Dynamics of a Higher-Order System of Difference Equations

Abstract EN

Consider the following system of difference equations: x n + 1 ( i ) = x n - m + 1 ( i ) / A i ∏ j = 0 m - 1 x n - j ( i + j + 1 ) + α i , x n + 1 ( i + m ) = x n + 1 ( i ) , x 1 - l ( i + l ) = a i , l , A i + m = A i , α i + m = α i , i , l = 1,2 , … , m ; n = 0,1 , 2 , … , where m is a positive integer, A i , α i , i = 1,2 , … , m , and the initial conditions a i , l , i , l = 1,2 , … , m , are positive real numbers.

We obtain the expressions of the positive solutions of the system and then give a precise description of the convergence of the positive solutions.

Finally, we give some numerical results.