AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 171624 10.1155/2014/171624 171624 Research Article Remarks on Some Recent Fixed Point Results on Quaternion-Valued Metric Spaces Agarwal Ravi P. 1, 2 Alsulami Hamed H. 3 Karapınar Erdal 3, 4 http://orcid.org/0000-0002-9300-8487 Khojasteh Farshid 5 Du Wei-Shih 1 Department of Mathematics Texas A&M University-Kingsville MSC 172, Rhode Hall 217B Kingsville TX 78363-8202 USA tamuk.edu 2 Department of Mathematics Faculty of Science King Abdulaziz University Jeddah 21589 Saudi Arabia kau.edu.sa 3 Nonlinear Analysis and Applied Mathematics Research Group (NAAM) King Abdulaziz University Jeddah Saudi Arabia kau.edu.sa 4 Department of Mathematics Atilim University Incek 06836 Ankara Turkey atilim.edu.tr 5 Department of Mathematics Islamic Azad University Arak Branch Arak Iran azad.ac.ir 2014 2152014 2014 17 03 2014 18 04 2014 21 5 2014 2014 Copyright © 2014 Ravi P. Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Very recently, Ahmed et al. introduced the notion of quaternion-valued metric as a generalization of metric and proved a common fixed point theorem in the context of quaternion-valued metric space. In this paper, we will show that the quaternion-valued metric spaces are subspaces of cone metric spaces. Consequently, the fixed point results in such spaces can be derived as a consequence of the corresponding existing fixed point result in the setting cone metric spaces.

1. Introduction

Recently, Azam et al.  introduced the notion of complex-valued metric space, as a generalization of Banach-valued metric space which is also known as a cone metric space. The authors  proved several fixed point theorems in the context of complex-valued metric space. Inspired from these results, Ahmed et al.  defined the concept of quaternion-valued metric space, as a generalization of complex-valued metric space, and proved a common fixed point theorem in the context of such spaces.

In this paper, we announce that the quaternion-valued metric spaces, introduced by Ahmed et al. , are subspaces of cone metric spaces. Consequently, the fixed point results in such spaces can be concluded from the classical versions in cone metric spaces. Consequently, the fixed point results in such spaces can be concluded from the classical versions in cone metric spaces. On the other hand, several results have been reported on the equivalence of cone metric space and metric space; see, for example, . In particular, by the help of scalarization function, Du  proved that several fixed point results in the context of cone metric spacecan be concluded from the existing associated results in the setting of metric space. Furthermore, if the cone is normal, then there is a metric induced by Banach-valued metric. Hence, most of the announced fixed point results in the setting cone metric space can be deduced from related existing results in the literature in the context of the metric space.

1.1. Complex-Valued Metric Spaces

First we recall the concept of complex-valued metric space which is given by Azam et al. in .

Let C be the set of complex numbers and z1,z2C. Define a partial order on C as follows: (1)z1z2iffRe(z1)Re(z2),Im(z1)Im(z2). It follows that (2)z1z2 if one of the following conditions is satisfied: (3)(h1)Re(z1)=Re(z2);Im(z1)<Im(z2),(h2)Re(z1)<Re(z2);Im(z1)=Im(z2),(h3)Re(z1)<Re(z2);Im(z1)<Im(z2),(h4)Re(z1)=Re(z2);Im(z1)=Im(z2). In particular, we will write z1z2 if z1z2 and one of (h1), (h2), and (h3) is satisfied and we will write z1z2 if only (h3) is satisfied. Note that (4)0z1z2|z1|<|z2|, where |·| represents modulus or magnitude of z, and (5)z1z2,z2z3z1z3.

Definition 1 (see [<xref ref-type="bibr" rid="B9">1</xref>]).

Let X be a nonempty set. A function d:X×XC is called a complex-valued metric on X, if it satisfies the following conditions:

0d(x,y) for all x,yX and d(x,y)=0, if and only if x=y,

d(x,y)=d(y,x), for all x,yX,

d(x,y)d(x,z)+d(y,z), for all x,y,zX.

