# New Results and Generalizations for Approximate Fixed Point Property and Their Applications

Authors: Wei-Shih Du; Farshid Khojasteh
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We first introduce the concept of manageable functions and then prove some new existence theorems related to approximate fixed point property for manageable functions andα-admissible multivalued maps. As applications of our results, some new fixed point theorems which
generalize and improve Du's fixed point theorem, Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, and Nadler's fixed point theorem and some well-known results in the literature are given.


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## Body

## 1.  Introduction and Preliminaries

In 1922, Banach established the most famous fundamental fixed point theorem (so-called the Banach contraction principle [1]) which has played an important role in various fields of applied mathematical analysis. It is known that the Banach contraction principle has been extended and generalized in many various different directions by several authors; see [2–40] and references therein. An interesting direction of research is the extension of the Banach contraction principle to multivalued maps, known as Nadler’s fixed point theorem [2], Mizoguchi-Takahashi’s fixed point theorem [3], and Berinde-Berinde’s fixed point theorem [5] and references therein.Let us recall some basic notations, definitions, and well-known results needed in this paper. Throughout this paper, we denote byℕ and ℝ the sets of positive integers and real numbers, respectively. Let (X,d) be a metric space. For each x∈X and A⊆X, let d(x,A)=inf⁡y∈Ad(x,y). Denote by 𝒩(X) the class of all nonempty subsets of X, 𝒞(X) the family of all nonempty closed subsets of X, and 𝒞ℬ(X) the family of all nonempty closed and bounded subsets of X. A function ℋ:𝒞ℬ(X)×𝒞ℬ(X)→[0,∞) defined by
(1)ℋ(A,B)=max⁡{sup⁡x∈Bd(x,A),sup⁡x∈Ad(x,B)}
is said to be the Hausdorff metric on 𝒞ℬ(X) induced by the metric d on X. A point v in X is a fixed point of a map T, if v=Tv (when T:X→X is a single-valued map) or v∈Tv (when T:X→𝒩(X) is a multivalued map). The set of fixed points of T is denoted by ℱ(T). The map T is said to have the approximate fixed point property [29–34] on X provided inf⁡x∈Xd(x,Tx)=0. It is obvious that ℱ(T)≠∅ implies that T has the approximate fixed point property, but the converse is not always true.Definition 1 (see [6, 13]).
A functionφ:[0,∞)→[0,1) is said to be an ℳ𝒯-function (or ℛ-function) if lim⁡sup⁡s→t+φ(s)<1 for all t∈[0,∞).It is evident that ifφ:[0,∞)→[0,1) is a nondecreasing function or a nonincreasing function, then φ is a ℳ𝒯-function. So the set of ℳ𝒯-functions is a rich class.Recently, Du [6] first proved the following characterizations of ℳ𝒯-functions which are quite useful for proving our main results.Theorem 2 (see [6]).
Letφ:[0,∞)→[0,1) be a function. Then the following statements are equivalent.(a)
φis an ℳ𝒯-function.(b)
For eacht∈[0,∞), there exist rt(1)∈[0,1) and εt(1)>0 such that φ(s)≤rt(1) for all s∈(t,t+εt(1)).(c)
For eacht∈[0,∞), there exist rt(2)∈[0,1) and εt(2)>0 such that φ(s)≤rt(2) for all s∈[t,t+εt(2)].(d)
For eacht∈[0,∞), there exist rt(3)∈[0,1) and εt(3)>0 such that φ(s)≤rt(3) for all s∈(t,t+εt(3)].(e)
For eacht∈[0,∞), there exist rt(4)∈[0,1) and εt(4)>0 such that φ(s)≤rt(4) for all s∈[t,t+εt(4)).(f)
For any nonincreasing sequence{xn}n∈ℕ in [0,∞), one has 0≤sup⁡n∈ℕφ(xn)<1.(g)
φ is a function of contractive factor [15]; that is, for any strictly decreasing sequence {xn}n∈ℕ in [0,∞), one has 0≤sup⁡n∈ℕφ(xn)<1.In 1989, Mizoguchi and Takahashi [3] proved a famous generalization of Nadler’s fixed point theorem which gives a partial answer of Problem 9 in Reich [4].Theorem 3 (Mizoguchi and Takahashi [3]).
