Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 135, pp. 1-17.
Title: Positive solutions for a system of second-order boundary-value
problems involving first-order derivatives
Authors: Kun Wang (Qingdao Technological Univ., China)
Zhilin Yang (Qingdao Technological Univ., China)
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; R_+^2-monotone matrix;
concave function.