Nonlinear Analysis64(2006)546–557
/locate/na
Fixed point theorems in metric spaces
Petko D.Proinov
Faculty of Mathematics and Informatics,University of Plovdiv,Plovdiv4000,Bulgaria
Received21February2005;accepted11April2005
Abstract
In this paper,we establish two general theorems for equivalence between the Meir–Keeler type contractive conditions and the contractive definitions involving gauge functions.One of the theorems is an extension of a recent result of Lim(On characterization of Meir–Keeler contractive maps, Nonlinear Anal.46(2001)113–120).
Also,we establish the following newfixed point theorem.Suppo :R+→R+is a contractive gauge func
tion in the n that for any >0there exists > such that <t< implies (t) , and suppo T is a continuous and asymptotically regular lfmapping on a complete metric space (X,d)satisfying the following:
(i)d(T x,T y) (D(x,y))for all x,y∈X,and
(ii)d(T x,T y)<D(x,y)for all x,y∈X with x=y,
where D(x,y)=d(x,y)+ .[d(x,T x)+d(y,T y)]with 0.Then T has a uniquefixed point and all of the Picard iterates of T converge to thisfixed point.
This result includes tho of Jachimski(Equivalent conditions and the Meir–Keeler type theorems, J.Math.Anal.Appl.194(1995)293–303),Matkowski(Fixed point theorems for contractive mappings in metric spaces,Cas.Pest.Mat.105(1980)341–344)and others.
᭧2005Elvier Ltd.All rights rerved.
Keywords:Fixed point theorems;Meir–Keeler conditions;Gauge functions;Equivalent conditions
1.Introduction
Many authors have extended the well-known Banach’sfixed point theorem in various ways.In this paper we shall consider two types of such generalizations.Thefirst type
E-mail address:proinov@pu.acad.bg.
0362-546X/$-e front matter᭧2005Elvier Ltd.All rights rerved.
doi:10.1016/j.na.2005.04.044
P.D.Proinov/Nonlinear Analysis64(2006)546–557547 involves the Meir–Keeler type conditions and the cond one involves contractive gauge functions.We establish two general theorems for equivalence between the two types of contractive definitions.Also,we prent a newfixed point theorem.
Definition1.1.Let T be a lfmapping on a metric space(X,d).Afixed point of T is said to be contractive[13]if all of the Picard iterates of T converge to thisfixed point.
Definition1.2.A lfmapping T on a metric space(X,d)is said to be
(i)contractive[8]if d(T x,T y)<d(x,y)for all x,y∈X with x=y;
(ii)asymptotically regular[4]if lim n→∞d(T n x,T n+1x)=0for each x∈X.
The following theorem of´Ciri´c[5]and Matkowski[12,Theorem1.5.1]1generalizes a well-knownfixed point theorem of Meir and Keeler[17].
Theorem1.3(´Ciri´c[5],Matkowski[12]).Suppo T is a contractive lfmapping on a complete metric space(X,d)which satisfies the following condition:for any >0there exists > such that <d(x,y)< implies d(T x,T y) .Then T has a contractivefixed point.
Suppo T is a lfmapping on a complete metric space(X,d)satisfying d(T x,T y) (d(x,y))for all x,y∈X(1.1) with a gauge function which maps the t R+of all nonnegative numbers into itlf. If is defined by (t)= t(0< <1),then T has a contractivefixed point.This is the famousfixed point theorem of Banach.Thefirst extensions of Banach’s theorem involving gauge functions are due to Rakotch[19],Browder[3],Boyd and Wong[2],Bianchini and Grandolfi[1],Furi[9],Zitarosa[22],Matkowski[15],and others.
In1995,Jachymski[11,Corollary]prented afixed point theorem involving gauge functions with much weaker conditions.
Theorem1.4(Jachymski[11]).Suppo T is a contractive lfmapping on a complete metric space(X,d)satisfying(1.1)with some gauge function which satisfies the following condition:for any >0there exists > such that <t< implies (t) .Then T has a contractivefixed point.
In Section3,we prove that Theorems1.3and1.4are equivalent.Moreover,we establish two general theorems(Theorems3.2and3.5)for equivalence between the Meir–Keeler type contractive conditions and the contractive conditions involving gauge functions.Theorem 3.5is an extension of a recent result of Lim[14,Theorem1].Note that other characterizations of the Meir–Keeler type conditions in terms of gauge functions are given in[10,21].
