Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1167.34004
Webb, J.R.L.; Infante, Gennaro; Franco, Daniel
Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions. (English)
[J] Proc. R. Soc. Edinb., Sect. A, Math. 138, No. 2, 427-446 (2008). ISSN 0308-2105; ISSN 1473-7124/e

Authors' abstract: We establish new existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam. We consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non-local boundary conditions, with a unified approach. Our method is to show that each boundary-value problem can be written as the same type of perturbed integral equation, in the space $C[0,1]$, involving a linear functional $\alpha[u]$ but, although we seek positive solutions, the functional is not assumed to be positive for all positive $u$. The results are new even for the classic boundary conditions of clamped or hinged ends when $\alpha[u]=0$, because we obtain sharp results for the existence of one positive solution; for multiple solutions we seek optimal values of some of the constants that occur in the theory, which allows us to impose weaker assumptions on the nonlinear term than in previous works. Our non-local boundary conditions contain multi-point problems as special cases and, for the first time in fourth-order problems, we allow coefficients of both signs.
[Pierpaolo Omari (Trieste)]

MSC 2000:: *34B18 Positive solutions of nonlinear boundary value problems
34B15 Nonlinear boundary value problems of ODE
34B09 Boundary value problems with an indefinite weight
34B10 Multipoint boundary value problems

Keywords: Existence; multiplicity; positive solution; fourth order equation; local and non-local boundary conditions

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]