Interval oscillation criteria for second-order forced impulsive differential equations with mixed nonlinearities. (English) Zbl 1233.34013
By using an arithmetic-geometric mean inequality and the Riccati transformation, interval oscillation criteria are established for second-order forced impulsive differential equations with mixed nonlinearities of the form
\[
\begin{cases} (r(t)\Phi_\alpha(x'(t)))'+p_0(t)\Phi_\alpha(x(t))+\sum_{i=1}^n p_i(t)\Phi_{\beta_i}(x(t))=e(t),\;t\neq\tau_k,\\ x(\tau_k^+)=a_kx(\tau_k),\quad x'(\tau_k^+)=b_kx'(\tau_k),\end{cases}
\]
where \(t\geq t_0\), \(k\in\mathbb{N}\); \(\Phi_*(u)=|u|^{*-1}u\), \(\{\tau_k\}\) is the impulse moments sequence with \(0\leq t_0=\tau_0<\tau_1<\dots<\tau_k<\dots\) and \(\lim_{k\to\infty}\tau_k=\infty\); \(\alpha=p/q\), \(p\), \(q\) are odd, and the exponents satisfy \(\beta_1>\dots>\beta_m>\alpha>\beta_{m+1}>\dots>\beta_n>0\). Some known results are generalized and improved. Examples are also given to illustrate the effectiveness and non-emptiness of the results.
Reviewer: Qingkai Kong (DeKalb)
MSC:
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 34A37 | Ordinary differential equations with impulses |
References:
| [1] | Agarwal, R. P.; Grace, S. R.; OʼRegan, D., Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations (2002), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 1073.34002 |
| [2] | Fite, W. B., Concerning the zeros of the solutions of certain differential equations, Trans. Amer. Math. Soc., 19, 341-352 (1918) · JFM 46.0702.02 |
| [3] | Wintner, A., A criterion of oscillatory stability, Quart. Appl. Math., 7, 115-117 (1949) · Zbl 0032.34801 |
| [4] | Hartman, P., On nonoscillatory linear differential equations of second order, Amer. J. Math., 74, 389-400 (1952) · Zbl 0048.06602 |
| [5] | Kamenev, I. V., An integral criterion for oscillation of nonlinear differential equations of second order, Mat. Zametki. Mat. Zametki, Math. Notes, 23, 136-138 (1978), translation in: · Zbl 0408.34031 |
| [6] | Philos, Ch. G., Oscillation theorems for linear differential equations of second order, Arch. Math., 53, 482-492 (1989) · Zbl 0661.34030 |
| [7] | Kwong, M. K.; Zettl, A., Integral inequalities and second linear oscillation, J. Differential Equations, 45, 16-33 (1982) · Zbl 0498.34022 |
| [8] | El-Sayed, M. A., An oscillation criteria for forced second order linear differential equations, Proc. Amer. Math. Soc., 118, 813-817 (1993) · Zbl 0777.34023 |
| [9] | Wong, J. S.W., Oscillation criteria for a forced second order linear differential equation, J. Math. Anal. Appl., 231, 235-240 (1999) · Zbl 0922.34029 |
| [10] | Kong, Q., Interval criteria for oscillation of second-order linear differential equation, J. Math. Anal. Appl., 229, 483-492 (1999) |
| [11] | Li, W. T.; Agarwal, R. P., Interval oscillation criteria for second order nonlinear differential equations with damping, Comput. Math. Appl., 40, 217-230 (2000) · Zbl 0959.34026 |
| [12] | Li, W. T.; Agarwal, R. P., Interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations, J. Math. Anal. Appl., 245, 171-188 (2000) · Zbl 0983.34020 |
| [13] | Çakmak, D., Interval Integral averaging technique for the interval oscillation criteria of certain second-order nonlinear differential equations, J. Math. Anal. Appl., 300, 408-425 (2004) · Zbl 1082.34030 |
| [14] | Wong, Q.; Yang, Q., Interval criteria for oscillation of second-order half-linear differential equations, J. Math. Anal. Appl., 291, 224-236 (2004) · Zbl 1053.34034 |
| [15] | Wong, Q., Interval criteria for oscillation of second-order nonlinear differential equations, J. Comput. Anal. Math., 205, 231-238 (2007) · Zbl 1142.34331 |
| [16] | Agarwal, R. P.; Anderson, D. R.; Zafer, A., Interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities, Comput. Math. Appl., 59, 977-993 (2010) · Zbl 1197.34117 |
| [17] | Beckenbach, E. F.; Bellman, R., Inequalities (1961), Springer: Springer Berlin · Zbl 0513.26003 |
| [18] | Sun, Y. G.; Wong, J. S.W., Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl., 334, 549-560 (2007) · Zbl 1125.34024 |
| [19] | Sun, Y. G.; Meng, F. W., Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput., 15, 375-381 (2008) · Zbl 1141.34317 |
| [20] | Nasr, A. H., Sufficient conditions for the oscillation of forced super-linear second order differential equations with oscillatory potential, Proc. Amer. Math. Soc., 126, 123-125 (1998) · Zbl 0891.34038 |
| [21] | Liu, X.; Xu, Z., Oscillation criteria for a forced mixed type Emden-Fowler equation with impulses, Appl. Math. Comput., 215, 283-291 (2009) · Zbl 1193.34067 |
| [22] | Yang, Q., Interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential, Appl. Math. Comput., 135, 49-64 (2003) · Zbl 1030.34034 |
| [23] | Hardy, G. H.; Littlewood, J. E.; Polya, G., Inequalities (1964), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0634.26008 |
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