# Coincidence Points for Expansive Mappings underc-Distance in Cone Metric Spaces

Authors: Hassen Aydi; Erdal Karapınar; Sirous Moradi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We establish some fixed (common fixed) and coincidence point results for mappings verifying some expansive type contractions in cone metric spaces with the help of the concept of ac-distance. Our results generalize, extend, and unify several well-known comparable results in the literature. Some examples are also presented.


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## Body

## 1. Introduction and Preliminaries

Huang and Zhang [1] reintroduced the notion of cone metric spaces and established fixed point theorems for mappings on this space. After that, many fixed point theorems have been proved in normal or nonnormal cone metric spaces by some authors (see e.g., [1–26] and references contained therein).We need to recall some basic notations, definitions, and necessary results from literature. Letℝ+ be the set of nonnegative real numbers. Let E be a real Banach space and 0E is the zero vector of E.Definition 1.1 (see [1]).
A nonempty subsetP of E is called a cone if the following conditions hold:(i)
P is closed and P≠{0E},(ii)
a,b∈ℝ,a,b≥0,x,y∈P⇒ax+by∈P,(iii)
x∈P,-x∈P⇒x=0E.
Given a coneP⊂E, a partial ordering ≤E with respect to P is naturally defined by x≤Ey if and only if y-x∈P, for x,y∈E. We will write x<Ey to indicate that x≤Ey but x≠y, while x≪y will stand for y-x∈intP, where intP denotes the interior of P. A cone P is said solid if intP is non-empty.Definition 1.2 (see [1]).
LetX be a non-empty set and d:X×X→P satisfies(i)
d(x,y)=0E if and only if x=y,(ii)
d(x,y)=d(y,x) for all x,y∈X,(iii)
d(x,y)≤Ed(x,z)+d(z,y) for all x,y,z∈E.Then, the mapping d is called a cone metric on X and the pair (X,d) is called a cone metric space.Definition 1.3 (see [1]).
Let(X,d) be a cone metric space, {xn} is a sequence in X and x∈X.(i)
If for everyc∈E with 0E≪Ec there is N∈ℕ such that d(xn,x)≪Ec for all n≥N, then {xn} is said to be convergent to x. This limit is denoted by limn→+∞xn=x or xn→x as n→+∞.(ii)
If for everyc∈E with 0E≪Ec, there is N∈ℕ such that d(xn,xm)≪Ec for all n,m>N, then {xn} is called a Cauchy sequence in X.(iii)
If every Cauchy sequence inX is convergent in X, then (X,d) is called a complete cone metric space.Definition 1.4.
Let(X,d) be a cone metric space and let T:X→X be a given mapping. We say that T is continuous on x0∈X if for every sequence {xn} is X, we have
(1.1)xn⟶x0asn⟶∞⟹Txn⟶Tx0asn⟶∞.
If T is continuous on each point x0∈X, then we say that T is continuous on X.In 2011, Cho et al. [11] and Wang and Guo [27] introduced a new concept of c-distance in cone metric spaces, which is a cone version of w-distance of Kada et al. [16] and proved some fixed point theorems for some contractive-type mappings in partially ordered cone metric spaces using the c-distance. For other results, see [13, 24].Definition 1.5 (see [11, 27]).
Let(X,d) be a cone metric space. Then, a function q:X→E is called a c-distance on X if the following are satisfied:(q1)
q(x,y)≥E0E for all x,y∈X,(q2)
q(x,z)≤Eq(x,y)+q(y,z) for all x,y,z∈X,(q3)
for eachx∈X and n≥1, if q(x,yn)≤Eu for some u=ux∈P, then q(x,y)≤Eu whenever {yn} is a sequence in X converging to a point y∈X,(q4)
for allc∈E with 0E≪c, there exists e∈E with 0E≪e such that q(z,x)≪e and q(z,y)≪e imply d(x,y)≪c.Remark 1.6 (see [11, 27]).
Thec-distance q is a w-distance on X if we take (X,d) is a metric space, E=ℝ+, P=[0,∞), and (q3) is replaced by the following condition. For any x∈X, q(x,·):X→ℝ+ is lower semicontinuous. Moreover, (q3) holds whenever q(x,·) is lower semicontinuous. Thus, if (X,d) is a metric space, E=ℝ+ and P=[0,∞), then every w-distance is a c-distance. But the converse is not true in general case. Therefore, the c-distance is a generalization of the w-distance.Example 1.7 (see [11, 27]).
