×

zbMATH — the first resource for mathematics

Boundary value problems for some fourth order ordinary differential equations. (English) Zbl 0682.34020
The following nonlinear beam equation is studied (1) \(u^{(4)}-\alpha (x)u+f(x,u)=h(x),\) \(x\in (0,\pi)\) under one of the boundary conditions (2) \(u(0)=u(\pi)=u''(0)=u''(\pi)=0\), or (3) \(u(0)=u(\pi)=u'(0)=u'(\pi)=0\). Here \(\alpha\) (x) is a bounded, nonnegative function and it is assumed that 0 is the first eigenvalue of the linear part of (1). The right-hand side h(x) is orthogonal to the first eigenfunction. Using variational methods, some existence and multiplicity results are obtained: (a) existence of at least one solution if f(x,u) satisfies an assumption which generalizes a “sign condition”; (b) existence of at least one solution if f(x,u) satisfies a Landesman- Lazer condition: (c) existence of at least three solutions near resonance (a variational approach to bifurcation from infinity), for the problem (1)-(2).
Reviewer: L.Sanchez

MSC:
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aftabizadeh A.R., J. Math.Anal. Appl 116 pp 415– (1986) · Zbl 0634.34009
[2] Gupta C.P., Applicable Analysis 26 pp 289– (1988) · Zbl 0611.34015
[3] Gupta C.P., J. Math. Anal. Appl 135 pp 208– (1988) · Zbl 0655.73001
[4] Usmani R.A., Proc. Amer.Math. Soc 77 pp 327– (1979)
[5] Yang Y., Proc. Amer. Math. Soc 104 pp 175– (1988)
[6] Chow S.N., J. Diff. Eq 14 pp 101– (1973) · Zbl 0286.34020
[7] Nonlinear Functional Analysis and its applications I (1986)
[8] Bahri A., these de 3eme cycle (1979)
[9] Sanchez L., Applicable Analysis 25 pp 275– (1987) · Zbl 0611.34017
[10] Ramos M., Proc. Royal Soc. Edinburgh 112 pp 177– (1989) · Zbl 0692.35048
[11] Ahmad S., Indiana Univ. Math.J 25 pp 933– (1976) · Zbl 0351.35036
[12] Mawhin J., Lecture Notes in Math 25 pp 269– (1986)
[13] Mawhin J., Proc. Amer. Math. Soc 93 pp 667– (1985)
[14] Mawhin J., Ann. Inst. Poinc 3 pp 431– (1986)
[15] Rabinowitz, P.H. 1986. Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conference Series in Mathematics. 1986. · Zbl 0609.58002
[16] Iannacci R., Memphis State Univ. Report Series (1986)
[17] Mawhin J., Pure et Appliquee 132 (1988)
[18] Figueiredo D . G . De, Lectures on the Ekeland variational principle with applications and detours (1989) · Zbl 0688.49011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.