SYSTEMS OF DIFFERENCE EQUATIONS WITH ASYMPTOTICALLY CONSTANT SOLUTIONS William F. Trench Trinity University, San Antonio, Texas, USA Nonlinear Analysis, Theory, Methods, and Applications 40 (2000) 611–615 Dedicated to Professor V. Lakshmikantham on his 75th birthday Key words: Difference equations, Asymptotically constant, Conditional convergence, Banach space, Schauder-Tychonoff theorem We consider the system x n D A n x n C f .n; x n /; (1) where x n and f are k-vectors (real or complex) and A n is a k  k matrix. We give conditions implying that (1) has a solution fO x n g such that lim n!1 O x n D c , a given constant vector. If u is a k-vector and B is a k  k matrix, then juj and jAj are the 1-norms of u and A. THEOREM 1 Let c be a given k-vector, and suppose there is a constant M>0 and an integer N such that f .n; x/ is continuous with respect to x and jf .n; x/  f .n; c/j R.n; jx  c j/ (2) on the set S Df.n; x/ j n  N; jx  c j M g; where R D R.n; / is defined on the set f.n; x/ j n  N; 0    M g and nondecreasing in  for each n, and 1 X nDN jR.n; M /j < 1: (3) Suppose that either 1 X nDN jA n j < 1 (4) or there is a positive integer q such that the sequences A .r/ n D 1 X mDn A .r 1/ mC1 A m ; r D 1;2;:::;q .with A .0/ m D I/ (5)