Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 163, pp. 1-9.
Title: Three positive solutions for a system of singular generalized
Lidstone problems
Authors: Jiafa Xu (Qingdao Technological Univ., China)
Zhilin Yang (Qingdao Technological Univ., China)
Abstract:
In this article, we show the existence of at least three positive
solutions for the system of singular generalized Lidstone boundary
value problems
$\displaylines{
(-1)^m x^{(2m)}=a(t)f_1(t,x,-x'',\dots,(-1)^{m-1}x^{(2m-2)},y,-y'',\cr
\dots,(-1)^{n-1}y^{(2n-2)}), \cr
(-1)^n y^{(2n)}=b(t)f_2(t,x,-x'',\dots,(-1)^{m-1}x^{(2m-2)},y,-y'',\cr
\dots,(-1)^{n-1}y^{(2n-2)}), \cr
a_1 x^{(2i)}(0)-b_1 x^{(2i+1)}(0)=c_1x^{(2i)}(1)+d_1 x^{(2i+1)}(1)=0,\cr
a_2y^{(2j)}(0)-b_2y^{(2j+1)}(0)=c_2y^{(2j)}(1)+d_2y^{(2j+1)}(1)=0.
}$$
The proofs of our main results are based on the Leggett-Williams
fixed point theorem. Also, we give an example to illustrate our
results.
Submitted October 7, 2009. Published December 21, 2009.
Math Subject Classifications: 34A34, 34B18, 45G15, 47H10.
Key Words: Singular generalized Lidstone problem; positive solution;
cone; concave functional.