, ,

(College of Mathematics and Systems Science, Shenyang Normal University, Shenyang 110034, China)

Abstract: The problem of endpoints is a new study line of the fixed point theory. The fixed point theory is very interesting and meaningful, which has a wide range of applications and profound theoretical value. The research area mainly studies the existence, uniqueness and solution method of fixed points of various mappings in different spaces. In 2007, Huang and Zhang introduced cone metrics paces, proved some fixed point theorems of contractive mappings on cone metric spaces. In recent years, the problem of developing different spaces and researching fixed points of various mappings in the developed spaces excited research enthusiasm of many scholars. In this paper we develop the metric space of partially ordered module with monotonic laws and the related convergence of sequences, which extend the cone metric space and the related convergence of sequences introduced by Huang and Zhang(2007). And establish three endpoint theorems for contractive multi-valued maps on such space, which cover some recent results of the fixed point theory. Our contributions not only vastly extend the range and the depth of the fixed point research area, but also strongly advance the mutual influences between the analysis and algebra.

Key words: endpoint; metric space; partially ordered module; monotonic law; topological structure; multi-valued map

0 Introduction

LetXbe a set andT:X→2Xbe a multi-valued(set-valued) map. A pointxis called a fixed point ofTifx∈Tx. Define Fix(T)={x∈X:x∈Tx}. An elementx∈Xis said to be an endpoint (or stationary point) ofTifTx={x}. We denote the set of all endpoints ofTby End(T).

In recent years, there has been an increasing interest in extending the study of fixed points. For example, Huang and Zhang[1](2007) replaced the real numbers by ordering Banach space and defined cone metric space. They also established some fixed point theorems for contractive type maps in a normal cone metric space. Subsequently, in the cone metric space, some other authors gave many results about fixed point, common fixed point and endpoint theory for maps and multi-valued maps, see e.g. [2-6]. In particular, by providing non-normal cones and omitting the assumption of normality, Rezapour and Hamlbarani[2](2008) generalize the major results of [1]. Lakshmikanthama and Ciric[7](2009) studied coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Zhang[8](2010) studied fixed point theorems for multi-valued monotone mappings in ordered metric spaces. Subba Rao and Pant[9](2011) studied fixed point theorems in Boolean vector spaces. A. Amini-Harandi[10](2011) studied fixed point theorems for generalized quasi-contraction maps in vector modular spaces. Among many other studies, see e.g. [11-18] and the references therein.

The investigation of endpoints of multi-valued mappings is an important extending of the study of fixed points, which was made as early as 30 years ago, and has received great attention in recent years, see e.g.[4,11-12] and the references therein.

Following the trend stated above, thepresent work introduces the metric space of partially ordered module with monotonic laws, which extends the cone metric space, and established a few of endpoint theorems for the contractive type multi-valued maps on the metric space introduced by us.

1 Preliminaries

This section proposes necessary preliminaries for our posterior discussions.

We first state, for a partial order ? of a set, we writeabto indicate thata?bbuta≠b, whereaandbare elements of the set.

Definition1 LetRbe an integral ring with partial order ≤, andMbeR-module with partial order?. Let also

M+={a∈M:a≥θ},M+={a∈M:a?θ},R+={r∈R:r≥0},R+={r∈R:r>0}.

Hereθand 0 are respectively the null elements ofMandR. Assume≤and?satisfy the following laws. (m1) 1>0, where 1 is the unit element ofR. (m2)

r≤s?r+t≤s+t,?r,s,t∈R; andr≤s?rt≤st,?r,s∈R, and?t∈R+.

(m3)a?b?a+c?b+c,?a,b,c∈M. (m4)a?b?ra?rb,?a,b∈Mand ?r∈R+; andr≤s?ra?sa,?r,s∈Rand?a∈M+. Then we callMisa(R,≤,?)-partially ordered module with monotonic laws.

Example1 LetEbe a Banach space of real fieldandPa subset ofE.Pis called a cone if and only if: 1)Pis closed, nonempty, andP≠{θ}; 2)r,s∈+,a,b∈P?ra+sb∈P; 3)a∈Pand -a∈P?a=θ. Here+denotes all the positive real numbers. For a given conePofE, define the partial order?byx?yif and only ify-x∈P, see [1]. Then it can be easily verified thatEis a (,≤,?)-partially ordered module with monotonic laws.

In the following we always supposeMisa(R,≤,?)-partially ordered module with monotonic laws.

