Abstract
We study nonoscillation/oscillation of the dynamic equation
where \({t_0 \in \mathbb{T}}\), \({{\rm sup} \mathbb{T} = \infty}\), \({r \in {\rm C}_{\rm rd}([t_0, \infty)_{\mathbb{T}}, \mathbb{R}^+)}\), \({p \in {\rm C}_{\rm rd}([t_0, \infty)_{\mathbb{T}}, {\mathbb{R}^+_0})}\). By using the Riccati substitution technique, we construct a sequence of functions which yields a necessary and sufficient condition for the nonoscillation of the equation. In addition, our results are new in the theory of dynamic equations and not given in the discrete case either. We also illustrate applicability and sharpness of the main result with a general Euler equation on arbitrary time scales. We conclude the paper by extending our results to the equation
which is extensively discussed on time scales.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bohner M., Erbe L., Peterson A.: Oscillation for nonlinear second order dynamic equations on a time scale. J. Math. Anal. Appl. 301, 491–507 (2005)
Bohner M., Peterson A.: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Boston (2001)
E. Braverman and B. Karpuz, Nonoscillation of second-order dynamic equations with several delays. Abstr. Appl. Anal. (2011), Art. ID 591254, 34 pp.
Erbe L.: Oscillation criteria for second order linear equations on a time scale. Canad. Appl. Math. Quart. 9, 345–375 (2001)
Erbe L., Peterson A., Řehák P.: Comparison theorems for linear dynamic equations on time scales. J. Math. Anal. Appl. 275, 418–438 (2002)
Erbe L., Peterson A., Saker S. H.: Oscillation criteria for second-order nonlinear dynamic equations on time scales. J. Lond. Math. Soc. (2) 67, 701–714 (2003)
Fite W. B.: Concerning the zeros of the solutions of certain differential equations. Trans. Amer. Math. Soc. 19, 341–352 (1918)
Guseinov G.: Integration on time scales. J. Math. Anal. Appl. 285, 107–127 (2003)
Guseinov G., Kaymakçalan B.: On a disconjugacy criterion for second order dynamic equations on time scales. J. Comput. Appl. Math. 141, 187–196 (2002)
Hille E.: Non-oscillation theorems. Trans. Amer. Math. Soc. 64, 234–252 (1948)
B. Karpuz, B. Kaymakçalan, Öcalan Ö.: A dynamic generalization for Opial’s inequality and its application to second-order dynamic equations. Differ. Equ. Dyn. Syst. 18, 11–18 (2010)
Kneser A.: Untersuchungen über die reellen Nullstellen der Integrale linearer Differentialgleichungen. Math. Ann. 42, 409–435 (1893)
Kwong M. K., Zettl A.: Asymptotically constant functions and second order linear oscillation. J. Math. Anal. Appl. 93, 475–494 (1983)
Opial Z.: Sur les intégrales oscillantes de l’équation différentielle \({u''+f(t)u = 0}\). Ann. Polon. Math. 4, 308–313 (1958)
P. Řehák, How the constants in Hille-Nehari theorems depend on time scales. Adv. Difference Equ. (2006), Art. ID 64534, 15 pp.
Wintner A.: A criterion of oscillatory stability. Quart. Appl. Math. 7, 115–117 (1949)
Wintner A.: On the non-existence of conjugate points. Amer. J. Math. 73, 368–380 (1951)
Tang X. H., Yu J. S., Peng D. H.: Oscillation and nonoscillation of neutral difference equations with positive and negative coefficients. Comput. Math. Appl. 39, 169–181 (2000)
J. R. Yan, Oscillatory properties of second-order differential equations with an “integralwise small” coefficient. Acta Math. Sinica 30 (1987), 206–215 (in Chinese).
Zafer A.: On oscillation and nonoscillation of second-order dynamic equations. Appl. Math. Lett. 22, 136–141 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Karpuz, B. Nonoscillation and oscillation of second-order linear dynamic equations via the sequence of functions technique. J. Fixed Point Theory Appl. 18, 889–903 (2016). https://doi.org/10.1007/s11784-016-0334-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-016-0334-8