A Note on tvs-G-Cone Metric Fixed Point Theory · In this paeper, for a tvs-G-cone metric space X,G...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2012, Article ID 407071, 10 pagesdoi:10.1155/2012/407071

Research ArticleA Note on tvs-G-Cone Metric Fixed Point Theory

Ing-Jer Lin,1 Chi-Ming Chen,2 Chao-Hung Chen,3and Tien-Ying Cheng2

1 Department of Mathematics, National Kaohsiung Normal University, Taiwan2 Department of Applied Mathematics, National Hsinchu University of Education, Taiwan3 Department of Applied Mathematics, Chung Yuan Christian University, Taiwan

Correspondence should be addressed to Chi-Ming Chen, [email protected]

Received 26 April 2012; Accepted 2 July 2012

Academic Editor: Yansheng Liu

Copyright q 2012 Ing-Jer Lin et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

For a tvs-G-cone metric space (X,G) and for the family A of subsets of X, we introduce a newnotion of the tvs-H-cone metric H with respect to G, and we get a fixed result for the CBW-tvs-G-cone-type function in a complete tvs-G-cone metric space (A,H). Our results generalize somerecent results in the literature.

1. Introduction and Preliminaries

In 2007, Huang and Zhang [1] introduced the concept of cone metric space by replacing theset of real numbers by an ordered Banach space, and they showed some fixed point theoremsof contractive-type mappings on cone metric spaces. The category of cone metric spaces islarger than metric spaces. Subsequently, many authors studied this subject and many resultson fixed point theory are proved (see, e.g., [2–15]). Recently, Du [16] introduced the conceptof tvs-cone metric and tvs-cone metric space to improve and extend the concept of conemetric space in the sense of Huang and Zhang [1]. Later, in the papers [16–21], the authorstried to generalize this approach by using cones in topological vector spaces tvs instead ofBanach spaces. However, it should be noted that an old result shows that if the underlyingcone of an ordered tvs is solid and normal, then such tvs must be an ordered normed space.Thus, proper generalizations when passing from norm-valued cone metric spaces to tvs-valued cone metric spaces can be obtained only in the case of nonnormal cones (for details,see [19]). Further, Radenovic et al. [22] introduced the concept of set-valued contraction ofNadler type in the setting of tvs-cone spaces and proved a fixed point theorem in the settingof tvs-cone spaces with respect to a solid cone.

2 Journal of Applied Mathematics

Definition 1.1 (see [22]). Let (X, d) be a tvs-cone metric space with a solid cone P , and let Abe a collection of nonempty subsets of X. A map H : A × A → E is called a tvs-H-conemetric with respect to d if for any A1, A2 ∈ A the following conditions hold:

(H1) H(A1, A2) = θ ⇒ A1 = A2,

(H2) H(A1, A2) = H(A2, A1),

(H3) ∀ε∈E,θ�ε ∀x∈A1 ∃y∈A2d(x, y) � H(A1, A2) + ε,

(H4) one of the following is satisfied:

(i) ∀ε∈E,θ�ε ∃x∈A1 ∀y∈A2H(A1, A2) � d(x, y) + ε,(ii) ∀ε∈E,θ�ε ∃y∈A2 ∀x∈A1H(A1, A2) � d(x, y) + ε.

Theorem 1.2 (see [22]). Let (X, d) be a tvs-cone complete metric space with a solid cone P and letA be a collection of nonempty closed subsets of X, A /=φ, and let H : A ×A → E be a tvs-H-conemetric with respect to d. If the mapping T : X → A the condition that exists a λ ∈ (0, 1) such thatfor all x, y ∈ X holds

H(Tx, Ty

)� λG

(x, y

)(1.1)

then T has a fixed point in X.

We recall some definitions and results of the tvs-cone metric spaces that introduced in[19, 23], which will be needed in the sequel.

Let E be be a real Hausdorff topological vector space (tvs for short) with the zerovector θ. A nonempty subset P of E is called a convex cone if P + P ⊆ P and λP ⊆ P for λ ≥ 0.A convex cone P is said to be pointed (or proper) if P ∩ (−P) = {θ}; P is normal (or saturated)if E has a base of neighborhoods of zero consisting of order-convex subsets. For a given coneP ⊆ E, we can define a partial ordering � with respect to P by x � y if and only if y − x ∈ P ;x ≺ y will stand for x � y and x /=y, while x � y will stand for y − x ∈ intP , where intPdenotes the interior of P . The cone P is said to be solid if it has a nonempty interior.