Let(X,d) be a complete metric space, let φ:[0,∞)→[0,1) be an ℳ𝒯-function, and let T:X→𝒞ℬ(X) be a multivalued map. Assume that
(2)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y),
for all x,y∈X. Then ℱ(T)≠∅.In 2007, M. Berinde and V. Berinde [5] proved the following interesting fixed point theorem which generalized and extended Mizoguchi-Takahashi’s fixed point theorem.Theorem 4 (M. Berinde and V. Berinde [5]).
Let(X,d) be a complete metric space, let φ:[0,∞)→[0,1) be an ℳ𝒯-function, let T:X→𝒞ℬ(X) be a multivalued map, and L≥0. Assume that
(3)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y)+Ld(y,Tx),
for all x,y∈X. Then ℱ(T)≠∅.In 2012, Du [6] established the following fixed point theorem which is an extension of Berinde-Berinde’s fixed point theorem and hence Mizoguchi-Takahashi’s fixed point theorem.Theorem 5 (Du [6]).
Let(X,d) be a complete metric space, let T:X→𝒞ℬ(X) be a multivalued map, let φ:[0,∞)→[0,1) be a ℳ𝒯-function, and let h:X→[0,∞) be a function. Assume that
(4)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y)+h(y)d(y,Tx)∀x,y∈X.
Then T has a fixed point in X.The paper is organized as follows. In Section2, we first introduce the concept of manageable function and give some examples of it. Section 3 is dedicated to the study of some new existence theorems related to approximate fixed point property for manageable functions and α-admissible multivalued maps. As applications of our results, some new fixed point theorems which generalize and improve Du’s fixed point theorem, Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, and Nadler’s fixed point theorem and some well-known results in the literature are given in Section 4. Consequently, some of our results in this paper are original in the literature, and we obtain many results in the literature as special cases.


## 2.  Manageable Functions

In this paper, we first introduce the concept of manageable functions.Definition 6.
A functionη:ℝ×ℝ→ℝ is called manageable if the following conditions hold:(η1)
η(t,s)<s-t for all s,t>0;(η2)
for any bounded sequence{tn}⊂(0,+∞) and any nonincreasing sequence {sn}⊂(0,+∞), it holds that
(5)lim⁡sup⁡n→∞tn+η(tn,sn)sn<1.
We denote the set of all manageable functions byMan(ℝ)^.Here, we give simple examples of manageable function.Example A.
Letγ∈[0,1). Then ηγ:ℝ×ℝ→ℝ defined by
(6)ηγ(t,s)=γs-t
is a manageable function.Example B.
Letf:ℝ×ℝ→ℝ be any function. Then the function η:ℝ×ℝ→ℝ defined by
(7)η(t,s)={ss+9ln⁡(s+10)-t,if(t,s)∈[0,+∞)×[0,+∞),f(t,s),otherwise,
is a manageable function. Indeed, let
(8)g(x)=ln⁡(x+10)x+9∀x>-9.
Then g(s)<1 for all s>0, and
(9)η(t,s)={sg(s)-t,if(t,s)∈[0,+∞)×[0,+∞),f(t,s),otherwise.
For any s,t>0, we have
(10)η(t,s)=sg(s)-t<s-t,
so (η1) holds. Let {tn}⊂(0,+∞) be a bounded sequence and let {sn}⊂(0,+∞) be a nonincreasing sequence. Then lim⁡n→∞sn=inf⁡n∈ℕsn=a for some a∈[0,+∞). Since g is continuous, we get
(11)lim⁡sup⁡n→∞tn+η(tn,sn)sn=lim⁡n→∞g(sn)=g(a)<1,
which means that (η2) holds. Hence, η∈Man(ℝ)^.Example C.
Letf:ℝ×ℝ→ℝ be any function and let φ:[0,∞)→[0,1) be an ℳ𝒯-function. Define η:ℝ×ℝ→ℝ by
(12)η(t,s)={sφ(s)-t,if(t,s)∈[0,+∞)×[0,+∞),f(t,s),otherwise.
