Reurings, A fixed point theorem in

partially ordered sets and some applications to matrix equations, Proc.

BJORNER, Shellable and Cohen-Macaulay partially ordered sets, Trans.

WALKER, Homotopy type and Euler characteristic of partially ordered sets, European J.

Rodriguez-Lopez: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22(2005), 223-239.

Reurings: A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc.

We refer the reader to [18, Chapter 3] and [9,19] for background on

partially ordered sets and the topology of simplicial complexes, respectively.

Sadarangani: Fixed point theorems for weakly contractive mappings in

partially ordered sets, Nonlinear Anal.

The existence of fixed point in

partially ordered sets was investigated by Ran and Reurings [22] and then by Nieto and Lopez [19].