Fixed point and common fixed point theorems in vector metric spaces

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
247
Fixed Point and Common Fixed Point Theorems in Vector Metric
Spaces
Renu Praveen Pathak* and Ramakant Bhardwaj**
*Department of Mathematics, Late G.N. Sapkal College of Engineering,Anjaneri Hills, Nasik (M.H.), India
** Department of Mathematics,Truba Institute of Engineering & Information Technology, Bhopal (M.P.), India
pathak_renu@yahoo.co.in, rkbhardwaj100@gmail.com
Abstract
In the present paper we prove fixed point and common fixed point theorems in two self mappings satisfy quasi
type contraction.
Keywords :Fixed point, Common Fixed point ,self mapping, vector Metric space , E-complete.
Mathematical Subject Classification: 47B60, 54H25.
Introduction and Preliminaries
Vector metric space, which is introduced in [6] by motivated this paper [7] , is generalisation of metric
space ,where the metric is Riesz space valued .Actually in both of them , the metric map is vector space valued.
One of the difference between our metric definition and Huang-Zhang’s metric definition is that there exist a
cone due to the natural existence of ordering on Riesz space .The other difference is that our definition
eliminates the requirement for the vector space to have a topological structure.
A Riesz space (or a vector lattice) is an ordered vector space and alattice. Let E be a Riesz space with the
positive E = x ∈ E ∶ x ≥ 0 . If a is a decreasing sequence in E such that inf a = a, we write a ↓ a
Definition 1. The Riesz space E is said to be Archimedean if a ↓ 0 holds for every a ∈ E .
Definition 2. A sequence (bn) is said to order convergent (or o-convergent) to b, if there is a sequence (an) in E
satisfying a ↓ 0 and |b − (b|( ≤ a for all n, and written b
*
+ b or
o-limb = b , where |a| = sup a, −a for any a ∈ E.
Definition3. A sequence (bn) is said to order Cauchy (or o-cauchy) if there exists a sequence (an) in E such that
a ↓ 0 and |b -b | ≤ a holds for n and p.
Definition4. The Riesz space E is said to be o-cauchy complete if every o-cauchy sequence is o-convergent.
VECTOR METRIC SPACES
Definition 4.Let X be anon empty set and E be a Riesz space .The function
d ∶ X × X + E is said to be avector metric (or E-metric) if it is satisfying the following properties :
(i) d1x, y2 = 0 if and only if x = y.
1ii2d1x, y2 ≤ d1x, z2 + d1y, z2
For all x, y, z ∈ X. Also the triple (X , d ,E) (briefly X with the default parameters omitted) is said to be vector
metric space. .
Main Results
Recently ,many authors have studied on common fixed point theorems for weakly compatible pairs (see [1], [3],
[4], [8], [9]). Let T and S be self maps of a set X. if y = Tx = Sx for some ,
x ∈ X, then y is said to be a point of coincidence and xis said to be coincidence point
of T and S. If T and S are weakly compatible ,that is they are commuting at their coincidence point on X, then
the point of coincidence y is the unique common fixed point of these maps [1].
Theorem 1 :Let X be a vector space with E is Archimedean. Suppose the mappings S, T ∶ X × X +
X satis7ies the following conditions
1i2for all x, y ∈ X, d1Tx, Ty2 ≤ ku1x, y2 (1)
where k ∈ 90,12is a constant and
u1x, y2 ∈ ;d1Sx, Sy2, d1Sx, Tx2, d1Sy, Ty2, d1Sx, Ty2, d1Sy, Tx2,
1
2
9d1Sx, Tx2 + d1Sy, Tx2=>
(ii) T1x2 ⊆ S1x2
(iii) S1x2 or T1x2is a E-Complete subspace of x.
Then T and S have a unique fixed point of coincidence in X . Moreover, if S and Tare weakly compatible ,then
they have a unique common fixed point in X.
Proof : Let x@, x ∈ X. De7ine the sequence 1x 2 by Sx = Tx = y for n ∈ N.
We first show that
d1y , y 2 ≤ k d1y - , y 2 (2)
For all n. we have that

248
d1y , y 2 = d1Tx , Tx 2 ≤ ku1x , x 2
Fir all n. Now we have to consider the following three cases:
If u1x , x 2 = d1y - , y 2 then clearly 122holds . If u1x , x 2 = d1y , y 2
Then according to Remark 1 d1y , y 2 = 0 and 122is immediate. Finally , suppose that
u1x , x 2 =
D
d1y - , y 2. Then,
d1y , y 2 ≤
k
2
d1y - , y 2 ≤
k
2
d1y - , y 2 +
1
2
d1y , y 2
Holds , and we prove (2).
