Existence Results for Some Fourth Order Boundary Value Problems With a Parameter

YANG Yang; ZHANG Ji-hui

doi:10.3879/j.issn.1000-0887.2010.03.010

Volume 31 Issue 3

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2010 > 31(3): 351-359

YANG Yang, ZHANG Ji-hui. Existence Results for Some Fourth Order Boundary Value Problems With a Parameter[J]. Applied Mathematics and Mechanics, 2010, 31(3): 351-359. doi: 10.3879/j.issn.1000-0887.2010.03.010

Citation:

PDF( 361 KB)

Existence Results for Some Fourth Order Boundary Value Problems With a Parameter

doi: 10.3879/j.issn.1000-0887.2010.03.010

YANG Yang^1,2,
ZHANG Ji-hui¹

1.
Institute of Mathematics, School of Mathematics Science, Nanjing Normal University, Nanjing 210097, P. R. China;

Received Date: 1900-01-01
Rev Recd Date: 2009-01-26
Publish Date: 2010-03-15

Abstract

Abstract

A sequel to Yang, Zhang [Nonlinear Anal, 2008, 69:1364-1375.] in which nontrivial solutions for the fourth order boundary value problems are studied. Now under the same conditions near in finity, but different from the conditions near zero, positive, negative, and sign-changing solutions are obtained by combining critical point theory, retracting property and invariant sets.
- boundary value problem,
- critical point,
- invariant sets,
- retracting property

FullText(HTML)

References(24)

References

[1]	Davis J M， Eloe P W， Henderson J. Triple positive solutions and dependence on higher order derivatives[J]. J Math Anal Appl, 1999, 237(2): 710-720. doi: 10.1006/jmaa.1999.6500
[2]	Davis J M， Henderson J， Wong P J Y. General Lidstone problems: multiplicity and symmetry of solutions[J]. J Math Anal Appl, 2000, 251(2): 527-548. doi: 10.1006/jmaa.2000.7028
[3]	Bai Z， Wang H. On positive solutions of some nonlinear fourth-order beam equations[J]. J Math Anal Appl, 2002, 270(2): 357-368. doi: 10.1016/S0022-247X(02)00071-9
[4]	Graef J R， Qian C， Yang B. Multiple symmetric positive solution of a class of boundary value problem for higher order ordinary differential equations[J]. Proc Amer Math Soc, 2003, 131(2): 577-585. doi: 10.1090/S0002-9939-02-06579-6
[5]	Li Y. Positive solutions of fourth-order periodic boundary problems[J]. Nonlinear Anal, 2003, 54(6): 1069-1078. doi: 10.1016/S0362-546X(03)00127-5
[6]	Yao Q. Positive solutions for eigenvalue problems of fourth-order elastic beam equations[J]. Appl Math Lett, 2004, 17(2): 237-243. doi: 10.1016/S0893-9659(04)90037-7
[7]	Ma R Y， Zhang J H， Fu S M. The method of lower and upper solutions for fourth-order two-point boundary value problems[J]. J Math Anal Appl, 1997, 215(2): 415-422. doi: 10.1006/jmaa.1997.5639
[8]	Bai Z B. The method of lower and upper solutions for a bending of an elastic beam equation[J]. J Math Anal Appl, 2000, 248(1): 195-202. doi: 10.1006/jmaa.2000.6887
[9]	Charkrit S， Kananathai A. Existence of solutions for some higher order boundary value problems[J]. J Math Anal Appl, 2007, 329(2): 830-850. doi: 10.1016/j.jmaa.2006.06.092
[10]	Li F， Liang Z P，Zhang Q. Existence and multiplicity of solutions of a kind of fourth-order boundary value problem[J]. Nonlinear Anal, 2005, 62(5): 803-816.
[11]	Liu X L， Li W T. Existence and multiplicity of solutions for fourth-order boundary value problems with parameters[J]. J Math Anal Appl, 2007, 327(1): 362-375. doi: 10.1016/j.jmaa.2006.04.021
[12]	Li F Y，Li Y H, Liang Z P. Existence of solutions to nonlinear Hammerstein integral equations and applications[J]. J Math Anal Appl, 2006, 323(1): 209-227. doi: 10.1016/j.jmaa.2005.10.014
[13]	Li F Y，Li Y H， Liang Z P. Existence and multiplicity of solutions to 2m th-order ordinary differential equations[J]. J Math Anal Appl, 2007, 331(2): 958-977. doi: 10.1016/j.jmaa.2006.09.025
[14]	Han G D，Li F Y. Multiple solutions of some fourth-order boundary value problems[J]. Nonlinear Anal, 2007, 66(11): 2591-2603. doi: 10.1016/j.na.2006.03.042
[15]	Han G D，Xu Z B. Multiple solutions of some nonlinear fourth-order beam equations[J]. Nonlinear Anal, 2008, 68(12): 3646-3656. doi: 10.1016/j.na.2007.04.007
[16]	Yang Y，Zhang J H. Existence of solutions for some fourth-order boundary value problems with parameters[J]. Nonlinear Anal, 2008, 69(4): 1364-1375. doi: 10.1016/j.na.2007.06.035
[17]	Yang Y， Zhang J H. Nontrivial solutions for some fourth order boundary value problems with parameters[J]. Nonlinear Anal, 2009, 70(11): 3966-3977. doi: 10.1016/j.na.2008.08.005
[18]	Li F Y， Zhang Y B，Li Y H. Sign-changing solutions on a kind of fourth order Neumann boundary value problem[J]. J Math Anal Appl, 2008, 344(1): 417-428. doi: 10.1016/j.jmaa.2008.02.050
[19]	Zhang Z T，Li S J. On sign-changing and multiple solutions of the p-Laplacian[J]. J Func Anal, 2003, 197(2): 447-468. doi: 10.1016/S0022-1236(02)00103-9
[20]	Liu Z L， Sun J X. Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations[J]. J Differential Equations, 2001, 172(2): 257-299. doi: 10.1006/jdeq.2000.3867
[21]	郭大钧，孙经先，抽象空间中的常微分方程[M].济南：山东科技出版社， 1989.
[22]	Dancer E N， Zhang Z T. Fucik spectrum，Sigh-changing and multiple solutions for semilinear elliptic boundary value problem with resonance at infinity[J]. J Math Anal Appl, 2000, 250(2): 449-464. doi: 10.1006/jmaa.2000.6969
[23]	Zhang Z T，Li X D. Sign-changing solutions and multiple solutions for semilinear elliptic boundary value problems with a retraction term nonzero at zero[J]. J Differential Equ, 2002, 178(2): 298-313. doi: 10.1006/jdeq.2001.4015
[24]	Chang K C. Infinite Dimensional Morse Theory and Multiple Solution Problems[M]. Boston: Birkhuser，1993.