PARTIAL CONE METRIC SPACE AND SOME FIXED POINT THEOREMS
D. DEY1, M. SAHA2 §
(Dedicated to the 71st birth anniversery of Professor A.P.Baisnab, F.N.A.Sc, India)
Abstract. In the present paper, we have proved some convergence properties of a sequence of elements in a partial cone metric space and thereby we have established some fixed point theorems on it.
Keywords: Partial metric space, partial cone metric space, fixed point. AMS Subject Classification: Primary 54A05; Secondary 40A05, 47H10.
1. Introduction
Partial metric space was originally developed by Matthews ([7], [8]) to provide mecha-nism generalizing metric space theories. He introduced the notion of partial metric space as a part of the study of denotational semantics of dataflow networks in computer science. In particular, he established the precise relationship between partial metric spaces and the so called weightable quasi-metric space (see [10]). In partial metric spaces, the distance of a point in the self may not be zero. Introducing partial metric space, Matthews proved Banach fixed point theorem [1] in this space.
Definition 1.1. (Partial metric space)A partial metric on a non-empty setXis a function p:X×X →R+ such that for all x, y, z∈X:
(p1): 0≤p(x, x)≤p(x, y),
(p2): x=y⇔p(x, x) =p(x, y) =p(y, y),
(p3): p(x, y) =p(y, x),
(p4): p(x, y)≤p(x, z) +p(z, y)−p(z, z).
A partial metric space is a pair (X, p) such thatX is a non-empty set andpis a partial metric on X. It is clear that if p(x, y) = 0, then from (p1) and (p2) we obtain x = y.
But if x=y, p(x, y) may not be 0. A basic example of a partial metric space is the pair (R+, p), where p(x, y) = max{x, y} for all x, y ∈ R+. Various application on this space has been extensively investigated by many authors (see [5], [13] for details).
In 1980, Rzepecki [11] introduced a generalized metric dE on a set X in a way that
dE :X×X→P, replacing the set of real numbers with a Banach space E in the metric
function whereP is a normal cone in E with a partial order≤. Seven years later, Lin [6]
1 Koshigram Union Institution, Koshigram-713150, Burdwan, West Bengal, India , e-mail: [email protected]
2 Department of Mathematics,The University of Burdwan, Burdwan-713104, West Bengal, India , e-mail: [email protected]
§ Manuscript received September 17, 2012.
TWMS Journal of Applied and Engineering Mathematics Vol.3 No.1 c⃝I¸sık University, Department of Mathematics 2013; all rights reserved.
considered the notion of cone metric spaces by replacing real numbers with a cone P in the metric function in which it is called aK-metric. On the otherhand, L.-G. Huang and X. Zhang [3] replaced real numbers with an ordering Banach space to define cone metric space, which is the same as either the definition of Rzepecki or of Lin. In the same paper, they proved some important fixed point theorems including Banach’s fixed point theorem in cone metric space.
LetE always be a real Banach space andP a subset ofE. P is called a cone if (C1): P is closed, non-empty and P ̸={0},
(C2): ax+by∈P for all x, y∈P and non-negative real numbersa, b,
(C3): P∩(−P) ={0}.
For a given cone P ⊆ E, we can define a partial ordering ≤ on E with respect to P by x≤y if and only if y−x ∈P. We write x < y to indicate that x ≤y butx ̸=y, while x≪ y will stand for y−x ∈ intP, intP denotes the interior of P. The cone P is called normal if there is a constant numberK >0 such that for all x, y∈E
θ≤x≤y implies ∥x∥ ≤K∥y∥.
The least positive number satisfying above is called the normal constant ofP. The coneP is called regular if every increasing sequence which is bounded from above is convergent. That is, if{xn}is sequence such thatx1 ≤x2≤...xn≤...≤y for somey∈E, then there
isx∈Esuch that∥xn−x∥ →0 asn→ ∞.Equivalently the coneP is regular if and only
if every decreasing sequence which is bounded from below is convergent. It is well known that a regular cone is a normal cone.
In the following we always suppose E is a Banach space, P a cone in E with intP ̸= Φ and≤ is partial ordering with respect toP.