We have
d1y , y 2 ≤ k d1y@, y 2
For all n and p,
dEy , y FG ≤ d1y , y 2 + d1y , y D2 + ⋯ + d1y F- , y F2
≤ 1k + k + k D
+ ⋯ + k F- 2 d1y@, y 2
dEy , y FG ≤
k
1 − k
d1y@, y 2
Holds .Now since E is Archimedean then (y@) is an E-cauchy sequence. Since the range of S contains the range
of T and the range of at least one is E-Complete , there exists a z ∈ S1X2
such that Sx
I.J
KL z. Hence there exists a sequence 1a 2in E such that a ↓ 0
and d1Sx , z2 ≤ a . On the other hand , we can 7ind w ∈ X such that Sw = z.
Let us show that Tw = z we have
d(Tw, z) ≤ d1Tw, Tx 2 + d1Tx , z2 ≤ ku1x , w2 + a
for all n . Since
u1x , w2 ∈ P
d1Sx , Sw2, d1Sx , Tx 2, d1Sw, Tw2, d1Sx , Tw2, d1Sw, Tx 2
,
1
2
9d1Sx , Tw2 + d1Sw, Tx 2=
Q
At least one of the following four cases holds for all n.
Case 1 :
d1Tw, z2 ≤ d1Sx , Sw2 + a ≤ a + a ≤ 2a
Case 2 :
d1Tw, z2 ≤ d1Sx , Tx 2 + a ≤ d1Sx , z2 + 2a ≤ 3a
Case 3:
d1Tw, z2 ≤ kd1Sw, Tw2 + a ≤ kd1Tw, z2 + a ,
that is d1Tw, z2 ≤
1
1 − k
a .
Case 4 :
d1Tw, z2 ≤ d1Sx , Tw2 + a ≤ d1Sx , z2 + d1Tw, z2 + 3a ≤ 4a
Case 5 :
d1Tw, z2 ≤ kd1Sw, Tx 2 + a ≤ k d1Tx , z2 + 2a ≤ 3a
Case 6 :
d1Tw, z2 ≤
D
9d1Sx , Tw2 + d1Sw, Tx 2= + a
≤
1
2
d1Sx , Tw2 +
3
2
a
≤
D
d1Sx , z2 +
D
d1Tw, z2 +
T
D
a
≤
1
2
d1Tw, z2 + 2a
That is , d1Tw, z2 ≤ 4a
Since the infimum of the sequence on the right side of last inequality are zero, then
d1Tw, z2 = 0, i. e. Tw = z . Therefore , z is a point of coincidence of T and S.
If z is another point of coincidence then there is w ∈ X with z = Tw = Sw .
Now from 112, it follows that
d (z, z ) =d1Tw, Tw 2 ≤ ku1w, w 2.
Where
u1w, w 2 ∈ U
d1Sw, Sw 2, d1Sw, Tw2, dESw ,Tw G, dESw ,Tw GdESw ,Tw G,
1
2
9d1Sw, Tw 2 + d1Sw , Tw2=
V
= 0, d1z, z 2 .

249
Hence d1z, z 2 = 0 that is z = z .
If S and T are weakly compatible then it is obvious that z is unique common fixed point of T and S by [1].
Theorem 2 : Let X be an vector metric space with E is Archimedean. Suppose the mappings S, T ∶ X +
X satis7ies the following conditions
1i2 for all x, y ∈ X,
d1Tx, Ty2 ≤ ku1x, y2 132
Where k ∈ 90,12is a constant and
u1x, y2 ∈ W
d1Sx, Sy2,
1
2
9d1Sx, Tx2 + d1Sy, Ty2=,
1
2
9d1Sx, Ty2 + d1Sy, Tx2=,
1
2
9d1Sx, Tx2 + d1Sx, Ty2=,
1
2
9d1Sy, Ty2 + d1Sy, Tx2=
X
1ii2 T1X2 ⊑ S1X2,
1iii2 S1X2 or T1X2 is E- Complete subspace of X.
Then T and S have a unique point of coincidence in X. Moreover, if S and T are weakly compatible, then they
have a unique common fixed point in X.