1Theorem1.5.1in Kuczma et al.[12]is due to Matkowski.
548P.D.Proinov/Nonlinear Analysis64(2006)546–557
In1995,Jachymski[11,Theorem2]replaced the distance function d(x,y)in´Ciri´c–Matkowski’s theorem by
群邑集团m(x,y)=max{d(x,y),d(x,T x),d(y,T y),[d(x,T y)+d(y,T x)]/2}.(1.2) We refer to this result of Jachymski as Jachymski–Matkowski’s theorem(e Corollary4.4) becau it is equivalent to a result of Matkowski[16,Theorem1].
In Section4,we establish a newfixed point theorem,replacing m(x,y)in Matkowski–Jachymski’s theorem by
D(x,y)=d(x,y)+ .[d(x,T x)+d(y,T y)]where 0,(1.3) and assuming that T is asymptotically regular.We formulate this result in both types of contractive conditions(Theorems4.1and4.2).Note that ourfixed point theorem generalizes Matkowski–Jachymski’s theorem([16,Theorem1;11,Theorem2])and some results of
´Ciri´c[6,7].
2.Gauge functions
In this ction we consider veral class of gauge functions and some of their properties. Definition2.1.Let
(i) 1denote the class of all functions :R+→R+satisfying:for any >0there exists
> such that <t< implies (t) ;
(ii) 2denote the class of all functions :R+→R+satisfying:for any >0there exists > such that t< implies (t) ;
(iii) denote the class of all nondecreasing functions :R+→R+satisfying:for any >0there exists > such that ( ) .
It is easy to e that 1⊃ 2⊃ .
The following uful property of the class 1improves a lemma of Taskovic [20,Lemma2].
Lemma2.2.Let T be a lfmapping of an arbitrary t X and let E:X→R+be a real-valued function defined on X.Suppo that the following conditions hold:
(i)There exists a function ∈ 1such that E(T x) (E(x))for all x∈X;
(ii)E(x)>0implies E(T x)<E(x)and E(x)=0implies E(T x)=0.
Then lim n→∞E(T n x)=0for each x∈X.
Proof.Take x∈X and put E n=E(T n x)for n∈N.If E n=0for some n∈N,then the conclusion is trivial.So,assume that E n>0for all n∈N.It follows from(ii)that(E n)is strictly decreasing,hence it converges to some E 0and E n>E for each n∈N.Suppo that E>0.Since ∈ 1,there exists >E such that E<t< implies (t) E.Since
P .D.Proinov /Nonlinear Analysis 64(2006)546–557549
E n ↓E and E < ,then there exists n ∈N such that E <E n < ,and so (E n ) E .Now,it follows from (i),that E n +1 (E n ) E which is a contradiction.Thus we get E =E n →0.
Lemma 2.3.Suppo :R +→R +.Then ∈ if and only if satisfies the following conditions :
(i) is nondecreasing and (t) t for all t 0;说唱的英文
(ii)if ( +)= for some >0,then there exists > such that ( )= .
Proof.Suppo ∈ .We shall prove only (ii).Let ( +)= for some >0.Since is nondecreasing,we deduce that (t) for all t > .On the other hand,by the definition of ,there exists > such that ( ) .Therefore, ( )= .Now suppo that satisfies the conditions (i),(ii)and suppo that >0.By (i)we get ( +) .If ( +)< ,then there exists > such that ( )< .If ( +)= ,then by (ii)there exists > such that ( )= .Therefore,in both cas ( ) ,and so ∈ .
Lemma 2.4.For any function ∈ there exists a right continuous function ∈ such that .Moreover ,one can choo to satisfy also the following condition :
(t)>0for all t >0.(2.1)Proof.The first part of the conclusion is trivial.Indeed,suppo ∈ and define the function :R +→R +by (t)= (t +)for all t 0.It is easy to show that ∈ , is right continuous and .Now we shall prove the cond part of the conclusion.If (t)>0for all t >0,we put = .If (t)=0for all t 0,we put (t)= t (0< <1).Now suppo that there exist t 1>0and t 2>0such that (t 1)=0and (t 2)>0.Define b >0by
b =sup {t >0: (t)=0}.
By the definition of b ,we have (t)=0for all t ∈[0,b),and (t)>0for all t ∈(b,+∞).Further,we shall consider two cas:
Ca 1.Let (b)>0.Choo a number such that 0< < (b)/b .Then define the func-tion :R +→R +by (t)= t for 0 t <b, (t)for t b.
