Common Fixed Points for Maps on Topological Vector Space Valued Cone Metric Spaces

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Volume 2009, Article ID 560264,8pages doi:10.1155/2009/560264

Research Article

Common Fixed Points for Maps on Topological

Vector Space Valued Cone Metric Spaces

Ismat Beg,

1

_{Akbar Azam,}

2

_{and Muhammad Arshad}

3 1_{Centre for Advanced Studies in Mathematics, Lahore University of Management Sciences,}

Lahore Cantt 54792, Pakistan

2_{Mathematics Department, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan}

3_{Mathematics Department, International Islamic University, Islamabad 44000, Pakistan}

Correspondence should be addressed to Ismat Beg,[email protected]

Received 31 August 2009; Revised 30 November 2009; Accepted 13 December 2009

Recommended by Yuri Latushkin

We introduced a notion of topological vector space valued cone metric space and obtained some common fixed point results. Our results generalize some recent results in the literature.

Copyrightq2009 Ismat Beg 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.

1. Introduction

Huang and Zhang 1 generalized the notion of metric space by replacing the set of real numbers by ordered Banach space, deﬃned a cone metric space, and established some fixed point theorems for contractive type mappings in a normal cone metric space. Subsequently, several other authors 2–5 studied the existence of common fixed point of mappings satisfying a contractive type condition in normal cone metric spaces. Afterwards, Rezapour and Hamlbarani6studied fixed point theorems of contractive type mappings by omitting the assumption of normality in cone metric spacessee also7–14. In this paper we obtain common fixed points for a pair of self-mappings satisfying a generalized contractive type condition without the assumption of normality in a class of topological vector space valued cone metric spaces which is bigger than that introduced by Huang and Zhang1.

LetE, τbe always a topological vector space andP a subset ofE. Then,Pis called a cone whenever

iPis closed, nonempty andP /{0},

iiaxby∈Pfor allx, y∈Pand nonnegative real numbersa, b,

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For a given coneP ⊆E, we can define a partial ordering≤with respect toPbyx≤y

if and only ify−x ∈ P. x < y will stand forx ≤ yandx /y, whilex y will stand for

y−x∈intP, where intPdenotes the interior ofP.

Definition 1.1. LetXbe a nonempty set. Suppose that 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 cone metric onXandX, dis called a topological vector space valued cone metric space.

Note that Huang and Zhang1notion of cone metric space is a special case of our notion of topological vector space valued cone metric space.

Example 1.2. LetX 0,1,and let Ebe the set of all real valued functions onX which also have continuous derivatives on X,then E is a vector space over R under the following operations:

fgt ft gt, αft αft, 1.1

for allf, g ∈ E, α ∈ R. Letτ be the strongest vectorlocally convextopology on E,then

X, τis a topological vector space which is not normable and is not even metrizablesee

15. Defined:X×X → Eas follows:

dx, yt x−yet_,

P{x∈E:xt0∀t∈X}. 1.2

ThenX, d is a topological vector space valued cone metric space.

Example 1.2 shows that this category of cone metric spaces is larger than that considered in1–8.

Definition 1.3. LetX, dbe a topological vector space valued cone metric space, and letx∈X

and{xn}n≥1be a sequence inX. Then

i{xn}n≥1 converges to xwhenever for everyc ∈ Ewith 0 c there is a natural

numberNsuch thatdxn, xcfor alln≥N. We denote this by limn→ ∞xnxorxn → x. ii{xn}n≥1 is a Cauchy sequence whenever for everyc ∈ Ewith 0 c there is a

natural numberNsuch thatdxn, xmcfor alln, m≥N.

iii X, d is a complete topological vector space valued cone metric space if every Cauchy sequence is convergent.

2. Fixed Point

In this section, we shall give some results which generalize6, Theorems 2.3, 2.6, 2.7, and 2.8

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Theorem 2.1. LetX, dbe a complete topological vector space valued cone metric space and let the self-mappingsS, T:X → Xsatisfy

dSx, Ty≤kdx, yldx, Tydy, Sx, 2.1

for allx, y∈X, wherek, l∈0,1withk2l <1. ThenSandThave a unique common fixed point.

