A discrete n-species Schoener competition system with time delays and feedback controls is proposed. By applying the comparison theorem of difference equation, sufficient conditions are obtained for the permanence of the system.

and Applied Analysis 3 In the sequel, let q > 1 4p and . . . , φ, ψ, φ, ψ, . . . the unique prime period-two solution of 1.1 with φ < ψ. Define f ∈ C φ, ψ × φ, ψ , φ, ψ by f ( x, y ) p y qx y 2.2 for any x, y ∈ φ, ψ and g ∈ C φ, ψ , φ, ψ by y∗ g ( y ) p y − y2 qy 2.3 for any y ∈ φ, ψ . Then

In this work we define Ψ -boundedness of sequences and discuss the existence of Ψ -bounded solutions for linear difference equations. We present a necessary and sufficient condition for the existence of Ψ -bounded solutions for the linear nonhomogeneous difference equation x(n + 1) = A(n)x(n)+ f (n). c © 2006 Elsevier Ltd. All rights reserved.