Positive Solutions and Mann Iterations of a Fourth Order Nonlinear Neutral Delay Differential Equation

Zeqing Liu,Jeong Sheok Ume,Jingjing Zhu,Shin Min Kang

doi:10.1155/2014/105719

Abstract

This paper deals with a fourth order nonlinear neutral delay differential equation. By using the Banach fixed point theorem, we establish the existence of uncountably many bounded positive solutions for the equation, construct several Mann iterative sequences with mixed errors for approximating these positive solutions, and discuss some error estimates between the approximate solutions and these positive solutions. Seven nontrivial examples are given.

Highlights

Introduction and PreliminariesThe oscillation, nonoscillation, and existence of solutions for various kinds of second order and third order neutral delay differential equations have been extensively studied over the last decades; for example, see [1,2,3,4,5,6,7,8,9,10,11,12]
Tang and Liu [9] studied the existence of bounded oscillation for the second order linear delay differential equation of unstable type x󸀠󸀠 (t) = p (t) x (t − τ), t ≥ t0, (2)
Using the Banach fixed point theorem, Kulenovicand Hadziomerspahic [4] deduced the existence of a nonoscillatory solution for the second order linear neutral delay differential equation with positive and negative coefficients (x (t) + cx (t − τ))󸀠󸀠 + Q1 (t) x (t − σ1) (3)

Summary

Introduction and Preliminaries

The oscillation, nonoscillation, and existence of solutions for various kinds of second order and third order neutral delay differential equations have been extensively studied over the last decades; for example, see [1,2,3,4,5,6,7,8,9,10,11,12]. Using the Banach fixed point theorem, Kulenovicand Hadziomerspahic [4] deduced the existence of a nonoscillatory solution for the second order linear neutral delay differential equation with positive and negative coefficients (x (t) + cx (t − τ))󸀠󸀠 + Q1 (t) x (t − σ1) (3). Yu and Wang [10] studied the existence of a nonoscillatory solution for the second order nonlinear neutral delay differential equations with positive and negative coefficients (r (t) (x (t) + P (t) x (t − τ))󸀠)󸀠 + Q1 (t) f (x (t − σ1)) (7). Liu and Kang [7] investigated the existence of nonoscillatory solutions of the second order nonlinear neutral delay differential equation (a (t) (x (t) + b (t) x (t − τ))󸀠)󸀠.

Uncountably Many Bounded Positive Solutions and Iterative Approximations

Examples