Asymptotic behaviour of the solutions of systems of partial linear homogeneous and nonhomogeneous difference equations

In this paper we consider the following system of partial linear homogeneous difference equations:xs(i+2,j)+asxs+1(i+1,j+1)+bsxs(i,j+2)=0,s=1,2,...,n-1,xn(i+2,j)+anx1(i+1,j+1)+bnxn(i,j+2)=0and the system of partial linear nonhomogeneous difference equations:ys(i+2,j)+asys+1(i+1,j+1)+bsys(i,j+2)=fs(i,j),s=1,2,...,n-1,yn(i+2,j)+any1(i+1,j+1)+bnyn(i,j+2)=fn(i,j)where n=2,3,..., xs(0,j)=phi s(j), j=2,3,..., xs(1,j)=psi s(j),j=1,2,... (resp. ys(0,j)=phi s(j), j=2,3,..., ys(1,j)=psi s(j),j=1,2,...) for the first system (resp. for the second system); as, bs, are real constants; fs:N2 -> R are known functions; phi s(j),psi s(j) are given sequences; and s=1,2,...,n and the domain of the solutions of the above systems are the sets Nm={(i,j),i+j=m}, m=2,3,.... More precisely, we find conditions so that every solution of the first system converges to 0 as i ->infinity 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 ->infinity uniformly with respect to j.