Definition 1.2. (Cone metric space)Let X be a non-empty set. Suppose X the mapping d:X×X→E satisfies such that :
(CM1): θ≤d(x, y) for allx, y∈X and d(x, y) =θ if and only if x=y,
(CM2): d(x, y) =d(y, x) for allx, y∈X,
(CM3): d(x, y)≤d(x, z) +d(z, y) for all x, y, z∈X.
Then dis called a cone metric on X, and (X, d) is called a cone metric space.
Motivated by this beautiful generalization of metric space, recently A.S¨onmez [12] intro-duced the notion of partial cone metric space and its topological characterization. He also developed some important fixed point theorems in this generalized setting. We now state the following definition of partial cone metric space due to A.S¨onmez [12].
Definition 1.3. (Partial cone metric space)A partial cone metric on a non-empty set X is a function p:X×X →R+ such that for all x, y, z∈X:
(PCM1): θ≤p(x, x)≤p(x, y),
(PCM2): x=y⇔p(x, x) =p(x, y) =p(y, y),
(PCM3): p(x, y) =p(y, x),
(PCM4): p(x, y)≤p(x, z) +p(z, y)−p(z, z)
A partial cone metric space is a pair (X, p) such that X is a non-empty set andp is a partial cone metric onX. It is clear that, ifp(x, y) =θ, then from (PCM1) and (PCM2),
x=y. But if x=y,p(x, y) may not be equal to θ.
A cone metric space is a partial cone metric space, but there exists partial cone metric spaces which are not cone metric space. The following examples supports our contention.
Example 1.1. (A.S¨onmez [12]) Let E =R2,
and X = R+ and p : X×X → E defined by p(x, y) = (max{x, y}, αmax{x, y}) where α≥0is a constant. Then (X, p) is a partial cone metric space which is not a cone metric space.
Example 1.2. (A.S¨onmez [12]) Let E =l1,
P ={{xn} ∈l1 :xn≥0}
Also let X = {(xn) : (xn) ∈ (R+)w,
∑
xn < ∞} where (R+)w be the set of all infinite
sequences over R+, and p:X×X →E defined by
p(x, y) = (x1∨y1, x2∨y2, ..., xn∨yn, ...)
where the symbol∨denotes the maximum, i.e. x∨y= max{x, y}. Then(X, p)is a partial cone metric space which is not a cone metric space.
In the following sequel, (X, p) will denote a partial cone metric space. We now invite the following definitions and theorems due to A.S¨onmez [12]
Theorem 1.1. (Theorem 1, [12]) Any partial cone metric space (X, p) is a topological space.
Theorem 1.2. (Theorem 2, [12]) Let (X, p) be a partial cone metric space and P be a normal cone with normal constantK, then (X, p) is T0.
Definition 1.4. (Definition 3, [12]) Let (X, p)be a partial cone metric space. Let (xn) be
a sequence in X and x ∈X. If for every c ∈intP, there is N such that for all n > N, p(xn, x)≪c+p(x, x), then(xn) is said to be convergent and (xn) converges to x, and x
is the limit of(xn). We denote this by lim
n→∞xn=x or, xn→x(n→ ∞)
Theorem 1.3. (Theorem 3, [12])Let(X, p)be a partial cone metric space,P be a normal cone with normal constantK. Let (xn) be a sequence in X. Then (xn) converges to x if
and only if p(xn, x)→p(x, x) (n→ ∞)
S¨onmez [12] also noted that if (X, p) is a partial cone metric space,P be a normal cone with normal constantK and p(xn, x) →p(x, x) (n→ ∞), then p(xn, xn)→p(x, x) (n→
∞).
Lemma 1.1. (Lemma 1, [12]) Let (xn) be a sequence in partial cone metric space (X, p).
If a pointx is the limit of (xn) andp(y, y) =p(y, x) theny is the limit point of (xn).
Definition 1.5. (Definition 4, [12]) Let (X, p) be a partial cone metric space. (xn) be a
sequence in X. (xn) is Cauchy sequence if there is a∈P such that for every ϵ >0 there
isN such that for all n, m > N
∥p(xn, xm)−a∥< ϵ.
