Electronic Journal of Differential Equations,
Vol. 2014 (2014), No. 71, pp. 1-17.

Title: Existence of solutions for an n-dimensional operator equation and
       applications to BVPs

Author: George L. Karakostas (Univ. of Ioannina, Greece)

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].

Submitted February 19, 2014. Published March 16, 2014.
Math Subject Classifications: 34B10, 34K10.
Key Words: Krasnoselskii's fixed point theorem; high-dimensional cones; 
           nonlocal and multipoint boundary value problems;
           system of differential equations.