Ca 2.Let (b)=0.Then there exists c >b such that (c)<b .Put = (c)/b and define the function :R +→R +by (t)= t for 0 t b, (c)for b t c, (t)for t c.
It is easy to verify that in both cas ∈ , is right continuous and satisfies (2.1).The inequality is obvious.
除法的初步认识
550P .D.Proinov /Nonlinear Analysis 64(2006)546–557
3.Equivalent conditions
Lemma 3.1.Let T be a lfmapping on an arbitrary t X and let E :X →R +and F :X →R +be two real-valued functions defined on X.Suppo that
E(T x) F (x)for any x ∈X .(3.1)Then the following statements are equivalent :
(i)There is a function ∈ 1such that E(T x) (F (x))for any x ∈X .
(ii))For any >0there is > such that <F (x)< implies E(T x) .
初战告捷In (i)one can choo to be also nondecreasing ,right continuous and satisfying (2.1).Proof.(i )⇒(ii ):Suppo (i)holds.Since ∈ 1,then for any >0there exists > such that <t < implies (t) .Therefore, <F (x)< implies E(T x) (F (x)) .(ii )⇒(i ):For a given t 0define the ts
A t ={x ∈X :F (x) t }and
B t ={E(T x):x ∈X,F (x) t }.
十八岁的天空2From (3.1),we get that F (x) t implies E(T x) t .Therefore,the t B t is nonempty and bounded if A t is nonempty.Now define :R +→R +by
新春年货(t)= sup B t if A t is nonempty ,0if A t is empty .
It is easy to prove that is nondecreasing and E(T x) (F (x))for any x ∈X .From (ii)and (3.1)it follows that for any >0there exists > such that F (x) implies E(T x) .Then by the definition of it follows that for any >0there exists > such that ( ) ,hence ∈ .By Lemma 2.4,it follows that one can choo to be also right continuous and satisfying (2.1).
The following theorem is a special ca of Lemma 3.1.
Theorem 3.2.Let T be a lfmapping on a metric space (X,d)and let F :X 2→R +be a real-valued function defined on X 2.Suppo that
d(T x,T y) F (x,y)for any x,y ∈X .(3.2)Then the following statements are equivalent :
(i)There exists ∈ 1such that d(T x,T y) (F (x,y))for any x,y ∈X .
(ii)For any >0there exists > such that <F (x,y)< implies d(T x,T y) .In (i)one can choo to be also
nondecreasing ,right continuous and satisfying (2.1).Remark 3.3.By Theorem 3.2,it follows that Theorem 1.3is equivalent to Theorem 1.4.垃圾分类图画
P.D.Proinov/Nonlinear Analysis64(2006)546–557551 Lemma3.4.Let T be a lfmapping on an arbitrary t X and let E:X→R+and F:X→R+be two real-valued functions defined on X.Suppo that
F(x)=0implies E(T x)=0.(3.3)
Then the following statements are equivalent:
(i)There exists a function ∈ 1such that E(T x)< (F(x))for any x∈X with
F(x)>0.
(ii)For any >0there is > such that F(x)< implies E(T x)< .
In(i)one can choo to be also nondecreasing,right continuous and satisfying(2.1). Proof.Since the implication(i)⇒(ii)is trivial,it suffices to prove(ii)⇒(i).Wefirst obrve that(ii)trivially implies that
E(T x)<F(x)for all x∈X with F(x)>0.
From this and(3.3)we get(3.1).Now,by Lemma3.1,there exists a right continuous function ∈ such that
E(T x) (F(x))for all x∈X.(3.4)
Define the t A by
A={ >0: ( )= }.
It is easy to e that the t A is at most countable.Further,we shall consider following two cas:
Ca1.Suppo A is nonempty.Then by Lemma2.3and(ii)there exists a function :A→(0,∞)such that ( )> for all ∈A and satisfying the conditions:
t< ( )implies (t)= (3.5)
and
F(x)< ( )implies E(T x)< .(3.6)
Take a function :A→(0,∞)satisfying the conditions.Obviously,for all ∈A the intervals[ , ( ))are mutual
ly disjoint.Denote
[ , ( )).
大豆卵磷脂的功效与作用B=
∈A
It is easy to prove that
E(T x)< (F(x))for all x∈X with F(x)∈B.(3.7)