Definition 1.6. A partial cone metric space(X, p) is said to be complete if every Cauchy sequence in(X, p) is convergent in (X, p).
Theorem 1.4. (Theorem 4, [12]) Let (X, p) be a partial cone metric space. If (xn) is a
Cauchy sequence in (X, p), then it is a Cauchy sequence in the cone metric space(X, d).
2. Some properties of partial cone metric space
We first prove an analogue of important lemma (Lemma 5, L-G Huang et al.[3]) in partial cone metric space.
Proposition 2.1. Let (X, p) be a partial cone metric space, P be a normal cone with normal constant K. Let (xn) and (yn) be two sequences in X such that xn → x, yn →
x (n→ ∞). Thenp(xn, yn)→p(x, y) (n→ ∞).
Proof. For every real ϵ >0, choosec∈intP with ∥c∥< 4Kϵ+2. As xn→x,yn→y, there
isN,for all n > N,p(xn, x)≪c+p(x, x) and p(yn, y)≪c+p(y, y).
Then we have for alln > N,
p(xn, yn) ≤ p(xn, x) +p(x, y) +p(yn, y)−p(x, x)−p(y, y)
≤ p(x, y) + 2c and
p(x, y) ≤ p(xn, x) +p(xn, yn) +p(yn, y)−p(xn, xn)−p(yn, yn)
≤ p(xn, yn) + 2c
Hence for alln > N,θ≤p(x, y) + 2c−p(xn, yn)≤4c
and so for n > N,
∥p(xn, yn)−p(x, y)∥ ≤ ∥p(x, y) + 2c−p(xn, yn)∥+∥2c∥
≤ (4K+ 2)∥c∥< ϵ
Thereforep(xn, yn)→p(x, y) (n→ ∞).
Definition 2.1. Let (X, p) be a partial cone metric space. If for any sequence (xn) in X,
there is a subsequence (xni) of (xn) such that (xni) is convergent inX. Then X is called
a sequentially compact partial cone metric space.
Proposition 2.2. Let (X, p) be a partial cone metric space, (xn) be a sequence in X. If
(xn) converges to x, then (xn) is a Cauchy sequence.
Proof. For any ϵ > 0, choose c ∈ intP with K∥c∥ < ϵ, there is N such that for all n, m > N,p(xn, x)≪ 2c+p(x, x) and p(xm, x)≪ c2 +p(x, x). Then forn, m > N,
p(xn, xm) ≤ p(xn, x) +p(x, xm)−p(x, x)
≤ c+p(x, x)
So∥p(xn, xm)−p(x, x)∥ ≤K∥c∥< ϵ. Therefore (xn) is a Cauchy sequence.
3. Fixed point theorems in partial cone metric space
Banach’s fixed point theorem [1] in partial cone metric space has been proved by A.S¨onmez [12]
Theorem 3.1. (Banach’s fixed point theorem ) Let (X, p) be a complete partial cone metric space, P be a normal cone with constantK. Suppose that the mappingT :X →X satisfies the contractive condition
p(T x, T y)≤αp(x, y) for allx, y∈X
whereα∈(0,1), is a constant. Then T has a unique fixed point inX, and for anyx∈X, the iterative sequence(Tnx) converges to the fixed point.
Theorem 3.2. (Kannan’s fixed point theorem ) Let (X, p) be a complete partial cone metric space, P be a normal cone with constantK. Supose that the mapping T :X→ X satisfies the contractive condition
p(T x, T y)≤b[p(x, T x) +p(y, T y)] for all x, y∈X
whereb∈(0,12), is a constant. Then T has a unique fixed point inX, and for anyx∈X, the iterative sequence(Tnx) converges to the fixed point.
Following theorem is an important result of L.-G. Huang et al. (Corollary 1, [3]) proved here in partial cone metric space.
Theorem 3.3. Let(X, p)be a complete partial cone metric space, P be a normal cone with constantK. For c∈intP andx0∈X, setB(x0, c) ={x∈X :p(x0, x)≤c}. Suppose the
mapping T :X→X satisfies the contractive condition
p(T x, T y)≤αp(x, y) for allx, y∈X
whereα∈[0,1), is a constant andp(T x0, x0)≤(1−α)c.ThenT has a unique fixed point
in B(x0, c).
