*(English)*Zbl 0634.34009

The differential equation (1) ${y}^{\left(IV\right)}=f(x,y,{y}^{\text{'}\text{'}})$ 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

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

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 ${y}^{\text{'}\text{'}}$.

##### MSC:

34B15 | Nonlinear boundary value problems for ODE |