Convergence of solutions of a system of recurrence equations

In this paper, we study the convergence of positive solutions to the close-to-symmetric system of higher-order difference equations x(n+ 1) = x(n)-(2k+1)/1 + y(n- k), y(n+1) = y(n)-(2k+1)/1 + x(n-k), n, k is an element of N-0, (1) where the initials values x-( 2k+1), x-2k,..., x(0), y-( 2k+1), y-2k,..., y(0) are positive real numbers. The obtained results, considerably extending some recent results in the literature.