Fixed Point in Topological Vector Space-Valued Cone Metric Spaces

Volume 2010, Article ID 604084,9pages doi:10.1155/2010/604084

Research Article

Fixed Point in Topological Vector Space-Valued Cone Metric Spaces

Akbar Azam,

Ismat Beg,

and Muhammad Arshad

1Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan

2Department of Mathematics, Centre for Advanced Studies in Mathematics, Lahore University of Management Sciences, Lahore, Pakistan

3Department of Mathematics, International Islamic University, Islamabad, Pakistan

Correspondence should be addressed to Ismat Beg,[email protected] Received 16 December 2009; Accepted 2 June 2010

Academic Editor: Jerzy Jezierski

Copyrightq2010 Akbar Azam 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.

We obtain common fixed points of a pair of mappings satisfying a generalized contractive type condition in TVS-valued cone metric spaces. Our results generalize some well-known recent results in the literature.

1. Introduction and Preliminaries

Many authors 1–16 studied fixed points results of mappings satisfying contractive type condition in Banach space-valued cone metric spaces. In a recent paper 17 the authors obtained common fixed points of a pair of mapping satisfying generalized contractive type conditions without the assumption of normality in a class of topological vector space-valued cone metric spaces which is bigger than that of studied in1–16. In this paper we continue to study fixed point results in topological vector space valued cone metric spaces.

LetE, τbe always a topological vector spaceTVSandP a subset ofE. Then,P is called a cone whenever

iPis closed, nonempty, andP /{0},

iiaxby∈Pfor allx, y∈Pand nonnegative real numbersa, b, iiiP∩−P {0}.

For a given coneP ⊆E, we can define a partial ordering≤with respect toPbyx≤y if and only ify−x ∈ P.x < ywill stand forx ≤ y andx /y, whilex ywill stand for y−x∈intP, where intPdenotes the interior ofP.

Definition 1.1. LetXbe a nonempty set. Suppose the mappingd:X×X → Esatisfies d10≤dx, yfor allx, y∈Xanddx, y 0 if and only ifxy,

d2dx, y dy, xfor allx, y∈X,

d3dx, y≤dx, z dz, yfor allx, y, z∈X.

Thendis called a topological vector space-valued cone metric onX, andX, dis called a topological vector space-valued cone metric space.

If Eis a real Banach space then X, d is called Banach space-valuedcone metric space9.

Definition 1.2. LetX, dbe a TVS-valued cone metric space,x∈Xand{xn}_n≥1a sequence in X. Then

i{xn}_n≥1 converges to xwhenever for every c ∈ Ewith 0 c there is a natural numberNsuch thatdxn, xcfor alln≥ N. We denote this by limn→ ∞xn x orxn → x.

ii{xn}_n≥1is a Cauchy sequence whenever for everyc∈Ewith 0cthere is a natural numberNsuch thatdxn, xmcfor alln, m≥N.

iii X, dis a complete cone metric space if every Cauchy sequence is convergent.

Lemma 1.3. LetX, dbe a TVS-valued cone metric space,P be a cone. Let{xn}be a sequence in X,and{an}be a sequence inPconverging to0. Ifdxn, xm≤anfor everyn∈Nwithm > n, then {xn}is a Cauchy sequence.

Proof. Fix0 ctake a symmetric neighborhoodV of 0 such thatcV ⊆intP. Also, choose a natural numbern0 such thatan ∈ V, for all n ≥ n0. Thendxn, xm ≤ an cfor every m, n≥n₀. Therefore,{xn}_n≥1is a Cauchy sequence.

Remark 1.4. LetA, B, C, D, Ebe nonnegative real numbers withABCDE <1, BC, orDE.IfF ABD1−C−D⁻¹andG ACE1−B−E⁻¹, thenFG <1. In fact, ifBCthen

FG ABD

1−C−D ·ACE

1−B−E ACD

1−B−E ·ABE

1−C−D <1, 1.1 and ifDE,

FG ABD

1−C−D ·ACE

1−B−E ABE

1−C−D ·ACD

1−B−E <1. 1.2

2. Main Results

The following theorem improves/generalizes the results of5, Theorems 1, 3, and 4and4, Theorems 2.3, 2.6, 2.7, and 2.8.

Theorem 2.1. LetX, d be a complete topological vector space-valued cone metric space,Pbe a cone andm, nbe positive integers. If a mappingT :X → X satisfies

d

T^mx, Tⁿy

≤Ad x, y

Bdx, T^mx Cd y, Tⁿy

Dd x, Tⁿy

Ed

y, T^mx

2.1 for allx, y∈X, whereA, B, C, D, Eare non negative real numbers withABCDE <1, BC, or DE.ThenT has a unique fixed point.