Here, the pair (X,d) is called a complex-valued metric space.

Let {xn} be a sequence in X and X. If for every cC, with 0c, there is n0N such that, for all n>n0, d(xn,x)c, then {xn} is said to be convergent, {xn} converges to x, and x is the limit point of {xn}. We denote this by limnxn=x, or xnx, as n. If for every cC with 0c there is n0N such that for all n>n0, d(xn,xn+m)c, then {xn} is called a Cauchy sequence in (X,d). If every Cauchy sequence is convergent in (X,d), then (X,d) is called a complete complex-valued metric space.

Lemma 2 (see [<xref ref-type="bibr" rid="B9">1</xref>, Lemma 2, Azam et al.]).

Let (X,d) be a complex-valued metric space and let {xn} be a sequence in. Then {xn} converges to x if and only if |d(xn,x)|0 as n.

Lemma 3 (see [<xref ref-type="bibr" rid="B9">1</xref>, Lemma 3, Azam et al.]).

Let (X,d) be a complex-valued metric space and let {xn} be a sequence in. Then {xn} is a Cauchy sequence if and only if |d(xn,xn+m)|0 as n.

1.2. Quaternion Metric Space

Now, we recollect the basic definitions and concept on quaternion-valued metric spaces.

The skew field of quaternion denoted by H means to write each element qH in the form q=x0+x1i+x2j+x3k; xnR, where 1, i, j, and k are the basis elements of H and n=1,2,3. For these elements we have the multiplication rules i2=j2=k2=-1, ij=-ji=k, kj=-jk=-i, and ki=-ik=j. The conjugate element q- is given by q-=x0-x1i-x2j-x3k.

The quaternion modulus has the form of |q|=x02+x12+x22+x32. A quaternion q may be viewed as a four-dimensional vector (x0,x1,x2,x3).

Define a partial order on H as follows.

q 1 q 2 if and only if Re(q1)Re(q2), Ims(q1)Ims(q2), q1,q2H, s=i,j,k where Imi=x1, Imj=x2, and Imk=x3. It follows that q1q2 if one of the following conditions holds:

Re(q1)=Re(q2); Ims1(q1)=Ims1(q2) where s1=j,k; Imi(q1)<Imi(q2),

Re(q1)=Re(q2); Ims2(q1)=Ims2(q2) where s2=i,k; Imj(q1)<Imj(q2),

Re(q1)=Re(q2); Ims3(q1)=Ims3(q2) where s3=i,j; Imk(q1)<Imk(q2),

Re(q1)=Re(q2); Ims1(q1)<Ims1(q2); Imi(q1)=Imi(q2),

Re(q1)=Re(q2); Ims2(q1)<Ims2(q2); Imj(q1)=Imj(q2),

Re(q1)=Re(q2); Ims3(q1)<Ims3(q2); Imk(q1)=Imk(q2),

Re(q1)=Re(q2); Ims(q1)<Ims(q2),

Re(q1)<Re(q2); Ims(q1)=Ims(q2),

Re(q1)<Re(q2); Ims1(q1)=Ims1(q2); Imi(q1)<Imi(q2),

Re(q1)<Re(q2); Ims2(q1)=Ims2(q2); Imj(q1)<Imj(q2),

Re(q1)<Re(q2); Ims3(q1)=Ims3(q2); Imk(q1)<Imk(q2),

Re(q1)<Re(q2); Ims1(q1)<Ims1(q2); Imi(q1)=Imi(q2),

Re(q1)<Re(q2); Ims2(q1)<Ims2(q2); Imi(q1)=Imi(q2),

Re(q1)<Re(q2); Ims3(q1)<Ims3(q2); Imi(q1)=Imi(q2),

Re(q1)<Re(q2); Ims(q1)<Ims(q2),

Re(q1)=Re(q2); Ims(q1)=Ims(q2).

Remark 4.

