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.