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.