Some Generalizations of Fixed Point Theorems in Cone Metric Spaces

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Volume 2009, Article ID 657914,10pages doi:10.1155/2009/657914

Review Article

Some Generalizations of Fixed Point Theorems in Cone Metric Spaces

J. O. Olaleru

Mathematics Department, University of Lagos, Yaba, Lagos, Nigeria

Correspondence should be addressed to J. O. Olaleru,[email protected] Received 17 March 2009; Revised 15 July 2009; Accepted 29 August 2009 Recommended by Mohamed A. Khamsi

We generalize, extend, and improve some recent fixed point results in cone metric spaces including the results of H. Guang and Z. Xian2007; P. Vetro 2007; M. Abbas and G. Jungck 2008;

Sh. Rezapour and R. Hamlbarani2008. In all our results, the normality assumption, which is a characteristic of most of the previous results, is dispensed. Consequently, the results generalize several fixed results in metric spaces including the results of G. E. Hardy and T. D. Rogers1973, R. Kannan1969, G. Jungck, S. Radenovic, S. Radojevic, and V. Rakocevic2009, and all the references therein.

Copyrightq2009 J. O. Olaleru. 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

The recently discovered applications of ordered topological vector spaces, normal cones and topical functions in optimization theory have generated a lot of interest and research in ordered topological vector spacese.g., see1,2. Recently, Huang and Zhang3introduced cone metric spaces, which is a generalization of metric spaces, by replacing the real numbers with ordered Banach spaces. They later proved some fixed point theorems for diﬀerent contractive mappings. Their results have been generalized by diﬀerent authorse.g. see4–

7. This paper generalizes, extends and improves the results of all those authors.

The following definitions are given in3.

LetEbe a real Banach space andP a subset ofE.Pis called a cone if and only if iPis closed, nonempty, andP /{0};

iia, b∈R,a, b≥0,x, y∈P ⇒axby∈P; 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.

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The coneP is callednormalif there isM > 0 such that for allx, y ∈ E, 0 ≤ x ≤ y impliesx ≤My.

The least positive numberMsatisfying the above is called the normal constant ofP.

The cone P is called regular if every increasing sequence which is bounded from above is convergent. That is, if{xn}_n≥1is a sequence such that x1 ≤ x2 ≤ · · · ≤y for some y∈E, then there isx∈Esuch that lim_n_{→ ∞}xn−x0. Equivalently, the conePis regular if and only if every decreasing sequence which is bounded from below is convergent. In5it was shown that every regular cone is normal.

In the sequel we will suppose thatEis a metrizable linear topological space whose topology is defined by a real-valued functionF : X → R calledF-normsee8. We will assume thatPis a cone inEwith intP /0 and≤is partial ordering with respect toP.

Metrizable linear topological spaces contain metrizable locally convex spaces and normed linear spaces9. Therefore ourEgeneralizes theEas a normed linear space used in all the previous results on cone metric spaces.

A coneP ⊆Eis therefore called normal if there is M >0 such that for allx, y∈E, 0≤ x≤yimpliesFx≤MFy.

Definition 1.1. LetXbe a nonempty set. Suppose thatd:X×X → Esatisfies i0≤dx, yfor allx, y∈Xanddx, y 0 if and only ifxy, iidx, y dy, xfor allx, y∈X,

iiidx, y≤dx, z dz, yfor allx, y, z∈X.

Thendis called a cone metric onX, andX, dis called a cone metric space.

Example 1.2see3. LetE R²,P {x, y ∈ E : x, y ≥ 0},X R,andd : X×X → E defined bydx, y |x−y|, α|x−y|, whereα≥0 is a constant. ThenX, dis a cone metric space.

Clearly, this example shows that cone metric spaces generalize metric spaces.

We now give another example whereEis a metrizable linear topological vector space that is not a normed linear space.

Example 1.3. LetE^p,0< p <1,P{{xn}_n≥1∈E:x_n≥0,for alln},X, ρa metric space andd:X×X → Edefined bydx, y {ρx, y/2ⁿ}_n≥1. ThenX, dis a cone metric space.

