Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 52, pp. 1-15.
Title: Positive solutions for a system of higher order boundary-value
problems involving all derivatives of odd orders
Authors: Kun Wang (Qingdao Technological Univ., China)
Zhilin Yang (Qingdao Technological Univ., China)
Abstract:
In this article we study the existence of positive
solutions for the system of higher order boundary-value problems
involving all derivatives of odd orders
$$\displaylines{
(-1)^mw^{(2m)}
=f(t, w, w',-w''',\dots, (-1)^{m-1}w^{(2m-1)}, z, z',-z''',\dots,
(-1)^{n-1}z^{(2n-1)}),
\cr
(-1)^nz^{(2n)}
=g(t, w, w',-w''',\dots, (-1)^{m-1}w^{(2m-1)}, z, z',-z''',\dots,
(-1)^{n-1}z^{(2n-1)}), \cr
w^{(2i)}(0)=w^{(2i+1)}(1)=0\quad (i=0,1,\dots, m-1),\cr
z^{(2j)}(0)=z^{(2j+1)}(1)=0\quad (j=0,1,\dots, n-1).
} $$
Here $f,g\in C([0,1]\times\mathbb{R}_+^{m+n+2},\mathbb{R}_+)$
$(\mathbb{R}_+:=[0,+\infty))$.
Our hypotheses imposed on the nonlinearities $f$ and $g$ are
formulated in terms of two linear functions $h_1(x)$ and $h_2(y)$.
We use fixed point index theory to establish our main results based
on a priori estimates of positive solutions achieved by utilizing
nonnegative matrices.
Submitted October 24, 2011. Published March 30, 2012.
Math Subject Classifications: 34B18, 45G15, 45M20, 47H07, 47H11.
Key Words: Systenm of higher order boundary value problem; positive solution;
nonnegative matrix; fixed point index.