In the sequel, Ewill be a locally convexHausdorff tvswith its zero vector θ, P a proper,closed, and convex pointed cone in E with intP /=φ and � a partial ordering with respect toP .

Definition 1.3 (see [16, 18, 19]). Let X be a nonempty set and (E, P) an ordered tvs. A vector-valued function d : X × X → E is said to be a tvs-cone metric, if the following conditionshold:

(C1) ∀x,y∈X,x /=y θ � d(x, y),

(C2) ∀x,y∈X d(x, y) = θ ⇔ x = y,

(C3) ∀x,y∈X d(x, y) = d(y, x),

(C4) ∀x,y,z∈X d(x, z) � d(x, y) + d(y, z).

Then the pair (X, d) is called a tvs-cone metric space.

Definition 1.4 (see [16, 18, 19]). Let (X, d) be a tvs-cone metric space, x ∈ X, and {xn} asequence in X.

Journal of Applied Mathematics 3

(1) {xn} tvs-cone converges to x whenever for every c ∈ E with θ � c, there existsn0 ∈ N such that d(xn, x) � c for all n ≥ n0. We denote this by cone-limn→∞xn = x;

(2) {xn} is a tvs-cone Cauchy sequence whenever for every c ∈ E with θ � c, thereexists n0 ∈ N such that d(xn, xm) � c for all n,m ≥ n0;

(3) (X, d) is tvs-cone complete if every tvs-cone Cauchy sequence in X is tvs-coneconvergent in X.

Remark 1.5. Clearly, a cone metric space in the sense of Huang and Zhang [1] is a special caseof tvs-cone metric spaces when (X, d) is a tvs-cone metric space with respect to a normal coneP .

Remark 1.6 (see [19, 22, 23]). Let (X, d) be a tvs-cone metric space with a solid cone P . Thefollowing properties are often used, particularly in the case when the underlying cone isnonnormal.

(p1) If u � v and v � w, then u � w,



(p4) If θ � u � c for each c ∈ intP , then u = θ,

(p5) If a � b + c for each c ∈ intP , then a � b,

(p6) If E is tvswith a cone P , and if a � λa where a ∈ P and λ ∈ [0, 1), then a = θ,

(p7) If c ∈ intP , an ∈ E, and an → θ in locally convex tvsE, then there exists n0 ∈ N

such that an � c for all n > n0.

Metric spaces are playing an important role in mathematics and the applied sciences.To overcome fundamental laws in Dhage’s theory of generalized metric spaces [24]. In 2003,Mustafa and Sims [25] introduced a more appropriate and robust notion of a generalizedmetric space as follows.

Definition 1.7 (see [25]). LetX be a nonempty set, and letG : X×X×X → [0,∞) be a functionsatisfying the following axioms:

(G1) ∀x,y,z∈X G(x, y, z) = 0 ⇔ x = y = z,

(G2) ∀x,y∈X,x /=y G(x, x, y) > 0,

(G3) ∀x,y,z∈X G(x, y, z) ≥ G(x, x, y),

(G4) ∀x,y,z∈X (x, y, z) = G(x, z, y) = G(z, y, x) = · · · (symmetric in all three variables),

(G5) ∀x,y,z,w∈X G(x, y, z) ≤ G(x,w,w) +G(w,y, z).

Then the function G is called a generalized metric, or, more specifically a G-metric on X, andthe pair (X,G) is called a G-metric space.

By using the notions of generalized metrics and tvs-cone metrics, we introduce thebelow notion of tvs-generalized-cone metrics.

Definition 1.8. Let X be a nonempty set and (E, P) an ordered tvs, and let G : X ×X ×X → Ebe a function satisfying the following axioms:


(G1) ∀x,y,z∈X G(x, y, z) = θ if and only if x = y = z,

(G2) ∀x,y∈X,x /=y θ � G(x, x, y),

(G3) ∀x,y,z∈X G(x, x, y) � G(x, y, z),

(G4) ∀x,y,z∈X G(x, y, z) = G(x, z, y) = G(z, y, x) = · · · (symmetric in all three variables),

(G5) ∀x,y,z,w∈X G(x, y, z) � G(x,w,w) +G(w,y, z).

Then the functionG is called a tvs-generalized-conemetric, or, more specifically, a tvs-G-conemetric on X, and the pair (X,G) is called a tvs-G-cone metric space.

Definition 1.9. Let (X,G) be a tvs-G-cone metric space, x ∈ X, and {xn} a sequence in X.