In particular, we write q1q2 if q1q2 and one from (I) to (XVI) is satisfied. Also, we will write q1<q2 if only (XV) is satisfied. It should be remarked that (6)q1q2|q1||q2|.

Ahmed et al.  introduced the definition of the quaternion-valued metric space as follows.

Definition 5.

Let X be a nonempty set. A function dH:X×XH is called a quaternion-valued metric on X, if it satisfies the following conditions:

0dH(x,y)0 for all x,yX and dH(x,y)=0, if and only if, x=y,

dH(x,y)=dH(y,x), for all x,yX,

dH(x,y)dH(x,z)+dH(y,z), for all x,y,zX.

Then, (X,dH) is called a quaternion-valued metric space.

Let {xn} be a sequence in X and X. If for every cC, with 0<c, there is n0N such that, for all n>n0, dH(xn,x)<c, then {xn} is said to be convergent, {xn} converges to x, and x is the limit point of {xn}. We denote this by limnxn=x, or xnx, as n. If for every cC with 0<c there is n0N such that, for all n>n0, dH(xn,xn+m)<c, then {xn} is called a Cauchy sequence in (X,dH). If every Cauchy sequence is convergent in (X,dH), then (X,dH) is called a complete quaternion-valued metric space.

Lemma 6 (see [<xref ref-type="bibr" rid="B10">2</xref>, Lemma 2.1, Ahmed et al.]).

Let (X,d) be a quaternion-valued metric space and let {xn} be a sequence in. Then {xn} converges to x if and only if |d(xn,x)|0 as n.

Lemma 7 (see [<xref ref-type="bibr" rid="B10">2</xref>, Lemma 2.2, Ahmed et al.]).

Let (X,dH) be a quaternion-valued metric space and let {xn} be a sequence in. Then {xn} is a Cauchy sequence if and only if |dH(xn,xn+m)|0 as n.

1.3. Cone Metric Space

Let E be a real Banach space. A subset P of E is called a cone, if and only if the following holds:

P is closed, nonempty, and P{0},

a,bR, a,b0, and x,yP imply that ax+byP,

xP and -xP imply that x=0.

Given a cone PE, we define a partial ordering with respect to P by xy, if and only if y-xP. We write x<y to indicate that xy but xy, while xy stands for y-xintP, where intP denotes the interior of P.

The cone P is called normal, if there exist a number K>0 such that 0xy implies xKy, for all x,yE. The least positive number satisfying this is called the normal constant . It is proved that the normal constant can not be less then 1 (see ). For more details on cone metric space, we refer, for example, to .

In this paper, E denotes a real Banach space, P denotes a cone in E with intP, and denotes partial ordering with respect to P.

Definition 8 (see [<xref ref-type="bibr" rid="B1">10</xref>]).

Let X be a nonempty set. A function d:X×XE is called a cone metric on X, if it satisfies the following conditions:

d(x,y)0 for all x,yX and d(x,y)=0, if and only if, x=y,

d(x,y)=d(y,x), for all x,yX,

d(x,y)d(x,z)+d(y,z), for all x,y,zX.

Then, (X,d) is called a cone metric space.

The following definitions and lemmas have been chosen from [10, 16].

Definition 9.

Let (X,d) be a cone metric space and let {xn}nN be a sequence in X and xX. If, for all cE with 0c, there is n0N such that for all n>n0, d(xn,x0)c, then {xn}nN is said to be convergent, {xn}nN converges to x, and x is the limit of {xn}nN.

Definition 10.

Let (X,d) be a cone metric space and let {xn}nN be a sequence in X. If for all cE with 0c, there is n0N such that, for all m,n>n0, d(xn,xm)c, then {xn}nN is called a Cauchy sequence in X.

Definition 11.

Let (X,d) be a cone metric space. If every Cauchy sequence is convergent in X, then X is called a complete cone metric space.

Definition 12.

Let (X,d) be a cone metric space. A self-map T on X is said to be continuous, if limnxn=x implies limnT(xn)=T(x) for all sequence {xn}nN in X.

Lemma 13.

