linear dynamic
equations on time scales and disconjugacy, Dyn. Cont. Discr. Impuls. Syst.
Ser. B Appl. Algorithms 18, 149--161, 2011.
Saker, S. H., Lyapunov type inequalities for a second order
differential equation with a damping term, Ann. Polon. Math. 103 (1),
37--57, 2012.
Saker, S. H., Opial's type inequalities on time scales and some
applications, Ann. Polon. Math. 104, 243--260, 2012.
Saker, S. H., Applications of Opial inequalities on time scales
on dynamic equations with damping terms, Math. Comp. Model.58, 1777--1790, 2013.
Saker, S. H., Osman, M. M., O'Regan, D., Agarwal, R. P., Some
new Opial dynamic inequalities with weighted functions on time scales, Math.
Ineq. Appl. 18 (3), 1171--1187, 2015.
Tiryaki, A., Recent development of Lyapunov-type inequalities,
Advances in Dyn. Syst. Appl. 5 (2), 231--248, 2010.
Wintner, A., On the nonexistence of conjugate points, Amer. J.
Math. 73, 368--380, 1951.
Yang, X., On inequalities of Lyapunov type, Appl. Math. Comp.
134, 293--300, 2003.
Lyapunov inequalities for dynamic equations via new Opial type inequalities
Year 2018,
Volume: 47 Issue: 6, 1537 - 1551, 12.12.2018
In this paper, we prove some new dynamic inequalities of Opial type on time scales. By employing these new inequalities we establish some new Lyapunov type inequalities for a second order dynamic equation with a damping term. These new Lyapunov inequalities give lower bounds on the distance between zeros of a solution and/or its derivative.
Harris, B. J., Kong, Q., On the oscillation of differential
equations with an oscillatory coefficient, Trans. Amer. Math. Soc. 347 (5),
1831-1839, 1995.
Hartman, P., Ordinary Differential Equations, Wiley, New
York, 1964 and Birkhuser, Boston, 1982.
Hartman, P., Wintner, A., On an oscillation criterion of de la
Vallée Poussin, The Quarterly J. Math. 13, 330--332, 1955.
Hilger, S., Analysis on measure chain--a unified approach to
continuous and discrete calculus, Results Math. 18, 18--56, 1990.
Karpuz, B., Kaymakçalan, B., Öcalan, Ö., A
generalization of Opial's inequality and applications to second order
dynamic equations, Diff. Eqns. Dyn. Syst. 18 (1--2), 11--18, 2010.
Kwong, M. K., On Lyapunov's inequality for disfocality, J. Math.
Anal. Appl. 83, 486-494, 1981.
Lee, C. F., Yeh, C. C., Hong, C. H., Agarwal, R. P., Lyapunov
and Wirtinger inequalities, Appl. Math. Lett. 17, 847--853, 2004.
Olech, Z., Asimple proof of a certain result of Z. Opial,
Ann. Polon. Math. 8, 61--63, 1960.
Opial, Z., Sur uné inegalité, Ann. Polon. Math.
8, 29--32, 1960.
Saker, S. H., Oscillation Theory of Dynamic Equations on
Time Scales: Second and Third Orders, Lambert Academic Publishing, Germany, 2010.
Saker, S. H., Lyapunov inequalities for half-linear dynamic
equations on time scales and disconjugacy, Dyn. Cont. Discr. Impuls. Syst.
Ser. B Appl. Algorithms 18, 149--161, 2011.
Saker, S. H., Lyapunov type inequalities for a second order
differential equation with a damping term, Ann. Polon. Math. 103 (1),
37--57, 2012.
Saker, S. H., Opial's type inequalities on time scales and some
applications, Ann. Polon. Math. 104, 243--260, 2012.
Saker, S. H., Applications of Opial inequalities on time scales
on dynamic equations with damping terms, Math. Comp. Model.58, 1777--1790, 2013.
Saker, S. H., Osman, M. M., O'Regan, D., Agarwal, R. P., Some
new Opial dynamic inequalities with weighted functions on time scales, Math.
Ineq. Appl. 18 (3), 1171--1187, 2015.
Tiryaki, A., Recent development of Lyapunov-type inequalities,
Advances in Dyn. Syst. Appl. 5 (2), 231--248, 2010.
Wintner, A., On the nonexistence of conjugate points, Amer. J.
Math. 73, 368--380, 1951.
Yang, X., On inequalities of Lyapunov type, Appl. Math. Comp.
134, 293--300, 2003.
Saker, S. H., Osman, M. M., O’regan, D., Agarwal, R. P. (2018). Lyapunov inequalities for dynamic equations via new Opial type inequalities. Hacettepe Journal of Mathematics and Statistics, 47(6), 1537-1551.
AMA
Saker SH, Osman MM, O’regan D, Agarwal RP. Lyapunov inequalities for dynamic equations via new Opial type inequalities. Hacettepe Journal of Mathematics and Statistics. December 2018;47(6):1537-1551.
Chicago
Saker, S. H., M. M. Osman, D. O’regan, and R. P. Agarwal. “Lyapunov Inequalities for Dynamic Equations via New Opial Type Inequalities”. Hacettepe Journal of Mathematics and Statistics 47, no. 6 (December 2018): 1537-51.
EndNote
Saker SH, Osman MM, O’regan D, Agarwal RP (December 1, 2018) Lyapunov inequalities for dynamic equations via new Opial type inequalities. Hacettepe Journal of Mathematics and Statistics 47 6 1537–1551.
IEEE
S. H. Saker, M. M. Osman, D. O’regan, and R. P. Agarwal, “Lyapunov inequalities for dynamic equations via new Opial type inequalities”, Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 6, pp. 1537–1551, 2018.
ISNAD
Saker, S. H. et al. “Lyapunov Inequalities for Dynamic Equations via New Opial Type Inequalities”. Hacettepe Journal of Mathematics and Statistics 47/6 (December 2018), 1537-1551.
JAMA
Saker SH, Osman MM, O’regan D, Agarwal RP. Lyapunov inequalities for dynamic equations via new Opial type inequalities. Hacettepe Journal of Mathematics and Statistics. 2018;47:1537–1551.
MLA
Saker, S. H. et al. “Lyapunov Inequalities for Dynamic Equations via New Opial Type Inequalities”. Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 6, 2018, pp. 1537-51.
Vancouver
Saker SH, Osman MM, O’regan D, Agarwal RP. Lyapunov inequalities for dynamic equations via new Opial type inequalities. Hacettepe Journal of Mathematics and Statistics. 2018;47(6):1537-51.