(1) {xn}tvs-G-cone converges to x whenever, for every c ∈ E with θ � c, there existsn0 ∈ N such that G(xn, xm, x) � c for all m,n ≥ n0. Here x is called the limit of thesequence {xn} and is denoted by G-cone-limn→∞xn = x;

(2) {xn} is a tvs-G-cone Cauchy sequence whenever, for every c ∈ E with θ � c, thereexists n0 ∈ N such that G(xn, xm, xl) � c for all n,m, l ≥ n0;

(3) (X,G) is tvs-G-cone complete if every tvs-G-cone Cauchy sequence in X is tvs-G-cone convergent in X.

Proposition 1.10. Let (X,G) be a tvs-G-cone metric space, x ∈ X, and {xn} a sequence in X. Thefollowing are equivalent:

(i) {xn} tvs-G-cone converges to x,

(ii) G(xn, xn, x) → θ as n → ∞,

(iii) G(xn, x, x) → as n → ∞,

(iv) G(xn, xm, x) → θ as n,m → ∞.

In this paper, we also introduce the below concept of the CBW-tvs-G-cone-typefunction.

Definition 1.11. One calls ϕ : intP ∪ {θ} → intP ∪ {θ} a CBW-tvs-G-cone-type function if thefunction ϕ satisfies the following condition

(ϕ1) ϕ(t) � t for all t � θ and ϕ(θ) = θ;

(ϕ2) limn→∞ϕn(t) = θ for all t ∈ intP ∪ {θ}.

In this paeper, for a tvs-G-cone metric space (X,G) and for the family A of subsetsof X, we introduce a new notion of the tvs-H-cone metric H with respect to G, and we geta fixed result for the CBW-tvs-G-cone-type function in a complete tvs-G-cone metric space(A,H). Our results generalize some recent results in the literature.

2. Main Results

Let E be a locally convex Hausdorff tvs with its zero vector θ, P a proper, closed and convexpointed cone in E with intP /=φ, and � a partial ordering with respect to P . We introduce thebelow notion of the tvs-H-cone metric Hwith respect to tvs-G-cone metric G.


Definition 2.1. Let (X,G) be a tvs-G-cone metric space with a solid cone P , and let A be acollection of nonempty subsets of X. A map H : A × A × A → E is called a tvs-H-conemetric with respect to G if for any A1, A2, A3 ∈ A the following conditions hold:

(H1) H(A1, A2, A3) = θ ⇒ A1 = A2 = A3,

(H2) H(A1, A2, A3) = H(A2, A1, A3) = H(A1, A3, A2) = · · · (symmetry in all variables),

(H3) H(A1, A1, A2) � H(A1, A2, A3),

(H4) ∀ε∈E,θ�ε ∀x∈A1,y∈A2 ∃z∈A3G(x, y, z) � H(A1, A2, A3) + ε,

(H5) one of the following is satisfied:

(i) ∀ε∈E,θ�ε ∃x∈A1 ∀y∈A2,z∈A3 H(A1, A2, A3) � G(x, y, z) + ε,

(ii) ∀ε∈E,θ�ε ∃y∈A2 ∀x∈A1,z∈A3 H(A1, A2, A3) � G(x, y, z) + ε,

(iii) ∀ε∈E,θ�ε ∃z∈A3 ∀y∈A2,x∈A1 H(A1, A2, A3) � G(x, y, z) + ε.

We will prove that a tvs-H-cone metric satisfies the conditions of (G1)–(G5).

Lemma 2.2. Let (X,G) be a tvs-G-cone metric space with a solid cone P , and letA be a collection ofnonempty subsets of X, A /=φ. If H : A × A × A → E is a tvs-H-cone metric with respect to G,then pair (A,H) is a tvs-G-cone metric space.

Proof. Let {εn} ⊂ E be a sequence such that θ � εn for all n ∈ N and G-cone-limn→∞εn = θ.Take any A1, A2, A3 ∈ A and x ∈ A1, y ∈ A2. From (H4), for each n ∈ N, there exists zn ∈ A3

such that

G(x, y, zn

)� H(A1, A2, A3) + εn. (2.1)

Therefore, H(A1, A2, A3) + εn ∈ P for each n ∈ N. By the closedness of P , we conclude thatθ � H(A1, A2, A3).

Assume that A1 = A2 = A3. From H5, we obtain H(A1, A2, A3) � εn for any n ∈ N. SoH(A1, A2, A3) = θ.