Let (X,d) be a normal cone metric space and let P be a normal cone. Let {xn}nN be a sequence in X. Then, {xn}nN converges to x, if and only if (7)limnd(xn,x)=0.

Lemma 14.

Let (X,d) be a cone metric space and let {xn}nN be a sequence in X. If {xn}nN is convergent, then it is a Cauchy sequence.

Lemma 15.

Let (X,d) be a cone metric space and let P be a normal cone in E. Let {xn}nN be a sequence in X. Then, {xn}nN is a Cauchy sequence, if and only if limm,nd(xm,xn)=0.

2. Main Result

Let (X,dH) be a quaternion-valued metric space where H is the skew field of quaternion number q; that is, (8)H={x0+x1i+x2j+x3k:(x0,x1,x2,x3)R4}. Define (9)PH={x0+x1i+x2j+x3k:x00,kkkx10,x20,x30}. It is apparent that PHH. Assume 0H is the zero of H from now on. Note that (H,|·|) is a real Banach space.

Lemma 16.

P H is a normal cone in real Banach space (H,|·|).

Proof.

Precisely, PH is nonempty, closed and PH(0H). Also for all α,βR+, and p,qPH we have αp+βqPH and PH(-PH)=(0H). Notice that the normality of the cone PH follows from Remark 4.

Lemma 17.

Any quaternion-valued metric space (X,dH) is a cone metric space.

Proof.

For all p,qH define (10)pqiffq-pPH. defines a partial ordered on H and one can easily verify that (X,dH) is a cone metric space with respect to .

Lemma 18.

The partial ordered defined in Lemma 17 is equivalent to .

Proof.

Assume p=p0+p1i+p2j+p3k and q=q0+q1i+q2j+q3k. pq, if and only if q-pPH, if and only if q0-p00, q1-p10, q2-p20, and q3-p30. In other words, Re(p)Re(q), Ims(p)Ims(q), s=i,j,k where Imi(p)=p1, Imj(p)=p2 and Imk(p)=p3 and Imi(q)=q1, Imj(q)=q2 and Imk(q)=q3, if and only if pq.

Lemma 19.

A sequence {xn} in (X,dH) is convergent in the context of quaternion-valued metric space if and only if {xn} is convergent in the setting of of cone metric space.

Proof.

Let {xn} be sequence in X. {xn} converges to xX as the concept of quaternion-valued metric space if and only if |d(xn,x)|0 as n (see Lemma 6) if and only if {xn} converges to x as the concept of cone metric space by considering H as the Banach space endowed with the cone PH (see Lemma 13).

Let (X,dC) be a complex-valued metric space where C is the skew field of complex number z; that is, (11)C={x+yi:(x,y)R2}. Define (12)PC={x+yi:x0,y0}. It is apparent that PCC. Assume 0C is the zero of C from now on. Note that (C,|·|) is a real Banach space.

Lemma 20.

P C is a normal cone in real Banach space (C,|·|).

Proof.

Precisely, PC is nonempty, closed and PC(0C). Also for all α,βR+, and p,qPC we have αp+βqPC and PC(-PC)=(0C).

Lemma 21.

Any complex-valued metric space (X,dC) is a cone metric space.

Proof.

For all p,qC define (13)pqiffq-pPC. defines a partial ordered on C and one can easily verify that (X,dC) is a cone metric space with respect to .

Lemma 22.

The partial ordered defined in Lemma 21 is equivalent to .

We omitted the proof of Lemma 22 since it is the mimic of the proof Lemma 18.

Lemma 23.

A sequence {xn} in (X,dC) is convergent as the concept of complex-valued metric space if and only if {xn} is convergent as the concept of cone metric space.

We omit the proof of Lemma 23 above due to Lemma 19.

Definition 24.

Let (X,d) be a complete cone metric space. For all x,yX. A cone metric space (X,d) is said to be metrically convex if X has the property that, for each x,yX with xy, there exists zX, xyz such that (14)d(x,z)+d(z,y)=d(x,y).