Let(X,d) be a cone metric space and let P be a normal cone. Define a mapping q:X×X→E by q(x,y)=d(x,y) for all x,y∈X. Then, q is a c-distance.Example 1.8 (see [11, 27]).
Let(X,d) be a cone metric space and let P be a normal cone. Define a mapping q:X×X→E by q(x,y)=d(u,y) for all x,y∈X, where u is a fixed point in X. Then, q is a c-distance.Example 1.9.
Let(X,d)be a cone metric space and letPbe a normal cone. Define a mappingq:X×X→Ebyq(x,y)=(d(x,u)+d(u,y))/2for allx,y∈X, whereuis a fixed point in X. Thenqis ac-distance.Remark 1.10 (see [11, 27]).
(1)q(x,y)=q(y,x)does not necessarily hold for allx,y∈X.
(2)q(x,y)=0E is not necessarily equivalent to x=y for all x,y∈X.Lemma 1.11 (see [11, 27]).
Let(X,d) be a cone metric space and let q be a  c-distance on  X. Let  {xn} and  {yn} be a sequences in  X and  x,y,z∈X. Suppose that  {un} is a sequence in  P converging to  0E. The following hold. (1)
Ifq(xn,y)≤Eun and q(xn,z)≤Eun, then y=z.(2)
Ifq(xn,yn)≤Eun and q(xn,z)≤Eun, then {yn} converges to z.(3)
Ifq(xn,xm)≤Eun for all m>n, then {xn} is a Cauchy sequence in X.(4)
Ifq(y,xn)≤Eun, then {xn} is a Cauchy sequence in X.Letf,g:X→X be two selfmaps on a nonempty set X. Recall that a point x∈X is called a coincidence point of the pair (f,g) if fx=gx. The point y=fx=gx is called a point of coincidence. If x=fx=gx, then x is said a common fixed point of f and g.The purpose of this paper is to give some common fixed and coincidence point theorems for mappings verifying some expansive type contractions on cone metric spaces via ac-distance. Also, some examples are presented.


## 2. Main results

First, we present the following useful lemma, which is a variant of (2.2) in Lemma 1.11.Lemma 2.1.
Let(X,d) be a cone metric space and let  q be a  c-distance on  X. Let  {xn} be a sequence in  X. Suppose that  {αn} and  {βn} are tow sequences in  P converging to  0E. If  q(xn,y)≤Eαn and  q(xn,z)≤Eβn, then  y=z.Proof.
Letc≫0E be arbitrary. Since αn→0E, so there exists N1∈ℕ such that αn≪c/2 for all n≥N1. Similarly, there exists N2∈ℕ such that βn≪c/2 for all n≥N1. Thus, for all N≥max{N1,N2}, we have
(2.1)q(xn,y)≪c2,q(xn,z)≪c2.
Take e=c/2, so by (q4), we get that d(y,z)≪c for each c≫0E, hence y=z.Now, we present coincidence point results in the frame work of cone metric spaces in terms of ac-distance. Note that (2.2) is called an expansive type contraction.Theorem 2.2.
Let(X,d) be a cone metric space and let  q be a  c-distance on  X. Let  f,T:X→X be two functions such that there are three nonnegative real numbers  a,  b, and  c with  a+b+c>1 such that
(2.2)q(Tx,Ty)≥Eaq(fx,fy)+bq(Tx,fx)+cq(Ty,fy).
Assume the following hypotheses: (1)
b<1 and a≠0,(2)
f(X)⊆T(X),(3)
(T(X),d) is a complete subset of (X,d).Then T and f have a coincidence point, say u∈X. One has q(Tu,Tu)=0E.
Also, ifa>1, then the point of coincidence x=Tu=fu is unique.Proof.
Letx0∈X. Since f(X)⊆T(X), we can choose x1∈X such that Tx1=fx0. Again since f(X)⊆T(X), we can choose x2∈X such that Tx2=fx1. Continuing this process, we can construct a sequence {xn} in X such that Txn=fxn-1 for all n≥1.