Definition2 Let ? be a relation ofM.? is called a permissible topological structure ofMif it is satisfies: (t1)a?b?ab,?a,b∈M; (t2) (m3) And (m4); (t3)θ?a?b,?b?θ?a=θ; and (t4)a?b,b≤c?a?c.

Remark1 In particular, when there exists nobsuch thatb?θ, we believe (t3) holds for everya∈M+, that is, in the case,M+={θ} actually.

‖u-(b-a)-(c-b)‖=‖u-(c-a)‖

This impliesu-(b-a)∈N((c-b),r)?P. On the other hand, (b-a)∈Pfora?b. Sou=u-(b-a)+(b-a)∈P. NamelyN((c-a),r)?P. Henceθ?c-a,e.g.a?c, that is, (t4) holds. Therefore, ? is a permissible topological structure ofM.

Definition3 Let ? be a permissible topological structure ofM. A sequence {an} ofMwithan≥θis said to be convergent toθ(in?) if ?ε?θ, there is a natural numberNsuch thatan?εfor alln>N, denoted byan→θ.

Remark2 For moduleM, it is clear thatis a permissible topological structure. Let ? be another permissible topological structure and {an} be a sequence inM+. Then we can easily know thatan→θin ? ifan→θin. That is, the convergence inis stronger than in ?. So, the convergence in ? can be regarded as a kind of weak convergence.

Remark3 For moduleEand the permissible topological structure ?of Example 2, let {an} be a sequence inE+andan→θin norm. Then,?ε?θ, there existsr∈+such thatN(ε,r)?P. Due toan→θin norm, there exists also a natural numberNsuch that ‖an‖N. Therefore, ‖(ε-an)-ε‖N. This implies thatan→θin? ifan→θin norm. Whetheran→θin norm ifan→θin ?is a topic for further research.

Malways associates with a permissible topological structure ? and the mentioned convergence is in ? are assumed below.

Lemma1 Let {an} and {bn} be two sequences ofMwithan→θandbn→θ. Thenan+bn→θ.

ProofIf there is a natural numberNsuch thatan=θfor alln>N, then the result holds of course. Otherwise, we can complete the proof as follows. Letε?θ. Then we can choose anη∈Msuch thatη>θandε-η>θ. Hence, there are natural numbersN1andN2such thatan?η,?n>N1andbn?ε-η,?n>N2. PutN=max{N1,N2}. We have:an+bn?η+ε-η=ε,?n>N. So,an+bn→θ.

Definition4 LetXbe a non-empty set. Suppose the mappingd:X×X→Msatisfies

(d1)d(x,y)≥θfor allx,y∈Xandd(x,y) =θif and only ifx=y;

(d2)d(x,y)=d(y,x) for allx,y∈X;

(d3)d(x,y)?d(x,z)+d(z,y) for allx,y,z∈X.

Thendis called a metric of partially ordered moduleM(with monotonic laws) onX, and (X,d) is called a metric space of moduleM(with monotonic laws).

Next we always assume(X,d) is a metric space of moduleM.

Definition5 Given (X,d), letx∈Xand {xn} be a sequence inX.

ⅱ) {xn} is a Cauchy sequence if and only ifd(xn,xm)→θ, that is, ?ε?θ, there is a natural numberNsuch thatd(xn,xm)?εfor alln,m≥N.

ⅲ) (X,d) is complete if and only if every Cauchy sequence is convergent.

2 Endpoint theory

Now we are ready to propose and prove our main results.

We first extend Banach’s Contraction Principle of single maps in the usual metric space to multi-valued maps in the metric space of moduleMby the following Theorem 1.

Lemma2 Assume (X,d) is complete andT:X→(2X-?) is a multi-valued map. Letx0∈X, andxn∈Txn-1for alln∈. If there exists ar∈Rwith 0≤r<1, such that

d(xn+1,xn)?rd(xn,xn-1),?n∈,

(1)

andrn·d(x1,x0)→θ, then the iterative sequence {xn} is convergent.

Proof. Firstly, by (1) we have

d(xn+1,xn)?rd(xn,xn-1)?…?rnd(x1,x0).

(2)

Next, for arbitraryn,m∈, assumingn

whereSm-n-1=1+r+…+rm-n-1. For 0≤r<1, this leads to

(1-r)d(xn,xm)?(1-r)Sm-n-1rnd(x1,x0).

(3)

On the other hand, fromSm-n-1=1+r+…+rm-n-1, we have

rSm-n-1=r+r2+…+rm-n,

further

(1-r)Sm-n-1=1-rm-n<1.

(4)

Combining (3) and (4), we obtain

(1-r)d(xn,xm)?rnd(x1,x0).

