Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 98, pp. 1-10.

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

Authors: Douglas R. Anderson (Concordia College, Moorhead, MN, USA)
         Feliz Minhos (Univ. of Evora, Portugal)

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.