Proof : Let us define the sequence 1x 2and 1y 2 as in the proof of Theorem 1 , we 7irst
Show that
d1y , y 2 ≤ k d1y - , y 2 (4)
For all n. Notice that
d1y , y 2 = d1Tx , Tx 2 ≤ ku1x , x 2,
For all n.
As in Theorem 1, we have to consider three cases: u1x , x 2 = d1y - , y 2,
u1x , x 2 =
D
9d1y - , y 2 + d1y , y 2= and u1x , x 2 =
D
d1y - , y 2.
First and third have been shown in the proof of Theorem 1. Consider only the second case.
If u1x , x 2 =
1
2
9d1y - , y 2 + d1y , y 2=, then from 132we have
d1y , y 2 ≤
1
2
9d1y - , y 2 + d1y , y 2= ≤
k
2
d1y - , y 2 +
1
2
d1y , y 2 .
Hence. (4) Holds.
In the proof of this Theorem 1 we illustrate that (y ) is an E-Cauchy sequence. Then there exist z ∈ S1X2, w ∈
X and 1a 2 in E such that Sw = z, d1Sx , z2 ≤ a and a ↓ 0
Now, we have to show that Tw = z. We have
d1Tw, z2 ≤ d1Tw, , Tx 2 + d1Tx , z2 ≤ u1x , w2 + a
For all n. since
u1x , w2 ∈ W
d1Sx , Sw2,
1
2
9d1Sx , Tx 2 + d1Sw, Tw2=,
1
2
9d1Sx , Tw2 + d1Sw, Tx 2=,
1
2
9d1Sx , Tx 2 + d1Sx , Tw2=,
1
2
9d1Sw, Tx 2 + d1Sw, Tw2=
X
At least three of the five holds for all n. Consider only the cases of
u1x , w2 =
1
2
9d1Sx , Tx 2 + d1Sw, Tw2=,
1
2
9d1Sx , Tx 2 + d1Sx , Tw2=,
1
2
9d1Sw, Tx 2 + d1Sw, Tw2=
Because the other four cases have shown that the proof of Theorem 1 . It is satisfied that
d1Tw, z2 ≤
1
2
9d1Sx , Tx 2 + d1Sw, Tw2= +
1
2
9d1Sx , Tx 2 + d1Sx , Tw2=
+
1
2
9d1Sw, Tx 2 + d1Sw, Tw2= + a
≤
1
2
9d1Sx , z2 + d1Tx , z2 + d1z, Tw2= +
1
2
9d1Sx , z2 + d1Tx , z2 + d1Sx , z2 + d1z, Tw2=
+
1
2
9d1Tx , z2 + d1z, Tw2= + a
≤
1
2
d1Sx , z2 +
1
2
d1Tx , z2 +
1
2
d1z, Tw2 + a
≤
1
2
a +
1
2
d1z, Tw2 +
3
2
a

250
d1Tw, z2 ≤
1
2
d1z, Tw2 + 2a
That is d1z, Tw2 ≤ 4a . Since 4a ↓ 0 then Tw = z . Hence z is a point of coincidence
of T and S. The uniqueness of z as in the proof of Theorem 1. Also, If S and T are weakly
compatible, then it is obvious that z is unique common 7ixed point of Tand S by 91=.
Theorem 3 :Let X be an vector space with E is Archimedean. Suppose the mappings
S, T: X + X satis7ies the following conditions
(i) for all x , y ∈ X
d1Tx, Ty2 ≤ α d1Sx, Tx2 + β d1Sy, Ty2 + γ d1Sx, Ty2 + δ d1Sy, Tx2 + η d1Sx, Sy2 +
1
2
μ d1Sx, Tx2 + d1Sy, Ty2
(ii) T1X2 ⊆ S1X2,
(iii) S(X) or T(X) is E-complete subspace of X
Then T and S have a unique point of coincidence in X .Moreover, If S and T are weakly compatible, then they
have a unique common fixed point in X .
Proof :Let us define the sequence 1x 2and 1y 2 as in the proof of Theorem 1 , we have to show that
d1y , y 2 ≤ k d1y - , y 2 (5)
For some k ∈ 90,12 and for all n. Consider Sx = Tx = y for all n. Then
d1y , y 2 ≤ 1α + η2d1y - , y 2 + β d1y , y 2 + γd1y - , y 2
+
1
2
μ9d1y - , y 2 + d1y , y 2=
And
d1y , y 2 ≤ α d1y , y 2 + 1β + η2d1y - , y 2 + δd1y - , y 2
For all n. Hence,
d1y , y 2 ≤
α + β + γ + δ + 2η + μ
2 − α + β + γ + δ
d1y - , y 2
If we choose k =
c d e f Dg μ
D-c d e f
, then k ∈ 90,12and 152is hold.