Proof. As a consequence of Theorem 1 of A.S¨onmez [12], we need only to prove that B(x0, c) is complete andT x∈B(x0, c) for allx∈B(x0, c).
Suppose (xn) is a Cauchy sequence in B(x0, c). Then (xn) is also a Cauchy sequence in
X. By the completeness of X, there isx∈X such thatxn→x(n→ ∞). Now
p(x0, x)≤p(xn, x0) +p(xn, x)−p(xn, xn) (1)
As
lim
n→∞p(xn, x)−nlim→∞p(xn, xn)
= lim
n→∞[p(xn, x)−p(xn, xn)]
= lim
n→∞∥p(xn, x)−p(xn, xn)∥
= 0
So [p(xn, x)−p(xn, xn)]→θ(n→ ∞). Then by taking n→ ∞ in (1) we getp(x0, x)≤c
which implies thatx∈B(x0, c). So B(x0, c) is closed and thus B(x0, c) is complete. Now
forx∈B(x0, c),
p(x0, T x) ≤ p(x0, T x0) +p(T x0, T x)−p(T x0, T x0)
≤ p(x0, T x0) +αp(x0, x)
≤ (1−α)c+αc = c
showing thatT x∈B(x0, c).
Next result is also a consequence of Theorem 1 of A.S¨onmez [12].
Theorem 3.4. Let (X, p) be a complete partial cone metric space, P be a normal cone with constantK. Suppose a mapping T :X→X satisfies for some positive integer n,
p(Tnx, Tny)≤αp(x, y) for all x, y∈X (2)
Proof. LetS =Tn, then from (2)
p(Sx, Sy)≤αp(x, y) for allx, y∈X
So by Theorem 3.1,Si.e. Tnhas a unique fixed pointx
0. ButTn(T x0) =T(Tnx0) =T x0.
SoT x0 is also a fixed point ofTn. HenceT x0 =x0 i.e. x0 is a fixed point of T. Since the
fixed point ofT is also fixed point ofTn, the fixed point ofT is unique.
Theorem 3.5. Let (X, p) be a sequentially compact partial cone metric space, P be a regular cone. Suppose the mapping T :X →X satisfies
p(T x, T y)< p(x, y) for all x, y∈X, x̸=y
ThenT has a unique fixed point in X.
Proof. Choose x0 ∈X. Set x1 =T x0, x2 =T x1 =T2x0, . . . , xn+1 =T xn=Tn+1x0, .
. . . If for somen,xn+1 =xn, thenxn is a fixed point of T, the proof is complete. So we
assume that for alln,xn+1 ̸=xn. Setdn=p(xn, xn+1), then
dn+1=p(xn+1, xn+2) =p(T xn, T xn+1)< p(xn, xn+1) =dn.
Therefore (dn) is a decreasing sequence bounded below by 0. Since P is regular, there is
d∗ ∈ E such that dn → d∗ (n → ∞). From the sequence compactness of X, there are
subsequence (xni) of (xn) andx∗∈X such thatxni →x∗ (i→ ∞). We have
p(T xni, T x∗)≤p(xni, x∗), i= 1,2, ..
So
∥p(T xni, T x∗)∥ ≤ ∥p(xni, x∗)∥ →0 (n→ ∞)
whereK is the normal constant of E. Hence T xni → T x∗ (i→ ∞). Similarly, T 2x
ni →
T2x∗ (i→ ∞). By using Proposition 2.1, we havep(T xni, xni)→p(T x∗, x∗) (i→ ∞) and
p(T2xni, T xni) → p(T
2x∗, T x∗) (i → ∞). It is obvious that p(T x
ni, xni) = dni → d∗ =
p(T x∗, x∗) (i→ ∞). Now we shall prove that T x∗ =x∗. If T x∗ ̸=x∗, thend∗ ̸= 0. We have
d∗ =p(T x∗, x∗)> p(T2x∗, T x∗) = lim
i→∞p(T
2x
ni, T xni) = limdni+1=d∗
which is a contradiction. SoT x∗ =x∗. That is x∗ is a fixed point of T . The uniqueness
of fixed point is obvious.
