Positive solutions for a system of higher order boundary-value problems involving all derivatives of odd orders

Wang, Kun ; Yang, Zhilin

Electronic Journal of Differential Equations, Tome 2012 (2012).

Résumé

Summary: 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 $$\displaylines{ (-1)^mw^{(2m)} =f(t, w, w',-w''',\dots, (-1)^{m-1}w^{(2m-1)}, z, z',-z''',\dots, (-1)^{n-1}z^{(2n-1)}), \cr (-1)^nz^{(2n)} =g(t, w, w',-w''',\dots, (-1)^{m-1}w^{(2m-1)}, z, z',-z''',\dots, (-1)^{n-1}z^{(2n-1)}), \cr w^{(2i)}(0)=w^{(2i+1)}(1)=0\quad (i=0,1,\dots, m-1),\cr z^{(2j)}(0)=z^{(2j+1)}(1)=0\quad (j=0,1,\dots, n-1). } $$ Here $f,g\in C([0,1]\times\mathbb{R}_+^{m+n+2},\mathbb{R}_+)(\mathbb{R}_+:=[0,+\infty))$. 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.

Détail

Classification : 34B18, 45G15, 45M20, 47H07, 47H11
Keywords: systenm of higher order boundary value problem, positive solution, nonnegative matrix, fixed point index