(5)

Finally, letε?θ. Then (1-r)ε?θalso. Thus, forrnd(x1,x0)→θ, there exists a natural numberNsuch that

rnd(x1,x0)?(1-r)ε

(6)

for alln>N. Combining (5) and (6), we have

(1-r)d(xn,xm)?(1-r)ε?d(xn,xm)?ε

for alln>m>N. That is, the sequence {xn} is a Cauchy sequence. Therefore, we can know that {xn} is convergent from (X,d) is complete. This ends the proof.

Theorem1 Given (X,d), assumeT:X→(2X-?) be a multi-valued map. If there exists ar∈Rwith 0≤r<1, such that

d(x′,y′)?rd(x,y),?x′∈Tx,?y′∈Ty,

(7)

for allx,y∈Xwithx≠y, then we have the following conclusions.

ⅰ)Thas one fixed point at most. (|End(T)|≤|Fix(T)|≤1. Here |End(T)| denotes the cardinal number of End(T)).

ⅱ) If(X,d) is complete andrn·a→θfor anya∈M+, then Thas a unique endpoint. (|End(T)|=1).

Proof. 1) To prove (ⅰ), letx,y∈Fix(T) andx≠y. Then, by (7), we have

d(x,y)?rd(x,y)?(1-r)d(x,y)?θ?d(x,y)?θ.

Ford(x,y)±θ, this impliesd(x,y)=θ. Hence we havex=yfrom (d1). So |End(T)|≤|Fix(T)|≤1. (ⅰ) holds.

2) To prove (ⅱ), we first assume|End(T)|<1. Then, for anyx0∈X, we can construct a sequence {xn} ofXsuch thatxn∈Txn-1andxn≠xn-1for alln∈. For the iterative sequence, by (7), we can easily know (1) holds. Hence, from Lemma 1, it converges to a pointxofXsince (X,d) is complete andrn·a→θfor anya∈M+. We showx∈End(T) next.

Sincexn≠xn-1for alln∈, we can choose a subsequence {xni} of {xn} such thatxni≠xfor alli∈. Hence, without loss of generality, we assumexn≠xfor alln∈. Letx′∈Tx. Then

Forxn→x, by Lemma 1, we haved(x,xn)+d(xn-1,x)→θ. Hence, from (8), we further know thatd(x,x′)?εfor anyε?θ. This yields tod(x,x′)=θdue tod(x,x′)±θ. That isx′=x. Sox∈End(T). For this contradicts |End(T)|<1, we have |End(T)|≥1.

Finally, in terms of |End(T)|≥1 and (ⅰ), we have |End(T)|=1. (ⅱ) holds.

The study to generalize and extend Banach’s Contraction Principle has been at the center of the research activity of fixed point theory for long time, and it has a wide range of applications in different areas such as nonlinear and adaptive control systems, fractal image decoding and convergence of recurrent networks. Recently, Huang and Zhang generalized the metric space by replacing real numbers with an ordered Banach space and obtained some fixed point theorems for mapping satisfying different contractive conditions, see[1]. Consequently, the research of fixed points in such spaces is followed by many other mathematicians, see e.g.[2-6] and the references therein. On the other hand, the investigations of endpoints of multi-valued maps, as is obviously an generalization of fixed points of single maps, have received much attention in recent years, see e.g.[11-12] and the references therein. Through the study of endpoints of multi-valued maps on the metric space of moduleM, we not only generalize some recent results, such as the Theorem 1 of [1] and the Theorem 2.3 of [2], but also extend the activity range of this research line.

Next, we extend Theorem 2.6 and Theorem 2.7 of[2] (which generalize Theorem 3 and Theorem 4 of[1], respectively) to the Theorem 2 below, which discusses about endpoints of other types of contractive maps.

Theorem2 Given (X,d), assumeT:X→(2X-?) be a multi-valued map. If there exists ak∈R+with 2k=(k+k)<1, such that

d(x′,y′)?k[d(x′,x)+d(y′,y)],?x′∈Tx,?y′∈Ty;

(9)

or

d(x′,y′)?k[d(x′,y)+d(y′,x)],?x′∈Tx,?y′∈Ty,

(10)

for allx,y∈Xwithx≠y. Then we have the following conclusions.

ⅰ)Thas one fixed point at most. (|End(T)|≤|Fix(T)|≤1)

ⅱ) If(X,d) is complete and (2k)n·a→θfor anya∈M+, thenThas a unique endpoint.

