Necessary and Sufficient Condition for Oscillation of Higher Order Nonlinear Delay Difference Equations

-This paper is concerned with the oscillation of solutions of higher order nonlinear delay difference equations with forcing terms of the form Anx(t) + f(t, x(t),x(a(t))) = h(t), t E I = {0, 1 . . . . }, where A is the forward difference operator defined by Ax(t) = x(t + 1) -x(t) and A'nx(t) = A(Am-lx(t)), m > 1. A necessary and sufficient condition is established under which every solution x(t) is oscillatory when n is even, and is either oscillatory or strongly monotone when n is odd. A sufficient condition for the oscillation of solutions of neutral type delay difference equations is also obtained. (~) 1998 Elsevier Science Ltd. All rights reserved. K e y w o r d s D e l a y difference equation, Oscillation, Forcing term. 1. I N T R O D U C T I O N The problem of oscillation and nonoscillation of solutions of difference equations has received a great deal of at tent ion during the last few years. Much of the work, however, has been restricted to first and second order homogeneous difference equations and provides only sufficient conditions for oscillation (see, e.g., [1-8] and the references cited therein). There is considerably a limited number of works concerning the oscillation of higher order difference equations. For some results, we refer in part icular to [9-14]. Here in this article we deal with higher order nonlinear delay difference equations with forcing terms of the form A n x ( t ) + f ( t , x ( t ) , x ( a ( t ) ) ) = h( t ) , t e I = {0, 1, 2 , . . . }, (1) and establish a necessary and sufficient condition for the oscillation. We also apply our technique to obtain new oscillation criteria for solutions of neutral type delay difference equations of the form An[x(t) + a ( t ) x ( v ( t ) ) ] + f ( t , x ( t ) , x ( a ( t ) ) ) = h( t ) , t e I = {0, 1 , 2 , . . . }. (2) Wi th regard to equations (1) and (2), the following conditions are assumed to hold without further mention: (a) a, r : I --* I , a ( t ) <_ t, r ( t ) < t, and limt-.oo a( t ) = limt-.oo r ( t ) = oo; (b) h : I --* R; Typeset by .Aj~S-TEX 125