Electron. J. Diff. Eqns., Vol. 2009(2009), No. 98, pp. 1-10.

Existence of positive solutions for a fourth-order multi-point beam problem on measure chains

Douglas R. Anderson, Feliz Minhos

Abstract:
This article concerns the fourth-order multi-point beam problem
$$\displaylines{ (EIW^{\Delta \nabla }) ^{\nabla \Delta }(x)=m(x)f(x,W(x)),\quad x\in [x_{1},x_{n}]_{\mathbb{X}} \cr W(\rho ^2(x_{1}))=\sum_{i=2}^{n-1}a_iW(x_i),\quad W^{\Delta}(\rho ^2(x_{1}))=0, \cr (EIW^{\Delta \nabla }) (\sigma (x_{n}))=0,\quad (EIW^{\Delta \nabla })^{\nabla }(\sigma(x_{n})) =\sum_{i=2}^{n-1}b_i(EIW^{\Delta \nabla })^{\nabla}(x_i). }$$
Under various assumptions on the functions $f$ and $m$ and the coefficients $a_i$ and $b_i$ we establish the existence of one or two positive solutions for this measure chain boundary value problem using the Green's function approach.

Submitted February 6, 2009. Published August 11, 2009.
Math Subject Classifications: 34B15, 39A10.
Key Words: Measure chains; boundary value problems; Green's function; fixed point; fourth order; cantilever beam.

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Douglas R. Anderson
Department of Mathematics, Concordia College
Moorhead, MN 56562 USA
email: andersod@cord.edu
Feliz Minhós
Department of Mathematics
University of Évora, Portugal
email: fminhos@uevora.pt

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