Then η is a manageable function. Indeed, one can verify easily that (η1) holds. Next, we verify that η satisfies (η2). Let {tn}⊂(0,+∞) be a bounded sequence and let {sn}⊂(0,+∞) be a nonincreasing sequence. Then lim⁡n→∞sn=inf⁡n∈ℕsn=a for some a∈[0,+∞). Since φ is an ℳ𝒯-function, by Theorem 2, there exist ra∈[0,1) and εa>0 such that φ(s)≤ra for all s∈[a,a+εa). Since lim⁡n→∞sn=inf⁡n∈ℕsn=a, there exists na∈ℕ, such that
(13)a≤sn<a+εa∀n∈ℕwithn≥na.
Hence, we have
(14)lim⁡sup⁡n→∞tn+η(tn,sn)sn=lim⁡sup⁡n→∞φ(sn)≤ra<1,
which means that (η2) holds. So we prove η∈Man(ℝ)^.The following result is quite obvious.Proposition 7.
Letζ:ℝ×ℝ→ℝ be a function. If there exists η∈Man(ℝ)^ such that ζ(t,s)≤η(t,s) for all s,t>0, then ζ∈Man(ℝ)^.Proposition 8.
Let{ηi}i∈ℕ⊂Man(ℝ)^. Then the following statements hold.(a)
For eachk∈ℕ, the function  η(k)min⁡:ℝ×ℝ→ℝ, defined by(15)η(k)min⁡(t,s)=min⁡{η1(t,s),η2(t,s),…,ηk(t,s)},is a manageable function  (i.e.,η(k)min⁡∈Man(ℝ)^foranyk∈ℕ).(b)
For eachk∈ℕ, the function  η(k)¯:ℝ×ℝ→ℝ, defined by(16)η(k)¯(t,s)=1k∑i=1k‍ηi(t,s),is a manageable function  (i.e.,η(k)¯∈Man(ℝ)^foranyk∈ℕ).Proof.
Sinceη(k)min⁡(t,s)≤η1(t,s) for all t,s>0, the conclusion (a) is a direct consequence of Proposition 7. Next, we prove the conclusion (b). Let k∈ℕ be given. It is obvious that η(k)¯(t,s)<s-t for all s,t>0. Let {tn}⊂(0,+∞) be a bounded sequence and let {sn}⊂(0,+∞) be a nonincreasing sequence. For any n∈ℕ, we have
(17)tn+η(k)¯(tn,sn)sn=1sn(tn+1k∑i=1k‍ηi(t,s))=1k∑i=1k‍tn+ηi(tn,sn)sn.
Because each ηi satisfies (η2), we get
(18)lim⁡sup⁡n→∞tn+η(k)¯(tn,sn)sn=1klim⁡sup⁡n→∞(∑i=1k‍tn+ηi(tn,sn)sn)≤1k∑i=1k‍(lim⁡sup⁡n→∞tn+ηi(tn,sn)sn)<1.
Hence, for each k∈ℕ, the function η(k)¯ is a manageable function.


## 3.  New Existence Results for Manageable Functions and Approximate Fixed Point Property

Recall that a multivalued mapT:X→𝒞ℬ(X) is called(1)
a Nadler’s type contraction (or a multivaluedk-contraction [3, 33]), if there exists a number 0<k<1 such that
(19)ℋ(Tx,Ty)≤kd(x,y)∀x,y∈X;(2)
a Mizoguchi-Takahashi’s type contraction [33], if there exists an ℳ𝒯-function α:[0,∞)→[0,1) such that
(20)ℋ(Tx,Ty)≤α(d(x,y))d(x,y)∀x,y∈X;(3)
a multivalued(θ,L)-almost contraction [28, 29, 33], if there exist two constants θ∈(0,1) and L≥0 such that
(21)ℋ(Tx,Ty)≤θd(x,y)+Ld(y,Tx)∀x,y∈X;(4)
a Berinde-Berinde’s type contraction [33] (or a generalized multivalued almost contraction [28, 29, 33]), if there exists an ℳ𝒯-function α:[0,∞)→[0,1) and L≥0 such that
(22)ℋ(Tx,Ty)≤α(d(x,y))d(x,y)+Ld(y,Tx)∀x,y∈X;(5)
a Du’s strong type contraction, if there exist anℳ𝒯-function α:[0,∞)→[0,1) and a function h:X→[0,∞) such that
(23)ℋ(Tx,Ty)≤α(d(x,y))d(x,y)+h(y)d(y,Tx)∀x,y∈X;(6)
a Du’s weak type contraction, if there exist anℳ𝒯-function α:[0,∞)→[0,1) and a function h:X→[0,∞) such that
(24)d(y,Ty)≤α(d(x,y))d(x,y)∀y∈Tx.Definition 9 (see [36–39]).
