Electronic Journal of Differential Equations (Mar 2014)
Existence of solutions for an n-dimensional operator equation and applications to BVPs
Abstract
By applying the Guo-Lakshmikantham fixed point theorem on high dimensional cones, sufficient conditions are given to guarantee the existence of positive solutions of a system of equations of the form $$ x_i(t)=\sum_{k=1}^n\sum_{j=1}^n\gamma_{ij}(t)w_{ijk}(\Lambda_{ijk} [x_k])+(F_ix)(t),\quad t\in[0,1],\quad i=1, \dots, n. $$ Applications are given to three boundary value problems: A 3-dimensional 3+3+3 order boundary value problem with mixed nonlocal boundary conditions, a 2-dimensional 2+4 order nonlocal boundary value problem discussed in [14], and a 2-dimensional 2+2 order nonlocal boundary value problem discussed in [35]. In the latter case we provide some fairly simpler conditions according to those imposed in [35].
