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Existence and uniqueness theorems for fourth-order boundary value problems. (English) Zbl 0634.34009

The differential equation (1) ${y}^{\left(IV\right)}=f\left(x,y,{y}^{\text{'}\text{'}}\right)$ is considered under the following types of boundary conditions:

(2) $y\left(0\right)={y}_{0}$, $y\left(1\right)={y}_{1}$, ${y}^{\text{'}\text{'}}\left(0\right)={\overline{y}}_{0}$, ${y}^{\text{'}\text{'}}\left(1\right)={\overline{y}}_{1}$;

(3) $y\left(0\right)={y}_{0}$, $y\left(1\right)={y}_{1}$, ${y}^{\text{'}\text{'}}\left(0\right)={\overline{y}}_{0}$, ${y}^{\text{'}\text{'}\text{'}}\left(1\right)={\overline{y}}_{1}$;

(4) $y\left(0\right)=y\left(1\right)={y}^{\text{'}\text{'}}\left(0\right)={y}^{\text{'}\text{'}}\left(1\right)=0$ or

(5) $y\left(0\right)=y\left(1\right)=0$; ${y}^{\text{'}\text{'}\text{'}}\left(0\right)-h{y}^{\text{'}\text{'}}\left(0\right)=0$, ${y}^{\text{'}\text{'}\text{'}}\left(1\right)+k{y}^{\text{'}\text{'}}\left(1\right)=0$, with $h,k\ge 0$, $h+k>0$.

Existence theorems for all of the boundary value problems (1-2)–(1-5) are obtained by application of Schauder’s fixed point theorem under continuity and boundedness hypotheses on $f$ and its partial derivative ${f}_{3}$. A uniqueness theorem is obtained for the problem (1-4) under the additional assumption of a bound on the partial derivative ${f}_{2}$. Further uniqueness results for the problem (1-2)–(1-5) are reduced to uniqueness questions for solutions of corresponding second order boundary value problems for the integrodifferential equation

${u}^{\text{'}\text{'}}=f\left(x,{y}_{0}+x\left({y}_{1}-{y}_{0}\right)+{\int }_{0}^{1}G\left(x,t\right)u\left(t\right)\phantom{\rule{0.166667em}{0ex}}dt,u\right)$

obtained by setting ${y}^{\text{'}\text{'}}=u$, with $G\left(x,t\right)$ the Green’s function for the problem

${u}^{\text{'}\text{'}}=0,\phantom{\rule{1.em}{0ex}}u\left(0\right)=u\left(1\right)=0·$

Boundary conditions of the form (2)–(5) have been less extensively studied than the familiar conjugate or focal-type problems. R. A. Usmani [Proc. Am. Math. Soc. 77, 329–335 (1979; Zbl 0424.34019)] has studied a problem of the form (1-2) in which the equation is linear and independent of ${y}^{\text{'}\text{'}}$.

Reviewer: L.J.Grimm

##### MSC:
 34B15 Nonlinear boundary value problems for ODE