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

Sun, Yuan Gong; Wong, James S. W.

J. Math. Anal. Appl. 334, No. 1, 549-560 (2007).

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.

Reviewer: Roman Hilscher (Brno)

MSC:

34C10

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

Keywords:

Forced oscillation; Nonlinear differential equation

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