By (2.2), we have
(2.3)q(fxn-1,fxn)=q(Txn,Txn+1)≥Eaq(fxn,fxn+1)+bq(Txn,fxn)+cq(Txn+1,fxn+1)=aq(fxn,fxn+1)+bq(fxn-1,fxn)+cq(fxn,fxn+1).
Therefore,
(2.4)(1-b)q(fxn-1,fxn)≤E(a+c)q(fxn,fxn+1).
Set λ=(1-b)/(a+c). By hypotheses, we have λ∈(0,1). Also,
(2.5)q(fxn,fxn+1)≤Eλq(fxn-1,fxn)∀n≥1.
By induction, we get that
(2.6)q(fxn,fxn+1)≤Eλnq(fx0,fx1)∀n≥0.
Let m>n. By (q2), we have
(2.7)q(fxn,fxm)≤E∑i=nm-1q(fxi,fxi+1)≤Eλn1-λq(fx0,fx1).
Since λ<1, so by Lemma 1.11(3), the sequence {Txn=fxn-1} is Cauchy in (T(X),d), which is complete, hence there exists u∈X such that Txn→Tu. Thus, fxn→Tu as n→∞, that is,
(2.8)lim⁡n→+∞d(Txn,Tu)=lim⁡n→+∞d(fxn,Tu)=0.
We claim that Tu=fu. Recall that q(fxn,fxm)≤E(λn/(1-λ))q(fx0,fx1) and fxn→Tu, so by (q3) and as m→∞, we get that
(2.9)q(fxn,Tu)≤Eλn1-λq(fx0,fx1).
From (2.2), we have
(2.10)q(fxn-1,Tu)=q(Txn,Tu)≥Eaq(fxn,fu)+bq(Txn,fxn)+cq(Tu,fu)≥Eaq(fxn,fu).
Since a≠0, so it follows that
(2.11)q(fxn,fu)≤E1aq(fxn-1,Tu).
By (2.9), we obtain
(2.12)q(fxn,fu)≤E1aq(fxn-1,Tu)≤E1aλn-11-λq(fx0,fx1).
Set αn=:(λn/(1-λ))q(fx0,fx1) and βn=:(1/a)(λn-1/(1-λ))q(fx0,fx1). Since λ<1, so αn,βn→0E as n→∞. Thus, by (2.9), (2.12), and Lemma 2.1, get that Tu=fu.
Using (2.2), we get
(2.13)q(Tu,Tu)≥E(a+b+c)q(Tu,Tu).
Since a+b+c>1, so q(Tu,Tu)=0E. Now, we prove that if a>1, then the point of coincidence x=Tu=fu is unique.
Letx and y be two points of coincidence of T and f, that is, there exist u,v∈X such that x=Tu=fu and y=Tv=fv. By the above, we have q(Tu,Tu)=q(Tv,Tv)=0E. By (2.2), the following holds:
(2.14)q(Tu,Tv)≥Eaq(fu,fv)+bq(Tu,fu)+cq(Tv,fv)=aq(Tu,Tv)+bq(Tu,Tu)+cq(Tv,Tv)=aq(Tu,Tv).
If a>1, we conclude that q(Tu,Tv)=0.
Letc≫0E be arbitrary. Take e=c≫0E. Since q(Tu,Tu)=q(Tu,Tv)=0E≪e, then by the condition (q4), we get that d(Tu,Tv)≪c. Thus, x=Tu=Tv=y. This completes the proof.Now, we state the following corollaries.Corollary 2.3.
Let(X,d) be a cone metric space and let  q be a  c-distance on  X. Let  f,T:X→X be two functions. Assume there exists  a>1 such that
(2.15)q(Tx,Ty)≥Eaq(fx,fy).
Assume the following hypotheses: (1)
f(X)⊆T(X),(2)
(T(X),d) is a complete subset of X.Then, T and f have a coincidence point, say u∈X. One has q(Tu,fu)=0E and x=Tu=fu is the unique point of coincidence of T and f.Proof.
It follows by takingb=c=0 in Theorem 2.2.Corollary 2.4.
