I. Kiguradze and B. Půža

On Some Boundary Value Problems for Fourth Order Functional Differential Equations

abstract:
Optimal in a sense sufficient conditions are established for the solvability and unique solvability of the boundary value problems of the type
\begin{gather*}
u^{(iv)}(t)=g(u)(t), \\
u(a)=0, \;\; u(b)=0, \;\;\; \sum_{k=1}^2
\big(\al_{ik}u^{(k)}(a)+\bt_{ik}u^{(k)}(b)\big)=0 \;\;
(i=1,2),
\end{gather*}
where $g: C^1([a,b];\bR)\to L([a,b];\bR)$ is a continuous operator, $\al_{ik}$ and $\bt_{ik}$ $(i,k=1,2)$ are real constants such that
$$ \sum_{i=1}^2 \Big|\sum_{k=1}^2
(\al_{ik}x_k+\bt_{ik}y_k)\Big|>0 \;\; \text{for} \;\;
x_1x_2<y_1y_2. $$

Mathematics Subject Classification: 34B15

Key words and phrases: Fourth order nonlinear functional differential equation, two-point boundary value problem, solvability, unique solvability