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Successively iterative technique of a classical elastic beam equation with Carathéodory nonlinearity. (English) Zbl 1188.34013

The author considers the following nonlinear fourth-order boundary-value problem with non-homogeneous boundary conditions:

$\begin{array}{cc}& {u}^{\left(4\right)}\left(t\right)=f\left(t,u\left(t\right),{u}^{\text{'}}\left(t\right),{u}^{\text{'}\text{'}}\left(t\right),{u}^{\text{'}\text{'}\text{'}}\left(t\right)\right),\phantom{\rule{1.em}{0ex}}\text{a.}\phantom{\rule{4.pt}{0ex}}\text{e.}\phantom{\rule{4.pt}{0ex}}t\in \left[0,1\right],\hfill \\ & u\left(0\right)=a,\phantom{\rule{4pt}{0ex}}{u}^{\text{'}}\left(1\right)=b,\phantom{\rule{4pt}{0ex}}{u}^{\text{'}\text{'}}\left(0\right)=c,\phantom{\rule{4pt}{0ex}}{u}^{\text{'}\text{'}\text{'}}\left(1\right)=d,\hfill \end{array}\phantom{\rule{2.em}{0ex}}\left(*\right)$

where $a,b,c,d$ are given constants and $f:\left[0,1\right]×{\left(-\infty ,\infty \right)}^{4}\to \left(-\infty ,\infty \right)$ 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

for a. e. $t\in \left[0,1\right]$, $f\left(t,·,·,·,·\right):{\left(-\infty ,\infty \right)}^{4}\to \left(-\infty ,\infty \right)$ is continuous,

for all $\left(u,v,w,z\right)\in {\left(-\infty ,\infty \right)}^{4}$, $f\left(·,u,v,w,z\right):\left[0,1\right]\to \left(-\infty ,\infty \right)$ is measurable and

for every $r>0$, there exists a non-negative function ${j}_{r}\in {L}^{1}\left[0,1\right]$ such that $|f\left(t,u,v,w,z\right)|\le {j}_{r}\left(t\right)$, $\left(t,u,v,w,z\right)\in \left[0,1\right]×{\left[-r,r\right]}^{4}$.

An example is presented to illustrate the result obtained.

##### MSC:
 34A45 Theoretical approximation of solutions of ODE 34B15 Nonlinear boundary value problems for ODE
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