Proof. Forx₀∈Xandk≥0, define

x_2k1T^mx_2k,

x_2k2Tⁿx_2k1. 2.2

Then

dx_2k1, x_2k2 dT^mx_2k, Tⁿx_2k1

≤Adx2k, x_2k1 Bdx2k, T^mx2k Cdx2k1, Tⁿx_2k1 Ddx2k, Tⁿx_2k1 Edx_2k1, T^mx_2k

≤ABdx2k, x_2k1 Cdx2k1, x_2k2 Ddx2k, x_2k2

≤ABDdx2k, x_2k1 CDdx2k1, x_2k2.

2.3

It implies that

1−C−Ddx2k1, x_2k2≤ABDdx2k, x_2k1. 2.4

That is,

dx_2k1, x_2k2≤Fdx2k, x_2k1, 2.5

whereF ABD/1−C−D.

Similarly,

dx2k2, x_2k3 dT^mx_2k2, Tⁿx_2k1

≤Adx_2k2, x_2k1 Bdx_2k2, T^mx_2k2 Cdx_2k1, Tⁿx_2k1 Ddx2k2, Tⁿx_2k1 Edx2k1, T^mx_2k2

≤Adx2k2, x_2k1 Bdx2k2, x_2k3 Cdx2k1, x_2k2 D dx_2k2, x_2k2 Edx_2k1, x_2k3

≤ACEdx2k1, x_2k2 BEdx2k2, x_2k3,

2.6

which implies

dx2k2, x_2k3≤Gdx2k1, x_2k2, 2.7 withG ACE/1−B−E.

Now by induction, we obtain for eachk0,1,2, . . . dx2k1, x_2k2≤F dx2k, x_2k1

≤FGdx2k−1, x2k

≤FFGdx_2k−2, x_2k−1

≤ · · · ≤FFG^kdx0, x1, dx_2k2, x_2k3≤Gdx_2k1, x_2k2

≤ · · · ≤FG^k1dx0, x1.

2.8

ByRemark 1.4, forp < qwe have d

x_2p1, x_2q1

≤d

x_2p1, x_2p2 d

x_2p2, x_2p3 d

x_2p3, x_2p4

· · ·d

x2q, x_2q1

≤

⎡

⎣F q−1 ip

FGⁱ q ip1

FGⁱ

⎤

⎦dx0, x₁

≤ FFG^p

1−FG FG^p1 1−FG

dx0, x1

≤1F

FG^p 1−FG

dx0, x1.

2.9

In analogous way, we deduced

d

x_2p, x_2q1

≤1F

FG^p 1−FG

dx0, x₁,

d x2p, x2q

≤1F

FG^p 1−FG

dx0, x1,

d

x_2p1, x_2q

≤1F

FG^p 1−FG

dx0, x₁.

2.10

Hence, for 0< n < m

dxn, x_m≤a_n, 2.11

wherea_n 1FFG^p/1−FGdx0, x₁withpthe integer part ofn/2.

Fix0cand choose a symmetric neighborhoodV of 0 such thatcV ⊆intP. Since a_n → 0 asn → ∞, byLemma 1.3, we deduce that{xn}is a Cauchy sequence. SinceXis a complete, there existsu∈Xsuch thatx_n → u. Fix0cand choosen₀ ∈Nbe such that

du, x2k c

3K, dx_2k−1, x_2k c

3K, du, x_2k−1 c

3K 2.12

for allk≥n₀, where

Kmax

1D

1−B−E, AE

1−B−E, C 1−B−E

. 2.13

Now,

du, T^mu≤du, x2k dx2k, T^mu

≤du, x2k dTⁿx_2k−1, T^mu

≤du, x2k Adu, x_2k−1 Bdu, T^mu Cdx_2k−1, Tⁿx_2k−1 Ddu, Tⁿx_2k−1 Edx2k−1, T^mu

≤du, x2k Adu, x2k−1 Bdu, T^mu Cdx2k−1, x2k Ddu, x2k Edx_2k−1, u Edu, T^mu

≤1Ddu, x2k AEdu, x2k−1 Cdx2k−1, x2k BEdu, T^mu.

2.14

So,

du, T^mu≤Kdu, x2k Kdu, x2k−1 Kdx2k−1, x2k c

3 c 3c

3 c. 2.15

Hence

du, T^mu c

p 2.16

for everyp∈N. From

c

p−du, T^mu∈intP 2.17

beingPclosed, asp → ∞, we deduce−du, T^mu∈Pand sodu, T^mu 0. This implies that uT^mu.

