# On Approximate Coincidence Point Properties and Their Applications to Fixed Point Theory

Authors: Wei-Shih Du
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We first establish some existence results concerning approximate coincidence point properties and approximate fixed point properties for various types of nonlinear
contractive maps in the setting of cone metric spaces and general metric spaces. From these results,
we present some new coincidence point and fixed point theorems which generalize Berinde-Berinde's
fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, and some well-known results in the
literature.


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## Body

## 1. Introduction

Let(X,d) be a metric space. For each x∈X and A⊆X, let d(x,A)=infy∈Ad(x,y). Denote by 𝒩(X) the class of all nonempty subsets of X, 𝒞(X) the collection 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(1.1)H(A,B)=max⁡{sup⁡x∈B⁡d(x,A),sup⁡x∈Ad(x,B)}
is said to be the Hausdorff metric on 𝒞ℬ(X) induced by the metric d on X. Let T:X→𝒩(X) be a multivalued map. A point x in X is a fixed point of T if x∈Tx. The set of fixed points of T is denoted by ℱ(T). Let g:X→X be a single-valued self-map and T:X→𝒩(X) be a multivalued map. A point x in X is said to be a coincidence point(see, for instance, [1, 2]) of g and T if gx∈Tx. The set of coincidence point of g and T is denoted by 𝒞𝒪𝒫(g,T). Throughout this paper, we denote by ℕ and ℝ the sets of positive integers and real numbers, respectively.Letf be a real-valued function defined on ℝ. For c∈ℝ, we recall that(1.2)limsup⁡x→cf(x)=inf⁡ɛ>0sup⁡0<|x-c|<ɛf(x),limsup⁡x→c+f(x)=inf⁡ɛ>0sup⁡0<x-c<ɛf(x).Definition 1.1 (see [2–9]).
A functionφ:[0,∞)→[0,1) is said to be an ℳ𝒯-function (or ℛ-function) if
(1.3)limsup⁡s→t+φ(s)<1,∀t∈[0,∞).
It is obvious that ifφ:[0,∞)→[0,1) is a nondecreasing function or a nonincreasing function, then φ is an ℳ𝒯-function. So the set of ℳ𝒯-functions is a rich class.
Very recently, Du [7] first proved some characterizations of ℳ𝒯-functions.Theorem D (see [7]).
Letφ:[0,∞)→[0,1) be a function. Then, the following statements are equivalent.(a)
φ is an ℳ𝒯-function.(b)
For eacht∈[0,∞), there exist rt(1)∈[0,1) and ɛt(1)>0 such that φ(s)≤rt(1) for all s∈(t,t+ɛt(1)).(c)
For eacht∈[0,∞), there exist rt(2)∈[0,1) and ɛt(2)>0 such that φ(s)≤rt(2) for all s∈[t,t+ɛt(2)].(d)
For eacht∈[0,∞), there exist rt(3)∈[0,1) and ɛt(3)>0 such that φ(s)≤rt(3) for all s∈(t,t+ɛt(3)].(e)
For eacht∈[0,∞), there exist rt(4)∈[0,1) and ɛt(4)>0 such that φ(s)≤rt(4) for all s∈[t,t+ɛt(4)).(f)
For any nonincreasing sequence{xn}n∈ℕ in [0,∞), we have 0≤supn∈ℕφ(xn)<1.(g)
φ is a function of contractive factor  [5]; that is, for any strictly decreasing sequence {xn}n∈ℕ in [0,∞), we have 0≤supn∈ℕφ(xn)<1.It is worth to mention that there exist functions which are notℳ𝒯-functions. For example, let φ:[0,∞)→[0,1) be defined by(1.4)φ(t):={sin⁡⁡tt,ift∈(0,π2],0,otherwise.
Since limsups→0+φ(s)=1, φ is not an ℳ𝒯-function.Letg:X→X be a single-valued self-map. Recall that a multivalued map T:X→𝒩(X) is called(1)
amultivaluedk-contraction[1, 10, 11], if there exists a number 0<k<1 such that
(1.5)H(Tx,Ty)≤kd(x,y),∀x,y∈X,(2)
amultivalued(θ,L)-almost  contraction[1, 12], if there exist two constants θ∈(0,1) and L≥0 such that(1.6)H(Tx,Ty)≤θd(x,y)+Ld(y,Tx),∀x,y∈X,(3)
ageneralized multivalued  almost  contraction[1, 12], if there exists an ℳ𝒯-function φ and L≥0 such that(1.7)H(Tx,Ty)≤φ(d(x,y))d(x,y)+Ld(y,Tx),∀x,y∈X,(4)
amultivalued  almostg-contraction[1, 12, 13], if there exist two constants θ∈(0,1) and L≥0 such that(1.8)H(Tx,Ty)≤θd(gx,gy)+Ld(gy,Tx),∀x,y∈X,(5)
ageneralized multivalued  almostg-contraction[1], if there exists an ℳ𝒯-function φ and L≥0 such that(1.9)H(Tx,Ty)≤φ(d(gx,gy))d(gx,gy)+Ld(gy,Tx),∀x,y∈X,In 1989, Mizoguchi and Takahashi [14] proved the following fixed point theorem which is a generalization of Nadler’s fixed point theorem [10] and the celebrated Banach contraction principle (see, e.g., [11]). It is worth to mention that Mizoguchi-Takahashi’s fixed point theorem gave a partial answer of Problem 9 in Reich [15] and it’s primitive proof is difficult. Recently, Suzuki [16] presented a very simple proof of Mizoguchi-Takahashi’s fixed point theorem.Theorem MT (Mizoguchi and Takahashi).