Let(X,d) be a metric space and let T:X→𝒩(X) be a multivalued map. One says that T is α-admissible, if there exists a function α:X×X→[0,+∞) such that for each x∈X and y∈Tx with α(x,y)≥1, one has α(y,z)≥1 for all z∈Ty.The following existence theorem is one of the main results of this paper.Theorem 10.
Let(X,d) be a metric space, let T:X→𝒩(X) be an α-admissible multivalued map, and η∈Man(ℝ)^. Let
(25)Ω={(α(x,y)d(y,Ty),d(x,y))∈[0,+∞)×[0,+∞):x∈X,y∈Tx}.
If η(t,s)≥0 for all (t,s)∈Ω and there exist x0∈X and x1∈Tx0 such that α(x0,x1)≥1, then the following statements hold.(a)
There exists a Cauchy sequence{wn}n∈ℕ in X such that
(i)
wn+1∈Twn for all n∈ℕ,(ii)
α(wn,wn+1)≥1 for all n∈ℕ,(iii)
lim⁡n→∞d(wn,wn+1)=inf⁡n∈ℕd(wn,wn+1)=0.(b)
inf⁡x∈Xd(x,Tx)=0; that is, T has the approximate fixed point property on X.Proof.
By our assumption, there existx0∈X and x1∈Tx0 such that α(x0,x1)≥1. If x1=x0, then x0∈Tx0 and
(26)inf⁡x∈Xd(x,Tx)≤d(x0,Tx0)≤d(x0,x0)=0,
which implies inf⁡x∈Xd(x,Tx)=0. Let wn=x0 for all n∈ℕ. Then {wn}n∈ℕ is a Cauchy sequence in X and
(27)lim⁡n→∞d(wn,wn+1)=inf⁡n∈ℕd(wn,wn+1)=d(x0,x0)=0.
Clearly, α(wn,wn+1)=α(x0,x1)≥1 for all n∈ℕ. Hence, the conclusions (a) and (b) hold in this case. Assume x1∉x0 or d(x0,x1)>0. If x1∈Tx1, then, following a similar argument as above, we can prove the conclusions (a) and (b) by taking a Cauchy sequence {wn}n∈ℕ with w1=x0 and wn=x1 for all n≥2. Suppose x1∉Tx1. Thus d(x1,Tx1)>0. Define λ:ℝ×ℝ→ℝ by
(28)λ(t,s)={t+η(t,s)s,if(t,s)∈Ω∖{(0,0)},0,otherwise.
By (η1), we know that
(29)0<λ(t,s)<1∀(t,s)∈Ω∖{(0,0)}.
Since η∈Man(ℝ)^ and η(t,s)≥0 for all (t,s)∈Ω, we have
(30)0<t≤sλ(t,s)∀(t,s)∈Ω∖{(0,0)}.
Clearly, (α(x0,x1)d(x1,Tx1),d(x0,x1))∈Ω∖{(0,0)}. So, by (29), we obtain
(31)0<λ(α(x0,x1)d(x1,Tx1),d(x0,x1))<1.
Let
(32)ϵ1=(α(x0,x1)λ(α(x0,x1)d(x1,Tx1),d(x0,x1))-1)×d(x1,Tx1).