LetA1, A2, A3, A4 ∈ A. Assume thatA1, A2, A3 satisfy condition (H5)(i). Then, for eachn ∈ N, there exists xn ∈ A1 such thatH(A1, A2, A3) � G(xn, y, z)+εn for all y ∈ A2 and z ∈ A3.From (H4), there exists a sequence {wn} ⊂ A4 satisfying G(xn,wn,wn) � H(A1, A4, A4) + εnfor every n ∈ N. Obviously, for any y ∈ A2 and any z ∈ A3 and n ∈ N, we have

H(A1, A2, A3) � G(xn, y, z

)+ εn

� G(xn,wn, wn) +G(wn, y, z

)+ εn.

(2.2)

Now for each n ∈ N, there exists yn ∈ A2, zn ∈ A3 such thatG(wn, yn, zn) � H(A4, A2, A3)+εn.


Consequently, we obtain that for each n ∈ N

H(A1, A2, A3) � H(A1, A4, A4) +H(A4, A2, A3) + 3εn. (2.3)

Therefore,

H(A1, A2, A3) � H(A1, A4, A4) +H(A4, A2, A3). (2.4)

In the case when (H5)(ii) or (H5)(iii) holds, we use the analogous method.

In the sequel, we denote by Θ the class of functions ϕ : intP ∪ {θ} → intP ∪ {θ}satisfying the following conditions:

(C1) ϕ is a CBW-tvs-G-cone-type-function;

(C2) ϕ is subadditive, that is, ϕ(u1 + u2) � ϕ(u1) + ϕ(u2) for all u1, u2 ∈ intP .

Our main result is the following.

Theorem 2.3. Let (X,G) be a tvs-G-cone complete metric space with a solid cone P , let A be acollection of nonempty closed subsets of X, A /=φ, and let H : A × A × A → E be a tvs-H-conemetric with respect to G. If the mapping T : X → A satisfies the condition that exists a ϕ ∈ Θ suchthat for all x, y, z ∈ X holds

H(Tx, Ty, Tz

)� ϕ

(G(x, y, z

)), (2.5)


Proof. Let us choose ε ∈ intP arbitrarily, and let εn ∈ E be a sequence such that θ � εnand εn � ε/3n. Let us choose x0 ∈ X arbitrarily and x1 ∈ Tx0. If G(x0, x0, x1) = θ, thenx0 = x1 ∈ T(x0), and we are done. Assume that G(x0, x0, x1) � θ. Taking into account (2.5)and (H4), there exists x2 ∈ Tx1 such that

G(x1, x1, x2) � H(Tx0, Tx0, Tx1) + ε1

� ϕ(G(x0, x0, x1)) + ε1.(2.6)

Taking into account (2.5), (2.6), and (H4) and since ϕ ∈ Θ, there exists x3 ∈ Tx2 such that

G(x2, x2, x3) � H(Tx1, Tx1, Tx2) + ε2

� ϕ(G(x1, x1, x2)) + ε2

� ϕ(ϕ(G(x0, x0, x1)) + ε1

)+ ε2

� ϕ(ϕ(G(x0, x0, x1))

)+ ϕ(ε1) + ε2

� ϕ2(G(x0, x0, x1)) + ε1 + ε2

� ϕ2(G(x0, x0, x1)) +ε

3+

ε

32.

(2.7)


We continue in this manner. In general, for xn, n ∈ N, xn+1 is chosen such that xn+1 ∈ Txn and

G(xn, xn, xn+1) � H(Txn−1, Txn−1, Txn) + εn

� ϕ(G(xn−1, xn−1, xn)) + εn

� ϕ2(G(xn−2, xn−2, xn−1)) + εn−1 + εn

� · · · · · ·

� ϕn(G(x0, x0, x1)) +n∑

i=1

εi

� ϕn(G(x0, x0, x1)) +n∑

i=1

ε

3i

� ϕn(G(x0, x0, x1)) +12ε.

(2.8)

Since ε is arbitrary, letting ε → θ and by the definition of the CBW-tvs-G-cone-type function,we obtain that

limn→∞

G(xn, xn, xn+1) = θ. (2.9)

Next, we let cm = G(xm, xm+1, xm+1), and we claim that the following result holds: foreach γ � θ, there is n0(ε) ∈ N such that for all m, n ≥ n0(γ),

G(xm, xm+1, xm+1) � γ, (2.10)

We will prove (2.10) by contradiction. Suppose that (2.10) is false. Then there exists someγ � θ such that for all p ∈ N, there are mp, np ∈ N withmp > np ≥ p satisfying

(i) mp is even and np is odd,

(ii) G(xmp , xnp , xnp) � γ , and

(iii) mp is the smallest even number such that conditions (i), (ii) hold.