Let(X,d) be a complete metric space, T:X→𝒞ℬ(X) be a multivalued map, and φ:[0,∞)→[0,1) be a ℳ𝒯-function. Assume that
(1.10)H(Tx,Ty)≤φ(d(x,y))d(x,y),∀x,y∈X,
Then, ℱ(T)≠∅.Since then a number of generalizations in various different directions of Mizoguchi-Takahashi’s fixed point theorem have been investigated by several authors in the past. In 2007, M. Berinde and V. Berinde [12] proved the following interesting fixed point theorem to generalize Mizoguchi-Takahashi’s fixed point theorem.Theorem BB (M. Berinde and V. Berinde).
Let(X,d) be a complete metric space, T:X→𝒞ℬ(X) be a multivalued map, φ:[0,∞)→[0,1) be a ℳ𝒯-function, and L≥0. Assume that
(1.11)H(Tx,Ty)≤φ(d(x,y))d(x,y)+Ld(y,Tx),∀x,y∈X,
that is, T is a generalized multivalued almost contraction. Then, ℱ(T)≠∅.In [3], the author established some new fixed point theorems for nonlinear multivalued contractive maps by using τ0-metrics (see [3, Def. 1.2]), τ0-metrics (see [3, Def. 1.3]), and ℳ𝒯-functions. Applying those results, the author gave the generalizations of Berinde-Berinde‘s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem, and Banach contraction principle, Kannan’s fixed point theorems, and Chatterjea’s fixed point theorems for nonlinear multivalued contractive maps in complete metric spaces; for more detail, one can refer to [3].Very recently, Du [17] first introduced the concepts of TVS-cone metric and TVS-cone metric space to improve and extend the concept of cone metric space in the sense of Huang and Zhang [18].Definition 1.2 (see [17]).
LetX be a nonempty set and Y a locally convex Hausdorff t.v.s. with its zero vector θ, K a proper, closed, convex, and pointed cone in Y, and ≾K a partial ordering with respect to K defined by
(1.12)x≾Ky⟺y-x∈K.
A vector-valued function p:X×X→Y is said to be a TVS-conemetric, if the following conditions hold:(C1)
θ≾Kp(x,y) for all x,y∈X and p(x,y)=θ if and only if x=y,(C2)
p(x,y)=p(y,x) for all x,y∈X,(C3)
p(x,z)≾Kp(x,y)+p(y,z) for all x,y,z∈X. The pair (X,p) is then called a TVS-cone  metric space.In this paper, we first establish some existence results concerning approximate coincidence point property and approximate fixed point property for various types of nonlinear contractive maps in the setting of cone metric spaces and general metric spaces. From these results, we present some new coincidence point and fixed point theorems which generalize Berinde-Berinde’s fixed point theorem and Mizoguchi-Takahashi’s fixed point theorem. Our results generalize and improve some recent results in [1–6, 10–19] and references therein.


## 2. Preliminaries

LetE be a topological vector space (t.v.s. for short) with its zero vector θE. A nonempty subset K of E is called a convex cone if K+K⊆K and λK⊆K for λ≥0. A convex cone K is said to be pointed if K∩(-K)={θE}. For a given proper, pointed, and convex cone K in E, we can define a partial ordering ≾K with respect to K by(2.1)x≾Ky⟺y-x∈K.x≺Ky will stand for x≾Ky and x≠y, while x≪Ky will stand for y-x∈intK, where intK denotes the interior of K.In the following, unless otherwise specified, we always assume thatY is a locally convex Hausdorff t.v.s. with its zero vector θ, K is a proper, closed, convex, and pointed cone in Y with intK≠∅, ≾K is a partial ordering with respect to K and e∈intK is fixed.Recall that the nonlinear scalarization functionξe:Y→ℝ is defined by(2.2)ξe(y)=inf⁡{r∈R:y∈re-K},∀y∈Y.Theorem 2.1 (see [6, 17, 20, 21]).
