Journal of Applied Analysis Vol. 6, No. 1 (2000), pp. 87–103 OSCILLATION CRITERIA OF COMPARISON TYPE FOR SECOND ORDER DIFFERENCE EQUATIONS S.R. GRACE and H.A. EL-MORSHEDY * Received March 23, 1998 Abstract. In this paper we investigate the oscillatory character of the second order nonlinear difference equations of the forms Δ(c n-1 Δ(x n-1 + p n x σn )) + q n f (x τn )=0, n =1, 2,... and the corresponding nonhomogeneous equation Δ(c n-1 Δ(x n-1 + p n x σn )) + q n f (x τn )= r n , n =1, 2,... via comparison with certain second order linear difference equations where the function f is not necessarily monotonic. The results of this paper are essentially new and can be extended to more general equa- tions. 1. Introduction In this article we are concerned with nonlinear difference equations of the forms Δ(c n−1 Δ(x n−1 + p n x σ n )) + q n f (x τ n )=0, n =1, 2,... (1.1) 1991 Mathematics Subject Classification. 39A10. Key words and phrases. Difference equations, forced equations, oscillation. * On leave from Department of Mathematics, Damietta Faculty of Science, New Damietta 34517, Egypt ISSN 1425-6908 c  Heldermann Verlag.