### Solvability of a fourth-order boundary value problem with periodic boundary conditions II

Let $f:\left[0,1\right]×{\mathbf{R}}^{4}\to \mathbf{R}$ be a function satisfying Caratheodory's conditions and $e\left(x\right)\in {L}^{1}\left[0,1\right]$ . This paper is concerned with the solvability of the fourth-order fully quasilinear boundary value problem $\frac{{d}^{4}u}{d{x}^{4}}+f\left(x,u\left(x\right),{u}^{\prime }\left(x\right),{u}^{″}\left(x\right),{u}^{‴}\left(x\right)\right)=e\left(x\right), 0 , with $u\left(0\right)-u\left(1\right)={u}^{\prime }\left(0\right)-{u}^{\prime }\left(1\right)={u}^{″}\left(0\right)-{u}^{″}\left(1\right)={u}^{‴}\left(0\right)-{u}^{‴}\left(1\right)=0$ . This problem was studied earlier by the author in the special case when $f$ was of the form $f\left(x,u\left(x\right)\right)$ , i.e., independent of ${u}^{\prime }\left(x\right)$ , ${u}^{″}\left(x\right)$ , ${u}^{‴}\left(x\right)$ . It turns out that the earlier methods do not apply in this general case. The conditions need to be related to both of the linear eigenvalue problems $\frac{{d}^{4}u}{d{x}^{4}}={\lambda }^{4}u$ and $\frac{{d}^{4}u}{d{x}^{4}}=-{\lambda }^{2}\frac{{d}^{2}u}{d{x}^{2}}$ with periodic boundary conditions.
Authors:
[1]
1. Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439-4801, USA
Publication Date:
OSTI Identifier:
1198035
Grant/Contract Number:
W-31-109-Eng-38
Type:
Published Article
Journal Name:
International Journal of Mathematics and Mathematical Sciences
Journal Volume: 14; Journal Issue: 1; Related Information: CHORUS Timestamp: 2016-08-23 12:12:35; Journal ID: ISSN 0161-1712
Publisher:
Hindawi Publishing Corporation
USDOE
Country of Publication:
Country unknown/Code not available
Language:
English