Positive solutions and iterative approximations of a third order nonlinear neutral delay difference equation

Abstract

Abstract This paper deals with the third order nonlinear neutral delay difference equation with a forced term Δ2(a(n)Δ(x(n)+c(n)x(n−τ)))+f(n,x(n−b1(n)),x(n−b2(n)),…,x(n−bk(n)))=d(n),n≥n0. $$\begin{aligned}& \Delta^{2} \bigl(a(n)\Delta \bigl(x(n)+c(n)x(n-\tau) \bigr) \bigr)+f \bigl(n,x \bigl(n-b_{1}(n) \bigr),x \bigl(n-b_{2}(n) \bigr), \ldots,x \bigl(n-b_{k}(n) \bigr) \bigr) \\& \quad =d(n),\quad n\geq n_{0}. \end{aligned}$$ Using the Banach fixed point theorem, we prove the existence of uncountably many bounded positive solutions for the equation, suggest some Mann iterative schemes and obtain the error estimates between these bounded positive solutions and the sequences generated by the iterative schemes. Five nontrivial examples are also included.

Keywords

• Uncountably many bounded positive solutions
• Third order nonlinear neutral delay difference equation with a forced term
• Banach fixed point theorem
• Mann iterative scheme
• Error estimate