Trichotomy of a system of two difference equations

In this paper we study the boundedness and the asymptotic behavior of the positive solutions of the system of difference equations xn+1=A+<NU>∑i=1kaixn−pi</NU>/∑j=1mbjyn−qj</fr>,<hsp sp="2">yn+1=B+<fr SHAPE="BUILT" ALIGN="C" STYLE="S"><NU>∑i=1kciyn−pi</NU>/∑j=1mdjxn−qj, yn+1=B+<NU>∑i=1kciyn−pi</NU>/∑j=1mdjxn−qj, where <f>k,m∈{1,2,…}</f>, <f>A,B,ai,ci,bj,dj</f>, <f>i∈{1,…,k}</f>, <f>j∈{1,…,m}</f>, are positive constants, <f>pi</f>, <f>qj</f>, <f>i∈{1,…,k}</f>, <f>j∈{1,…,m}</f>, are positive integers such that <f>p1<p2<⋯<pk</f>, <f>q1<q2<⋯<qm</f> and the initial values <f>xi,yi</f>, <f>i∈{−π,−π+1,…,0}</f>, <f>π=max{pk,qm}</f> are positive numbers. [Copyright &y& Elsevier]