Successively iterative technique of a classical elastic beam equation with Carathéodory nonlinearity
type: Item
instance of: scholarly article
title: Successively iterative technique of a classical elastic beam equation with Carathéodory nonlinearity
zbMATH Open document ID: 1188.34013
published in: Acta Applicandae Mathematicae
publication date: 2009-11-16T00:00:00Z
review text: The author considers the following nonlinear fourth-order boundary-value problem with non-homogeneous boundary conditions: \[ \begin{aligned} &u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)),\quad\text{a. e. }t\in[0,1],\\ & u(0)=a,\;u'(1)=b,\;u''(0)=c,\;u'''(1)=d, \end{aligned}\tag{*} \] where \(a, b, c, d\) are given constants and \(f:[0,1]\times(-\infty,\infty)^4\to(-\infty,\infty)\) is a given continuous function. In mechanics, the problem (\(*\)) describes the equilibrium state of an elastic beam simply supported at left and clamped at right by sliding clamps. An improved iterative sequence is constructed by the help of monotonic technique which approximates successively the solution of the problem (\(*\)) under suitable assumptions. The function \(f\) is said to be Carathéodory if {\parindent8mm \begin{itemize}\item[(i)] for a. e. \(t\in[0,1]\), \(f(t,\cdot,\cdot,\cdot,\cdot):(-\infty,\infty)^4\to(-\infty,\infty)\) is continuous, \item[(ii)] for all \((u,v,w,z)\in(-\infty,\infty)^4\), \(f(\cdot,u,v,w,z):[0,1]\to(-\infty,\infty)\) is measurable and \item[(iii)] for every \(r > 0\), there exists a non-negative function \(j_r\in L^1[0,1]\) such that \(|f(t,u,v,w,z)|\leq j_r(t)\), \((t,u,v,w,z)\in[0,1]\times[-r,r]^4\). \end{itemize}} An example is presented to illustrate the result obtained.
Mathematics Subject Classification ID: 34A45
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 5633625
zbMATH Keywords: fourth order ordinary differential equations
zbMATH Keywords: non-homogeneous boundary-value problem
zbMATH Keywords: iterative technique
zbMATH Keywords: successive approximation
zbMATH Keywords: classical elastic beam equation
zbMATH Keywords: Caratheodory functions
author: Qingliu Yao
MaRDI profile type: Publication
full work available at URL: s10440-008-9317-0
OpenAlex ID: W2087079031
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DOI: 10.1007/S10440-008-9317-0
description: scientific article; zbMATH DE number 5633625