Electron. J. Diff. Equ., Vol. 2012 (2012), No. 52, pp. 1-15.

Positive solutions for a system of higher order boundary-value problems involving all derivatives of odd orders

Kun Wang, Zhilin Yang

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.

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Kun Wang
Department of Mathematics
Qingdao Technological University
Qingdao, 266033, China
email: wangkun880304@163.com
Zhilin Yang
Department of Mathematics
Qingdao Technological University
Qingdao, 266033, China
email: zhilinyang@sina.com

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