For eachr∈ℝ and y∈Y, the following statements are satisfied:(i)
ξe(y)≤r⇔y∈re-K,(ii)
ξe(y)>r⇔y∉re-K,(iii)
ξe(y)≥r⇔y∉re-intK,(iv)
ξe(y)<r⇔y∈re-intK,(v)
ξe(·) is positively homogeneous and continuous on Y,(vi)
ify1∈y2+K (i.e., y2≲Ky1), then ξe(y2)≤ξe(y1),(vii)
ξe(y1+y2)≤ξe(y1)+ξe(y2) for all y1,y2∈Y.
Clearly,ξe(θ)=0. Notice that the reverse statement of (vi) in Theorem 2.1 (i.e., ξe(y2)≤ξe(y1)⇒y1∈y2+K) does not hold in general. We illustrate the truth with the following simple example.Example A.
LetY=ℝ2, K=ℝ+2={(x,y)∈ℝ2:x,y≥0}, and e=(1,1). Then K is a proper, closed, convex, and pointed cone in Y with intK={(x,y)∈ℝ2:x,y>0}≠∅ and e∈intK. For r=1, it is easy to see that y1=(10,-30)∉re-intK, and y2=(0,0)∈re-intK. By applying (iii) and (iv) of Theorem 2.1, we have ξe(y2)<1≤ξe(y1) while y1∉y2+K.Definition 2.2 (see [17]).
Let(X,p) be a TVS-cone metric space, x∈X, and {xn}n∈ℕ a sequence in X.(i)
{xn} is said to TVS-cone converge to x if, for every c∈Y with θ≪Kc, there exists a natural number ℕ0 such that p(xn,x)≪Kc for all n≥ℕ0. We denote this by cone-limn→∞xn=x or xn→conex as n→∞ and call x the TVS-cone limit of {xn}.(ii)
{xn} is said to be a TVS-cone  Cauchy  sequence if, for every c∈Y with θ≪Kc, there is a natural number ℕ0 such that p(xn,xm)≪Kc for all n,m≥ℕ0.(iii)
(X,p) is said to be TVS-cone complete if every TVS-cone Cauchy sequence in X is TVS-cone convergent in X.
In [17], the author proved the following important results.Theorem 2.3 (see [17]).
Let(X,p) be a TVS-cone  metric spaces. Then, dp:X×X→[0,∞) defined by dp:=ξe∘p is a metric.Example B.
LetX=[0,1], Y=ℝ2, K=ℝ+2={(x,y)∈ℝ2:x,y≥0}, e=(1,1), and θ=(0,0). Define p:X×X→Y by
(2.3)p(x,y)=(|x-y|,8|x-y|).
Then, (X,p) is a TVS-cone complete metric space and
(2.4)dp(x,y)=ξe(p(x,y))=inf⁡{r∈R:p(x,y)∈re-K}=8|x-y|.
So dp is a metric on X and (X,dp) is a complete metric space.Theorem 2.4 (see [17]).
Let(X,p) be a TVS-cone  metric space, x∈X, and {xn}n∈ℕ a sequence in X. Then, the following statements hold.(a)
If{xn}  TVS-cone  converges to x(i.e.,xn→conex as n→∞), then dp(xn,x)→0 as n→∞(i.e., xn→dpx as n→∞).(b)
If{xn} is a TVS-cone  Cauchy sequence in (X,p), then {xn} is a Cauchy sequence (in usual sense) in (X,dp).Definition 2.5 (see [1, 22]).
Let(X,d) be a metric space. A multivalued map T:X→𝒩(X) is said to have an approximate fixed point property provided
(2.5)inf⁡x∈Xd(x,Tx)=0,
or, equivalently, for any ϵ>0, there exists z∈X such that
(2.6)d(z,Tz)≤ϵ,
or, equivalently, for any ϵ>0, there exists xϵ∈X such that
(2.7)T(xϵ)∩B(xϵ,ϵ)≠∅,
whereB(x,r) denotes a closed ball of radius r centered at x.
It is known that every generalized multivalued almost contraction in a metric space(X,d) has the approximate fixed point property (see [1, Lemma 2.2]).Remark 2.6.
It is obvious thatℱ(T)≠∅ implies that T has the approximate fixed point property.Definition 2.7 (see [1]).
Let(X,d) be a metric space, g:X→X a single-valued self-map, and T:X→𝒩(X) a multivalued map. The maps g and T are said to have an approximate coincidence point property provided
(2.8)inf⁡x∈Xd(gx,Tx)=0,
or, equivalently, for any ϵ>0, there exists z∈X such that
(2.9)d(gz,Tz)≤ϵ.
It is known that every generalized multivalued almostg-contraction in a metric space (X,d) has the approximate coincidence point property provided each Tx is g-invariant (i.e., g(Tx)⊆Tx) for each x∈X (see [1, Theorem 2.7]).


