ON THE EXISTENCE OF WEIGHTED ASYMPTOTICALLY CONSTANT SOLUTIONS OF VOLTERRA DIFFERENCE EQUATIONS OF NONCONVOLUTION TYPE

A Volterra difference equation of the form x(n + 2) = a(n) + b(n)x(n + 1) + c(n)x(n) + Sigma K-n+1(i=1)(n, i)x(i) where a, b, c, x: N --> R and K: N x N --> R is studied. For every admissible constant C is an element of R, sufficient conditions for the existence of a solution x: N --> R of the above equation such that x(n) similar to C n beta(n), where beta(n) = 1/2(n) Pi(n-1)(j=1)b(j), are presented. As a corollary of the main result, sufficient conditions for the existence of an eventually positive, oscillatory, and quickly oscillatory solution x of this equation are obtained. Finally, a conditions under which considered equation possesses an asymptotically periodic solution are given.