Similarly, by using the inequality,

du, Tⁿu≤du, x_2k1 dx_2k1, Tⁿu, 2.18

we can show that u Tⁿu, which in turn implies that u is a common fixed point of T^m, Tⁿ and, that is,

uT^muTⁿu. 2.19

Now using the fact that

dTu, u dTT^mu, Tⁿu dT^mTu, Tⁿu

≤AdTu, u BdTu, T^mTu Cdu, Tⁿu DdTu, Tⁿu Edu, T^mTu

≤AdTu, u BdTu, Tu Cdu, u DdTu, u Edu, Tu ADEdTu, u.

2.20

We obtainuis a fixed point ofT. For uniqueness, assume that there exists another pointu^∗ inXsuch thatu^∗Tu^∗for someu^∗inX. From

du, u^∗ dT^mu, Tⁿu^∗

≤Adu, u^∗ Bdu, T^mu Cdu^∗, Tⁿu^∗ Ddu, Tⁿu^∗ Edu^∗, T^mu

≤Adu, u^∗ Bdu, u Cdu^∗, u^∗ Ddu, u^∗ Edu, u^∗

≤ADEdu, u^∗,

2.21

we obtain thatu^∗u.

Huang and Zhang 9 proved Theorem 2.1 by using the following additional assumptions.

aEBanach Space.

bP is normali.e., there is a numberκ≥ 1 such that for allx, y,∈E,0 ≤x≤ y ⇒ x ≤κy.

cmn1.

dOne of the following is satisfied:

iBCDE0 withA <15, Theorem 1, iiADE0 withBC <1/25, Theorem 3, iiiABC0 withDE <1/25, Theorem 4.

Azam and Arshad4improved these results of Huang and Zhang5by omitting the assumptionb.

Theorem 2.2. LetX, d be a complete topological vector space-valued cone metric space,Pbe a cone andm, nbe positive integers. If a mappingT :X → X satisfies:

d

Tx, Ty

≤Ad x, y

Bdx, Tx Cd y, Ty

Dd x, Ty

Ed y, Tx

2.22 for allx, y ∈X, whereA, B, C, D, Eare non negative real numbers withABCDE <1.

ThenT has a unique fixed point.

Proof. The symmetric property ofdand the above inequality imply that

d

Tx, Ty

≤Ad x, y

BC 2

dx, Tx d y, Ty

DE 2

d x, Ty

d y, Tx

. 2.23

By substituting T^m Tⁿ T in the Theorem 2.1, we obtain the required result. Next we present an example to supportTheorem 2.2.

Example 2.3. X 0,1, Ebe the set of all complex-valued functions onXthenE is a vector space overRunder the following operations:

fg

t ft gt, αf

t αft 2.24 for allf, g ∈E, α ∈R. Letτ be the topology onEdefined by the the family{px :x∈X}of seminorms onE, where

p_x f

fx 2.25 then X, τ is a topological vector space which is not normable and is not even metrizable see18,19. Defined:X×X → Eas follows:

d x, y

t x−y,3x−y3^t,

P{x∈E:xt0∀t∈X}. 2.26

ThenX, d is a topological vector space-valued cone metric space. DefineT : X → X as Tx x²/9, then all conditions ofTheorem 2.2are satisfied.

Corollary 2.4. LetX, dbe a complete Banach space-valued cone metric space,Pbe a cone, andm, n be positive integers. If a mappingT :X → X satisfies

d

T^mx, Tⁿy

≤Ad x, y

Bdx, T^mx Cd y, Tⁿy

Dd x, Tⁿy

Ed

y, T^mx

2.27 for allx, y∈X, whereA, B, C, D, Eare non negative real numbers withABCDE <1, BC, orDE.ThenT has a unique fixed point.

Next we present an example to show that corollary 2.4 is a generalization of the results 9, Theorems 1, 3, and 4and15, Theorems 2.3, 2.6, 2.7, and 2.8.

Example 2.5. LetX{1,2,3},BR², andP {x, y∈ B |x, y≥0} ⊂R². Defined:X×X → R²as follows:

d x, y

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

0,0, ifxy, 5

7,5

, ifx /y, x, y∈X− {2}, 1,7, ifx /y, x, y∈X− {3}, 4

7,4

, ifx /y, x, y∈X− {1}.

2.28

Define the mapping T :X → X as follows:

Tx

⎧⎨

⎩

1, if x /2,

3, if x2. 2.29

Note that the assumptionsdof results9, Theorems 1, 3, and 4and15, Theorems 2.3, 2.6, 2.7, and 2.8are not satisfied to find a fixed point of T. In order to apply inequality 2.1 consider mappingT²x 1 for eachx∈X,then forABCD0, E5/7, T², andT satisfy all the conditions ofCorollary 2.4and we obtainT1 1.

Acknowledgment

The authors are thankful to referee for precise remarks to improve the presentation of the paper.

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