The following lemma finds immediate applications which is straightforward from .

Lemma 25.

Let (X,d) be a metrically convex quaternion-valued metric space, and let K be a nonempty closed subset of X. If xK and yK, then there exists a point zK (where K stands for the boundary of K) such that (15)d(x,y)=d(x,z)+d(z,y).

Definition 26.

Let K be a nonempty subset of a cone metric space (X,d) and F,T:KX. The pair (F,T) is said to be weakly commuting if, for each x,yK such that x=Fy and TyK, we have (16)d(Tx,FTy)d(Ty,Fy) (see also [12, Hadžić and Gajić]).

Denote R by the collection of all continuous and increasing mappings such that φ:[0,+)[0,+) such that φ-1(0)={0}.

Lemma 27.

Let φ:[0,+)[0,+) be an increasing function. Then, (17)φ(tn)0impliestn0.

Proof.

Suppose that φ(tn)0 and tn0. Then there exists n0>0 and δ>0 such that 0<δ<tn, for all nn0. Since φ is increasing, we have (18)0<φ(δ)φ(tn) and this is a contradiction since φ(tn)0.

Definition 28.

Let K be a nonempty subset of a cone metric space (X,d) and let F,T:KX be two mappings. We say that F is generalized T-contractive if (19)φ(d(Fx,Fy))b[φ(d(Tx,Fx))+φ(d(Ty,Fy))]+cmin{φ(d(Tx,Fy)),kkkkkkkkkφ(d(Ty,Fx))}. For all x,yK, with xy, b,c0, 2b+c<1, and let φR.

Proposition 29.

Let (X,d) be a complete Banach-valued metric space, which is metrically onvex. Let K be a nonempty closed subset of X and φR, and let F,T:KX be such that F is generalized T-contractive. Suppose also we have

KTK and FKTK,

TxKFxK,

F and T are weakly commuting,

T is continuous on K.

Then, there exists a unique common fixed point z in K such that z=Tz=Fz.

Proof.

We construct the sequences {xn} and {yn} in the following way.

Let xK. Then there exists a point x0K such that x=Tx0 as KTK. From Tx0K and the implication TxKFxK, we conclude that Fx0KFKTK. Now, let x1K be such that (20)y1=Tx1=Fx0K. Let y2=Fx1 and assume that y2K, and then (21)y2KFKTK which implies that there exists a point x2K such that y2=Tx2. Suppose y2K, and then there exists a point pK (using Lemma 25), such that (22)d(Tx1,p)+d(p,y2)=d(Tx1,y2). Since pKTK, there exists a point x2K such that p=Tx2 and so (23)d(Tx1,Tx2)+d(Tx2,y2)=d(Tx1,y2). Let y3=Fx2. Thus, repeating the forgoing arguments, we obtain two sequences {xn} and {yn} such that

yn+1=Fxn,

ynKyn=Txn,

ynKTxnK, and (24)d(Txn-1,Txn)+d(Txn,yn)=d(Txn-1,yn).

Denote (25)P={Txi{Txn}:Txi=yi},Q={Txi{Txn}:Txiyi}. Obviously, the two consecutive terms of {Txn} cannot lie in Q. Let us denote rn=d(Txn;Txn+1). We have the following three cases.

Case 1. If Txn,Txn+1P.

Case 2. If TxnP and Txn+1Q.

Case 3. If TxnQ  Txn+1P and so Txn-1P.

Proving the above cases are similar to [2, Theorem 3.1]. Also we see that for all nN we get (26)φ(tn)(b+c1-b)nφ(t0). Letting n, we have φ(tn)0. Since φR, we have tn0. So that {Txn} is a Cauchy sequence and hence it converges to a point zK. Now there exists a subsequence {Txnk} of {Txn} which is contained in P. Without loss of generality, we may denote {Txnk}={Txn}. Since T is continuous, {TTxn} converges to Tz. We are going now to show that T and F have common fixed point (Tz=Fz). Using the weak commutativity of T and F, we obtain that (27)Txn=Fxn-1,Txn-1K, and then (28)d(TTxn,FTxn-1)d(Txn-1,Fxn-1)=d(Txn,Txn-1). This implies that (29)d(TTxn,FTxn-1)d(Txn,Txn-1). On letting n, we obtain (30)d(Tz,FTxn-1)0. It means that FTxn-1Tz.

