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

Abstract

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.

Authors:
 [1]
  1. 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}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
DOI: 10.1155/S0161171291000121

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