Positive Solutions and Mann Iterative Algorithms for a Second-Order Nonlinear Difference Equation. (3rd April 2016) - Liu, Zeqing; Li, Xin; Kang, Shin Min; Kwun, Young Chel
Abstract : This paper deals with the second-order nonlinear neutral delay difference equation Δ ( a n h ( x n - τ 1 n, x n - τ 2 n, …, x n - τ m n ) Δ ( x n - q n x n - τ 0 ) ) + f ( n, x n - σ 1 n, x n - σ 2 n, …, x n - σ k n ) = b n, n ≥ n 0 . Using the Banach fixed point theorem, Mann iterative method with errors, and some new techniques, we prove the existence of uncountably many positive solutions and the convergence of the sequences generated by the Mann iterative method with errors relative to these solutions for the above equation. Six examples are included. Our results extend and improve essentially the known results in this field.