# New Existence Results and Generalizations for Coincidence Points and Fixed Points without Global Completeness

Authors: Wei-Shih Du
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

Some new existence theorems concerning approximate coincidence point property and approximate fixed point property for nonlinear maps in metric spaces without global completeness are established in this paper. By exploiting these results, we prove some new coincidence point and fixed point theorems which generalize and improve Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, Kikkawa-Suzuki's fixed point theorem, and some well known results in the literature. Moreover, some applications of our results to the existence of coupled coincidence point and coupled fixed point are also presented.


---

## Body

## 1. Introduction

Let us begin with some basic definitions and notations that will be needed in this paper. The symbolsℕ and ℝ are used to denote the sets of positive integers and real numbers, respectively. Let (X,d) be a metric space. Denote by 𝒩(X) the family of all nonempty subsets of X, C(X) the class of all nonempty closed subsets of X, and CB(X) the family of all nonempty closed and bounded subsets of X. For each x∈X and A⊆X, let d(x,A)=infy∈Ad(x,y). A function ℋ:CB(X)×CB(X)→[0,∞) defined by
(1)ℋ(A,B)=max{supx∈Bd(x,A),supx∈Ad(x,B)}
is said to be the Hausdorff metric on CB(X) induced by the metric d on X.Letg:X→X be a self-map and T:X→𝒩(X) be a multivalued map. A point v in X is said to be a coincidence point (see, for instance, [1–4]) of g and T if gv∈Tv. The set of coincidence points of g and T is denoted by COP(g,T). If v∈Tv, then v is called a fixed point of T. The set of fixed points of T is denoted by ℱ(T). The maps g and T are said to have an approximate coincidence point property [1, 4] on X provided infx∈Xd(gx,Tx)=0. The map T is said to have the approximate fixed point property [1–5] on X provided infx∈Xd(x,Tx)=0.It is obvious thatCOP(g,T)≠∅ (resp., ℱ(T)≠∅) implies that T has the approximate coincidence point property (resp., T has the approximate fixed point property). Hussain et al. [1, Theorem 2.6] showed that a generalized multivalued almost contraction T in a metric space (X,d) have 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. In [1, Lemma 2.2], the authors had also shown that every generalized multivalued almost contraction in a metric space (X,d) has the approximate fixed point property.The rapid growth of fixed point theory and its applications over the past decades has led to a number of scholarly essays that examine its nature and its importance in nonlinear analysis, applied mathematical analysis, economics, game theory, and so forth; see [1–47] and references therein. Many authors devoted their attention to investigate its generalizations in various different directions of the celebrated Banach contraction principle. In 2008, Suzuki [6] presented a new type of generalization of the celebrated Banach contraction principle and does characterize the metric completeness.Theorem 1 (Suzuki [6]).
Define a nonincreasing functionθ from [0,1) onto (1/2,1] by
(2)θ(r)={1,if0≤r≤12(5-1),1-rr2,if12(5-1)≤r≤12,11+r,if12≤r<1.
Then for a metric space (X,d), the following are equivalent:(1)
Xis complete.(2)
Every mappingT on X satisfying the following has a fixed point:
there existsr∈[0,1) such that θ(r)d(x,Tx)≤d(x,y) implies d(Tx,Ty)≤rd(x,y) for all x,y∈X.(3)
There existsr∈[0,1) such that every mapping T on X satisfying the following has a fixed point:
(1/10000)d(x,Tx)≤d(x,y) implies d(Tx,Ty)≤rd(x,y) for all x,y∈X.Remark 2 (see [6]).
For everyr∈[0,1), θ(r) is the best constant.Later, Kikkawa and Suzuki [8] proved an interesting generalization of both Theorem 1 and the Nadler fixed point theorem [9] which is an extension of the Banach contraction principle to multivalued maps.Theorem 3 (Kikkawa and Suzuki [8]).
Define a strictly decreasing functionη from [0,1) onto (1/2,1] by
(3)η(r)=11+r.
Let (X,d) be a complete metric space and let T be a map from X into CB(X). Assume that there exists r∈[0,1) such that
(4)η(r)d(x,Tx)≤d(x,y)impliesℋ(Tx,Ty)≤rd(x,y)
for all x,y∈X. Then ℱ(T)≠∅.Letf be a real-valued function defined on ℝ. For c∈ℝ, we recall that
(5)limsupx→c+f(x)=infε>0supc<x<ε+cf(x).Definition 4 (see [3, 4, 10–19]).
A functionφ:[0,∞)→[0,1) is said to be an ℳ𝒯-function (or ℛ-function) if limsups→t+φ(s)<1 for all 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.In 1989, Mizoguchi and Takahashi [19] proved a famous generalization of Nadler’s fixed point theorem which gives a partial answer of Problem 9 in Reich [20]. It is worth to mention that the primitive proof of Mizoguchi-Takahashi’s fixed point theorem is quite difficult. Recently, Suzuki [21] gave a very simple proof of Mizoguchi-Takahashi’s fixed point theorem.Theorem 5 (Mizoguchi and Takahashi [19]).
Let(X,d) be a complete metric space, α:[0,∞)→[0,1) be a ℳ𝒯-function and T:X→CB(X) be a multivalued map. Assume that
(6)ℋ(Tx,Ty)≤α(d(x,y))d(x,y),
for all x,y∈X. Then ℱ(T)≠∅.In 2007, M. Berinde and V. Berinde [22] proved the following interesting fixed point theorem which generalized and improved Mizoguchi-Takahashi’s fixed point theorem.Theorem 6 (M. Berinde and V. Berinde [22]).
Let(X,d) be a complete metric space, α:[0,∞)→[0,1) be a ℳ𝒯-function, T:X→CB(X) be a multivalued map and L≥0. Assume that
(7)ℋ(Tx,Ty)≤α(d(x,y))d(x,y)+Ld(y,Tx),
for all x,y∈X. Then ℱ(T)≠∅.Very recently, Du et al. [4] studied the existence of the approximate coincidence point property and the approximate fixed point property for some new nonlinear maps and applied them to metric fixed theory. Some new generalizations of Kikkawa-Suzuki’s fixed point theorem, Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, and some well-known results in the literature were established in [4]; for more detail, one can refer to [4].The paper is organized as follows. In Section3, we first present some new existence theorems concerning approximate coincidence point property, approximate fixed point property, coincidence point and fixed point for various types of nonlinear maps in metric spaces without global completeness. Section 4 is dedicated to the study of some new coincidence point, and fixed point theorems given by exploiting our results. We establish some generalizations of Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem and others. Some applications of our results to a generalizations of Kikkawa-Suzuki’s fixed point theorem and the existence of coupled coincidence point and coupled fixed point are also given in Section 5. Consequently, in this paper, some of our results are original in the literature and we obtain many results in the literature as special cases; see for example, [4–10, 13, 14, 17–23, 30] and references therein.