We now prove the fixed point theorem due to Chatterjea [2] in in partial cone metric space.
Theorem 3.6. (Chatterjea fixed point theorem ) Let (X, p) be a complete partial cone metric space, P be a normal cone with constantK. Supose that the mapping T :X→ X satisfies the contractive condition
p(T x, T y)≤b[p(T x, y) +p(T y, x)] for all x, y∈X
whereb∈(0,12), is a constant. Then T has a unique fixed point inX, and for anyx∈X, the iterative sequence(Tnx) converges to the fixed point.
Proof. Choosex0∈X. Define the sequence (xn) asx1=T x0, x2=T x1=T2x0, ..., xn+1 =
T xn=Tn+1x0,... We have
p(xn+1, xn) = p(T xn, T xn−1)
≤ b[p(T xn, xn−1) +p(T xn−1, xn)]
So
p(xn+1, xn) ≤ b[p(xn+1, xn) +p(xn, xn−1)−p(xn, xn)] +bp(xn, xn)
≤ b[p(xn+1, xn) +p(xn, xn−1)]
which imlies that
p(xn+1, xn)≤
b
1−bp(xn, xn−1) =hp(xn, xn−1) whereh= 1−bb <1.
Form > n,
p(xm, xn) ≤ p(xm, xm−1) +p(xm−1, xm−2) +...+p(xn+1, xn)− m−∑n−1
r=1
p(xm−r, xm−r)
≤ p(xm, xm−1) +p(xm−1, xm−2) +...+p(xn+1, xn)
≤ (hm−1+hm−2+...+hn)p(x1, x0)
≤ hn
1−hp(x1, x0) and
∥p(xm, xn)∥ ≤
hn
1−hK∥p(x1, x0)∥
This implies that p(xm, xn) → 0 (n, m → ∞). Hence (xn) is a Cauchy sequence. By
completeness ofX, there is u∈X such that xn→u (n→ ∞). Therefore
lim
n→∞p(xn, u) =p(u, u) = limn→∞p(xn, xn) = 0.
Since
p(T u, u) ≤ p(T xn, T u) +p(T xn, u)−p(T xn, T xn)
≤ p(T xn, T u) +p(T xn, u)
≤ b[p(T xn, u) +p(T u, xn)] +p(xn+1, u)
≤ b[p(xn+1, u) +p(T u, u) +p(u, xn)−p(u, u)] +p(xn+1, u)
≤ b[p(xn+1, u) +p(T u, u) +p(u, xn)] +p(xn+1, u)
gives
p(T u, u)≤ 1
1−b[b(p(xn+1, u) +p(u, xn)) +p(xn+1, u)] and
∥p(T u, u)∥ ≤b. 1
1−b[b(∥p(xn+1, u)∥+∥p(u, xn)∥) +∥p(xn+1, u)∥]→0. So∥p(T u, u)∥= 0, which implies thatT u=u and so u is a fixed point ofT.
Let v be another fixed point of T, then p(u, v) = p(T u, T v) ≤ b[p(T u, v) +p(T v, u)] = 2bp(u, v)⇒(1−2b)p(u, v)≤0. Since b∈[0,12),p(u, v) = 0 i.e. u=v.So the fixed point
is unique.
Following is the Reich’s [9] type contraction mapping considered here to prove an an-other fixed point theorem in partial cone metric space.
Theorem 3.7. Let (X, p) be a complete partial cone metric space, P be a normal cone with constantK. Supose that the mapping T :X→X satisfies the contractive condition
where0≤α+β+γ <1and α, β, γ are non-negative. ThenT has a unique fixed point in X, and for any x∈X, the iterative sequence (Tnx) converges to the fixed point.
Proof. Choosex0∈X. Define the sequence (xn) asx1=T x0, x2=T x1=T2x0, ..., xn+1 =
T xn=Tn+1x0,... We have
p(xn+1, xn) = p(T xn, T xn−1)
≤ αp(T xn, xn) +βp(T xn−1, xn−1) +γp(xn, xn−1)
≤ αp(xn+1, xn) +βp(xn, xn−1) +γp(xn, xn−1)
which implies that
p(xn+1, xn)≤
β+γ
1−αp(xn, xn−1) =hp(xn, xn−1)
where h = β1−+αγ < 1. Proceeding as previous theorem, we see that (xn) is a Cauchy
sequence. By completeness ofX, there is u∈X such that xn→u (n→ ∞). Therefore
lim
n→∞p(xn, u) =p(u, u) = limn→∞p(xn, xn) = 0.