In the proof of Theorem 1 we illustrate that 1y 2 is an E-Cauchy sequence .then there exist z ∈ s1X2, w ∈
X and 1a 2 in E such that Sw = z , d1Sx , z2 ≤ a and a ↓ 0 .
Let us show that Tw = z .we have
d1Tw, z2 ≤ d1Tw, Tx 2 + d1Tx , z2
≤ 1α + δ + μ2d1Tw, z2 + 1β + δ + η2d1Sx , z2 + 1β + γ + 12d1Tx , z2
≤ 1α + δ + μ2 d1Tw, z2 + 1β + δ + η2a + 1β + γ + 12 + a
≤ 1α + δ + μ2 d1Tw, z2 + 12β + γ + δ + η + 12a
That is d1Tw, z2 ≤
1Dd e f g 2
-1c f μ2
a for all n.
Then d1Tw, z2 = 0 , i. e. Tw = z . Hence ,
Z is a point of coincidence of T and S. The uniqueness of z is easily seen. Also ,If S and T are weakly
compatible , then it is obvious that z is unique common fixed point of T and S by [1].
Corrollary 1: Let X be an vector space with E is Archimedean. Suppose the mappings
S, T: X + X satis7ies the following conditions
(i) For all x , y ∈ X ,
d1Tx, Ty2 ≤ k d1Sx, Sy2 (6)
Where k< 1
(ii) T1X2 ⊑ S1X2,
(iii) S1X2 or T1X2 is E- Complete subspace of X.
Then T and S have a unique point of coincidence in X .Moreover, If S and T are weakly compatible, then they
have a unique common fixed point in X .
Now we give an illustrative example
Example :Let E = ℝD
with coordinatwise ordering 1since ℝD
is not Archmedean with
lexicogra7ical ordering ,then we can not use this ordering ), X= ℝD
, d1x, y2 = 1|x − y |, α|x − y |2, α > 0, Tx =
2xD
+ 1 and Sx = 4xD
. Then , for all x, y ∈ X we have
d1Tx, Ty2 =
1
2
d1Sx, Sy2 ≤ k d1Sx, Sy2
For k ∈ l
D
, 1m,
T1X2 = 91, ∞2 ⊆ 90, ∞2 = S1X2

251
And T(X) is E-complete subspace of X. Therefore all conditions of corollary 1 are satisfied. Thus T and S have a
unique point of coincidence in X .
References
[1] M. Abbas, G. Jungck, “Common fixed point results for no commuting mappings without continuity in cone
metric spaces”, J. Math. Anal.Appl, 341(2008), 416-420.
[2] C.D Apliprantis, K. C. Border, “Infinite Dimensional Analysis”, Springer-Verlag, Berlin, 1999.
[3] I. Altun,D. Turkoglu,“Some fixed point thermos for weakly compactible mappings satisfying an implicit
relation”, Taiwanese J. Math,13(4)(2009),1291-1304.
[4] I Altun,D.Turkoglu, B.E> Rhoades, “Fixed points of weakly compactible maps satisfying a general
contractive condition of integral type”, Fixed Point Theory Appl.,2007,Art.ID 17301,9 pp.
[5] IshakAltun and CiineytCevi’k, “some common fixed point theorem in vector metric space”,Filomat 25:1
(2011) 105,113 DOI: 10.2298/FIL1101105A.
[6] C. Cevik, I. Altun, “Vector metric spaces and some properties”, Topo.Met.Nonlin. Anal, 34(2) (2009) .375-
382.
[7] L.G .Huang, X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings”, J.
Math.Anal.Appl. ,332(2007), 1468-1476.
[8] G. Jungck, S. Radenovic, V. Rakocevic, “Common fixed point Theorems on weakly compatible pairs on
cone metric spaces”, Fixed Point Theory Appl. ,2009, Art. ID 643840, 13 pp.
[9] Z. Kadelburg, S. Radenovic, V. Rakocevic, “Topological vector space-valued cone metric spaces and fixed
point theorems”, Fixed Point Theory Appl. , 2010, Art. ID 170253, 17 pp.

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