## 2. Preliminaries

Recall that a functionp:X×X→[0,∞) is called a w-distance [5, 7, 10, 14, 15, 24–29, 38–40], if the following are satisfied:(w1)
p(x,z)≤p(x,y)+p(y,z) for any x,y,z∈X;(w2)
for anyx∈X, p(x,·):X→[0,∞) is l.s.c.;(w3)
for anyε>0, there exists δ>0 such that p(z,x)≤δ and p(z,y)≤δ imply d(x,y)≤ε.A functionp:X×X→[0,∞) is said to be a τ-function [5, 10, 14, 15, 27–29], first introduced and studied by Lin and Du, if the following conditions hold:(τ1)
p(x,z)≤p(x,y)+p(y,z) for all x,y,z∈X;(τ2)
ifx∈X and {yn} in X with limn→∞yn=y such that p(x,yn)≤M for some M=M(x)>0, then p(x,y)≤M;(τ3)
for any sequence{xn} in X with limn→∞sup{p(xn,xm):m>n}=0, if there exists a sequence {yn} in X such that limn→∞p(xn,yn)=0, then limn→∞d(xn,yn)=0;(τ4)
forx,y,z∈X, p(x,y)=0 and p(x,z)=0 imply y=z.Note that not either of the implicationsp(x,y)=0⇔x=y necessarily holds and p is nonsymmetric in general. It is well known that the metric d is a w-distance and any w-distance is a τ-function, but the converse is not true; see [27, 29] for more detail.Definition 7 (see [4]).
Let(X,d) be a metric space, p be a τ-function, g:X→X be a single-valued self-map and T:X→𝒩(X) be a multivalued map.(1)
The mapsg and T are said to have the p-approximate coincidence point property on X provided
(8)infx∈Xp(gx,Tx)=0.(2)
The mapT is said to have the p-approximate fixed point property on X provided
(9)infx∈Xp(x,Tx)=0.The following results are crucial in this paper.Lemma 8 (see [29, Lemma 2.1]).
Let(X,d) be a metric space and p:X×X→[0,∞) be a function. Assume that p satisfies the condition (τ3). If a sequence {xn} in X with limn→∞sup{p(xn,xm):m>n}=0, then {xn} is a Cauchy sequence in X.For eachx∈X and A⊆X, we denote p(x,A)=infy∈Ap(x,y).Lemma 9 (see [10]).
LetA be a closed subset of a metric space (X,d) and p:X×X→[0,∞) be any function. Suppose that p satisfies (τ3) and there exists u∈X such that p(u,u)=0. Then p(u,A)=0  if and only ifu∈A.The concepts ofτ0-functions and τ0-metrics were introduced in [10] as follows.Definition 10 (see [10]).
Let(X,d) be a metric space. A function p:X×X→[0,∞) is called a τ0-function if it is a τ-function on X with p(x,x)=0 for all x∈X.Remark 11.
Ifp is a τ0-function, then, from (τ4), p(x,y)=0 if and only if x=y.Example 12 (see [10]).
LetX=ℝ with the metric d(x,y)=|x-y| and 0<a<b. Define the function p:X×X→[0,∞) by
(10)p(x,y)=max{a(y-x),b(x-y)}.
Then p is nonsymmetric and hence p is not a metric. It is easy to see that p is a τ0-function.Definition 13 (see [10]).
Let(X,d) be a metric space and p be a τ0-function. For any A, B∈CB(X), define a function 𝒟p:CB(X)×CB(X)→[0,∞) by
(11)𝒟p(A,B)=max{δp(A,B),δp(B,A)},
where δp(A,B)=supx∈Ap(x,B), then 𝒟p is said to be the τ0-metric on CB(X) induced by p.Clearly, any Hausdorff metric is aτ0-metric, but the reverse is not true.Lemma 14 (see [10]).
Let(X,d) be a metric space and 𝒟p be a τ0-metric on CB(X) induced by a τ0-function p. Then every τ0-metric 𝒟p is a metric on CB(X).The following characterizations ofℳ𝒯-functions is quite useful for proving our main results.Lemma 15 (see [18]).
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 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≤supn∈ℕφ(xn)<1.(g)
φ is afunction of contractive factor [12]; that is, for any strictly decreasing sequence {xn}n∈ℕ in [0,∞), one has 0≤supn∈ℕφ(xn)<1.


