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:x s ( i + 2 , j ) + a sx s + 1 ( i + 1 , j + 1 ) + b sx s ( i , j + 2 ) = 0 ,s = 1 , 2 , . . . , n − 1 ,x n ( i + 2 , j ) + a nx 1 ( i + 1 , j + 1 ) + b nx n ( i , j + 2 ) = 0and the system of partial linear nonhomogeneous difference equations:y s ( i + 2 , j ) + a sy s + 1 ( i + 1 , j + 1 ) + b sy s ( i , j + 2 ) = f s ( i , j ) ,s = 1 , 2 , . . . , n − 1 ,y n ( i + 2 , j ) + a ny 1 ( i + 1 , j + 1 ) + b ny n ( i , j + 2 ) = f n ( i , j )where n = 2 , 3 , . . . ,  x s ( 0 , j ) = ϕ s ( j ) ,  j = 2 , 3 , . . . ,  x s ( 1 , j ) = ψ s ( j ) , j = 1 , 2 , . . .  (resp.y s ( 0 , j ) = ϕ s ( j ) ,  j = 2 , 3 , . . . ,  y s ( 1 , j ) = ψ s ( j ) , j = 1 , 2 , . . . ) for the first system (resp. for the second system);a s ,b s , are real constants;f s : N 2 → R are known functions;  ϕ s ( j ) , ψ s ( j ) are given sequences; and   s = 1 , 2 , . . . , n and the domain of the solutions of the above systems are the setsN m = { ( 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 → ∞ uniformly with respect to j . Moreover, we study the asymptotic stability of the trivial solution of the first system. In addition, under some conditions onf s , we prove that every solution of the second system is bounded, and finally, we find conditions onf sso that every solution of the second system converges to 0 as i → ∞ uniformly with respect to j .