## 3. Main Results

For any locally convex Hausdorff t.v.s.Y with its zero vector θ, let τ denote the topology of Y and let 𝒰τ be the base at θ consisting of all absolutely convex neighborhood of θ. Let(3.1)L={l:lisaMinkowskifunctionalofUforU∈Uτ}.
Then, ℒ is a family of seminorms on Y. For each ℓ∈ℒ, put(3.2)V(l)={y∈Y:l(y)<1}
and denote(3.3)UL={U:U=r1V(l1)∩r2V(l2)∩⋯∩rnV(ln),rk>0,lk∈L,1≤k≤n,n∈N}.
Then, 𝒰ℒ is a base at θ and the topology Γℒ generated by 𝒰ℒ is the weakest topology for Y such that all seminorms in ℒ are continuous and τ=Γℒ. Moreover, given any neighborhood 𝒪θ of θ, there exists U∈𝒰ℒ such that θ∈U⊂𝒪θ; for more detail, we refer the reader to ([23, Theorem 12.4 in II.12, Page 113]).The following lemmas are very crucial to our main results.Lemma 3.1.
LetOθ be a neighborhood of θ, ω∈K, and {αn}⊂[0,∞) with limn→∞αn=0. Suppose that U∈𝒰ℒ satisfies θ∈U⊂Oθ, then there exists n0∈ℕ, such that ±αnω∈U for all n≥n0.Proof .
Clearly, ifω=θ, then ±αnω=θ∈U for all n∈ℕ, and we are done. Suppose ω∈K∖{θ}. Since U∈𝒰ℒ,
(3.4)U=r1V(l1)∩r2V(l2)∩⋯∩rsV(ls),
for some ri>0, ℓi∈ℒ and 1≤i≤s. Let
(3.5)δ=min⁡{ri:1≤i≤s}>0,ρ=max⁡{li(ω):1≤i≤s}.
We need consider two possible cases.Case 1.
Ifρ=0, since each ℓi is a seminorm, we have ℓi(ω)=0 and
(3.6)li(±αnω)=αnli(ω)=0<ri
for all 1≤i≤s and all n∈ℕ.Case 2.
Ifρ>0, since limn→∞αn=0, there exists n0∈ℕ such that αn<δ/ρ for all n≥n0. So, for each i∈{1,2,…,s} and any n≥n0, we obtain
(3.7)li(±αnω)=αnli(ω)<δρli(ω)≤δ≤ri.
Hence, by Cases1 and 2, we get that, for any n≥n0, ±αnω∈riV(ℓi) for all 1≤i≤s. Therefore, ±αnω∈U for all n≥n0.Lemma 3.2 (see [24, Lemma 2.5]).
LetE be a t.v.s., K a convex cone with intK≠∅ in E, and a,b,c∈E. Then, the following statements hold.(a)
intK+K⊆intK.(b)
Ifa≾Kb and b≪Kc, then a≪Kc.In this section, we first establish an existence theorem related to approximate coincidence point property for maps inTVS-cone complete metric space which is one of the main results of this paper. It will have many applications to study metric fixed point theory.It is worth observing that the following existence theorem does not require theTVS-cone completeness assumption on TVS-cone metric space (X,p).Theorem 3.3.
Let(X,p) be a TVS-cone  metric space, T:X→𝒩(X) a multivalued map, h:X→X a self-map, and dp:=ξe∘p. Suppose that(D1)
there exists anℳ𝒯-function μ:[0,∞)→[0,1) such that, for each x∈X, if y∈Tx with y≠x then there exists z∈Ty such that(3.8)p(hy,hz)≾Kμ(dp(hx,hy))p(hx,hy),(D2)
Tx is h-invariant (i.e., h(Tx)⊆Tx) for each x∈X. Then, the following statements hold.(a)
h and T have the dp-approximate coincidence point property on X (i.e., infx∈Xdp(hx,Tx)=0).(b)
There exists a sequence{zn} in (X,p) such that {zn} is a TVS-cone  Cauchy sequence and limn→∞dp(zn,zn+1)=infx∈Xdp(hx,Tx)=0.Proof.
By Theorem2.3, we know that dp is a metric on X. Let x1∈X and x2∈Tx1. If x1=x2, then, by (D2), we have hx1∈Tx1. Since
(3.9)inf⁡x∈Xdp(hx,Tx)≤dp(hx1,Tx1)=0,
we have infx∈Xdp(hx,Tx)=0. Let zn=hx1 for all n∈ℕ. Then, {zn}⊂X is a TVS-cone Cauchy sequence and limn→∞dp(zn,zn+1)=0=infx∈Xdp(hx,Tx). Hence, all conclusions are proved in the case of x1=x2. If x2≠x1 or p(x1,x2)≠θ, then, by (D1), there exists x3∈Tx2 such that
(3.10)p(hx2,hx3)≾Kμ(dp(hx1,hx2))p(hx1,hx2).
