##
**Interval oscillation criteria for second-order dynamic equations with nonlinearities given by Riemann-Stieltjes integrals.**
*(English)*
Zbl 1226.26020

Summary: By using a generalized arithmetic-geometric mean inequality on time scales, we study the forced oscillation of second-order dynamic equations with nonlinearities given by Riemann-Stieltjes integrals of the form

\[ [p(t)\varphi_\alpha(x^\Delta(t))]^\Delta+ q(t)\varphi_\alpha(x(\tau(t)))+ \int^{\sigma(b)}_a r(t, s)\varphi_{\gamma(s)}(x(g(t, s))) \Delta\xi(s) = e(t), \]

where \(t\in [t_0, \infty)_{\mathbb T}= [t_0, \infty)\cap \mathbb T\), \(\mathbb T\) is a time scale which is unbounded from above; \(\varphi_*(u)= |u|^*\operatorname{sgn}u\); \(\gamma: [a,b]_{\mathbb T_1}\rightarrow \mathbb R\) is a strictly increasing right-dense continuous function; \(p,q,e: [t_0,\infty)_{\mathbb T} \rightarrow\mathbb R\), \(r:[t_0, \infty)_{\mathbb T}\times [a,b]_{\mathbb T_1}\rightarrow \mathbb R\), \(\tau:[t_0, \infty)_{\mathbb T}\rightarrow [t_0,\infty)_{\mathbb T}\), and \(g:[t_0,\infty)_{\mathbb T}\times [a, b]_{\mathbb T_1}\rightarrow [t_0, \infty)_{\mathbb T}\) are right-dense continuous functions; \(\xi:[a, b]_{\mathbb T_1} \rightarrow \mathbb R\) is strictly increasing. Some interval oscillation criteria are established in both cases of delayed and advanced arguments. As a special case, the work in this paper unifies and improves many existing results in the literature for equations with a finite number of nonlinear terms.

\[ [p(t)\varphi_\alpha(x^\Delta(t))]^\Delta+ q(t)\varphi_\alpha(x(\tau(t)))+ \int^{\sigma(b)}_a r(t, s)\varphi_{\gamma(s)}(x(g(t, s))) \Delta\xi(s) = e(t), \]

where \(t\in [t_0, \infty)_{\mathbb T}= [t_0, \infty)\cap \mathbb T\), \(\mathbb T\) is a time scale which is unbounded from above; \(\varphi_*(u)= |u|^*\operatorname{sgn}u\); \(\gamma: [a,b]_{\mathbb T_1}\rightarrow \mathbb R\) is a strictly increasing right-dense continuous function; \(p,q,e: [t_0,\infty)_{\mathbb T} \rightarrow\mathbb R\), \(r:[t_0, \infty)_{\mathbb T}\times [a,b]_{\mathbb T_1}\rightarrow \mathbb R\), \(\tau:[t_0, \infty)_{\mathbb T}\rightarrow [t_0,\infty)_{\mathbb T}\), and \(g:[t_0,\infty)_{\mathbb T}\times [a, b]_{\mathbb T_1}\rightarrow [t_0, \infty)_{\mathbb T}\) are right-dense continuous functions; \(\xi:[a, b]_{\mathbb T_1} \rightarrow \mathbb R\) is strictly increasing. Some interval oscillation criteria are established in both cases of delayed and advanced arguments. As a special case, the work in this paper unifies and improves many existing results in the literature for equations with a finite number of nonlinear terms.

### MSC:

26E70 | Real analysis on time scales or measure chains |

PDF
BibTeX
XML
Cite

\textit{Y. Sun}, Abstr. Appl. Anal. 2011, Article ID 719628, 14 p. (2011; Zbl 1226.26020)

Full Text:
DOI

### References:

[1] | S. Hilger, “Analysis on measure chains-A unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18-56, 1990. · Zbl 0722.39001 |

[2] | M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2001. · Zbl 0993.39010 |

[3] | M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. · Zbl 1025.34001 |

[4] | Y. G. Sun and J. S. W. Wong, “Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 549-560, 2007. · Zbl 1125.34024 |

[5] | M. A. El-Sayed, “An oscillation criterion for a forced second order linear differential equation,” Proceedings of the American Mathematical Society, vol. 118, no. 3, pp. 813-817, 1993. · Zbl 0777.34023 |

[6] | Y. G. Sun and J. S. W. Wong, “Note on forced oscillation of nth-order sublinear differential equations,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 114-119, 2004. · Zbl 1064.34020 |

[7] | A. H. Nasr, “Sufficient conditions for the oscillation of forced super-linear second order differential equations with oscillatory potential,” Proceedings of the American Mathematical Society, vol. 126, no. 1, pp. 123-125, 1998. · Zbl 0891.34038 |

[8] | Ch. G. Philos, “Oscillation theorems for linear differential equations of second order,” Archiv der Mathematik, vol. 53, no. 5, pp. 482-492, 1989. · Zbl 0661.34030 |

[9] | Y. G. Sun, “A note on Nasr’s and Wong’s papers,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 363-367, 2003. · Zbl 1042.34096 |

[10] | Y. G. Sun and F. W. Meng, “Interval criteria for oscillation of second-order differential equations with mixed nonlinearities,” Applied Mathematics and Computation, vol. 198, no. 1, pp. 375-381, 2008. · Zbl 1141.34317 |

[11] | Q. Kong, “Interval criteria for oscillation of second-order linear ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 229, no. 1, pp. 258-270, 1999. · Zbl 0924.34026 |

[12] | R. P. Agarwal and A. Zafer, “Oscillation criteria for second-order forced dynamic equations with mixed nonlinearities,” Advances in Difference Equations, vol. 2009, Article ID 938706, 20 pages, 2009. · Zbl 1181.34099 |

[13] | R. P. Agarwal, D. R. Anderson, and A. Zafer, “Interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 977-993, 2010. · Zbl 1197.34117 |

[14] | Y. Sun and Q. Kong, “Interval criteria for forced oscillation with nonlinearities given by RiemannStieltjes integrals,” Computers and Mathematics with Applications, vol. 62, no. 1, pp. 243-252, 2011. · Zbl 1228.34055 |

[15] | Y. G. Sun and F. W. Meng, “Oscillation of second-order delay differential equations with mixed nonlinearities,” Applied Mathematics and Computation, vol. 207, no. 1, pp. 135-139, 2009. · Zbl 1171.34338 |

[16] | D. R. Anderson, “Oscillation of second-order forced functional dynamic equations with oscillatory potentials,” Journal of Difference Equations and Applications, vol. 13, no. 5, pp. 407-421, 2007. · Zbl 1123.34051 |

[17] | A. Zafer, “Interval oscillation criteria for second order super-half linear functional differential equations with delay and advanced arguments,” Mathematische Nachrichten, vol. 282, no. 9, pp. 1334-1341, 2009. · Zbl 1180.34070 |

[18] | D. R. Anderson and A. Zafer, “Interval criteria for second-order super-half-linear functional dynamic equations with delay and advance arguments,” Journal of Difference Equations and Applications, vol. 16, no. 8, pp. 917-930, 2010. · Zbl 1205.34126 |

[19] | S. Sun, T. Li, Z. Han, and Y. Sun, “Oscillation of second-order neutral functional differential equations with mixed nonlinearities,” Abstract and Applied Analysis, vol. 2011, Article ID 927690, 15 pages, 2011. · Zbl 1210.34094 |

[20] | Y. Bai and L. Liu, “New oscillation criteria for second-order delay differential equations with mixed nonlinearities,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 796256, 2010 pages, 2010. · Zbl 1205.34079 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.