Electronic Journal of Differential Equations (Mar 2012)
Positive solutions for a system of higher order boundary-value problems involving all derivatives of odd orders
Abstract
In this article we study the existence of positive solutions for the system of higher order boundary-value problems involving all derivatives of odd orders $$displaylines{ (-1)^mw^{(2m)} =f(t, w, w',-w''',dots, (-1)^{m-1}w^{(2m-1)}, z, z',-z''',dots, (-1)^{n-1}z^{(2n-1)}), cr (-1)^nz^{(2n)} =g(t, w, w',-w''',dots, (-1)^{m-1}w^{(2m-1)}, z, z',-z''',dots, (-1)^{n-1}z^{(2n-1)}), cr w^{(2i)}(0)=w^{(2i+1)}(1)=0quad (i=0,1,dots, m-1),cr z^{(2j)}(0)=z^{(2j+1)}(1)=0quad (j=0,1,dots, n-1). } $$ Here $f,gin C([0,1]imesmathbb{R}_+^{m+n+2},mathbb{R}_+)$ $(mathbb{R}_+:=[0,+infty))$. Our hypotheses imposed on the nonlinearities $f$ and $g$ are formulated in terms of two linear functions $h_1(x)$ and $h_2(y)$. We use fixed point index theory to establish our main results based on a priori estimates of positive solutions achieved by utilizing nonnegative matrices.