Since cm ↓ θ, by (ii), we have that limp→∞G(xmp , xnp , xnp) = γ and

γ � G(xmp , xnp , xnp

)

� G(xmp , xmp+1, xmp+1

)+G

(xmp+1, xnp+1, xnp+1

)+G

(xnp+1, xnp , xnp

).

(2.11)

It follows from (H4); let us choose ε ∈ E arbitrarily such that

G(xnp+1, xnp+1, xmp+1

)� H

(Txnp+1, Txnp+1, Txmp+1

)+ ε. (2.12)


Taking into account (2.5), (2.11), and (2.12), we have that

γ � G(xmp , xnp , xnp

)


)+H

(Txnp+1, Txnp+1, Txmp+1

)+ ε +G

(xnp+1, xnp , xnp

)


)+ ϕ

(G(xnp , Gxnp , Gxmp

))+ ε +G

(xnp+1, xnp , xnp

)


)+G

(xnp , Gxnp , Gxmp

)+ ε +G

(xnp+1, xnp , xnp

).

(2.13)

Since ε is arbitrarily, letting ε → θ and by letting p → ∞, we have

γ � θ + limp→∞

G(xmp , xnp , xnp

)+ θ + θ = γ, (2.14)

a contradiction. So {xn} is a tvs-G-cone Cauchy sequence. Since (X,G) is a tvs-G-conecomplete metric space, {xn} is tvs-G-cone convergent in X and G-cone-limn→∞xn = x. Thus,for every τ ∈ intP and sufficiently large n, we have that

H(Txn, Txn, Tx) � ϕ(G(xn, xn, x)) � G(xn, xn, x) � τ

3. (2.15)

Since for n ∈ N ∪ {0}, xn+1 ∈ Txn, by (H4) we obtain that for all n ∈ N there exists yn ∈ Txsuch that

G(xn+1, xn+1, yn+1

)� H(Txn, Txn, Tx) + εn+1

� ϕ(G(xn, xn, x)) +ε

3n+1

� G(xn, xn, x) +ε

3n+1.

(2.16)

Since ε/3n+1 → θ, then for sufficiently large n, we obtain that

G(yn+1, x, x

)� G

(yn+1, xn+1, xn+1

)+G(xn+1, x, x) � 2τ

3+τ

3= τ, (2.17)

which implies G-cone-limn→∞yn = x. Since Tx is closed, we obtain that x ∈ Tx.

For the case ϕ(t) = kt, k ∈ (0, 1), then ϕ ∈ Θ and it is easy to get the following corollary.

Corollary 2.4. Let (X,G) be a tvs-G-cone complete metric space with a solid cone P , let A be acollection of nonempty closed subsets of X, A /=φ, and let H : A × A × A → E be a tvs-H-cone


metric with respect to G. If the mapping T : X → A satisfies the condition that exists k ∈ (0, 1) suchthat for all x, y, z ∈ X holds

H(Tx, Ty, Tz

)� k ·G(

x, y, z), (2.18)


Remark 2.5. Following Corollary 2.4, it is easy to get Theorem 1.2. So our results generalizesome recent results in the literature (e.g., [22]).

Acknowledgment

The authors would like to thank referee(s) for many useful comments and suggestions forthe improvement of the paper.

References

[1] L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,”Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468–1476, 2007.

[2] M. Abbas and G. Jungck, “Common fixed point results for noncommuting mappings withoutcontinuity in cone metric spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1,pp. 416–420, 2008.

[3] S. Jankovic, Z. Kadelburg, S. Radonevic, and B. E. Rhoades, “Assad-Kirktype fixed point theoremsfor a pair of nonself mappings on cone metric spaces,” Fixed Point Theory and Applications, vol. 2009,Article ID 761086, 16 pages, 2009.

[4] Sh. Rezapour and R. Hamlbarani, “Some notes on the paper: ‘Cone metric spaces and fixed pointtheorems of contractive mappings’,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2,pp. 719–724, 2008.

[5] M. Arshad, A. Azam, and P. Vetro, “Some common fixed point results in cone metric spaces,” FixedPoint Theory and Applications, vol. 2009, Article ID 493965, 11 pages, 2009.

[6] A. Azam and M. Arshad, “Common fixed points of generalized contractive maps in cone metricspaces,” Iranian Mathematical Society. Bulletin, vol. 35, no. 2, pp. 255–264, 2009.

