## Oscillation criteria for a forced mixed type Emden-Fowler equation with impulses.(English)Zbl 1193.34067

The authors study the oscillation of the following forced mixed type Emden-Fowler equation with impulses
$\begin{cases} (r(t)x'(t))'+p(t)x(t)+\displaystyle\sum_{i=1}^np_i(t)|x(t)|^{\alpha_i-1}x(t)=e(t), \;t\neq\tau_k,\\ x(\tau_k^{+})=a_kx(\tau_k), \;x'(\tau_k^{+})=b_kx'(\tau_k). \end{cases}\tag{*}$
By employing the arithmetic-geometric mean inequality, the authors obtain some oscillation criteria of for (*). When the impulses are dropped, the results of this paper extend some known results.

### MSC:

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

### Keywords:

Oscillation; impulses; Emden–Fowler equations
Full Text:

### References:

 [1] 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 Dordrecht · Zbl 1073.34002 [2] El-Sayed, E.A., An oscillation criteria for forced second order linear differential equations, Proc. am. math. soc., 11, 813-817, (1993) · Zbl 0777.34023 [3] Cakmak, D.; Tiryaki, A., Oscillation criteria for certain forced second order nonlinear differential equations, Appl. math. lett., 17, 275-279, (2003) · Zbl 1061.34017 [4] Beckenbach, E.F.; Bellman, R., Inequalities, (1961), Springer Berlin · Zbl 0206.06802 [5] Lakshimikantham, V.; Bainov, D.D.; Simieonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Publishers Singapore/New Jersey/London [6] Leighton, W., Comparison theorems for linear differential equation of second order, Proc. am. math. soc., 13, 603-610, (1962) · Zbl 0118.08202 [7] Li, W.T.; Agarwal, R.P., Interval oscillation criteria for a forced second order nonlinear differential equations, Appl. anal., 75, 3-4, 247-341, (2000) [8] Kartsatos, A.G., Maintenance of oscillations under the effect of a periodic differential equations, Proc. am. math. soc., 33, 377-383, (1972) · Zbl 0234.34040 [9] Kong, Q., Interval criteria for oscillation of second-order linear differential equations, J. math. anal. appl., 229, 483-492, (1999) [10] Nasr, A.H., Sufficient conditions for the oscillation of forced super-linear second order differential equations with oscillatory potential, Proc. am. math. soc., 126, 123-125, (1998) · Zbl 0891.34038 [11] Philos, Ch.G., Oscillation theorems for linear differential equations of second order, Arch. math., 53, 482-492, (1989) · Zbl 0661.34030 [12] 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 [13] 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 [14] Wong, J.S.W., Second order nonlinear forced oscillations, SIAM math. anal., 19, 667-675, (1988) · Zbl 0655.34023 [15] 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 [16] 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
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.