Successively iterative technique of a classical elastic beam equation with Carathéodory nonlinearity
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.
fourth order ordinary differential equations
scientific article; zbMATH DE number 5633625