## 3. New Nonlinear Conditions forp-Approximate Coincidence Point Property

In Section3, we will establish some new existence theorems concerning approximate coincidence point property, approximate fixed point property, coincidence point and fixed point for various types of nonlinear maps in metric spaces without global completeness.Theorem 16.
Let(X,d) be a metric space, p be a τ0-function, T:X→𝒩(X) be a multivalued map, and f:X→X be a self-map. Suppose that(S1)
there exist a nondecreasing functionτ:[0,∞)→[0,∞) and an ℳ𝒯-function φ:[0,∞)→[0,1) such that for each x∈X, if y∈X with fy≠fx and fy∈Tx, then it holds
(12)p(fy,Ty)≤φ(τ(p(fx,fy)))p(fx,fy).(S2)
T(X)=⋃x∈X‍T(x)⊆f(X).
Then the following statements hold.(a)
There exists a sequence{xn}n∈ℕ in X such that
(13)infn∈ℕp(fxn,fxn+1)=limn→∞p(fxn,fxn+1)=limn→∞d(fxn,fxn+1)=infn∈ℕd(fxn,fxn+1)=0.(b)
infx∈Xp(fx,Tx)=infx∈Xd(fx,Tx)=0; that is, f and T have the p-approximate coincidence point property and approximate coincidence point property on X.(c)
If one further assumes the following conditions hold:
(L1)
f(X) is a complete subspace of X,(L2)
for each sequence{xn} in X with fxn+1∈Txn, n∈ℕ and limn→∞fxn=fw, one has Tw as a closed subset of X and limn→∞p(fxn,Tw)=0,thenCOP(f,T)≠∅.Proof.
Letx1∈X. By (S2), there exists x2∈X such that fx2∈Tx1. If fx1=fx2, then fx1∈Tx1 and so
(14)infx∈Xp(fx,Tx)≤p(fx1,Tx1)≤p(fx1,fx1)=0,
which implies infx∈Xp(fx,Tx)=0. Clearly, infx∈Xd(fx,Tx)=0. Let wn=x1 for all n∈ℕ. Then
(15)limn→∞p(fwn,fwn+1)=infn∈ℕp(fwn,fwn+1)=p(fx1,fx1)=0,limn→∞d(fwn,fwn+1)=infn∈ℕd(fwn,fwn+1)=d(fx1,fx1)=0.
So, the conclusions (a) and (b) hold in this case. Otherwise, if fx2≠fx1, since p is a τ0-function, p(fx1,fx2)>0. Let μ:[0,∞)→[0,1) be defined by μ(t)=(1+φ(t))/2. Clearly, 0≤φ(t)<μ(t)<1 for all t∈[0,∞). By [3, Lemma 2.1], we know that μ is also an ℳ𝒯-function. From (S1), we get
(16)p(fx2,Tx2)≤φ(τ(p(fx1,fx2)))p(fx1,fx2)<μ(τ(p(fx1,fx2)))p(fx1,fx2).
Since μ(τ(p(fx1,fx2)))p(fx1,fx2)>0, there exists ξ∈Tx2 such that
(17)p(fx2,ξ)<μ(τ(p(fx1,fx2)))p(fx1,fx2).
Using (S2) again, there exists x3∈X such that fx3=ξ∈Tx2. Hence, from (17), we have
(18)p(fx2,fx3)<μ(τ(p(fx1,fx2)))p(fx1,fx2).
If fx2=fx3∈Tx2, then, following a similar argument as above, we can prove the conclusions (a) and (b). Otherwise, if fx3≠fx2, then there exists x4∈X such that fx4∈Tx3 and
(19)p(fx3,fx4)<μ(τ(p(fx2,fx3)))p(fx2,fx3).
By induction, we can obtain a sequences {xn} in X satisfying
(20)fxn+1∈Txn,(21)p(fxn+1,fxn+2)<μ(τ(p(fxn,fxn+1)))p(fxn,fxn+1),p(fxn+1,fxn+2)<μ(τ(p(fxn,fxn+1)))foreachn∈ℕ.
Since μ(t)<1 for all t∈[0,∞), we deduces from the inequality (21) that the sequence {p(fxn, fxn+1)}n∈ℕ is strictly decreasing in [0,∞). Hence
(22)limn→∞p(fxn,fxn+1)=infn∈ℕp(fxn,fxn+1)≥0exists.
Since τ is nondecreasing, {τ(p(fxn,fxn+1))}n∈ℕ is a nonincreasing sequence in [0,∞). Since μ is an ℳ𝒯-function, by (f) of Lemma 15, we have
(23)0≤supn∈ℕμ(τ(p(fxn,fxn+1)))<1.
Let γ:=supn∈ℕμ(τ(p(fxn,fxn+1))). So γ∈[0,1). Put λ:=(1+γ)/2. Then 0≤γ<λ<1. By (21), we get
(24)p(fxn+1,fxn+2)<μ(τ(p(fxn,fxn+1)))p(fxn,fxn+1)≤γp(fxn,fxn+1)<λp(fxn,fxn+1)<⋯<λnp(fx1,fx2)foreachn∈ℕ.
Since λ∈(0,1), limn→∞λn=0 and hence it follows from (24) that
(25)limn→∞p(fxn,fxn+1)=0.
According to (22) and (25), we obtain
(26)infn∈ℕp(fxn,fxn+1)=limn→∞p(fxn,fxn+1)=0.
Next, we verify that{fxn} is a Cauchy sequence in f(X). Let vn:=fxn for all n∈ℕ. We claim that limn→∞sup{p(vn,vm):m>n}=0. Put(27)αn=(λn-11-λ)p(v1,v2),n∈ℕ.
Thenαn>0 for all n∈ℕ. For m,n∈ℕ with m>n, by (24), we have
(28)p(vn,vm)≤∑j=nm-1‍p(vj,vj+1)<αn.
Since λ∈(0,1), limn→∞αn=0 and hence
(29)limn→∞sup{p(vn,vm):m>n}=0.
By Lemma 8, {vn} is a Cauchy sequence in f(X). Hence
(30)limn→∞d(fxn,fxn+1)=limn→∞d(vn,vn+1)=0.
