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=1)Sigma(n) qi (t)vertical bar x vertical bar(alpha i) sgn x = e(t), where p(t), q(t), q(i)(t), e(t) is an element of C[Q, infinity), p(t) is positive and differentiable, alpha(1) > center dot center dot center dot > alpha(m) > 1 > alpha(m+1)> center dot center dot center dot alpha(n). No restriction is imposed on the forcing term e(t) to be the second derivative of an oscillatory function. When n = 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 lin-ear 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. (c) 2006 Elsevier Inc. All rights reserved.