Taking into account α(x0,x1)≥1, d(x1,Tx1)>0, and the last inequality, we get ϵ1>0. Since
(33)d(x1,Tx1)<d(x1,Tx1)+ϵ1=α(x0,x1)λ(α(x0,x1)d(x1,Tx1),d(x0,x1))×d(x1,Tx1),
there exists x2∈Tx1 such that x2≠x1 and
(34)d(x1,x2)<α(x0,x1)λ(α(x0,x1)d(x1,Tx1),d(x0,x1))×d(x1,Tx1).
If x2∈Tx2, then the proof can be finished by a similar argument as above. Otherwise, we have d(x2,Tx2)>0. Since T is α-admissible, we obtain α(x1,x2)≥1. By taking
(35)ϵ2=(α(x1,x2)λ(α(x1,x2)d(x2,Tx2),d(x1,x2))-1)×d(x2,Tx2),
then there exists x3∈Tx2 with x3≠x2 such that
(36)d(x2,x3)<α(x1,x2)λ(α(x1,x2)d(x2,Tx2),d(x1,x2))×d(x2,Tx2).
By induction, if xk-1,xk,xk+1∈X is known satisfying xk-1∈Txk, xk+1∈Txk+2, d(xk,Txk)>0, α(xk-1,xk)≥1, and
(37)0<d(xk,xk+1)<α(xk-1,xk)λ(α(xk-1,xk)d(xk,Txk),d(xk-1,xk))×d(xk,Txk),k∈ℕ,
then, by taking
(38)ϵk=(α(xk-1,xk)λ(α(xk-1,xk)d(xk,Txk),d(xk-1,xk))-1)×d(xk,Txk),
one can obtain xk+2∈Txk+1 with xk+2≠xk+1 such that
(39)d(xk+1,xk+2)<α(xk,xk+1)λ(α(xk,xk+1)d(xk+1,Txk+1),d(xk,xk+1))×d(xk+1,Txk+1).
Hence, by induction, we can establish sequences {xn} in X satisfying, for each n∈ℕ,
(40)xn∈Txn-1,d(xn-1,xn)>0,d(xn,Txn)>0,α(xn-1,xn)≥1,d(xn,xn+1)<α(xn-1,xn)λ(α(xn-1,xn)d(xn,Txn),d(xn-1,xn))×d(xn,Txn).
By (30), we have
(41)α(xn-1,xn)d(xn,Txn)≤d(xn-1,xn)λ(α(xn-1,xn)d(xn,Txn),d(xn-1,xn))nnnnnnnnnnnnnnnnnnnnnnnnnnnnforeachn∈ℕ.
Hence, for each n∈ℕ, by combining (40) and (41), we get
(42)d(xn,xn+1)<λ(α(xn-1,xn)d(xn,Txn),d(xn-1,xn))×d(xn-1,xn),
which means that the sequence {d(xn-1,xn)}n∈ℕ is strictly decreasing in (0,+∞). So
(43)γ:=lim⁡n→∞d(xn,xn+1)=inf⁡n∈ℕd(xn,xn+1)≥0exists.
By (41), we have
(44)α(xn-1,xn)d(xn,Txn)≤d(xn-1,xn)∀n∈ℕ,
which means that {α(xn-1,xn)d(xn,Txn)}n∈ℕ is a bounded sequence. By (η2), we have
(45)lim⁡sup⁡n→∞λ(α(xn-1,xn)d(xn,Txn),d(xn-1,xn))<1.
Now, we claim γ=0. Suppose γ>0. Then, by (45) and taking lim sup in (42), we get
(46)γ≤lim⁡sup⁡n→∞λ(α(xn-1,xn)d(xn,Txn),d(xn-1,xn))γ<γ,
a contradiction. Hence we prove
(47)lim⁡n→∞d(xn,xn+1)=inf⁡n∈ℕd(xn,xn+1)=0.
To complete the proof of (a), it suffices to show that {xn}n∈ℕ is a Cauchy sequence in X. For each n∈ℕ, let
(48)ρn:=λ(α(xn-1,xn)d(xn,Txn),d(xn-1,xn)).
