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-τ))(m)](n-m)+[h(t,x(h1(t)),…,x(hl(t)))](i)+f(t,x(f1(t)),…,x(fl(t)))=g(t),for allt≥t0. 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.

The Study of the Solution to a Generalized KdV-mKdV Equation

“…where ( = 1, 2, 3) are real constants. Substituting (4) and (6) into (3) and equating the coefficients of all powers of ( ) and…”

“…Many powerful techniques have been established during the past decades for the study of the nonlinear dispersive partial differential equations [1][2][3][4][5][6][7][8][9]. The inverse scattering method, the Backlund transformation, the Darboux transformation, the Painleve' analysis, the pseudospectral method, the finite differences method, and the sine-cosine ansatz are used to acquire solitary wave solutions and compactons solutions for some nonlinear equations.…”

“…In this paper, we further develop the work in [15] for the study of (1). By using a mathematical technique different from those in previous works [1][2][3][4][5][6][7][8][9][10], we obtain the exact travelling wave solutions including Jacobi elliptic function solutions, degenerated soliton solutions, and triangle function solutions for (1) with exponent . Many of the solutions obtained are different from those presented in Zhang et al 's work [15].…”