The existence of positive solutions and oscillation of solutions of higher-order difference equations with forcing terms

Abstract

Sufficient conditions are established for the existence of positive solutions and oscillation of all bounded solutions of the neutral difference equation Δ p [ x n − c x n − 1 ] + q n f ( x n − k ) = h n , n ≥ n 0 where Δ is the forward difference operator Δ x n = x n+1 − x n , l and k are integers, c ≠ ± 1 is a real number, and “ q n ” and “ h n ” are real sequences. It is also shown that some of these sufficient conditions are necessary.