If x2=x3, then the conclusions also hold by following a similar argument as above. If x3≠x2, then there exists x4∈Tx3 such that
(3.11)p(hx3,hx4)≾Kμ(dp(hx2,hx3))p(hx2,hx3).
By induction, we can obtain a sequence {xn} in X satisfying the following: for each n∈ℕ,
(3.12)xn+1∈Txn,p(hxn+1,hxn+2)≾Kμ(dp(hxn,hxn+1))p(hxn,hxn+1).
Applying (v) and (vi) of Theorem 2.1, the inequality (3.12) implies
(3.13)dp(hxn+1,hxn+2)=ξe(p(hxn+1,hxn+2))≤ξe(μ(dp(hxn,hxn+1))p(hxn,hxn+1))≤μ(dp(hxn,hxn+1))ξe(p(hxn,hxn+1))=μ(dp(hxn,hxn+1))dp(hxn,hxn+1).
Since μ(t)<1 for all t∈[0,∞), the sequence {dp(hxn,hxn+1)}n∈ℕ is strictly decreasing in [0,∞). Since μ is an ℳ𝒯-function, by (g) of Theorem D,
(3.14)0≤sup⁡n∈Nμ(dp(hxn,hxn+1))<1.
Let λ:=supn∈ℕμ(dp(hxn,hxn+1)). So λ∈[0,1). From (3.12), we get
(3.15)p(hxn+1,hxn+2)≾Kμ(dp(hxn,hxn+1))p(hxn,hxn+1)≾Kλp(hxn,hxn+1).
It follows from (3.15) that
(3.16)p(hxn+1,hxn+2)≾Kλp(hxn,hxn+1)≾K⋯≾Kλnp(hx1,hx2)foreachn∈N.
Let zn:=hxn, n∈ℕ. Applying (v) and (vi) of Theorem 2.1, the inequality (3.16) implies
(3.17)dp(zn+1,zn+2)≤λndp(z1,z2)foreachn∈N.
Since λ∈[0,1), by (3.17), we have limn→∞dp(zn,zn+1)=0. By (3.12) and (D2), we obtain zn+1=hxn+1∈Txn, n∈ℕ. It implies that
(3.18)inf⁡x∈Xdp(hx,Tx)≤dp(hxn,Txn)≤dp(zn,zn+1),∀n∈N.
Since dp(zn,zn+1)→0 as n→∞, it follows from (3.18) that infx∈Xdp(hx,Tx)=0 and the conclusion (a) is proved.
To see (b), it suffices to prove that{zn} is a TVS-cone Cauchy sequence in (X,p). For m,n∈ℕ with m>n, it follows from (3.16) that
(3.19)p(zn,zm)≾K∑j=nm-1p(zj,zj+1)≾Kλn-11-λp(z1,z2).
Given c∈Y with θ≪Kc (i.e, c∈intK=int(intK)), there exists a neighborhood Nθ of θ such that c+Nθ⊆intK. Therefore, there exists Uc∈𝒰ℒ with Uc⊆Nθ such that
(3.20)c+Uc⊆c+Nθ⊆int⁡K.
Let ω=p(z1,z2) and αn=λn-1/(1-λ), n∈ℕ. Since λ∈[0,1), we have {αn}⊂[0,∞) and limn→∞αn=0. Applying Lemma 3.1, there exists n0∈ℕ, such that -αnω∈Uc for all n≥n0. So, by (3.20), we obtain
(3.21)c-αnω∈c+Uc⊆int⁡K
or
(3.22)αnω≪Kc
for all n≥n0. For m,n∈ℕ with m>n≥n0, since p(zn,zm)≾Kαnω from (3.19) and αnω≪Kc, it follows from Lemma 3.2 that
(3.23)p(zm,zn)≪Kc.
Hence, we prove that {zn} is a TVS-cone Cauchy sequence in (X,p). The proof is completed.Theorem 3.4.
Let(X,p) be a TVS-cone  metric space, T:X→𝒩(X) a multivalued map, and dp:=ξe∘p. Suppose that(D3)
there exists anℳ𝒯-function μ:[0,∞)→[0,1) such that, for each x∈X, if y∈Tx with y≠x, then there exists z∈Ty such that(3.24)p(y,z)≾Kμ(dp(x,y))p(x,y).
Then, there exists a nonempty proper subset V of X, such that infx∈Xdp(x,Tx)=infx∈Vdp(x,Tx)=0 (i.e., T has the dp-approximate fixed point property on V and X.)Proof.
Leth≡id be the identity map on X. Then, the conditions (D1) and (D2) as in Theorem 3.3 hold. Following the same argument as in the proof of Theorem 3.3, we obtain two sequences {xn} and {zn} in X such that(i)
xn+1∈Txn, zn=hxn and zn+1∈Txn for each n∈ℕ,(ii)
limn→∞dp(zn,zn+1)=0.
