In this paper we consider the following system of partial linear homogeneous difference equations:
xsfalse(i+2,jfalse)+asxs+1false(i+1,j+1false)+bsxsfalse(i,j+2false)=0,s=1,2,...,n−1,xnfalse(i+2,jfalse)+anx1false(i+1,j+1false)+bnxnfalse(i,j+2false)=0
and the system of partial linear nonhomogeneous difference equations:
ysfalse(i+2,jfalse)+asys+1false(i+1,j+1false)+bsysfalse(i,j+2false)=fsfalse(i,jfalse),s=1,2,...,n−1,ynfalse(i+2,jfalse)+any1false(i+1,j+1false)+bnynfalse(i,j+2false)=fnfalse(i,jfalse)
where
n=2,3,...,
xsfalse(0,jfalse)=ϕsfalse(jfalse),
j=2,3,...,
xsfalse(1,jfalse)=ψsfalse(jfalse),1emj=1,2,... (resp.
ysfalse(0,jfalse)=ϕsfalse(jfalse),
j=2,3,...,
ysfalse(1,jfalse)=ψsfalse(jfalse),1emj=1,2,...) for the first system (resp. for the second system);
as,
bs, are real constants;
fs:double-struckN2→double-struckR are known functions;
ϕsfalse(jfalse),ψsfalse(jfalse) are given sequences; and
s=1,2,...,n and the domain of the solutions of the above systems are the sets
Nm=false{false(i,jfalse),1emi+j=mfalse},
m=2,3,.... More precisely, we find conditions so that every solution of the first system converges to 0 as
i→∞ uniformly with respect to
j. Moreover, we study the asymptotic stability of the trivial solution of the first system. In addition, under some conditions on
fs, we prove that every solution of the second system is bounded, and finally, we find conditions on
fs so that every solution of the second system converges to 0 as
i→∞ uniformly with respect to
j.