Definition 1.4. LetX, dbe a cone metric space. Let{xn}be a sequence inX. If for everyc∈E with 0cthere isNsuch that for alln > N,dxn, xc, then{xn}is said to be convergent tox∈X, that is, limn→ ∞xnx.

Definition 1.5. LetX, dbe a cone metric space. Let{xn}be a sequence inX. If for everyc∈E with 0cthere isNsuch that for alln, m > N,dxn, x_mc, then{xn}is called a Cauchy sequence inX.

It is shown in3that a convergent sequence in a cone metric spaceX, dis a Cauchy sequence.

Definition 1.6. Let X, d be a cone metric space. If for any sequence {xn} in X, there is a subsequence{xni}of{xn}such that{xni}is convergent inX, thenXis called a sequentially

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compact metric space. Furthermore,X is compact if and only ifX is sequentially compact.

2. Theorems on Single Maps

Theorem 2.1. LetX, dbe a complete cone metric space andf:X → Xbe mappings such that

d

fx, f y

≤a₁d

fx, x a₂d

f y

, y a₃d

f y

, x a₄d

fx, y a₅d

y, x 2.1

for allx, y∈Xwherea₁, a₂, a₃, a₄, a₅∈0,1and a₁a₂a₃a₄a₅ <1. Then the mappingsf have a unique fixed point. Moreover, for anyx∈X, the sequence{fⁿx}converges to the fixed point.

Proof. We adapt the technique in13. Without loss of generality we may assume that a1a2

anda₃a₄so that from2.1, we have

d

fx, f y

≤ a1a2

2 d

fx, x d

f y

, y

a3a4

2 d

f y

, x d

fx, y a₅d

y, x .

2.2

Setyfxin2.1and simplify to obtain

d

fx, f²x

≤ a1a5

1−a₂ d

x, fx a3

1−a₂d

x, f²x

. 2.3

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By the triangle inequality,dfx, f²x≥df²x, x−dfx, xand so from2.3we get

d

f²x, x

−d

fx, x

≤ a₁a₅ 1−a₂ d

x, fx a₃

1−a₂d

x, f²x

, 2.4

which on simplifying gives

d

f²x, x

≤ 1a1a5−a2

1−a2−a3 d

x, fx

. 2.5

Substituting2.5into2.3we obtain

d

fx, f²x

≤ a₁a₃a₅ 1−a2−a3

d

x, fx

, 2.6

and by symmetry, we may exchangea₁witha₂anda₃witha₄in2.6to obtain

d

fx, f²x

≤ a₂a₄a₅ 1−a1−a4

d

x, fx

. 2.7

Ifαmin{a1a3a5/1−a2−a3,a2a4a5/1−a1−a4}, then d

fx, f²x

≤αd

x, fx

, 2.8

whereα∈0,1. Letm > n, then in view of2.8, we obtain d

f^mx, fⁿx

≤d

f^mx, f^m−1x

· · ·d

fⁿ¹x, fⁿx

≤αⁿ

1α· · ·α^m−n d

x, fx

≤ αⁿ 1−αd

x, fx .

2.9

Let 0cbe given and choose a natural numberN₁such thatαⁿ/1−αdx, fxcfor alln≥N₁. Thus,

d

f^mx, fⁿx

c 2.10

forn > m. Therefore,{fⁿx}_n≥1is a Cauchy sequence inX, d. SinceX, dis complete, there existsx^∗∈Xsuch thatfⁿx → x^∗. Choose a natural numberN₂such that for alln≥N₂,

d

fⁿx, x^∗

c1−a2a3 2a1a₄1 , d

fⁿ⁻¹x, x^∗

c1−a2a3 2a1a₃a₅.