Then ρn∈(0,1) for all n∈ℕ. By (42), we obtain
(49)d(xn,xn+1)<ρnd(xn-1,xn)∀n∈ℕ.
From (45), we have lim⁡sup⁡n→∞ρn<1, so there exist c∈[0,1) and n0∈ℕ, such that
(50)ρn≤c∀n∈ℕwithn≥n0.
For any n≥n0, since ρn∈(0,1) for all n∈ℕ and c∈[0,1), taking into account (49) and (50) concludes that
(51)d(xn,xn+1)<ρnd(xn-1,xn)<⋯<ρnρn-1ρn-2⋯ρn0d(x0,x1)≤cn-n0+1d(x0,x1).
Put αn=(cn-n0+1/(1-c))d(x0,x1), n∈ℕ. For m,n∈ℕ with m>n≥n0, we have from the last inequality that
(52)d(xn,xm)≤∑j=nm-1‍d(xj,xj+1)<αn.
Since c∈[0,1), lim⁡n→∞αn=0. Hence
(53)lim⁡n→∞sup⁡{d(xn,xm):m>n}=0.
So {xn} is a Cauchy sequence in X. Let wn=xn-1 for all n∈ℕ. Then {wn}n∈ℕ is the desired Cauchy sequence in (a).
To see (b), sincexn∈Txn-1 for each n∈ℕ, we have
(54)inf⁡x∈Xd(x,Tx)≤d(xn,Txn)≤d(xn,xn+1)∀n∈ℕ.
Combining (47) and (54) yields
(55)inf⁡x∈Xd(x,Tx)=0.
The proof is completed.Applying Theorem10, we can establish the following new existence theorem related to approximate fixed point property for α-admissible multivalued maps.Theorem 11.
Let(X,d) be a metric space and let T:X→𝒩(X) be an α-admissible multivalued map. Suppose that there exists an ℳ𝒯-function φ:[0,∞)→[0,1) such that
(56)α(x,y)d(y,Ty)≤φ(d(x,y))d(x,y)∀y∈Tx.
If there exist x0∈X and x1∈Tx0 such that α(x0,x1)≥1, then the following statements hold.(a)
There existsη∈Man(ℝ)^ such that η(t,s)≥0 for all (t,s)∈Ω, where
(57)Ω={(α(x,y)d(y,Ty),d(x,y))∈[0,+∞)×[0,+∞):x∈X,y∈Tx}.(b)
There exists a Cauchy sequence{wn}n∈ℕ in X such that
(i)
wn+1∈Twn for all n∈ℕ,(ii)
α(wn,wn+1)≥1 for all n∈ℕ,(iii)
lim⁡n→∞d(wn,wn+1)=inf⁡n∈ℕd(wn,wn+1)=0.(c)
inf⁡x∈Xd(x,Tx)=0; that is, T has the approximate fixed point property on X.Proof.
Defineη:ℝ×ℝ→ℝ by
(58)η(t,s)={sφ(s)-t,if(t,s)∈[0,+∞)×[0,+∞),0,otherwise.
By Example C, we know η∈Man(ℝ)^. By (56), we obtain η(t,s)≥0 for all (t,s)∈Ω. Therefore (a) is proved. It is obvious that the desired conclusions (b) and (c) follow from Theorem 10 immediately.The following interesting results are immediate from Theorem11.Corollary 12.
Let(X,d) be a metric space and let T:X→𝒞ℬ(X) be an α-admissible multivalued map. Assume that one of the following conditions holds.(L1)
There exist anℳ𝒯-function φ:[0,∞)→[0,1) and a function h:X→[0,∞) such that
(59)α(x,y)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y)+h(y)d(y,Tx)∀x,y∈X;(L2)
there exist anℳ𝒯-function φ:[0,∞)→[0,1) and L≥0 such that
(60)α(x,y)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y)+Ld(y,Tx)∀x,y∈X;(L3)
there exist two constantsθ∈(0,1) and L≥0 such that
(61)α(x,y)ℋ(Tx,Ty)≤θd(x,y)+Ld(y,Tx)∀x,y∈X;(L4)
there exists anℳ𝒯-function φ:[0,∞)→[0,1) such that
(62)α(x,y)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y)∀x,y∈X;(L5)
there exists a number0<k<1 such that
(63)α(x,y)ℋ(Tx,Ty)≤kd(x,y)∀x,y∈X.
