Digital Versions of Fixed Point Theorem in Partially Ordered Sets and Application to Simplicial Complex

ZHANG Cong-lei,ZHANG Zhi-guo

1.College of Arts and Sciences,Shanxi Agricultural University,Jinzhong 030801,Shanxi,China;2.College of Mathematics and Computer Science,Shanxi Normal University,Linfen 041000，Shanxi,China;3.School of Mathematical Sciences,Hebei Normal University,Shijiazhuang 050024,Hebei,China

Abstract：The present paper states and proves analogues of Banach's fixed point theorem in partially ordered sets which weaken the requirements on the contraction conditions.Fixed point theorem in partially ordered sets and digital versions of fixed point theorem are refined and improved.Then,an application to simplicial complex model is discussed.

Key words:digital metric space;partially ordered set;fixed point theorem;simplicial complex.

0 Introduction

Fixed point theory plays an important role in many areas of mathematics,in terms of the influences it has had in the developments of different branches of mathematics and of physical science in general[1],which leads to various applications in mathematics and applied mathematics such as computer,engineering,image processing and so on.In metric spaces,fixed point theory is well recognized to have been originated in the work of Banach.It not only guarantees the existence and uniqueness of fixed points,and but also provides a method of finding those fixed points.Hence the Banach fixed point theorem becomes an important tool and the most significant test for solution of some problems in mathematics and applied mathematics.In recent times,several developments of fixed point theory have been occured.Fixed point theory in partially ordered metric spaces[2～8]is of recent origin.

Digital topology focuses on studying topological properties ofn-Ddigital images[9，10],which have contributed to some developments of image processing,computer graphics and so forth.To establish digital versions of the Banach fixed point theorem,the recent paper studied “Banach fixed point theorem for digital images”[10]and “Banach fixed point theorem from the viewpoint of digital topology”[11].However,the papers[10] and [11] have some things to be refined.The present paper studies an analogue of Banach's fixed point theorem in partially ordered sets and some applications to simplicial complex.We show that relevant conclusions still hold under the weaker conditions than original.

The rest of the paper is organized as follows:In Section 1,we give some basic and required definitions and results.In Section 2,we state and prove main results on fixed point theorem in partially ordered sets and its digital versions.More precisely,fixed point theorem in partially ordered sets and its digital versions are refined and improved.In Sections 3，we give an important application to simplicial complex model.

1 Preliminaries

Before referring to the works,first of all,we need to recall some basic nomenclatures,notions and results from digital topology.

LetZnbe the set of points in then-DEuclidean spaceRnwith integer coordinates.LetNrepresents the set of natural numbers andN*represents the set of positive natural numbers.

Two distinct pointsp=(p1,p2,...,pn)andq=(q1,q2,...,qn)∈Znarek(t,n)-(k-,for short)adjacent[11～13]if at most of their coordinates differs by ±1,and all others coincide,wheret,n∈Nand 1≤t≤n.Thesek(t,n)-adjacent relations ofZnare determined according to the following:

A digital interval is the set [a,b]z={n∈Z|a≤n≤b} with 2-adjacency,fora,b∈Zwitha

Definition1[11]LetX⊂Zn,(X,d)is a metric space inherited from the metric space (Rn,d)with the standard Euclidean metric functiondonRnand (X,k)is a digital image (i.e.a setXwith ak-adjacency relation),we call (X,d,k)a digital metric space.

A sequence {xn} of points of a digital metric space (X,d,k)converges to a limitL∈Xif for allε∈N*,there isα∈Nsuch that for alln>α,thend(xn,L)<ε[11].And we can conclude that a sequence {xn} of points of a digital metric space (X,d,k)converges to a limitL∈Xif there isα∈Nsuch that for alln>α,thenxn=L.

Definition2[10，11]A digital metric space (X,d,k)is complete if any Cauchy sequence {xn} (if for allε∈N*,there isα∈Nsuch that for alln,m>α,thend(xn,xm)<ε.)converges to a pointLof (X,d,k).