2.11

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Then d

fx^∗, x^∗

≤d

fⁿx, fx^∗ d

fⁿx, x^∗

≤a₁d

fⁿx, fⁿ⁻¹x a₂d

fx^∗, x^∗ a₃d

fx^∗, fⁿ⁻¹x a₄d

fⁿx, x^∗ a₅d

fⁿ⁻¹x, x^∗ d

fⁿx, x^∗

≤a₁d

fⁿx, x^∗ a₁d

fⁿ⁻¹x, x^∗ a₂d

fx^∗, x^∗ a₃d

fⁿ⁻¹x, x^∗ a₄d

fⁿx, x^∗ a5d

fⁿ⁻¹x, x^∗ d

fⁿx, x^∗

≤ a₁a₃a₅ 1−a2a3d

fⁿ⁻¹x, x^∗

a₁a₄1 1−a2a3d

fⁿx, x^∗ c

2 c 2 c.

2.12

Thus,dfx^∗, x^∗ c/m, for allm ≥ 1. Soc/m−dfx^∗, x^∗ ∈ P, for allm ≥ 1. Since c/m → 0 as m → ∞, and P is closed, −dfx^∗, x^∗ ∈ P. Butdfx^∗, x^∗ ∈ P and so dfx^∗, x^∗ 0. Hencefx^∗ x^∗. The uniqueness follows from the contractive definition of fin2.1.

Remark 2.2. The theorem is valid if we replace the completeness ofXwith the condition that fXis complete. IfEis restricted to a normed linear space anda₁ a₂ a₃ a₄ 0 in Theorem 2.1we have5, Theorem 2.3; ifa₃ a₄ a₅ 0 inTheorem 2.1, we obtain5, Theorem 2.6; ifa1 a2 a5 0, we obtain5, Theorem 2.7and ifa1 a2 a3 0, we obtain5, Theorem 2.8. Furthermore, if we add the normality assumption toTheorem 2.1, then3, Theorems 1, 2, and 4there are special cases ofTheorem 2.1.

Thus Theorem 2.1 is both an extension generalization and an improvement of the results of3,5.

We now consider the situation wherea₁a₂a₃a₄a₅1 inTheorem 2.1.

Theorem 2.3. LetX, dbe a sequentially compact cone metric space andf :X → Xbe a continuous mapping such that

d

fx, f y

< a1d

fx, x a2d

f y

, y a3d

f y

, x a4d

fx, y a₅d

y, x

, 2.13

for allx, y ∈ X, x /y where a1, a2, a3, a4, a5 ∈ 0,1and a1 a2 a3a4a5 1. Then the mappingsfhave a unique fixed point.

Proof. We follow the same argument as Theorem 2.1. Without loss of generality, we may assume thata1a4anda2a3are less than 1. Hence2.8becomes

d

fx, f²x

< d

x, fx

. 2.14

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SinceX is sequentially compact, then it is compact10. The fact thatf is continuous and X is compact implies that fXis compact and hence inf{dx, fx : x ∈ X} exists and inf{dx, fx: x ∈X} dy, fyfor somey ∈ X. From2.14, it can be infered thatyis fixed underfand uniqueness follows from2.13.

Remark 2.4. If a₁ a₂ a₃ a₄ 0, with the additional assumption thatP is a regular cone inTheorem 2.3, we obtain3, Theorem 2. ThusTheorem 2.3is both an extension and improvement of3, Theorem 2.

3. Common Fixed Points

Theorem 3.1. LetX, dbe a cone metric space and letf, g:X → Xbe mappings such that

d

fx, f y

≤a1d

fx, gx a2d

f y

, g y

a3d f

y , gx a₄d

fx, g y

a₅d g

y

, gx 3.1

for allx, y ∈ X wherea₁, a₂, a₃, a₄, a₅ ∈0,1and a₁a₂a₃ a₄a₅ < 1. Supposef and g are weakly compatible andfX ⊂gXsuch thatfXorgXis a complete subspace ofX, then the mappingsf andg have a unique common fixed point. Moreover, for any x_o ∈ X, the sequence {xn} ⊂Xdefined bygxn fxn−1for alln, converges to the fixed point.

