Existence and Iterative Algorithms of Positive Solutions for a Higher Order Nonlinear Neutral Delay Differential Equation

Abstract

This paper is concerned with the higher order nonlinear neutral delay differential equation $[a(t)(x(t)+b(t)x(t-\tau ){)}^{(m)}{]}^{(n-m)}+[h(t,x(h_1(t)),\dots ,x(h_l(t))){]}^{(i)}+f(t,x(f_1(t)),\dots ,x(f_l(t)))=g(t),$ for all $t\ge t_0$. Using the Banach fixed point theorem, we establish the existence results of uncountably many positive solutions for the equation, construct Mann iterative sequences for approximating these positive solutions, and discuss error estimates between the approximate solutions and the positive solutions. Nine examples are included to dwell upon the importance and advantages of our results.