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

## Abstract

Let $f:\left[0,1\right]\times {\mathbf{R}}^{4}\to \mathbf{R}$ 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 $\frac{{d}^{4}u}{d{x}^{4}}+f\left(x,u\left(x\right),{u}^{\prime}\left(x\right),{u}^{\u2033}\left(x\right),{u}^{\u2034}\left(x\right)\right)=e\left(x\right),\mathrm{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0}<x<1$ , with $u\left(0\right)-u\left(1\right)={u}^{\prime}\left(0\right)-{u}^{\prime}\left(1\right)={u}^{\u2033}\left(0\right)-{u}^{\u2033}\left(1\right)={u}^{\u2034}\left(0\right)-{u}^{\u2034}\left(1\right)=0$ . 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 ${u}^{\prime}\left(x\right)$ , ${u}^{\u2033}\left(x\right)$ , ${u}^{\u2034}\left(x\right)$ . 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:

- Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439-4801, USA

- Publication Date:

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1198035

- Grant/Contract Number:
- W-31-109-Eng-38

- Resource Type:
- Published Article

- Journal Name:
- International Journal of Mathematics and Mathematical Sciences

- Additional Journal Information:
- Journal Name: International Journal of Mathematics and Mathematical Sciences Journal Volume: 14 Journal Issue: 1; Journal ID: ISSN 0161-1712

- Publisher:
- Hindawi Publishing Corporation

- Country of Publication:
- Country unknown/Code not available

- Language:
- English

### Citation Formats

```
Gupta, Chaitan P. Solvability of a fourth-order boundary value problem with periodic boundary conditions II. Country unknown/Code not available: N. p., 1991.
Web. doi:10.1155/S0161171291000121.
```

```
Gupta, Chaitan P. Solvability of a fourth-order boundary value problem with periodic boundary conditions II. Country unknown/Code not available. doi:10.1155/S0161171291000121.
```

```
Gupta, Chaitan P. Tue .
"Solvability of a fourth-order boundary value problem with periodic boundary conditions II". Country unknown/Code not available. doi:10.1155/S0161171291000121.
```

```
@article{osti_1198035,
```

title = {Solvability of a fourth-order boundary value problem with periodic boundary conditions II},

author = {Gupta, Chaitan P.},

abstractNote = {Let f : [ 0 , 1 ] × R 4 → R be a function satisfying Caratheodory's conditions and e ( x ) ∈ L 1 [ 0 , 1 ] . This paper is concerned with the solvability of the fourth-order fully quasilinear boundary value problem d 4 u d x 4 + f ( x , u ( x ) , u ′ ( x ) , u ″ ( x ) , u ‴ ( x ) ) = e ( x ) , 0 < x < 1 , with u ( 0 ) − u ( 1 ) = u ′ ( 0 ) − u ′ ( 1 ) = u ″ ( 0 ) - u ″ ( 1 ) = u ‴ ( 0 ) - u ‴ ( 1 ) = 0 . This problem was studied earlier by the author in the special case when f was of the form f ( x , u ( x ) ) , i.e., independent of u ′ ( x ) , u ″ ( x ) , u ‴ ( x ) . 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 d 4 u d x 4 = λ 4 u and d 4 u d x 4 = − λ 2 d 2 u d x 2 with periodic boundary conditions.},

doi = {10.1155/S0161171291000121},

journal = {International Journal of Mathematics and Mathematical Sciences},

number = 1,

volume = 14,

place = {Country unknown/Code not available},

year = {1991},

month = {1}

}

DOI: 10.1155/S0161171291000121