## Oscillation of solutions of second order mixed nonlinear differential equations under impulsive perturbations.(English)Zbl 1217.34055

Summary: New oscillation criteria are obtained for second order forced mixed nonlinear impulsive differential equations of the form $(r(t)\Phi_{\alpha}(x'))'q(t)\Phi_{\alpha}(x)+\sum_{k=1}^nq_k(t)\Phi_{\beta_k(x)}=e(t),~t\neq\theta_i$
$x(\theta_i^+)=a_ix(\theta),~x'(\theta_i^+)=b_ix'(\theta_i)$ where $$\Phi _{\gamma }(s):=|s|^{\gamma - 1}s$$ and $$\beta_{1}>\beta_{2}>\dots>\beta_{m}>\alpha >\beta_{m+1}>\dots>\beta_{n}>0$$. If $$\alpha=1$$ and the impulses are dropped, then the results obtained by Y. G. Sun and J. S. W. Wong [J. Math. Anal. Appl. 334, No. 1, 549–560 (2007; Zbl 1125.34024)] are recovered. Examples are given to illustrate the results.

### MSC:

 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34A37 Ordinary differential equations with impulses

Zbl 1125.34024
Full Text:

### References:

 [1] Ballinger, G.; Liu, X., Permanence of population growth models with impulsive effects, Math. comput. modelling, 26, (1997) · Zbl 1185.34014 [2] Lu, Z.; Chi, X.; Chen, L., Impulsive control strategies in biological control of pesticide, Theor. popul. biol., 64, (2003) · Zbl 1100.92071 [3] Sun, J.; Qiao, F.; Wu, Q., Impulsive control of a financial model, Phys. lett. A, 335, 282-288, (2005) · Zbl 1123.91325 [4] Tang, S.; Chen, L., Global attractivity in a “food-limited” population model with impulsive effect, J. math. anal. appl., 292, 211-221, (2004) · Zbl 1062.34055 [5] Tang, S.; Xiao, Y.; Clancy, D., New modelling approach concerning integrated disease control and cost-effectivity, Nonlinear anal., 63, 439-471, (2005) · Zbl 1078.92059 [6] Zhang, Y.; Xiu, Z.; Chen, L., Dynamics complexity of a two-prey one-predator system with impulsive effect, Chaos solitons fractals, 26, 131-139, (2005) · Zbl 1076.34055 [7] Bainov, D.D.; Domshlak, Yu.I.; Simeonov, P.S., Sturmian comparison theory for impulsive differential inequalities and equations, Arch. math. (basel), 67, 35-49, (1996) · Zbl 0856.34033 [8] Chen, Y.S.; Feng, W.Z., Oscillations of second order nonlinear ODE with impulses, J. math. anal. appl., 210, 150-169, (1997) · Zbl 0877.34014 [9] Gopalsamy, K.; Zhang, B.G., On delay differential equations with impulses, J. math. anal. appl., 139, 110-122, (1989) · Zbl 0687.34065 [10] Luo, J., Second-order quasilinear oscillation with impulses, Comput. math. appl., 2-3, 46, 279-291, (2003) · Zbl 1063.34004 [11] Özbekler, A.; Zafer, A., Sturmian comparison theory for linear and half-linear impulsive differential equations, Nonlinear anal., 63, 289-297, (2005) · Zbl 1159.34306 [12] Özbekler, A.; Zafer, A., Picone’s formula for linear non-selfadjoint impulsive differential equations, J. math. anal. appl., 319, 410-423, (2006) · Zbl 1100.34012 [13] Özbekler, A.; Zafer, A., Forced oscillation of super-half-linear impulsive differential equations, Comput. math. appl., 54, 785-792, (2007) · Zbl 1141.34024 [14] Özbekler, A.; Zafer, A., Interval criteria for forced oscillation of super-half-linear differential equations under impulse effect, Math. comput. modelling, 50, 59-65, (2009) · Zbl 1185.34039 [15] Shen, J., Qualitative properties of solutions of second-order linear ODE with impulses, Math. comput. modelling, 40, 337-344, (2004) · Zbl 1061.34022 [16] Liu, X.; Xu, Z., Oscillation of a forced super-linear second order differential equations with impulses, Comput. math. appl., 53, 1740-1749, (2007) · Zbl 1152.34331 [17] 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 [18] El-Sayed, M.A., An oscillation criterion for a forced-second order linear differential equation, Proc. amer. math. soc., 3, 118, 813-817, (1993) · Zbl 0777.34023 [19] Nasr, A.H., Sufficient conditions for the oscillation of forced super-linear second order differential equations with oscillatory potential, Proc. amer. math. soc., 1, 126, 123-125, (1998) · Zbl 0891.34038 [20] Sun, Y.G.; Wong, J.S.W., Note on forced oscillation of $$n$$th-order sublinear differential equations, J. math. anal. appl., 298, 114-119, (2004) · Zbl 1064.34020 [21] Wong, J.S.W., Oscillation criteria for forced second-order linear differential equation, J. math. anal. appl., 231, 235-240, (1999) · Zbl 0922.34029 [22] Agarwal, R.P.; Grace, S.R.; O’Regan, D., Oscillation theory for second order linear, half-linear, superlinear and sublinear dynamic equations, (2002), Kluwer Academic Publishers Netherlands · Zbl 1073.34002 [23] Beckenbach, E.F.; Bellman, R., Inequalities, (1961), Springer Berlin · Zbl 0206.06802
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.