Now
p(T u, u) ≤ p(T xn, T u) +p(T xn, u)−p(T xn, T xn)
≤ p(T xn, T u) +p(T xn, u)
≤ αp(T xn, xn) +βp(T u, u) +γp(xn, u) +p(xn+1, u)
≤ αp(xn+1, xn) +βp(T u, u) +γp(xn, u) +p(xn+1, u)
gives
p(T u, u)≤ 1
1−β(αp(xn+1, xn) +γp(xn, u) +p(xn+1, u)) and
∥p(T u, u)∥ ≤K 1
1−β(α∥p(xn+1, xn)∥+γ∥p(xn, u)∥+∥p(xn+1, u)∥)→0 So∥p(T u, u)∥= 0, which implies thatT u=u and so u is a fixed point ofT.
Let v be another fixed point of T, then p(u, v) = p(T u, T v) ≤ αp(T u, u) +βp(T v, v) + γp(u, v) =γp(u, v) gives u=v. So the fixed point is unique.
4. Conclusion
Some additional properties of partial cone metric space have been established in this paper. We have generalized some more fixed point theorems due to Chatterjea [2] and Reich [9] in partial cone metric space apart from A.S¨onmez’s [12] work. However, these results have vast potential in solving various nonlinear problems in functional analysis, differential and integral equation and of course, in computer science and engineering.
References
[1] Banach, S., (1922), Sur les operations dans les ensembles abstraits et leur application aux quations intgrales, Fund. Math., 3 ,133-181 (French).
[2] Chatterjee, S. K., (1972), Fixed point theorems, Rend. Acad. Bulgare Sc., 25, 727-730.
[3] Huang, L.-G. and Zhang, X., (2007), Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332(2), 1468-1476.
[4] Kannan, R., (1968), Some results on fixed points, Bull. Calcutta Math. Soc., 60, 71-76.
[5] K¨unzi, H.P.A., (Dordrecht, 2001), Nonsymmetric distances and their associated topologies: About the origins of basic ideas in the area of asymmetric topology, Handbook of the History of General Topology (eds. C.E. Aull and R. Lowen), Kluwer Acad. Publ., 3, 853-968.
[6] Lin, S.D., (1987), A common fixed point theorem in abstract spaces, Indian Journal of Pure and Applied Mathematics, 18(8), 685-690.
[7] Matthews, S., (1994), Partial Metric Topology, Proceedings of the 8th Summer Conference on Topol-ogy and its Applications, Annals of The New york Academy of Sciences, 728, 183-197.
[8] Bukatin, M., Kopperman, R., Matthews, S. and Pajoohesh, H., (october, 2009), Partial Metric Spaces, The Mathematical Association of America, 116, 708-718.
[9] Reich, S., (1971), Kannan’s fixed point thorem, Boll. Un. Math. Ital., 4, 1-11.
[10] Romaguera, S. and Schellekens, M., (2003), Weightable quasi-metric semigroups and semilattices, Proc. MFCSIT2000, Electronic Notes in Theoretical Computer Science, 40, 12 pages.
[11] Rzepecki, B., (1980), On fixed point theorems of Maia type, Publications de lInstitut Mathematique, 28(42), 179-186.
[12] S¨onmez, A., (14 Jan, 2011), Fixed point theorems in partial cone metric spaces, arXiv:1101.2741v1 [math.GN].
[13] Valero, O., (2005), On Banach fixed point theorems for partial metric spaces, Appl. Gen. Top, 6(2), 229-240.
Debashis Deyreceived his M.Sc and Ph.D degree from The University of Burdwan, Burdwan-713104, West Bengal, India. His major research interests include fixed point theory in different topological spaces, fuzzy metric space and random fixed point theory. Dr. Dey has published several research papers on fixed point theory in various international journals of repute.