Since infn∈ℕd(fxn,fxn+1)≤d(fxm,fxm+1) for all m∈ℕ and limm→∞d(fxm,fxm+1)=0, one also obtain
(31)limn→∞d(fxn,fxn+1)=infn∈ℕd(fxn,fxn+1)=0.
Hence (a) is proved. To see (b), since fxn+1∈Txn for each n∈ℕ, we have
(32)infx∈Xp(fx,Tx)≤p(fxn,Txn)≤p(fxn,fxn+1),infx∈Xd(fx,Tx)≤d(fxn,Txn)≤d(fxn,fxn+1),
for all n∈ℕ. Since limn→∞p(fxn,fxn+1)=limn→∞d(fxn,fxn+1)=0, combining (32), we get
(33)infx∈Xp(fx,Tx)=infx∈Xd(fx,Tx)=0.
Moreover, if we further assume that conditions (L1) and (L2) hold, we want to show COP(f,T)≠∅. Since {vn} is a Cauchy sequence in f(X), by (L1), there exists v^∈X such that vn≡fxn→fv^ as n→∞. So, by (L2), we have Tv^ is a closed subset of X and limn→∞p(vn,Tv^)=0. Since limn→∞sup{p(vn,vm):m>n}=0, there exists {an}⊂{vn} with limn→∞sup{p(an,am):m>n}=0 and {bn}⊂Tv^ such that limn→∞p(an,bn)=0. By (τ3), limn→∞d(an,bn)=0. Since an→fv^ as n→∞ and d(bn,fv^)≤d(bn,an)+d(an,fv^), it implies bn→fv^ as n→∞. By the closedness of Tv^, we have fv^∈Tv^ or v^∈COP(f,T). The proof is completed.Theorem 17.
In Theorem16, if f≡id is the identity map on X, then the following statements hold.(a)
There exists a sequence{xn}n∈ℕ in X such that
(34)infn∈ℕp(xn,xn+1)=limn→∞p(xn,xn+1)=limn→∞d(xn,xn+1)=infn∈ℕd(xn,xn+1)=0.(b)
infx∈Xp(x,Tx)=infx∈Xd(x,Tx)=0; that is, T has the p-approximate fixed point property and approximate fixed point property on X.(c)
If one further assumes the following conditions hold:
(L1)
f(X) is a complete subspace of X,(L3)
for each sequence{xn} in X with xn+1∈Txn, n∈ℕ and limn→∞xn=w, one has Tw as a closed subset of X and limn→∞p(xn,Tw)=0,thenℱ(T)≠∅.As an application of Theorems16, we can establish the following new existence of approximate coincidence point property easily.Theorem 18.
Let(X,d) be a metric space, p be a τ0-function, 𝒟p be a τ0-metric on CB(X) induced by p, T:X→CB(X) be a multivalued map and f:X→X be a self-map. Suppose that (S2) as in Theorem 16 is satisfied and T further satisfies one of the following conditions:(H1)
there exist a nondecreasing functionτ:[0,∞)→[0,∞), an ℳ𝒯-function φ:[0,∞)→[0,1) and a function L:X×X→[0,∞) such that
(35)𝒟p(Tx,Ty)≤φ(τ(p(fx,fy)))p(fx,fy)+L(fx,fy)p(fy,Tx)∀x,y∈X,(H2)
there exist a nondecreasing functionτ:[0,∞)→[0,∞), an ℳ𝒯-function φ:[0,∞)→[0,1) and a function L:X×X→[0,∞) such that
(36)p(fy,Ty)≤φ(τ(p(fx,fy)))p(fx,fy)+L(fx,fy)p(fy,Tx)∀x,y∈X.
Then the following statements hold.(a)
There exists a sequence{xn}n∈ℕ in X such that
(37)infn∈ℕp(fxn,fxn+1)=limn→∞p(fxn,fxn+1)=limn→∞d(fxn,fxn+1)=infn∈ℕd(fxn,fxn+1)=0.(b)
infx∈Xp(fx,Tx)=infx∈Xd(fx,Tx)=0.If one further assumes the following conditions hold:
(L1)
f(X) is a complete subspace of X,(L2)
for each sequence{xn} in X with fxn+1∈Txn, n∈ℕ and limn→∞fxn=fw, one has Tw as a closed subset of X and limn→∞p(fxn,Tw)=0,thenCOP(f,T)≠∅.Proof.
Suppose that (H1) holds. We first notice that for eachx∈X, by (S2), the set {y∈X:fy∈Tx}≠∅. Let x∈X be given and let y∈X with fy∈Tx be arbitrary. Since p(fy,Tx)=0, we have
(38)p(fy,Ty)≤𝒟p(Tx,Ty)≤φ(τ(p(fx,fy)))p(fx,fy).
Hence (H1) implies (S1). Therefore the conclusion follows from Theorem 16. Similarly, we can prove that (H2) implies (S1) and the desired result follows also from Theorem 16.Theorem 19.
In Theorem18, if f≡id is the identity map on X, then the conclusion following statements hold.(a)
There exists a sequence{xn}n∈ℕ in X such that
(39)infn∈ℕp(xn,xn+1)=limn→∞p(xn,xn+1)=limn→∞d(xn,xn+1)=infn∈ℕd(xn,xn+1)=0.(b)
infx∈Xp(x,Tx)=infx∈Xd(x,Tx)=0.(c)
If one further assumes the following conditions hold:
(L1)
f(X) is a complete subspace of X,(L3)
for each sequence{xn} in X with xn+1∈Txn, n∈ℕ and limn→∞xn=w, one has Tw as a closed subset of X and limn→∞p(xn,Tw)=0,thenℱ(T)≠∅.