Sinceh≡id, {xn} and {zn} are identical, we have limn→∞dp(xn,xn+1)=0. Put V={xn}n∈ℕ⊂X. Since
(3.25)inf⁡x∈Xdp(x,Tx)≤inf⁡x∈Vdp(x,Tx)≤dp(xn,Txn)≤dp(xn,xn+1),∀n∈N,
we obtain infx∈Xdp(x,Tx)=infx∈Vdp(x,Tx)=0. Hence, T has the dp-approximate fixed point property on V and X.New, we establish the following approximate coincidence point property for maps in general metric spaces by applying Theorem3.3.Theorem 3.5.
Let(X,d) be a metric space, T:X→𝒩(X) a multivalued map, and h:X→X a self-map. Suppose that(A1)
there exists anℳ𝒯-function μ:[0,∞)→[0,1) such that, for each x∈X, if y∈Tx with y≠x, then there exists z∈Ty such that(3.26)d(hy,hz)≤μ(d(hx,hy))d(hx,hy),(A2)
Tx is h-invariant (i.e., h(Tx)⊆Tx) for each x∈X.Then, h and T have the approximate coincidence point property on X.Proof .
In Theorem3.3, let Y=ℝ, K=[0,∞)⊂ℝ, and e=1. Therefore, the conclusion is immediate from Theorem 3.3.The following approximate fixed point property for maps is immediate from Theorem3.4.Theorem 3.6.
Let(X,d) be a metric space and T:X→𝒩(X) a multivalued map. Suppose that(A3)
there exists anℳ𝒯-function μ:[0,∞)→[0,1) such that, for each x∈X, if y∈Tx with y≠x, then there exists z∈Ty such that(3.27)d(y,z)≤μ(d(x,y))d(x,y).
Then, T has the approximate fixed point property on X.Theorem 3.7.
In Theorem3.6, if the condition (A3) is replaced by (A3)H, where(A3)H
there exists anℳ𝒯-function μ:[0,∞)→[0,1) such that, for each x∈X,(3.28)d(y,Ty)≤μ(d(x,y))d(x,y),∀y∈Tx,
then, T has the approximate fixed point property on X.Proof.
Defineτ:[0,∞)→[0,1) by τ(t)=(1+μ(t))/2. Then, by [3, Lemma 2.1], τ is also an ℳ𝒯-function. For each x∈X, let y∈Tx with y≠x. Then, d(x,y)>0. By (A3)H, we have
(3.29)d(y,Ty)<τ(d(x,y))d(x,y).
It follows that there exists z∈Ty such that
(3.30)d(y,z)<τ(d(x,y))d(x,y),
which shows that (A3) holds. Therefore, the conclusion follows from Theorem 3.6.Theorem 3.8.
Theorems3.6 and 3.7 are equivalent.Proof.
We have shown that Theorem3.6 implies Theorem 3.7. So it suffices to prove that Theorem 3.7 implies Theorem 3.6. If (A3) holds, then it is easy to verify that (A3)H also holds. Hence, Theorem 3.7 implies Theorem 3.6 and we get the desired result.Remark 3.9 (see [1, Lemma 2.2]).
Is a special cases of Theorems3.6 and 3.7.
Let(X,p) be a TVS-cone metric space. By Theorem 2.3, we know that dp:=ξe∘p is a metric on X. So we can obtain the topology Γdp on X induced by dp and hence define dp-open subsets, dp-closed subsets, and dp-compact subsets of X.
Here, we denote𝒞dp(X) by the collection of all nonempty dp-closed subsets of X.Theorem 3.10.
Let(X,p) be a TVS-cone  complete metric space, T:X→𝒞dp(X) a multivalued map, h:X→X a self-map, and dp:=ξe∘p. Suppose that the conditions (D1) and (D2) as in Theorem 3.3 hold and further assume one of the following conditions hold:(L1)
(X,dp) is dp-compact and the function f:X→[0,∞) defined by f(x)=dp(hx,Tx) is l.s.c.,(L2)
𝒢={(hx,y)∈X×X:x∈X,y∈Tx} is a dp-closed subset of X×X.Then, the following statements hold.(a)
There exists a nonempty subsetℳ of X, such that (ℳ,dp) is a complete metric space.(b)
𝒞𝒪𝒫(h,T)≠∅.Proof.
Following the same argument as in the proof of Theorem3.3, one can obtain that there exist two sequences {xn} and {zn} in X such that for any n∈ℕ, zn=hxn, zn+1∈Txn, {zn} is a TVS-cone Cauchy sequence in (X,p) and limn→∞dp(zn,zn+1)=infx∈Xdp(hx,Tx)=0. By the TVS-cone completeness of (X,p), there exists v∈X, such that {zn}  TVS-cone converges to v. On the other hand, applying Theorem 2.4, we obtain that {zn} is a Cauchy sequence in (X,dp) and dp(zn,v)→0 or zn→dpv as n→∞. Let ℳ={zn}n∈ℕ∪{v}. Then (ℳ,dp) is a complete metric space and the conclusion (a) holds.