[7] C. Di Bari and P. Vetro, “φ-pairs and common fixed points in conemetric spaces,” Rendiconti del CircoloMatematico di Palermo, vol. 57, no. 2, pp. 279–285, 2008.

[8] C. Di Bari and P. Vetro, “Weakly φ-pairs and common fixed points in cone metric spaces,” Rendicontidel Circolo Matematico di Palermo, vol. 58, no. 1, pp. 125–132, 2009.

[9] J. Harjani and K. Sadarangani, “Generalized contractions in partially ordered metric spaces andapplications to ordinary differential equations,” Nonlinear Analysis. Theory, Methods & Applications,vol. 72, no. 3-4, pp. 1188–1197, 2010.

[10] R. H. Haghi and Sh. Rezapour, “Fixed points of multifunctions on regular cone metric spaces,”Expositiones Mathematicae, vol. 28, no. 1, pp. 71–77, 2010.

[11] D. Klim andD.Wardowski, “Dynamic processes and fixed points of set-valued nonlinear contractionsin cone metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 11, pp. 5170–5175, 2009.

[12] Z. Kadelburg, S. Radenovic, and V. Rakocevic, “Remarks on ‘Quasi-contraction on a cone metricspace’,” Applied Mathematics Letters, vol. 22, no. 11, pp. 1674–1679, 2009.

[13] Z. Kadelburg, M. Pavlovic, and S. Radenovic, “Common fixed point theorems for orderedcontractions and quasicontractions in ordered cone metric spaces,” Computers & Mathematics withApplications, vol. 59, no. 9, pp. 3148–3159, 2010.

[14] S. Rezapour, H. Khandani, and S. M. Vaezpour, “Efficacy of cones on topological vector spaces andapplication to common fixed points of multifunctions,” Rendiconti del Circolo Matematico di Palermo,vol. 59, no. 2, pp. 185–197, 2010.


[15] Sh. Rezapour and R. H. Haghi, “Fixed point of multifunctions on cone metric spaces,” NumericalFunctional Analysis and Optimization, vol. 30, no. 7-8, pp. 825–832, 2009.

[16] W.-S. Du, “A note on cone metric fixed point theory and its equivalence,” Nonlinear Analysis. Theory,Methods & Applications, vol. 72, no. 5, pp. 2259–2261, 2010.

[17] A. Azam, I. Beg, andM. Arshad, “Fixed point in topological vector space-valued cone metric spaces,”Fixed Point Theory and Applications, vol. 2010, Article ID 604084, 9 pages, 2010.

[18] I. Beg, A. Azam, and M. Arshad, “Common fixed points for maps on topological vector space valuedcone metric spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID560264, 8 pages, 2009.

[19] Z. Kadelburg, S. Radenovic, and V. Rakocevic, “Topological vector space-valued cone metric spacesand fixed point theorems,” Fixed Point Theory and Applications, vol. 2011, Article ID 170253, 17 pages,2010.

[20] Z. Kadelburg and S. Radenovic, “Coupled fixed point results under tvs–cone metric and w-cone-distance,” Advanced in Fixed Point Theory, vol. 2, no. 1, pp. 29–46, 2012.

[21] L. J. Ciric, H. Lakzian, and V. Rakocevic, “Fixed point theorems for w-cone distance contractionmappings in tvs-cone metric spaces,” Fixed Point Theory and Applications, vol. 2012, p. 3, 2012.

[22] S. Radenovic, S. Simic, N. Cakic, and Z. Golubovic, “A note on tvs-cone metric fixed point theory,”Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2418–2422, 2011.

[23] Z. Kadelburg, S. Radenovic, and V. Rakocevic, “A note on the equivalence of some metric and conemetric fixed point results,” Applied Mathematics Letters, vol. 24, no. 3, pp. 370–374, 2011.

[24] B. C. Dhage, “Generalized metric space and mapping with fixed point,” Bulletin of the CalcuttaMathematical Society, vol. 84, pp. 329–336, 1992.

[25] Z. Mustafa and B. Sims, “A new approach to generalized metric spaces,” Journal of Nonlinear andConvex Analysis, vol. 7, no. 2, pp. 289–297, 2006.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

A Note on tvs-G-Cone Metric Fixed Point Theory · In this paeper, for a tvs-G-cone metric space X,G...

Documents

Transcript of A Note on tvs-G-Cone Metric Fixed Point Theory · In this paeper, for a tvs-G-cone metric space X,G...