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

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 $f,gin C([0,1]imes mathbb{R}_+^{4}, mathbb{R}_+)(mathbb{R}_+:=[0,infty))$. 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.

Keywords

• System of second-order boundary-value problems
• positive solution
• first-order derivative
• fixed point index
• R_+^2-monotone matrix
• concave function