Asymptotic behavior of nonoscillatory solutions of neutral difference equations
The authors consider the m(th)-order neutral difference equation Dm (y (n) + p (n) y (n - k)) + q (-) f (y (a (-))) = e (n), where m greater than or equal to 1, {p(n)}, {q(n)}, {e(n)}, and {a(1)(n)}, {a(2)(n)},..., {a(m-1)(n)} are real sequences, a(i)(n) > 0 for i = 1, 2...., m-1, a(m)(n) equivalent to 1, D(0)z(n) = y(n)+p(n)y(n-k), D(i)z(n) = a(i)(n)DeltaD(i-1)z(n) for i = 1, 2,... m, k is a positive integer, {sigma(n)} --> infinity is a sequence of positive integers, and f : R --> R is continuous with uf(u) > 0 for u not equal 0. In the case where {q(n)} is allowed to oscillate, they obtain sufficient conditions for all bounded nonoscillatory solutions to converge to zero, and if {q(n)} is a nonnegative sequence, they establish sufficient conditions for all nonoscillatory solutions to converge to zero. Examples illustrating the results are included throughout the paper. (C) 2003 Elsevier Science Ltd. All rights reserved.
asymptotic behavior difference equations higher order neutral equations non-oscillatory solutions