Proof. Observe that iffsatisfies3.1, it also satisfies

d

fx, f y

≤kd

fx, gx kd

f y

, g y

ld f

y , gx ld

fx, g y

md g

y

, gx 3.2

for allx, y∈Xwherek, l, m∈0,1and 2k2lm <1,2ka₁a₂,2la₃a₄, a₅m.

If fxn fx_n−1 for all n ∈ N, then {fxn} is a Cauchy sequence. Suppose fxn/fx_n−1 for all n ∈ N. Using3.2 and the fact thatgxn fx_n−1 for all n, we have

d

fxn1, fxn

≤kd

fxn1, fxn kd

fxn, fxn−1 ld

fxn, fxn ld

fxn1, fxn ld

fxn, fx_n−1 md

fx_n−1, fxn

≤ klm 1−kld

fx_n−1, fxn .

3.3

Consequently

d

fxn1, fxn

≤ klm 1−kl

n

d

fxo, fx1

. 3.4

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Now, for allm, n∈N, withn > m, we have d

fxn, fxm

≤kd

fxn, fx_n−1 kd

fx_n−1, fx_n−2

· · ·d

fx_m1, fxm

kⁿ⁻¹kⁿ⁻²· · ·k^m d

fxo, fx1

≤ k^m

1−kdfxo, fx1,

3.5 wherek klm/1−kl∈0,1.

Let 0cbe given and choose a natural numberN₁such thatk^m/1−kdx, fx cfor allm≥N₁. Thus,

d

fxm, fxn

c 3.6

forn > m. Therefore,{fxn}_n≥1is a Cauchy sequence. SincefXorgXis complete, then there existsx^∗∈gXsuch thatfxn → x^∗andgxn → x^∗. Lety∈Xsuch thatgy x^∗. We claim thatfy gy. From3.2, we have

d

fxn, f y

≤kd

fxn, gxn kd

f y

, g y

ld f

y , gxn ld

fxn, g y

md g

y , gxn

. 3.7

Asn → ∞we obtain d

x^∗, f y

≤kd f

y , g

y ld

f y

, x^∗ ld

x^∗, g y

md g

y , x^∗ kld

x^∗, f y

, and hencex^∗f y

g y

. 3.8

Sincefy gyandfandgare weakly compatible, then fx^∗ f

g y

g g

y

gx^∗. 3.9

Next we show thatx^∗fx^∗ gx^∗. Supposefx^∗/x^∗, from3.2, we have d

fx^∗, f y

≤kd

fx^∗, gx^∗ kd

f y

, g y

ld f

y , gx^∗ ld

fx^∗, g y

md g

y , gx^∗ 2ld

f y

, gx^∗ 2ld

f y

, fx^∗ .

3.10

This is a contradiction and hence fx^∗ x^∗ gx^∗. Thusx^∗is a common fixed point off andg. The uniqueness follows from3.1.

Remark 3.2. iIfa₃a₄0 andEis restricted to normed linear spaces inTheorem 3.1, with the additional normality assumption, we obtain the common fixed point Theorem of Vetro 7.

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ii Suppose E is restricted to normed linear spaces, with the additional normality assumption, ifa₁ a₂ a₃ a₄ 0, thenTheorem 3.1gives4, Theorem 2.1; if a₃ a₄ a₅0, we obtain4, Theorem 2.3, and ifa₁ a₂ a₅ 0, we obtain4, Theorem 2.4. Thus our theorem is both an extension, generalization and an improvement of the results of4,7.

iiiIfEis restricted to normed linear spaces,Theorem 3.1reduces to 14, Theorem 2.8.

ivIf inTheorem 3.1we choose chooseg IX the identity mapping onX, we have Theorem 2.1.

Open Question

Theorem 2.3 was proved for the usual metric space by the author in 15 without the assumptions thatf is continuous and X is compact. Is the aboveTheorem 2.3still valid if we remove the assumption thatfis continuous andXis compact?.

Acknowledgments

The author is grateful to the referees for careful readings and corrections. He is also grateful to Professor Stojan Radenvonic for giving him all the papers on cone metric spaces used in this paper and the African Mathematics Millennium Science InitiativeAMMSIfor financial support.

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