## 4. New Generalizations of Berinde-Berinde’s Fixed Point Theorem and Mizoguchi-Takahashi’s Fixed Point Theorem

In this section, we will establish some new coincidence point theorems which generalize and improve Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem and some main results in [10, 14, 17–19, 22].Theorem 20.
Let(X,d) be a metric space, p be a τ0-function, 𝒟p be a τ0-metric on CB(X) induced by p, T:X→CB(X) be a multivalued map and f:X→X be a self-map. Suppose that(i)
T(X)=⋃x∈X‍T(x)⊆f(X),(ii)
f(X) is a complete subspace of X,(iii)
there exists a nondecreasing functionτ:[0,∞)→[0,∞), an ℳ𝒯-function φ:[0,∞)→[0,1) and a function h:X→[0,∞) such that
(40)𝒟p(Tx,Ty)≤φ(τ(p(fx,fy)))p(fx,fy)+h(y)d(fy,Tx)∀x,y∈X,then COP(f,T)≠∅.Moreover, if we further assume thatfw=ffw for any w∈COP(f,T), then T and f have a common fixed point in X.Proof.
Letx∈X be given. Let y∈X with fy∈Tx be arbitrary. Since d(fy,Tx)=0, we have
(41)p(fy,Ty)≤𝒟p(Tx,Ty)≤φ(τ(p(fx,fy)))p(fx,fy).
Hence (iii) implies (S1). So, following the same argument as the proof of Theorem 16, we can obtain two sequences {xn} and {vn} in X satisfying the following:(i)
fxn+1∈Txn for each n∈ℕ;(ii)
vn:=fxn for all n∈ℕ;(iii)
p(v1,v2)>0;(iv)
There existsλ∈(0,1) such that p(vn+1,vn+2)≤λnp(v1,v2) for each n∈ℕ;(v)
p(vn,vm)<αn for m,n∈ℕ with m>n, where αn=(λn-1/(1-λ))p(v1,v2)>0 for n∈ℕ;(vi)
limn→∞sup{p(vn,vm):m>n}=0;(vii)
{vn} is a Cauchy sequence in f(X).
By (ii), there existsv-∈X such that fxn≡vn→fv- as n→∞ or limn→∞d(vn,fv-)=0. Since p(vn,vm)<αn for m,n∈ℕ with m>n, from (τ2), we have
(42)p(fxn,fv-)=p(vn,fv-)≤αn∀n∈ℕ.
So, for each n∈ℕ, it follows from (40) and (42) that
(43)p(vn+1,Tv-)=p(fxn+1,Tv-)≤supy∈Txnp(y,Tv-)≤𝒟p(Txn,Tv-)≤φ(τ(p(fxn,fv-)))p(fxn,fv-)+h(v-)d(fv-,Txn)<p(fxn,fv-)+h(v-)d(fv-,fxn+1)≤αn+h(v-)d(fv-,fxn+1).
The last inequality implies that there exists yn+1∈Tv- such that
(44)p(vn+1,yn+1)<αn+h(v-)d(fv-,fxn+1)foreachn∈ℕ.
Since limn→∞αn=limn→∞d(fv-,fxn+1)=0, we get limn→∞p(vn+1,yn+1)=0. By (τ3), we have limn→∞d(vn+1,yn+1)=0. Since
(45)0≤d(yn+1,fv-)≤d(yn+1,vn+1)+d(vn+1,fv-)∀n∈ℕ,
we obtain limn→∞d(yn+1,fv-)=0 or yn+1→fv- as n→∞. Since yn+1∈Tv- for all n∈ℕ and Tv- is closed, we have fv-∈Tv- which means that v-∈COP(f,T). So COP(f,T)≠∅.
Moreover, if we further assume thatfw=ffw for all w∈COP(f,T), then we have fv-=ffv-. For each n∈ℕ, by (40) and (42) again, we have
(46)p(vn+1,Tfv-)=p(fxn+1,Tfv-)≤𝒟p(Txn,Tfv-)≤φ(τ(p(fxn,ffv-)))p(fxn,ffv-)+h(fv-)d(ffv-,Txn)<p(fxn,fv-)+h(fv-)d(fv-,fxn+1)≤αn+h(fv-)d(fv-,fxn+1).
Therefore, there exists zn+1∈Tfv- such that
(47)p(vn+1,zn+1)<αn+h(fv-)d(fv-,fxn+1)∀n∈ℕ.
By (47) and limn→∞αn=limn→∞d(fv-,fxn+1)=0, we have limn→∞p(vn+1,zn+1)=0. By (τ3), we get limn→∞d(vn+1,zn+1)=0. Since
(48)d(fv-,zn+1)≤d(fv-,vn+1)+d(vn+1,zn+1),
we have limn→∞d(fv-,zn+1)=0. Since Tfv- is closed and zn∈Tfv- for all n∈ℕ, we get fv-∈Tfv-. Therefore, fv-=ffv-∈Tfv-, which means that fv- is a common fixed point of f and T in X. The proof is completed.Corollary 21.
Let(X,d) be a metric space, T:X→CB(X) be a multivalued map and f:X→X be a self-map. Suppose that(i)
T(X)=⋃x∈X‍T(x)⊆f(X),(ii)
f(X) is a complete subspace of X,(iii)
there exists a nondecreasing functionτ:[0,∞)→[0,∞), an ℳ𝒯-function φ:[0,∞)→[0,1) and a function h:X→[0,∞) such that
(49)ℋ(Tx,Ty)≤φ(τ(d(fx,fy)))d(fx,fy)+h(y)d(fy,Tx)∀x,y∈X,then COP(f,T)≠∅.The following results are immediate consequences of Theorem20. They are generalizations of Berinde-Berinde’s fixed point and Mizoguchi-Takahashi’s fixed point theorem.Theorem 22.
Let(X,d) be a complete metric space, p be a τ0-function, 𝒟p be a τ0-metric on CB(X) induced by p, T:X→CB(X) be a multivalued map, τ:[0,∞)→[0,∞) be a nondecreasing function, φ:[0,∞)→[0,1) be an ℳ𝒯-function, and h:X→[0,∞) be a function. Suppose that
(50)𝒟p(Tx,Ty)≤φ(τ(p(x,y)))p(x,y)+h(y)d(y,Tx)∀x,y∈X,