If there existx0∈X and x1∈Tx0 such that α(x0,x1)≥1, then the following statements hold.(a)
There existsη∈Man(ℝ)^ such that η(t,s)≥0 for all (t,s)∈Ω, where
(64)Ω={(α(x,y)d(y,Ty),d(x,y))∈[0,+∞)×[0,+∞):x∈X,y∈Tx}.(b)
There exists a Cauchy sequence{wn}n∈ℕ in X such that
(i)
wn+1∈Twn for all n∈ℕ,(ii)
α(wn,wn+1)≥1 for all n∈ℕ,(iii)
lim⁡n→∞d(wn,wn+1)=inf⁡n∈ℕd(wn,wn+1)=0.(c)
inf⁡x∈Xd(x,Tx)=0; that is, T has the approximate fixed point property on X.Proof.
It suffices to verify the conclusion under (L1). Note first that, for eachx∈X, d(y,Tx)=0 for all y∈Tx. So, for each x∈X, by (L1), we obtain
(65)α(x,y)d(y,Ty)≤φ(d(x,y))d(x,y)∀y∈Tx,
which means (56) holds. Therefore, the conclusion follows from Theorem 11.In Corollary12, if we take α:X×X→[0,+∞) by α(x,y)=1 for all x,y∈X, then we obtain the following existence theorem.Corollary 13.
Let(X,d) be a metric space and let T:X→𝒞ℬ(X) be a multivalued map. Assume that one of the following conditions holds.(1)
T is a Du’s weak type contraction;(2)
T is a Du’s strong type contraction;(3)
T is a Berinde-Berinde’s type contraction;(4)
T is a multivalued (θ,L)-almost contraction;(5)
T is a Mizoguchi-Takahashi’s type contraction;(6)
T is a Nadler’s type contraction.
Then the following statements hold.(a)
There existsη∈Man(ℝ)^ such that η(t,s)≥0 for all (t,s)∈𝒟, where
(66)𝒟={(d(y,Ty),d(x,y))∈[0,+∞)×[0,+∞):x∈X,y∈Tx}.(b)
There exists a Cauchy sequence{wn}n∈ℕ in X such that
(i)
wn+1∈Twn for all n∈ℕ,(ii)
lim⁡n→∞d(wn,wn+1)=inf⁡n∈ℕd(wn,wn+1)=0.(c)
inf⁡x∈Xd(x,Tx)=0; that is, T has the approximate fixed point property on X.


## 4. Some Applications to Fixed Point Theory

Definition 14 (see [36–39]).
Let(X,d) be a metric space and let α:X×X→[0,+∞) be a function. α is said tohavethe property (B) if any sequence {xn} in X with α(xn,xn+1)≥1 for all n∈ℕ and lim⁡n→∞xn=v, we have α(xn,v)≥1 for all n∈ℕ.Theorem 15.
Let(X,d) be a complete metric space and let T:X→𝒞ℬ(X) be an α-admissible multivalued map. Suppose that there exists an ℳ𝒯-function φ:[0,∞)→[0,1) such that
(67)α(x,y)d(y,Ty)≤φ(d(x,y))d(x,y)∀y∈Tx.
If there exist x0∈X and x1∈Tx0 such that α(x0,x1)≥1, and one of the following conditions is satisfied:(H1)
T is ℋ-continuous (i.e., xn→v implies ℋ(Txn,Tv)→0 as n→∞);(H2)
T is closed (i.e., GrT:={(x,y)∈X×X:y∈Tx}; the graph of T is a closed subset of X×X);(H3)
the mapg:X→[0,∞) defined by g(x)=d(x,Tx) is l.s.c.;(H4)
for any sequence{zn} in X with α(zn,zn+1)≥1, zn+1∈Tzn, n∈ℕ, and lim⁡n→∞zn=c, one has lim⁡n→∞d(zn,Tc)=0, then T admits a fixed point in X.Proof.
