Article history: Received 5 October 2010 Received in revised form 28 March 2011 Accepted 14 April 2011 Available online 20 April 2011

Article history: Received 8 April 2013 Received in revised form 2 June 2014 Accepted 18 June 2014 Available online 8 July 2014 Communicated by H. Prodinger

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.