Electron. J. Diff. Equ., Vol. 2012 (2012), No. 135, pp. 1-17.

Positive solutions for a system of second-order boundary-value problems involving first-order derivatives

Kun Wang, Zhilin Yang

Abstract:
In this article we study the existence and multiplicity of positive solutions for the system of second-order boundary value problems involving first order derivatives
$$\displaylines{ -u''=f(t, u, u', v, v'),\cr -v''=g(t, u, u', v, v'),\cr u(0)=u'(1)=0,\quad v(0)=v'(1)=0. }$$
Here $f,g\in C([0,1]\times \mathbb{R}_+^{4}, \mathbb{R}_+)(\mathbb{R}_+:=[0,\infty))$. We use fixed point index theory to establish our main results based on a priori estimates achieved by utilizing Jensen's integral inequality for concave functions and $\mathbb{R}_+^2$-monotone matrices.

Submitted May 31, 2012. Published August 17, 2012.
Math Subject Classifications: 34B18, 45G15, 45M20, 47H07, 47H11.
Key Words: System of second-order boundary-value problems; positive solution; first-order derivative; fixed point index; $\mathbb{R}_+^2$-monotone matrix; concave function.

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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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