Now, we verify the conclusion (b). Suppose that (L1) holds. Then the infimuminfx∈Xf(x) is attained. Since infx∈Xf(x)=infx∈Xdp(hx,Tx)=0, there exists ζ∈X such that
(3.31)dp(hζ,Tζ):=f(ζ)=inf⁡x∈Xf(x)=0.
Since Tζ is a dp-closed subset of X, it implies from (3.31) that hζ∈Tζ. Hence, ζ∈𝒞𝒪𝒫(h,T).
Suppose that (L2) holds. For anyn∈ℕ, since zn+1∈Txn, we know (zn,zn+1)∈𝒢. Since 𝒢 is dp-closed in X×X and zn→dpv as n→∞, we have (v,v)∈𝒢. Hence, there exists a∈X such that v=ha and v∈Ta, which say that a∈𝒞𝒪𝒫(h,T). The proof is completed.Theorem 3.11.
Let(X,p) be a TVS-cone  complete metric space, T:X→𝒞dp(X) a multivalued map, and dp:=ξe∘p. Suppose that the condition (D3) as in Theorem 3.4 holds and further assume one of the following conditions hold:(H1)
T is dp-closed; that is, GrT:={(x,y)∈X×X:y∈Tx}, the graph of T, is a dp-closed subset of X×X,(H2)
the functionf:X→[0,∞) defined by f(x)=dp(x,Tx) is l.s.c.,(H3)
inf{dp(x,z)+dp(x,Tx):x∈X}>0 for every z∉ℱ(T). Then, there exists a nonempty subset W of X, such that(a)
(W,dp) is a complete metric space,(b)
ℱ(T)∩W≠∅.Proof.
Following a similar argument as in the proof of Theorem3.3, we can obtain a sequence {xn} in X such that, for any n∈ℕ,(i)
xn+1∈Txn,(ii)
p(xn+1,xn+2)≾Kγp(xn,xn+1), where γ:=supn∈ℕμ(dp(xn,xn+1))∈[0,1),(iii)
p(xn,xm)≾K(γn-1/(1-γ))p(x1,x2), for m,n∈ℕ with m>n.(iv)
{xn} is a TVS-cone Cauchy sequence in (X,p).
Applying Theorem2.1, the inequality (iii) implies that, for m,n∈ℕ with m>n, we have
(3.32)dp(xn,xm)=ξe(p(xn,xm))≤ξe(γn-11-γp(x1,x2))=γn-11-γdp(x1,x2).
Since limn→∞γn-1/(1-γ)=0, it follows from (3.32) that {xn} is a Cauchy sequence in (X,dp). By the TVS-cone completeness of (X,p) and (iv), there exists v∈X, such that {xn}  TVS-cone converges to v. Applying Theorem 2.4, we have dp(xn,v)→0 or xn→dpv as n→∞. Let W={xn}n∈ℕ∪{v}. Then, (W,dp) is a complete metric space and the conclusion (a) holds.
Finally, in order to complete the proof, it suffices to show thatv∈ℱ(T). Suppose that (H1) holds. Since xn+1∈Txn, we have (xn,xn+1)∈GrT for each n∈ℕ. By (H1) and xn→dpv as n→∞, we get v∈ℱ(T).
If (H2) holds, by the lower semicontinuity off and (i), we obtain
(3.33)dp(v,Tv)=f(v)≤liminf⁡n→∞f(xn)≤liminf⁡n→∞dp(xn,Txn)≤lim⁡n→∞dp(xn,xn+1)=0,
which implies dp(v,Tv)=0. Since Tv is a dp-closed subset of X, v∈ℱ(T).
Let (H3) holds. Supposev∉ℱ(T). Since dp is a metric on X and xm→dpv as m→∞, by (3.32), we get
(3.34)dp(xn,v)≤γn-11-γdp(x1,x2)for  anyn∈N.
From (3.32) and (3.34), we have
(3.35)0<inf⁡x∈X{dp(x,v)+dp(x,Tx)}≤inf⁡n∈N{dp(xn,v)+dp(xn,Txn)}≤inf⁡n∈N{dp(xn,v)+dp(xn,xn+1)}≤lim⁡n→∞2γn-11-γdp(x1,x2)=0,
which leads a contradiction. Therefore, v∈ℱ(T). The proof is completed.The following result is immediate from Theorem3.10.Theorem 3.12.
