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

Abstract

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 , p x,y,p , and q q , we obtain the existence results of positive solutions. For the super-linear growth case, the Nagumo condition ( F 3 ) \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 p p and q q . The super-linear growth or sub-linear growth of the nonlinear terms f f and g g is described by related inequality conditions instead of the usual independent inequality conditions about f f and g g . The discussion is based on the fixed point index theory in cones.