Let(X,d) be a cone metric space and let  q be a  c-distance on  X. Let  f,T:X→X be two functions. Assume there exist two nonnegative real numbers  a and  b, with  a+b>1 such that
(2.16)q(Tx,Ty)≥Eaq(fx,fy)+bq(Tx,fx).
Assume the following hypotheses: (1)
b<1,(2)
f(X)⊆T(X),(3)
(T(X),d) is a complete subset of X.Then, T and f have a coincidence point, say u∈X. One has q(Tu,Tu)=0E and x=Tu=fu is the unique point of coincidence of T and f.Proof.
It follows by takingc=0 in Theorem 2.2.Corollary 2.5.
Let(X,d) be a complete cone metric space and let  q be a  c-distance on  X. Let  T:X→X be a surjective function. Assume there are three nonnegative real numbers  a,  b and  c with  a+b+c>1 such that
(2.17)q(Tx,Ty)≥Eaq(x,y)+bq(Tx,x)+cq(Ty,y).
If b<1 and a≠0, then T has a fixed point, say u∈X. One has q(Tu,Tu)=0E. Also if a>1, then the fixed point u is unique.Proof.
It follows by takingf=IdX, the identity on X, in Theorem 2.2. Note that when T is surjective, so T(X)=X, that is, the hypothesis (2) in Theorem 2.2 holds.The next result is similar to Theorem2.2, except that the contractive condition (2.2) is replaced by (2.18)q(Ty,Tx)≥Eaq(fy,fx)+bq(Tx,fx)+cq(Ty,fy).
Note that this contractive condition is studied since it is different to (2.2) because of Remark 1.10.(2). Its proof is essentially the same as for Theorem 2.2 and so is omitted.Theorem 2.6.
Let(X,d) be a cone metric space and let  q be a  c-distance on  X. Let  f,g:X→X be two functions such that there are three nonnegative real numbers  a,  b, and  c with  a+b+c>1 such that
(2.19)q(Ty,Tx)≥Eaq(fy,fx)+bq(Tx,fx)+cq(Ty,fy).
Assume the following hypotheses: (1)
c<1 and a≠0,(2)
f(X)⊆T(X),(3)
(T(X),d) is a complete subset of X.Then, T and f have a coincidence point say u∈X. One has q(Tu,Tu)=0E. If a>1, x=Tu=fu is the unique point of coincidence of T and f.Remark 2.7.
Let(X,d) be a cone metric space and let P be a normal cone. Take in Theorem 2.2 or Theorem 2.6 the c-distance q:X×X→E defined by q(x,y)=d(x,y) for all x,y∈X. Then, the inequalities (2.2) and (2.19) correspond to the contractive condition given in Theorem 2.1 of Shatanawi and Awawdah [28]. Thus, our results (Theorems 2.2 and 2.6) extend and generalize the results in [28].Remark 2.8.
Some similar results as above corollaries could be derived from Theorem2.6.Now, we present the following example.Example 2.9.
LetE=Cℝ1[0,1] with ∥x∥E=∥x∥∞+∥x′∥∞ and P={x∈E,x(t)≥0,t∈[0,1]}. Let X=[0,∞) and let d:X×X→E be defined by d(x,y)(t)=|x-y|2t. Then, (X,d) is a cone metric space. Let, further, q:X×X→E be defined by q(x,y)(t)=2ty. It is easy to check that q is a c-distance. Consider the mappings f,T:X→X defined by
(2.20)Tx=34x,fx=920x.
Take a=4/3 and b=c=0 (we have a+b+c>1, b<1 and a>1). For all x,y∈X, we have
(2.21)q(Tx,Ty)=2t34y≥2t35y=aq(fx,fy).
All hypotheses of Corollary 2.3 are satisfied, and u=0 is a coincidence point of f and T. Also, q(Tu,Tu)=q(0,0)=0 and x=Tu=T0=0 is the unique point of coincidence of T and f.Next, we present a common fixed point theorem for two maps involving some expansive type contractions given by the conditions (2.22).Theorem 2.10.