Let(X,d) be a complete metric space, T:X→𝒞(X) a multivalued map, and h:X→X a self-map. Suppose that the conditions (A1) and (A2) as in Theorem 3.5 hold and further assume one of the following conditions hold:(i)
(X,d) is compact and the function f:X→[0,∞) defined by f(x)=d(hx,Tx) is l.s.c.;(ii)
{(hx,y)∈X×X:x∈X,y∈Tx} is a closed subset of X×X.Then, 𝒞𝒪𝒫(h,T)≠∅.Theorem 3.13.
Let(X,d) be a complete metric space and T:X→𝒞(X) a multivalued map. Suppose that the condition (A3) or (A3)H  holds and further assume one of the following conditions hold:(h1)
T is closed,(h2)
the functionf:X→[0,∞) defined by f(x)=d(x,Tx) is l.s.c.,(h3)
inf{d(x,z)+d(x,Tx):x∈X}>0 for every z∉ℱ(T),(h4)
for each sequence{xn} in X with xn+1∈Txn, n∈ℕ and limn→∞xn=v, we have limn→∞ℋ(Txn,Tv)=0.Then, ℱ(T)≠∅.Proof.
The conclusion is immediate from Theorem3.11 if Y=ℝ, K=[0,∞)⊂ℝ, e=1, and one of conditions (h1), (h2), and (h3) holds. Suppose that (h4) holds. Following a similar argument as in the proof of Theorem 3.3, we can construct a Cauchy sequence {xn} in (X,d) such that xn+1∈Txn, n∈ℕ, and {xn} converge to some point v∈X. Since the function x↦d(x,Tv) is continuous, xn+1∈Txn and xn→dv as n→∞, by (h4), we get
(3.36)d(v,Tv)=lim⁡n→∞d(xn+1,Tv)≤lim⁡n→∞H(Txn,Tv)=0,
which implies v∈ℱ(T). The proof is completed.Remark 3.14.
(a)
LetA and B be two topological vector spaces and T:A→2B a multivalued map. Recall that T is u.s.c. if and only if for any open set V in B, T+(V):={x∈A:T(x)⊆V} is open in A. It is known that if T is u.s.c. with closed values, then T is closed (see [25]). Hence, Theorem 3.13 is true if (X,∥·∥) is a Banach space and T:X→C(X) is u.s.c.;(b)
letG be a nonempty subset of a metric space (X,d) and T:X→𝒞(X) u.s.c. Then, the function ϕ:G→[0,∞) defined by ϕ(x)=d(x,Tx) is l.s.c.; for detail, see [26, Lemma 2] or [27, Lemma 3.1];(c)
it is known that any single-valued map of Kannan’s type or Chatterjea’s type satisfies (h3); for more detail, one can see [28, Corollary 3] or [3, Remark 3.1].
Applying Theorem3.13, we can prove the following generalization of Berinde-Berinde’s fixed point theorem [12].Theorem 3.15.
Let(X,d) be a complete metric space, T:X→𝒞ℬ(X) a multivalued map, and G:X→[0,∞) a function. Suppose that there exists an ℳ𝒯-function μ:[0,∞)→[0,1) such that
(3.37)H(Tx,Ty)≤μ(d(x,y))d(x,y)+G(y)d(y,Tx),∀x,y∈X.
Then, ℱ(T)≠∅.Proof.
Letx∈X. If y∈Tx, then d(y,Tx)=0. So (3.37) implies the inequality
(3.38)d(y,Ty)≤μ(d(x,y))d(x,y),∀y∈Tx.
Hence, the condition (A3)H of Theorem 3.13 holds. Let {xn} in X with xn+1∈Txn, n∈ℕ, and limn→∞xn=v. By (3.37), we obtain
(3.39)lim⁡n→∞H(Txn,Tv)≤lim⁡n→∞{φ(d(xn,v))d(xn,v)+G(v)d(v,xn+1)}=0,
which says that the condition (h4) of Theorem 3.13 also holds. Therefore, the conclusion follows from Theorem 3.13.Remark 3.16.
(a)
Theorems3.11, 3.13, and 3.15 all generalize Berinde-Berinde’s fixed point theorem [12].(b)
In Theorem3.15, if G(x)=0 for all x∈X, then we can obtain Mizoguchi-Takahashi’s fixed point theorem [14].(c)
In Theorem3.15, if G(x)=0 for all x∈X, and μ:[0,∞)→[0,1) is defined by μ(t)=γ for some γ∈[0,1), then we can obtain Nadler’s fixed point theorem [10].(d)
[1, Theorem 2.6] is a special case of Theorem 3.15.(e)
Notice that, in [1, Theorem 2.6], the authors showed that a generalized multivalued almost contraction T in a metric space (X,d) has ℱ(T)≠∅ provided either (X,d) is compact and the function f(x)=d(x,Tx) is l.s.c. or T is closed and compact. But reviewing Theorem 3.15, we know that the conditions in [1, Theorem 2.6] are redundant.



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*Source: 302830-2012-05-13.xml*