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Zbl 0713.34025
Gupta, Chaitan P.
Existence and uniqueness theorems for some fourth order fully quasilinear boundary value problems. (English)
[J] Appl. Anal. 36, No.3-4, 157-169 (1990). ISSN 0003-6811; ISSN 1563-504X/e

The paper deals with the problem of existence and uniqueness of solutions for the boundary value problems $$ (1)\quad \frac{d\sp 4u}{dx\sp 4}+f(x,u(x),\quad u'(x),\quad u''(x),\quad u'''(x))=e(x),\quad 0<x<1, $$ $$ (2)\quad u(0)=u(1)=u''(0)=u''(1)=0\quad or\quad (3)\quad u(0)=u'(1)=u''(0)=u'''(1)=0 $$ where f: [0,1]$\times R\sp 4\to R$ is a function satisfying Caratheodory's conditions and e: [0,1]$\to R$ is in $L\sp 1(0,1)$. There are 4 theorems concerning the existence of the solutions and 2 theorems concerning the uniqueness of the solutions. We give the following theorem as an example: \par Theorem 5. Suppose that there exist functions a(x), b(x), c(x) and d(x) in $L\sp{\infty}[0,1]$ such that $(f(x,u\sb 1,v\sb 1,w\sb 1,y\sb 1)- f(x,u\sb 2,v\sb 2,w\sb 2,y\sb 2)$. $$ (w\sb 1-w\sb 2)\le a(x)(w\sb 1-w\sb 2)\sp 2+b(x)\vert u\sb 1-u\sb 2\vert \vert w\sb 1-w\sb 2\vert +c(x)\vert v\sb 1-v\sb 2\Vert w\sb 1-w\sb 2\vert +d(x)\vert y\sb 1-y\sb 2\vert \quad \vert w\sb 1-w\sb 2\vert$$ for all $(u\sb i,v\sb i,w\sb i,y\sb i)\in R\sp 4$, $i=1,2$ and a.e. $x\in [0,1]$. Suppose further that there exist an $L\sp 2$-Caratheodory function $\beta:[0,1]\times R\sp 3\to R$ and $\gamma (x)\in L\sp 1[0,1]$ such that $\vert f(x,u,v,w,y)\vert \le \beta (x,u,v,w)\vert y\vert +\gamma (x)$ for all $(u,v,w)\in R\sp 3$, $y\in R$ and a.e. $x\in [0,1]$. Then for $e(x)\in L\sp 1[0,1]:$ \par 1. (1),(2) has exactly one solution if $\pi \Vert a\Vert\sb{\infty}+\Vert b\Vert\sb{\infty}+\pi \Vert c\Vert\sb{\infty}+\pi\sp 2\Vert d\Vert\sb{\infty}<\pi\sp 3;$ \par 2. (1), (3) has exactly one solution if $4\pi\sp 2\Vert a\Vert\sb{\infty}+16\Vert b\Vert\sb{\infty}+8\pi \Vert c\Vert\sb{\infty}+2\pi\sp 3\Vert d\Vert\sb{\infty}<\pi\sp 4.$ \par We note that the proof of the existence of the solutions was made by applying the version of Leray-Schauder continuation theorem given by Mawhin. The paper is a continuation and complementation of an earlier paper of {\it C. P. Gupta} [Appl. Anal. 26, No.4, 289-304 (1988; Zbl 0611.34015)].
[M.Švec]

MSC 2000:: *34B15 Nonlinear boundary value problems of ODE
34G20 Nonlinear ODE in abstract spaces

Keywords: linear eigenvalue problem; elastic beam; Leray-Schauder continuation theorem

Citations: Zbl 0611.34015

Cited in: Zbl 0838.34022

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