And it is easy to gain that a digital metric space (X,d,k)is complete[11].

2 A fixed point theorem in partially ordered sets

In this section we will present the following digital versions of the Banach fixed point theorem in partially ordered sets.

First of all,we recall the notion of partially ordered sets as follows:

Let us now discuss our fixed point theorems.

Theorem1 Letx⊂Znbe a partially ordered set such thatXhas a lower bound and an upper bound.Moreover,letdbe a metric onXsuch that (X,d)is a digital metric space.Iffis a monotone (i.e.either order-preserving or order-reversing)self-map on (X,d,k)such that

thenfhas a unique fixed pointx*. Furthermore,for ∀x∈X

ProofThe proof of this theorem can be broken up into the following steps:

Existence:

d(fn(x0),fn+1(x0))≤γd(fn-1(x0),fn(x0))

Hence,by finite iterations,induction,we gain

d(fn(x0),fn+1(x0))≤γnd(fn(x0),fn(x0))

wheref0(x0)=x0.

Letn

Uniqueness:

Therefore,we get

(2)Since every digital metric space is proved complete,the hypothesis of “complete”in the original Banach fixed point theorem is redundant.

(3)Recall that the original Banach fixed point theorem statesfhas a unique fixed point iffis a continuous self-map on a complete metric space (X,d)and satisfies

∃0≤γ<1:d(f(x),f(y))≤γd(x,y) ∀x,y∈X

This condition is obviously stronger than conditions in Theorem 1.

In fact,we can replace the condition “LetX⊂Znbe a partially ordered set such thatXhas a lower bound and an upper bound”with a weaker condition as follows.

Theorem2 LetXbe a partially ordered set and (X,d)is a complete metric space.Iffis a continuous,monotone (i.e.either order-preserving or order-reversing)self-map on (X,d)such that

(iii)Every pairx,x0has a lower bound and an upper bound for ∀x∈X,

thenfhas a unique fixed pointx*. Moreover,for ∀x∈X,

ProofBy using the method similar to the proof of Theorem 1,we can prove the results.

Corollary1 Letfis a continuous,monotone (i.e.either order-preserving or order-reversing)self-map on [a,b] such thatf(x)is not differentiable at most finite points and |f′(x)|<1 for ∀x∈Xandfis differentiable atx,thenfhas a unique fixed pointx*.Moreover,for ∀x∈X,

Corollary2 LetX⊂Znbe a partially ordered set and (X,d,k)is a digital metric space.Iffis a monotone (i.e.either order-preserving or order-reversing)self-map on (X,d,k)such that

(iii)Every pairx,x0has a lower bound and an upper bound for ∀x∈X,

thenfhas a unique fixed pointx*. Moreover,for ∀x∈X,

3 Application to simplicial complex model

In this section,we will apply Theorem 2 to simplicial complex model.

An incidence structureG=[S,I,dim] is defined by a countable setSof nodes,an incidence relationIonSthat is reflexive and symmetric,and a functiondimdefined onSand into a finite set {0,1,...,m} of natural numbers[14].Suppose a simplexsn,ai0,ai1,ai2,...,air∈En(0≤r≤n)arer+1 vertices of the simplexsn,andai0,ai1,ai2,...,airare linearly independent.Then the simplexsr,determined by them is called subsimplex of the simplexsn,denote bysr

For an incident relation ofK,a simple incident path withl+1 elements inKis an injective sequence (si)i∈[0,l]z⊂Ksuch thatsiandsjare incident if and only if |i-j|=1. Ifs0=xandsl=y,then the length of the shortest simple incident path,denoted bylI(x,y),is the numberl.

ProofBy using the method similar to the proof of Theorem 1,we can prove the results.

Acknowledgements

We would like to express our gratitude to the anonymous reviewers and editors for their valuable comments and suggestions which have improved the quality of the present paper.