Proof. Forx0∈Xandn≥0, definex2n1Sx2nandx2n2Tx2n1. Then,

dx2n1, x2n2 dSx2n, Tx2n1

kdx2n, x2n1 ldx2n, Tx2n1 dx2n1, Sx2n kdx2n, x2n1 ldx2n, Tx2n1

kdx2n, x2n1 ldx2n, x2n1 dx2n1, x2n2

kldx2n, x2n1 ldx2n1, x2n2.

2.2

It implies thatdx2n1, x2n2kl/1−ldx2n, x2n1. Similarly,

dx2n2, x2n3 dSx2n2, Tx2n1

kdx2n2, x2n1 ldx2n2, Tx2n1 dx2n1, Sx2n2

kdx2n2, x2n1 ldx2n2, x2n3 dx2n1, x2n2

kldx2n1, x2n2 ldx2n2, x2n3.

2.3

Hence,dx2n2, x2n3≤kl/1−ldx2n1, x2n2. Thus,

dxn, xn1λndx0, x1, 2.4

for alln≥0, whereλ kl/1−l<1. Now, for n > mwe have

dxn, xmdxn, xn−1 dxn−1, xn−2 · · ·dxm1, xm

λn−1λn−2· · ·λmdx0, x1

λm

1−λdx0, x1.

2.5

Let 0 c. Take a symmetric neighborhood V of 0 such that cV ⊆ intP. Also, choose a natural numberN1 such thatλm/1−λdx1, x0 ∈ V, for allm ≥ N1. Then, λm/1−

λdx1, x0c, for allm≥N1. Thus,

dxn, xm≤ λ m

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for alln > m. Therefore,{xn}n≥1 is a Cauchy sequence inX, d. SinceX is complete, there

existsu ∈ X such thatxn → u. Choose a natural numberN2 such thatdxn, u c1−

l/21lfor allnN2. Thus,

du, Tudu, x2n1 dx2n1, Tu

du, x2n1 dSx2n, Tu

du, x2n1 kdu, x2n ldu, Sx2n dx2n, Tu

du, x2n1 kdu, x2n ldu, x2n1 dx2n, u du, Tu

1ldu, x2n1 kldu, x2n ldu, Tu.

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So,

du, Tu

1l

1−l

du, x2n1

kl

1−l

du, x2n

1l

1−l

du, x2n1

1l

1−l

du, x2n

c

2

c

2 c,

2.8

for alln≥N2. Therefore,du, Tuc/ifor alli1. Hence,c/i−du, Tu∈Pfor all i1.

SincePis closed,−du, Tu∈Pand sodu, Tu 0. Hence,uis a fixed point ofT. Similarly, we can show thatu Su. Now, we show thatSandT have a unique fixed point. For this, assume that there exists another pointu∗inXsuch thatu∗Tu∗Su∗. Then,

du, u∗ dSu, Tu∗

kdu, u∗ ldu, Tu∗ du∗, Su kdu, u∗ ldu, u∗ du∗, u k2ldu, u∗.

2.9

Sincek2l <1, du, u∗ 0 and souu∗.

The following corollary generalizes 6, Theorems 2.3, 2.7, and 2.8 and so 1, Theorems 1 and 4.

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Proof. The symmetric property ofdand the above inequality imply that

dTx, Tyadx, y bc

2 d

x, Tydy, Tx. 2.10

By substitutingSTakandbc/2linTheorem 2.1, we obtain the required result.

Theorem 2.3. LetX, dbe a complete topological vector space valued cone metric space and let the self-mappingsS, T:X → Xsatisfy

dSx, Ty≤kdx, yldx, Sx dy, Ty, 2.11

for allx, y∈X, wherek, l∈0,1withk2l <1. ThenSandThave a unique common fixed point.