Now, consider (31)φ(d(FTxn-1,Fz))b[φ(d(TTxn-1,FTxn-1))kkkk+φ(d(Tz,Fz))]+cmin{φ(d(TTxn-1,Fz)),kkkikkkiφ(d(Tz,FTxn-1))}. Taking limit on both sides of (31) yields (32)φ(d(Tz,Fz))bφ(d(Tz,Fz)), which is a contradiction, thus giving φ(||d(Tz,Fz)||)=0 which implies ||d(Tz,Fz)||=0, so that d(Tz,Fz)=0 and hence Tz=Fz.

To show that Tz=z, consider (33)φ(d(Txn,Tz))=φ(d(Fxn-1,Fz))b[φ(d(Txn-1,Fxn-1))kkkk+φ(d(Tz,Fz))]+cmin{φ(d(Txn-1,Fz)),kkkkkkkkkkkφ(d(Tz,Fxn-1))}. Taking limit on both sides of (33) yields (34)φ(d(Tz,z)cφ(d(Tz,z)), which is a contradiction, thereby giving φ(d(z,Tz))=0 which implies d(z,Tz)=0, so that d(z,Tz)=0 and hence z=Tz.

Thus, we have shown that z=Tz=Fz, so z is a common fixed point of F and T. To show that z is unique, let w be another fixed point of F and T, and then (35)φ(d(w,z))=φ(d(Tw,Tz))=φ(d(Fw,Fz))b[φ(d(Tz,Fz))+φ(d(Tw,Fw))]+cmin{φ(d(Tz,Fw)),kkkkkkkkkφ(d(Tw,Fz))}=cφ(d(w,z)), which is a contradiction, therefore giving φ(d(w,z))=0 which implies that d(w,z)=0, so that d(w,z)=0; thus, w=z.

Theorem 30 (see [<xref ref-type="bibr" rid="B10">2</xref>]).

Let (X,dH) be a complete quaternion-valued metric space which is metrically convex and K a nonempty closed subset of X and φR. Let F,T:KX be such that F is generalized T-contractive satisfying the following conditions:

KTK and FKTK,

TxKFxK,

F and T are weakly commuting,

T is continuous on K,

and then there exists a unique common fixed point z in K such that z=Tz=Fz.

Theorem 31.

Theorem 30 is a consequence of Proposition 29.

Proof.

Let (X,dH) be a complete quaternion-valued metric space and K a nonempty closed subset of. Then (X,dH) is a complete cone-valued metric space with cone PH={x0+x1i+x2j+x3k:x00,x10,x20,x30}. Further, we have that F is generalized T-contractive in cone metric space if and only if F is generalized T-contractive in quaternion metric space. The rest follows from Proposition 29.

3. Further Comment on Cone Metric Spaces 3.1. By Using a Scalarization Function

In 2010, Du  introduced the notion of TVS-valued metric space, also known as TVS-cone metric space (TVS-CMS), as a real generalization of Banach-valued metric space. Let Y be a locally convex Hausdorff t.v.s. with its zero vector θ, K a proper, closed, convex pointed cone in Y with K, eint(K), and K a partial ordering with respect to K.

Definition 32 (see [<xref ref-type="bibr" rid="B14">3</xref>]).

Let X be a nonempty set. Suppose that a vector-valued function p:X×XY satisfies the following:

θKp(x,y) for all x,yX and p(x,y)=θ if and only if x=y,

p(x,y)=p(y,x) for all x,yX,

p(x,y)Kp(x,z)+p(z,y), forall x,y,zX.

Then, the function p is called TVS-cone metric on X. Furthermore, the pair (X,p) is called a TVS-cone metric space (in short, TVS-CMS).

