Positive solutions for boundary value problems of a class of second-order differential equation system

Abstract

This article discusses the existence of positive solutions for the system of second-order ordinary differential equation boundary value problems −u″(t)=f(t,u(t),v(t),u′(t)),t∈[0,1],−v″(t)=g(t,u(t),v(t),v′(t)),t∈[0,1],u(0)=u(1)=0,v(0)=v(1)=0,\left\{\begin{array}{l}-{u}^{^{\prime\prime} }\left(t)=f\left(t,u\left(t),v\left(t),u^{\prime} \left(t)),\hspace{1em}t\in \left[0,1],\\ -{v}^{^{\prime\prime} }\left(t)=g\left(t,u\left(t),v\left(t),v^{\prime} \left(t)),\hspace{1em}t\in \left[0,1],\\ u\left(0)=u\left(1)=0,\hspace{1em}v\left(0)=v\left(1)=0,\end{array}\right. where f,g:[0,1]×R+×R+×R→R+f,g:\left[0,1]\times {{\mathbb{R}}}^{+}\times {{\mathbb{R}}}^{+}\times {\mathbb{R}}\to {{\mathbb{R}}}^{+} are continuous. Under the related conditions that the nonlinear terms f(t,x,y,p)f\left(t,x,y,p) and g(t,x,y,q)g\left(t,x,y,q) may be super-linear growth or sub-linear growth on x,y,px,y,p, and qq, we obtain the existence results of positive solutions. For the super-linear growth case, the Nagumo condition (F3)\left(F3) is presented to restrict the growth of f(t,x,y,p)f\left(t,x,y,p) and g(t,x,y,q)g\left(t,x,y,q) on pp and qq. The super-linear growth or sub-linear growth of the nonlinear terms ff and gg is described by related inequality conditions instead of the usual independent inequality conditions about ff and gg. The discussion is based on the fixed point index theory in cones.

Keywords

• second-order differential equation system
• positive solution
• cone
• fixed point index
• super-linear growth
• 34a34
• 34b18
• 47h11