Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities

We present new oscillation criteria for the second order forced ordinary differential equation with mixed nonlinearities:(p(t)x′)′+q(t)x+∑i=1nqi(t)|x|αisgnx=e(t), where p(t),q(t),qi(t),e(t)∈C[0,∞), p(t)p(t) is positive and differentiable, α1>⋯>αm>1>αm+1>⋯>αnα1>⋯>αm>1>αm+1>⋯>αn. No restriction is imposed on the forcing term e(t)e(t) to be the second derivative of an oscillatory function. When n=1n=1, our results reduce to those of El-Sayed [M.A. El-Sayed, An oscillation criterion for a forced second order linear differential equation, Proc. Amer. Math. Soc. 118 (1993) 813–817], Wong [J.S.W. Wong, Oscillation criteria for a forced second linear differential equations, J. Math. Anal. Appl. 231 (1999) 235–240], Sun, Ou and Wong [Y.G. Sun, C.H. Ou, J.S.W. Wong, Interval oscillation theorems for a linear second order differential equation, Comput. Math. Appl. 48 (2004) 1693–1699] for the linear equation, Nazr [A.H. Nazr, Sufficient conditions for the oscillation of forced super-linear second order differential equations with oscillatory potential, Proc. Amer. Math. Soc. 126 (1998) 123–125] for the superlinear equation, and Sun and Wong [Y.G. Sun, J.S.W. Wong, Note on forced oscillation of nth-order sublinear differential equations, J. Math. Anal. Appl. 298 (2004) 114–119] for the sublinear equation.