then ℱ(T)≠∅.Corollary 23.
Let(X,d) be a complete metric space, T:X→CB(X) be a multivalued map, τ:[0,∞)→[0,∞) be a nondecreasing function, φ:[0,∞)→[0,1) be an ℳ𝒯-function, and h:X→[0,∞) be a function. Suppose that
(51)ℋ(Tx,Ty)≤φ(τ(d(fx,fy)))d(x,y)+h(y)d(y,Tx)∀x,y∈X,
then ℱ(T)≠∅.Remark 24.
(a) It is worth to mention that Theorem22 is different from [10, Theorem 2.3]. Theorem 22 is comparable to [10, Theorem 2.3] in the following aspects.(1)
In [10, Theorem 2.3], the map T was assumed to satisfy
(52)𝒟p(Tx,Ty)≤φ(p(x,y))p(x,y)+Lp(y,Tx)∀x,y∈Xwhereφ is an ℳ𝒯-function and L is a given nonnegative real number. But in Theorem 22, we assume that
(53)𝒟p(Tx,Ty)≤φ(τ(p(x,y)))p(x,y)+h(y)d(y,Tx)∀x,y∈X,whereτ is a nondecreasing function, φ is an ℳ𝒯-function and h:X→[0,∞) is any function.(2)
Notice that in [10, Theorem 2.3], the author assumed that T further satisfies one of conditions (D1), (D2), (D3), (D4), and (D5), where
(D1)T is closed;(D2) the mapf:X→[0,∞) defined by f(x)=p(x,Tx) is l.s.c.;(D3) the mapg:X→[0,∞) defined by g(x)=d(x,Tx) is l.s.c.;(D4) for any sequence{xn} in X with xn+1∈Txn, n∈ℕ and limn→∞xn=v, we have limn→∞p(xn,Tv)=0;(D5)inf{p(x,z)+p(x,Tx):x∈X}>0 for every z∉ℱ(T).But Theorem22 does not require the conditions (D1)–(D5).
(b) If we takeτ(t)=t, t∈[0,∞) and h(t)=0 for all x∈X in Theorem 22, then we obtain [17, Theorem 3.1].
(c) Theorems20 and 22, and Corollary 21 all generalize Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, Nadler’s fixed point theorem, Banach contraction principle and some main results in [10, 14, 18, 19, 22] and references therein.
Here, we give a simple example illustrating Theorem20.Example 25.
LetX=(-9876,-10)∪[0,∞) with the metric d(x,y)=|x-y| for x,y∈X. Then (X,d) is a metric space. Let p:X×X→[0,∞) be defined by
(54)p(x,y)=max{2(x-y),3(y-x)},
for all x, y∈X. By Example 12, we know that p is a τ0-function. Let φ:[0,∞)→[0,1) and τ:[0,∞)→[0,∞) be defined by φ(t)=1/2 and τ(t)=5, respectively. Then φ is an ℳ𝒯-function and τ is a nondecreasing function. Let f:X→X be defined by
(55)f(x)≔{0,ifx∈(-9876,-10),2x,ifx∈[0,∞).
Then f(X)=[0,∞) is a proper complete subspace of X. Define T:X→CB(X) by
(56)T(x)≔{{0},ifx∈(-9876,-10),[0,x],ifx∈[0,∞).
Clearly, T(X)=⋃x∈X‍T(x)⊆f(X). Let h:X→[0,∞) be defined by h(x)=ex. We claim that
(*)𝒟p(Tx,Ty)≤φ(τ(p(fx,fy)))p(fx,fy)+h(y)d(fy,Tx),
for all x,y∈X. We consider the following six possible cases.Case 1.
Clearly, inequality(*) holds for x=y∈X.Case 2.
Ifx,y∈(-9876,-10), then 𝒟p(Tx,Ty)=0=φ(τ(p(fx,fy)))p(fx,fy)+h(y)d(fy,Tx).Case 3.
Ifx∈(-9876,-10) and y∈[0,∞), then Tx={0} and Ty=[0,y]. So
(57)p(fx,fy)=max{2(fx-fy),3(fy-fx)}=max{-4y,6y}=6y,𝒟p(Tx,Ty)=max{supz∈Txp(z,Ty),supz∈Typ(z,Tx)}=2y=13p(fx,fy)<φ(τ(p(fx,fy)))p(fx,fy)+h(y)d(fy,Tx).Case 4.
Ifx∈[0,∞) and y∈(-9876,-10), then Tx=[0,x], Ty={0} and p(fy,Tx)=0. Hence, we have
(58)p(fx,fy)=max{2(fx-fy),3(fy-fx)}=max{4x,-6x}=4x,𝒟p(Tx,Ty)=max{supz∈Txp(z,Ty),supz∈Typ(z,Tx)}=2x=12p(fx,fy)=φ(τ(p(fx,fy)))p(fx,fy)+h(y)d(fy,Tx).Case 5.
If0≤x<y, then Tx=[0,x] and Ty=[0,y]. So, we have
(59)p(fx,fy)=max{2(fx-fy),3(fy-fx)}=6(y-x),𝒟p(Tx,Ty)=max{supz∈Txp(z,Ty),supz∈Typ(z,Tx)}=2(y-x)<12p(fx,fy)≤φ(τ(p(fx,fy)))p(fx,fy)+h(y)d(fy,Tx).Case 6.
If0≤y<x, then Tx=[0,x], Ty=[0,y]. Thus we obtain
(60)p(fx,fy)=max{2(fx-fy),3(fy-fx)}=4(x-y),𝒟p(Tx,Ty)=max{supz∈Txp(z,Ty),supz∈Typ(z,Tx)}=2(x-y)=12p(fx,fy)≤φ(τ(p(fx,fy)))p(fx,fy)+h(y)d(fy,Tx).By Cases1–6, we verify that inequality (*) holds for all x,y∈X. So all the hypotheses of Theorem 20 are fulfilled. It is therefore possible to apply Theorem 20 to get COP(f,T)≠∅. In fact, COP(f,T)=(-9876,-10)∪{0}.Moreover, sincefw=ffw=0 for any w∈COP(f,T), by Theorem 20 again, we know that f and T have a common fixed point in X (precisely speaking, 0 is the unique common fixed point of f and T).


