Electron. J. Diff. Equ., Vol. 2014 (2014), No. 71, pp. 1-17.

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

George L. Karakostas

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.

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George L. Karakostas
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece
email: gkarako@uoi.gr

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