Journal of Dynamics and Differential Equations, Vol. 10, No. 4, 1998 Bendixson-Dulac Criteria for Difference Equations C. Connell McCluskey1 and James S. Muldowney1,2 Received March 7, 1996 Conditions are given which preclude the existence of a nontrivial periodic orbit for a difference equation in Rn. The conditions are analogous to those of Bendixson and Dulac for autonomous planar differential equations. KEY WORDS: Difference equation; periodic orbit. 1991 MATHEMATICS SUBJECT CLASSIFICATIONS: primary, 39A10; secondary, 58F08, 58F20. 1. INTRODUCTION Consider a difference equation where f is a C1 function with domain and range in Rn and Z+ is the set of nonnegative integers. A solution of (1) is any sequence t x1 which satisfies (1) on a maximal domain [0, ]CZ + . Clearly xt = f'(x0) where f°(x)=x and f't+1=f°ft(x) if ftU) is in the domain of f The orbit of x is the set {f t (x): t € [0, ]}. If 0<w€Z + and f w (x) = x, then .v is a W-periodic point and it is a proper w-periodic point if w is its least period: fr(x) x for each ,0< <w. A 1-periodic point is an equilibrium and a periodic point is nontrivial if it is not an equilibrium. In this paper we obtain conditions which preclude the existence of nontrivial periodic points. These are analogues for difference equations of a result for planar systems of ordinary differential equations obtained by Bendixson (1901), extended by Dulac (1937) and later by Lloyd (1979). 1 Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1. 2 To whom correspondence should be addressed. 567 1040-7294/98/1000-0567$15.00/0 © 1998 Plenum Publishing Corporation