## 5. Some Applications to New Coupled Coincidence Point Theorems and a Generalization of Kikkawa-Suzuki’s Fixed Point Theorem

Let(X,d) be a metric space. We endow the product space X×X with the metric ρ defined by
(61)ρ((x,y),(u,v))=d(x,u)+d(y,v)forany(x,y),(u,v)∈X×X.Letg:X→X be a self-map. Recall that an element (x,y)∈X×X is called a coupled coincidence point of the maps F and g if
(62)F(x,y)=gx,F(y,x)=gywhenF:X×X→Xisasingle-valuedmap
or
(63)gx∈F(x,y),gy∈F(y,x)whenF:X×X→Xisamultivaluedmap.
In particular, if we take g≡id (the identity map) in (62) and (63), then (x,y) is called a coupled fixed point of F. The existence of coupled coincidence point and coupled fixed point has been investigated by several authors recently in [4, 11, 12, 23, 30–37] and references therein.As an interesting application of Corollary21 (or Theorem 20), we establish the following new coupled coincidence point theorems. It is worth to mention that the technique in the proof of Theorem 26 is quite different in the well known literature.Theorem 26.
Let(X,d) be a metric space, F:X×X→CB(X) be a multivalued map and f:X→X be a self-map satisfying ⋃x∈X‍F(x,x)⊆f(X) and f(X) is a complete subspace of X. Assume that there exists an ℳ𝒯-function φ:[0,∞)→[0,1) such that for any (x,y),(u,v)∈X×X,
(64)ℋ(F(x,y),F(u,v))≤12φ(ρ((fx,fy),(fu,fv)))ρ((fx,fy),(fu,fv)),
where ρ((x,y),(u,v))=d(x,u)+d(y,v)for (x,y),(u,v)∈X×X. Then there exists v∈X such that fv∈F(v,v); that is, (v,v)∈X×X is a coupled coincidence point of F and f.Proof.
DefineT:X→CB(X) by
(65)T(x)=F(x,x).
Then T(X)=⋃x∈X‍T(x)⊆f(X). Let τ(t)=2t, ∀t∈[0,∞) and h(x)=0, ∀x∈X. So the inequality (64) implies
(66)ℋ(T(x),T(u))≤φ(2d(fx,fu))d(fx,fu)=φ(τ(d(fx,fu)))d(fx,fu)+h(u)d(fu,Tx)∀x,u∈X.
Therefore, all the assumptions of Corollary 21 are fulfilled. Hence we can apply Corollary 21 to show that there exists v∈X such that fv∈Tv=F(v,v).The following conclusion is immediate from Theorem26 with f≡id (the identity map).Corollary 27 (see [4]).
Let(X,d) be a complete metric space and F:X×X→CB(X) be a multivalued map. Assume that there exists an ℳ𝒯-function φ:[0,∞)→[0,1) such that for any (x,y), (u,v)∈X×X,
(67)ℋ(F(x,y),F(u,v))≤12φ(ρ((x,y),(u,v)))ρ((x,y),(u,v)),
where ρ((x,y),(u,v))=d(x,u)+d(y,v)for (x,y),(u,v)∈X×X. Then there exists v∈X such that v∈F(v,v); that is, (v,v)∈X×X is a coupled fixed point of F.Applying Theorem26, we obtain the following coupled coincidence point theorem.Theorem 28.
Let(X,d) be a metric space, F:X×X→X be a map and f:X→X be a self-map satisfying ⋃x∈X‍F(x,x)⊆f(X) and f(X) is a complete subspace of X. Assume that there exists an ℳ𝒯-function φ:[0,∞)→[0,1) such that for any (x,y),(u,v)∈X×X,
(68)d(F(x,y),F(u,v))≤12φ(ρ((fx,fy),(fu,fv)))ρ((fx,fy),(fu,fv)).
Then there exists v∈X such that (v,v) is the unique coupled coincidence point of F and f.Proof.
Applying Theorem26, there exists v^∈X such that fv^=F(v^,v^). To complete the proof, it suffices to show the uniqueness of the coupled coincidence point of F and f. On the contrary, suppose that there exists (x^,y^)∈X×X, such that fx^=F(x^,y^) and fy^=F(y^,x^). By (68), we have
(69)d(fv^,fx^)=d(F(v^,v^),F(x^,y^))<12[d(fv^,fx^)+d(fv^,fy^)],d(fv^,fy^)=d(F(v^,v^),F(y^,x^))<12[d(fv^,fy^)+d(fv^,fx^)].
So, combining (69), we get
(70)d(fv^,fx^)+d(fv^,fy^)<d(fv^,fx^)+d(fv^,fy^),
which leads a contradiction. The proof is completed.Corollary 29 (see [4]).
Let(X,d) be a complete metric space and F:X×X→X be a map. Assume that there exists an ℳ𝒯-function φ:[0,∞)→[0,1) such that for any (x,y),(u,v)∈X×X,