Applying Theorem11, there exists a Cauchy sequence {wn}n∈ℕ in X such that
(68)wn+1∈Twn,α(wn,wn+1)≥1∀n∈ℕ.
By the completeness of X, there exists v∈X such that wn→v as n→∞.
Now, we verifyv∈ℱ(T). If (H1) holds, since T is ℋ-continuous on X, wn+1∈Twn for each n∈ℕ, and wn→v as n→∞, we get
(69)d(v,Tv)=lim⁡n→∞d(wn+1,Tv)≤lim⁡n→∞ℋ(Twn,Tv)=0,
which implies d(v,Tv)=0. By the closeness of Tv, we have v∈Tv. If (H2) holds, since T is closed, wn+1∈Twn for each n∈ℕ, and wn→v as n→∞, we have v∈ℱ(T). Suppose that (H3) holds. Since {wn}n∈ℕ is convergent in X, we have
(70)lim⁡n→∞d(wn,wn+1)=0.
Since
(71)d(v,Tv)=g(v)≤lim⁡⁡inf⁡n→∞g(wn)≤lim⁡n→∞d(wn,wn+1)=0,
we obtain d(v,Tv)=0, and hence v∈ℱ(T). Finally, assume (H4) holds. Then we obtain
(72)d(v,Tv)=lim⁡n→∞d(wn,Tv)=0.
Hence v∈Tv. Therefore, in any case, we prove v∈ℱ(T). This completes the proof.Theorem 16.
Let(X,d) be a complete metric space and let T:X→𝒞ℬ(X) be an α-admissible multivalued map. Suppose that there exist an ℳ𝒯-function φ:[0,∞)→[0,1) and a function h:X→[0,∞) such that
(73)α(x,y)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y)+h(y)d(y,Tx)∀x,y∈X.
If there exist x0∈X and x1∈Tx0 such that α(x0,x1)≥1, and one of the following conditions is satisfied:(S1)
T is ℋ-continuous;(S2)
T is closed;(S3)
the mapg:X→[0,∞) defined by g(x)=d(x,Tx) is l.s.c.;(S4)
the functionα has the property (B), then T admits a fixed point in X.Proof.
It is obvious that (73) implies (67). If one of the conditions (S1), (S2), and (S3) is satisfied, then the desired conclusion follows from Theorem 15 immediately. Suppose that (S4) holds. We claim that (H4) as in Theorem 15 is satisfied. Let {zn} be in X with α(zn,zn+1)≥1, zn+1∈Tzn, n∈ℕ, and lim⁡n→∞zn=c. Since α has the property (B), α(zn,c)≥1 for all n∈ℕ. So, it follows from (73) that
(74)lim⁡n→∞d(zn+1,Tc)≤lim⁡n→∞ℋ(Tzn,Tc)≤lim⁡n→∞α(zn,c)ℋ(Tzn,Tc)≤lim⁡n→∞{φ(d(zn,c))d(zn,c)+h(c)d(c,zn+1)}=0,
which implies lim⁡n→∞d(zn,Tc)=0. Hence (H4) holds. By Theorem 15, we also prove ℱ(T)≠∅. The proof is completed.Applying Theorem16, we can give a short proof of Du’s fixed point theorem.Corollary 17 (Du [[6]).
Let(X,d) be a complete metric space, let T:X→𝒞ℬ(X) be a multivalued map, let φ:[0,∞)→[0,1) be a ℳ𝒯-function, let and h:X→[0,∞) be a function. Assume that
(75)ℋ(Tx,Ty)≤φ(d(x,y))d(x,y)+h(y)d(y,Tx)∀x,y∈X.
Then ℱ(T)≠∅.Proof.
Takeα:X×X→[0,+∞) by α(x,y)=1 for all x,y∈X. Then (75) implies (73). Moreover, T is an α-admissible multivalued map and the function α has the property (B). Therefore the conclusion follows from Theorem 16.Remark 18.
Theorems15 and 16 and Corollary 17 all generalize and improve Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem, and Banach contraction principle.



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*Source: 581267-2014-03-04.xml*