Let(X,d) be a complete cone metric space and let  q be a  c-distance on  X. Let  T,S:X→X be two mappings. Suppose that  T and  S satisfy the following inequalities:
(2.22)q(T(Sx),Sx)+kq(T(Sx),x)≥Eaq(Sx,x),q(S(Tx),Tx)+kq(S(Tx),x)≥Ebq(Tx,x)
for all x∈X and some nonnegative real numbers a, b, and k with a>1+2k and b>1+2k. If T and S are continuous and surjective, then T and S have a common fixed point.Proof.
Letx0 be an arbitrary point in X. Since T is surjective, there exists x1∈X such that x0=Tx1. Also, since S is surjective, there exists x2∈X such that x2=Sx1. Continuing this process, we construct a sequence (xn) in X such that x2n=Tx2n+1 and x2n+1=Sx2n+2 for all n∈ℕ. Now, for n∈ℕ, we have
(2.23)q(T(Sx2n+2),Sx2n+2)+kq(T(Sx2n+2),x2n+2)≥Eaq(Sx2n+2,x2n+2).
Thus, we have
(2.24)q(x2n,x2n+1)+kq(x2n,x2n+2)≥Eaq(x2n+1,x2n+2).
By (q2), we have q(x2n,x2n+1)+q(x2n+1,x2n+2)≥Ep(x2n,x2n+2). Hence, we get that
(2.25)q(x2n,x2n+1)+kq(x2n,x2n+1)+kq(x2n+1,x2n+2)≥Eaq(x2n+1,x2n+2).
Therefore,
(2.26)q(x2n+1,x2n+2)≤E1+ka-kq(x2n,x2n+1).
On other hand, we have
(2.27)q(S(Tx2n+1),Tx2n+1)+kq(S(Tx2n+1),x2n+1)≥Ebq(Tx2n+1,x2n+1).
Thus,
(2.28)q(x2n-1,x2n)+kq(x2n-1,x2n+1)≥Ebq(x2n,x2n+1).
Again, using (q2), we have
(2.29)q(x2n-1,x2n)+kq(x2n-1,x2n)+kq(x2n,x2n+1)≥Ebq(x2n,x2n+1).
Hence,
(2.30)q(x2n,x2n+1)≤E1+kb-kq(x2n-1,x2n).
Let
(2.31)λ=max⁡{1+ka-k,1+kb-k}.
Then, by combining (2.26) and (2.30), we have
(2.32)q(xn,xn+1)≤Eλq(xn-1,xn)∀n≥1.
Repeating (2.32) n-times, we get
(2.33)q(xn,xn+1)≤Eλnq(x0,x1).
Thus, for m>n, we have
(2.34)q(xn,xm)≤Eq(xn,xn+1)⋯+q(xm-1,xm)≤E(λn+⋯+λm-1)q(x0,x1)≤Eλn1-λq(x0,x1).
By assumption, we get that 0≤λ<1. By Lemma 1.11(3), {xn} is a Cauchy sequence in the complete cone metric space (X,d). Then, there exists v∈X such that xn→v as n→+∞. Since x2n+1→v and x2n→v as n→+∞, so clearly, the fact that T and S are continuous and uniqueness of limit yields that v=Tv=Sv, that is, v is a common fixed point of T and S.Corollary 2.11.
Let(X,d) be a complete cone metric space and let  q be a  c-distance on  X. Let  T:X→X be a continuous surjective mapping satisfying
(2.35)q(T(Tx),Tx)+kq(T(Tx),x)≥Eaq(Tx,x)
for all x∈X and some nonnegative real numbers a and k with a>1+2k. Then, T has a fixed point.Proof.
It follows from Theorem2.10 by taking S=T and b=a.Remark 2.12.
Corollary 4.1 of [29], Theorem 4 of [30], and Corollary 2.8 of [28] are particular cases of Corollary 2.11.We give the following examples illustrating our result obtained by Theorem2.10.Example 2.13.
LetE=ℝ with ∥x∥E=|x|, P=[0,∞), and X=[0,∞). Let d,q:X×X→E be defined by q(x,y)=d(x,y)=|x-y|. Then, obviously, (X,d) is a cone metric space and q is a c-distance. Consider two mappings S,T:X→X defined by Tx=2x and Sx=3x. For every k>0, take a=(2+5k)/2 and b=4+5k. We have a,b>1+2k and the conditions (2.22) hold. So T and S have a (unique) common fixed point, which is v=0.



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