Proof. Forx0∈Xandn≥0, definex2n1Sx2nandx2n2Tx2n1. Then,

dx2n1, x2n2 dSx2n, Tx2n1

kdx2n, x2n1 ldx2n, Sx2n dx2n1, Tx2n1 kdx2n, x2n1 ldx2n, x2n1 dx2n1, x2n2

kldx2n, x2n1 ldx2n1, x2n2.

2.12

It implies thatdx2n1, x2n2kl/1−ldx2n, x2n1. Similarly,

dx2n2, x2n3 dSx2n2, Tx2n1

kdx2n2, x2n1 ldx2n2, Sx2n2 dx2n1, Tx2n1 kdx2n2, x2n1 ldx2n2, x2n3 dx2n1, x2n2

kldx2n1, x2n2 ldx2n2, x2n3.

2.13

Hence,dx2n2, x2n3≤kl/1−ldx2n1, x2n2. Thus,

dxn, xn1λndx0, x1, 2.14

for alln≥0, whereλ kl/1−l<1. Now, for n > mwe have

dxn, xmdxn, xn−1 dxn−1, xn−2 · · ·dxm1, xm

λn−1_λn−2_{· · ·}_λm_d_x 0, x1

λm

1−λdx0, x1.

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Let 0 c. Take a symmetric neighborhood V of 0 such that cV ⊆ intP. Also, choose a natural numberN1 such thatλm/1−λdx1, x0 ∈ V, for allm ≥ N1. Then, λm/1−

λdx1, x0c, for allm≥N1. Thus,

dxn, xm≤ λ m

1−λdx1, x0c, 2.16

for alln > m. Therefore,{xn}n≥1 is a Cauchy sequence inX, d. SinceX is complete, there

existsu ∈ X such thatxn → u. Choose a natural numberN2 such thatdxn, u c1−

l/21lfor allnN2. Thus,

du, Tudu, x2n1 dx2n1, Tu

du, x2n1 dSx2n, Tu

du, x2n1 kdu, x2n ldu, Tu dx2n, Sx2n

du, x2n1 kdu, x2n ldu, x2n1 dx2n, u du, Tu

1ldu, x2n1 kldu, x2n ldu, Tu.

2.17

So,

du, Tu

1l

1−l

du, x2n1

kl

1−l

du, x2n

1l

1−l

du, x2n1

1l

1−l

du, x2n

c

2

c

2 c,

2.18

for alln≥N2. Therefore,du, Tuc/ifor alli1. Hence,c/i−du, Tu∈Pfor alli1.

SincePis closed,−du, Tu∈Pand sodu, Tu 0. Hence,uis a fixed point ofT. Similarly, we can show thatu Su. Now, we show thatSandT have a unique fixed point. For this, assume that there exists another pointu∗inXsuch thatu∗Tu∗Su∗. Then,

du, u∗ dSu, Tu∗

kdu, u∗ ldu, u∗ du∗, u

kdu, u∗.

2.19

Sincek <1, du, u∗ 0 and souu∗.

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Corollary 2.4. LetX, dbe a complete topological vector space valued cone metric space and let the self-mappingT :X → X satisfydTx, Tyadx, y bdx, Tx cdy, Tyfor allx, y ∈X, wherea, b, c∈0,1withabc <1. ThenThas a unique fixed point.

Proof is similar to the proof ofCorollary 2.2.

Example 2.5. Let X, d be a topological vector space valued cone metric space of

Example 1.2. DefineS, T :X → X as follows:

St Tt ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ t

3 if x /1, 1

6 if x1.

2.20

Then,

_Sx₋_Ty_et_≤_k_x₋_y_et_l _|_x₋_Sx_|_et_y₋_Ty_et_, _2.21

ifk1/6, l5/18. Hence all conditions ofTheorem 2.3are satisfied.

Acknowledgment

The present version of the paper owes much to the precise and kind remarks of the learned referees.

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