(71)d(F(x,y),F(u,v))≤12φ(ρ((x,y),(u,v)))ρ((x,y),(u,v)).
Then there exists v∈X such that (v,v) is the unique coupled fixed point of F.Applying Theorem18, we can prove the following coincidence point theorem which is a generalization of Kikkawa-Suzuki’s fixed point theorem.Theorem 30.
Define a strictly decreasing functionη from [0,1) onto (1/2,1] by
(72)η(r)=11+r.
Let (X,d) be a metric space, T:X→CB(X) be a multivalued map, and f:X→X be an injective self-map such that T(X)=⋃x∈X‍T(x)⊆f(X). Assume that(KS)
there existsγ∈[0,1) such that
(73)η(γ)d(fx,Tx)≤d(fx,fy)impliesℋ(Tx,Ty)≤γd(fx,fy),for all x,y∈X.Thenf and T have the approximate coincidence property on X.Moreover, if we further assumef(X) is a complete subspace of X, then COP(f,T)≠∅.Proof.
Letx∈X be arbitrary. If fx∈Tx, then x∈COP(f,T) and hence f and T have the approximate coincidence property. So the theorem is finished in this case. Suppose fx∉Tx. Let τ:[0,∞)→[0,∞) be defined by τ(t)=t. Then τ is a nondecreasing function. By (KS), we can define an ℳ𝒯-function φ:[0,∞)→[0,1) by φ(t)=γ, ∀t∈[0,∞). Let y∈X with fy≠fx and fy∈Tx. Since
(74)η(γ)d(fx,Tx)≤d(fx,Tx)≤d(fx,fy).
by (KS) again, we have
(75)d(fy,Ty)≤ℋ(Tx,Ty)≤γd(fx,fy)=φ(τ(d(fx,fy)))d(fx,fy),
which shows that (H1) as in Theorem 18 holds. Therefore T have the approximate coincidence property by applying Theorem 18.
Now, suppose thatf(X) is a complete subspace of X. We shall prove COP(f,T)≠∅. It suffices to prove that the condition (L2) as in Theorem 18 holds. Let {xn} in X with fxn+1∈Txn, n∈ℕ and limn→∞fxn=fw. Clearly, Tw is a closed subset of X. We will proceed with the following claims to prove limn→∞d(fxn,Tw)=0.
Claim 1.d(fw,Tx)≤γd(fw,fx) for all x∈X with fx≠fw.
Forx∈X with fx≠fw, since limn→∞fxn=fw, there exists n0∈ℕ, such that
(76)d(fw,fxn)≤13d(fx,fw)∀n∈ℕwithn≥n0.
For n∈ℕ with n≥n0, it follows from (76) that
(77)η(γ)d(fxn,Txn)≤d(fxn,Txn)≤d(fxn,fxn+1)≤d(fxn,fw)+d(fw,fxn+1)≤23d(fx,fw)=d(fx,fw)-13d(fx,fw)≤d(fx,fw)-d(fxn,fw)≤d(fxn,fx).
By our hypothesis (KS), we have
(78)d(fxn+1,Tx)≤ℋ(Txn,Tx)≤γd(fxn,fx)∀n∈ℕwithn≥n0.
Since limn→∞fxn=fw, the last inequality implies
(79)d(fw,Tx)≤γd(fw,fx)∀x∈Xwithfx≠fw.
Claim 2.ℋ(Tx,Tw)≤γd(fx,fw) for all x∈X.
Letx∈X. It is quite obvious that the desired inequality holds if x=w. Suppose x≠w. Since f is one-to-one, we have fx≠fw. For every n∈ℕ, there exists yn∈Tx such that
(80)d(fw,yn)<d(fw,Tx)+1nd(x,w).
Using (80) and Claim 1, we obtain
(81)d(fx,Tx)≤d(fx,yn)≤d(fx,fw)+d(fw,yn)<d(fx,fw)+d(fw,Tx)+1nd(x,w)≤d(fx,fw)+γd(fw,fx)+1nd(x,w)=(1+γ)d(fx,fw)+1nd(x,w)∀n∈ℕ.
The last inequality implies
(82)η(γ)d(fx,Tx)=11+γd(fx,Tx)≤d(fx,fw).
By (KS), we get
(83)ℋ(Tx,Tw)≤γd(fx,fw)∀x∈X.
Claim 3.limn→∞d(fxn,Tw)=0.
Sincefxn+1∈Txn, n∈ℕ and limn→∞fxn=fw, applying Claim 2, we have
(84)limn→∞d(fxn+1,Tw)≤limn→∞ℋ(Txn,Tw)≤limn→∞γd(fxn,fw)=0,
which implies limn→∞d(fxn,Tw)=0.
By our Claims, we prove that (L2) holds. Therefore, by applying Theorem18, we prove COP(f,T)≠∅.Takef≡id (the identity map) in Theorem 30, we obtain the following existence theorem which is also a generalized Kikkawa-Suzuki’s fixed point theorem.Corollary 31.
Define a strictly decreasing functionη from [0,1) onto (1/2,1] by
(85)η(r)=11+r.
Let (X,d) be a metric space and let T be a map from X into CB(X). Assume that there exists γ∈[0,1) such that
(86)η(γ)d(x,Tx)≤d(x,y)impliesℋ(Tx,Ty)≤γd(x,y),
for all x,y∈X. Then T has the approximate fixed property.Moreover, if we further assumeX is complete, then ℱ(T)≠∅.



---

*Source: 214230-2013-02-27.xml*