(|End(T)|=1)}

Proof1) To prove (ⅰ), letx,y∈Fix(T) andx≠y. Then, from (9) or (10), we have

d(x,y)?k[d(x,x)+d(y,y)]=θ

or

d(x,y)?k[d(x,y)+d(x,y)]=2kd(x,y)?(1-2k)d(x,y)?θ?d(x,y)?θ.

(Note 2k<1.) Hencex=y. (ⅰ) holds.

2) To prove (ⅱ), we first assume|End(T)|<1. Then, for anyx0∈X, we can construct a sequence {xn} ofXsuch thatxn∈Txn-1andxn≠xn-1for all n∈. From (9) or (10), we have

d(xn+1,xn)?k[d(xn+1,xn)+d(xn,xn-1)]

(11a)

or

d(xn+1,xn)?k[d(xn+1,xn-1)+d(xn,xn)]?k[d(xn+1,xn)+d(xn,xn-1)].

(11b)

This leads to

(1-k)d(xn+1,xn)?kd(xn,xn-1)

(12)

Note thatk+k<1, that is,k<1-k. From (12) we obtain

kd(xn+1,xn)?kd(xn,xn-1)?d(xn+1,xn)?d(xn,xn-1).

(13)

Combine (11) and (13), we have

d(xn+1,xn)?k[d(xn,xn-1)+d(xn,xn-1)]=2kd(xn,xn-1).

(14)

Note that (X,d) is complete and (2k)na→θfor anya∈M+. From (14) and Lemma 2, {xn} converges to a pointxofX. We showx∈End(T) next.

Without loss of generality, assumexn≠xfor alln∈. Letx′∈Tx.

Then, from (9) or (10), we have

or

Sincexn→xand 0≤2k<1, this implies tod(x,x′)=θ. That is,x′=x, or sayx∈End(T). So|End(T)|≥1.

Finally, in terms of |End(T)|≥1 and (ⅰ), we have |End(T)|=1. (ⅱ) holds.

Finally, we further extend Theorem 2.8 of [2] to the Theorem 3 below.

Theorem3 Given (X,d), assumeT:X→(2X-?) be a multi-valued map. If there existr,s∈R+withr+s<1, such that

d(x′,y′)?rd(x′,y)+sd(x,y′),?x′∈Tx,?y′∈Ty

(15)

for allx,y∈Xwithx≠y. Then we have the following conclusions.

ⅰ)Thas at most one fixed point. (|End(T)|≤|Fix(T)|≤1)

ⅱ) If (X,d) is complete and (r+s)n·a→θfor anya∈M+, thenThas a unique endpoint.

(|End(T)|=1)

Proof1) To prove (ⅰ), letx,y∈Fix(T) andx≠y. Then, by (15), we have

d(x,y)?rd(x,y)+sd(x,y)=(r+s)d(x,y).

Since (r+s)<1, this impliesd(x,y)=θ. (ⅰ) holds.

2) To prove (ⅱ), we first assume |End(T)|<1. Then, for anyx0∈X, we can construct a sequence {xn} ofXsuch thatxn∈Txn-1andxn≠xn-1for alln∈. By (15), we have

On the other hand,

d(xn+1,xn)=d(xn,xn+1)?rd(xn,xn)+sd(xn+1,xn-1)?(1-s)d(xn+1,xn)?sd(xn,xn-1).

(17)

Combining (16) and (17), we obtain

{1+[1-(r+s)]}d(xn+1,xn)?(r+s)d(xn,xn-1).

Since 0≤r+s<1, this implies

d(xn+1,xn)?(r+s)d(xn,xn-1).

(18)

Note that (X,d) is complete and (r+s)n·a→θfor anya∈M+. From (18) and Lemma 2, we know that {xn} converges to a pointxofX. Without loss of generality, assumexn≠xfor alln∈. Letx′∈Tx.

Then

Sincexn→xand 0≤r+s<1, we can easily knowd(x,x′)=θ, that is,x=x′. This implies |End(T)|≥1. Finally, we obtain |End(T)|=1 from (ⅰ). (ⅱ) holds.

Remark4 Basing on the observation of Example 1 and Example 2, we can immediately derive the major results of[1] and[2] from our Theorem 1, Theorem 2 and Theorem 3, respectively. For example, we can immediately derive Theorem 2.3 of[2], which generalize Theorem 1 of[1], from our Theorem 1. So those are extended, or say are covered, by our results.

On the other hand, we must state, the extension is never trivial. It can been easily seen from our results and their proofs.

Finally, we hope that the present work will stimulate more contributions in the research area of the fixed point theory.

Acknowledgements

The author cordially thanks the anonymous referees for their valuable comments which lead to the improvement of this paper.