## Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities.(English)Zbl 1125.34024

The authors introduce general oscillation criteria for the second order ordinary differential equation $\big(p(t)\,x'\big)'+q(t)\,x+\sum_{i=1}^nq_i(t)\,| x| ^{\alpha_1}\,\mathrm{sgn}\,x=e(t),$ where $$p,q,q_i,e\in C[0,\infty)$$, $$p(t)>0$$ and differentiable (but this assumption on the existence of $$p'(t)$$ is apparently not needed), $$\alpha_1>\dots>\alpha_m>1>\alpha_{m+1}>\dots>\alpha_n$$, and no restriction is invoked on the forcing term $$e(t)$$. Note that the equation contains both sublinear and superlinear terms due to the assumptions on the exponents $$\alpha_i$$. The main results (Theorems 1–3) are derived via the Riccati technique and generalize several results in the literature.

### MSC:

 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

### Keywords:

Forced oscillation; Nonlinear differential equation
Full Text:

### References:

 [1] Agarwal, R.P.; Grace, S.R., Oscillation theory for difference and functional differential equations, (2002), Kluwer Academic Dordrecht · Zbl 1061.34047 [2] 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 [3] Beckenbach, E.F.; Bellman, R., Inequalities, (1961), Springer Berlin · Zbl 0513.26003 [4] Butler, G.J., Oscillation theorems for a nonlinear analogue for Hill’s equation, Q. J. math. (Oxford), 27, 159-171, (1976) · Zbl 0341.34018 [5] Butler, G.J., Integral averages and oscillation of second order nonlinear differential equations, SIAM J. math. anal., 11, 190-200, (1980) · Zbl 0424.34033 [6] Coffman, C.V.; Wong, J.S.W., Oscillation and nonoscillation of solutions of generalized emden – fowler equations, Trans. amer. math. soc., 167, 399-434, (1972) · Zbl 0278.34026 [7] El-Sayed, M.A., An oscillation criterion for a forced second order linear differential equation, Proc. amer. math. soc., 118, 813-817, (1993) · Zbl 0777.34023 [8] Kartsatos, A.G., On the maintenance of oscillation of nth order equations under the effect of a small forcing term, J. differential equations, 10, 355-363, (1971) · Zbl 0211.11902 [9] Kartsatos, A.G., Maintenance of oscillations under the effect of a periodic forcing term, Proc. amer. math. soc., 33, 377-383, (1972) · Zbl 0234.34040 [10] Keener, M.S., Solutions of a certain linear nonhomogeneous second order differential equations, Appl. anal., 1, 57-63, (1971) · Zbl 0215.43802 [11] Kwong, M.K.; Wong, J.S.W., Linearization of second order nonlinear oscillation theorems, Trans. amer. math. soc., 279, 705-722, (1983) · Zbl 0544.34024 [12] Nazr, 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 [13] Ou, C.H.; Wong, J.S.W., Forced oscillation of nth order functional differential equations, J. math. anal. appl., 262, 722-731, (2001) · Zbl 0997.34059 [14] Philos, Ch.G., Oscillation theorems for linear differential equations, J. math. anal. appl., 53, 483-492, (1989) · Zbl 0661.34030 [15] Rankin, S.M., Oscillation theorems for second order nonhomogeneous linear differential equations, J. math. anal. appl., 53, 550-553, (1976) · Zbl 0328.34033 [16] Skidmore, A.; Bowers, J.J., Oscillatory behaviour of solutions of $$y'' + p(x) y = f(x)$$, J. math. anal. appl., 49, 317-323, (1975) · Zbl 0312.34025 [17] Skidmore, A.; Leighton, W., On the differential equation $$y'' + p(x) y = f(x)$$, J. math. anal. appl., 43, 45-55, (1973) · Zbl 0287.34031 [18] Sun, Y.G., A note on nazr’s and Wong’s papers, J. math. anal. appl., 286, 363-367, (2003) · Zbl 1042.34096 [19] Sun, Y.G.; Wong, J.S.W., Note on forced oscillation of nth-order sublinear differential equations, J. math. anal. appl., 298, 114-119, (2004) [20] Sun, Y.G.; Ou, C.H.; Wong, J.S.W., Interval oscillation theorems for a linear second order differential equation, Comput. math. appl., 48, 1693-1699, (2004) · Zbl 1069.34049 [21] Sun, Y.G.; Agarwal, R.P., Forced oscillation of nth order nonlinear differential equations, J. funct. differ. equ., 9, 587-596, (2004) · Zbl 1060.34019 [22] Sun, Y.G.; Agarwal, R.P., Interval oscillation criteria for higher order forced nonlinear differential equations, Nonlinear funct. anal. appl., 9, 441-449, (2004) · Zbl 1075.34031 [23] Teufel, H., Forced second order nonlinear oscillations, J. math. anal. appl., 40, 148-152, (1972) · Zbl 0211.12001 [24] Wong, J.S.W., Second order nonlinear forced oscillations, SIAM J. math. anal., 19, 667-675, (1988) · Zbl 0655.34023 [25] Wong, J.S.W., Oscillation criteria for a forced second linear differential equations, J. math. anal. appl., 231, 235-240, (1999) · Zbl 0922.34029 [26] Yang, Q., Interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential, Appl. math. comput., 136, 49-64, (2003) · Zbl 1030.34034 [27] Yang, X., Forced oscillation of nth order nonlinear differential equations, Appl. math. comput., 134, 301-305, (2003) · Zbl 1033.34046
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.