Global attractivity in nonlinear difference equations of higher order with a forcing term

Consider the following nonlinear difference equation of order k + 1 with a forcing term (0.1) x n + 1 - a n x n + b n f ( x n - k ) = r n , n = 0 , 1 , ? where {an} is a positive sequence in (0, 1], {bn} is a positive sequence, {rn} is a real sequence, k is a nonnegative integer, and f: (?, ∞) ? (?, ∞) is a continuous function with -∞ ? ? ? 0. We establish a sufficient condition for every solution of Eq. (0.1) to converge to zero as n ? ∞. Several new global attractivity results are obtained for some special cases of Eq. (0.1) which have been studied widely in the literature. Our results can be applied to some difference equations derived from mathematical biology.

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