On his paper, Du  concluded that, for a TVS-valued metric space (X,p), one can define a function dp:X×X[0,) by dp=ξep forming a metric, where ξe:YR, defined by (36)ξe(y)=inf{rR:yre-K},yY, is a nonlinear scalarization function (see e.g., ). In this part, whenever we write , we mean K.

Proposition 33 (see [<xref ref-type="bibr" rid="B14">3</xref>, <xref ref-type="bibr" rid="B15">4</xref>]).

Let (X,p) be a TVS-CMS, xX, and {xn}nN a sequence in X. Set dp=ξep. Then the following statements hold:

{xn}nN converges to x in TVS-CMS (X,p) if and only if dp(xn,x)0 as n,

{xn}nN is a Cauchy sequence in TVS-CMS (X,p) if and only if {xn}nN is a Cauchy sequence in (X,dp),

(X,p) is a complete TVS-CMS if and only if (X,dp) is a complete metric space.

From Proposition 33, the following result was derived easily.

Proposition 34 (see [<xref ref-type="bibr" rid="B14">3</xref>]).

Let (X,p) is complete TVS-CMS and T:XX satisfies the contractive condition: (37)p(Tx,Ty)kp(x,y) for all x,yX and 0k<1. Then, T has a unique fixed point in X. Moreover, for each xX, the iterative sequence {Tnx}n=1 converges to fixed point.

Proposition 35 (see [<xref ref-type="bibr" rid="B14">3</xref>]).

The Banach contraction principle and Proposition 34 are equivalent.

Definition 36.

Let K be a nonempty subset of a metric space (X,d) and let F,T:KX be two mappings. We say that F is generalized T-contractive of type A if (38)φ(d(Fx,Fy))b[φ(d(Tx,Fx))+φ(d(Ty,Fy))]+cmin{φ(d(Tx,Fy)),φ(d(Ty,Fx))}. For all x,yK, with xy, b,c0, 2b+c<1, and let φR.

Proposition 37.

Let (X,d) be a complete metric space, let K be a nonempty closed subset of X and φR, and let F,T:KX be such that F is generalized T-contractive of type A. If the following are satisfied:

KTK and FKTK,

TxKFxK,

F and T are weakly commuting,

T is continuous on K,

then there exists a unique common fixed point z in K such that z=Tz=Fz.

We skip the proof of Proposition 37 since it can be derived by the mimic of Proposition 29. On the other hand, regarding Proposition 35, we can conclude that Proposition 37 implies Proposition 29.

3.2. By Using a Metric-Type Space Definition 38 (see, e.g., [<xref ref-type="bibr" rid="B17">5</xref>]).

Let X be a set. Let D:X×X[0,) be a function which satisfies

D(x,y)=0 if and only if x=y,

D(x,y)=D(y,x), for any x,yX,

D(x,y)K(D(x,z1)+D(z2,z3)++D(zn-1,zn)+D(zn,y)) for any x,y,z1,,znX,

for some constant K>0. The pair (X,D) is called a metric-type space.

Proposition 39.

Let (X,d) be a metric cone over the Banach space E with the cone P which is normal with the normal constant K. The mapping D:X×X[0,) defined by D(x,y)=d(x,y) is a function which satisfies

D(x,y)=0 if and only if x=y,

D(x,y)=D(y,x), for any x,yX,

D(x,y)K(D(x,z1)+D(z2,z3)++D(zn-1,zn)+D(zn,y)) for any x,y,z1,,znX.

Remark 40.

In Definition 38, (3), the term zi needs not to be distinct. Hence, metric type space turns into b-metric space when we deal with cone metric space (see, e.g., [21, 22]).

Remark 41.

Furthermore, by Lemma 20, PH is a normal cone. Hence, some resuls of  and Theorem 30, are equivalent to the corresponding results in the context of metric-type space (see also ).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

This research was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors thank the anonymous referees for their remarkable comments, suggestions, and ideas that helped to improve this paper.