An Arbitrary Starting Simplicial Algorithm for Constructively Proving Tarski's Fixed Point Theorem on an n-dimensional Box

The well-known Tarski's fixed point theorem asserts that an increasing mapping from an n-dimensional box to itself has a fixed point. In this paper, a constructive proof of this theorem is obtained from an application of the (n + l)-ray arbitrary starting simplicial algorithm. The algorithm assigns an integer label to each point of the box and employs a triangulation to subdivide the box into simplices. For any given mesh size of the triangulation, starting from an arbitrary interior point of the box, the algorithm generates within a finite number of iterations a complete n-dimensional simplex, any point of which yields an approximate fixed point. If the accuracy is not good enough, the mesh size of the triangulation is refined and the algorithm is restarted. When the mesh size goes to zero sequentially, one will obtain a sequence of approximate fixed points satisfying that every limit point of the sequence is a fixed point.