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

Abstract

We present new oscillation criteria for the second order forced ordinary differential equation with mixed nonlinearities: ( p ( t ) x ′ ) ′ + q ( t ) x + ∑ i = 1 n q i ( t ) | x | α i sgn x = e ( t ) , where p ( t ) , q ( t ) , q i ( t ) , e ( t ) ∈ C [ 0 , ∞ ) , p ( t ) is positive and differentiable, α 1 > ⋯ > α m > 1 > α m + 1 > ⋯ > α 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 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.