Journal of Applied Mathematics

VolumeΒ 2012, Article IDΒ 407071, 10 pages

http://dx.doi.org/10.1155/2012/407071

## A Note on *tvs-G*-Cone Metric Fixed Point Theory

^{1}Department of Mathematics, National Kaohsiung Normal University, Taiwan^{2}Department of Applied Mathematics, National Hsinchu University of Education, Taiwan^{3}Department of Applied Mathematics, Chung Yuan Christian University, Taiwan

Received 26 April 2012; Accepted 2 July 2012

Academic Editor: YanshengΒ Liu

Copyright Β© 2012 Ing-Jer Lin 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.

#### Abstract

For a *tvs-G*-cone metric space and for the family
of subsets of *X*, we introduce a new notion of the *tvs- β*-cone metric

*with respect to*

*β**G*, and we get a fixed result for the

*-tvs-G*-cone-type function in a complete

*tvs-G*-cone metric space . Our results generalize some recent results in the literature.

#### 1. Introduction and Preliminaries

In 2007, Huang and Zhang [1] introduced the concept of cone metric space by replacing the set of real numbers by an ordered Banach space, and they showed some fixed point theorems of contractive-type mappings on cone metric spaces. The category of cone metric spaces is larger than metric spaces. Subsequently, many authors studied this subject and many results on fixed point theory are proved (see, e.g., [2β15]). Recently, Du [16] introduced the concept of -cone metric and -cone metric space to improve and extend the concept of cone metric space in the sense of Huang and Zhang [1]. Later, in the papers [16β21], the authors tried to generalize this approach by using cones in topological vector spaces instead of Banach spaces. However, it should be noted that an old result shows that if the underlying cone 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 -valued cone metric spaces can be obtained only in the case of nonnormal cones (for details, see [19]). Further, RadenoviΔ et al. [22] introduced the concept of set-valued contraction of Nadler type in the setting of -cone spaces and proved a fixed point theorem in the setting of *tvs*-cone spaces with respect to a solid cone.

*Definition 1.1 (see [22]). *Let be a -cone metric space with a solid cone , and let be a collection of nonempty subsets of . A map is called a --cone metric with respect to if for any the following conditions hold:,,, one of the following is satisfied:(i),(ii).

Theorem 1.2 (see [22]). *Let be a -cone complete metric space with a solid cone and let be a collection of nonempty closed subsets of , , and let be a --cone metric with respect to . If the mapping the condition that exists a such that for all holds
**
then has a fixed point in .*

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

Let be be a real Hausdorff topological vector space ( for short) with the zero vector . A nonempty subset of is called a convex cone if and for . A convex cone is said to be pointed (or proper) if ; is normal (or saturated) if has a base of neighborhoods of zero consisting of order-convex subsets. For a given cone , we can define a partial ordering with respect to by if and only if ; will stand for and , while will stand for , where denotes the interior of . The cone is said to be solid if it has a nonempty interior.

In the sequel, will be a locally convex Hausdorff with its zero vector , a proper, closed, and convex pointed cone in with and a partial ordering with respect to .

*Definition 1.3 (see [16, 18, 19]). *Let be a nonempty set and an ordered . A vector-valued function is said to be a -cone metric, if the following conditions hold:,,,. Then the pair is called a -cone metric space.

*Definition 1.4 (see [16, 18, 19]). *Let be a -cone metric space, , and a sequence in .(1)-cone converges to whenever for every with , there exists such that for all . We denote this by cone-;(2) is a -cone Cauchy sequence whenever for every with , there exists such that for all ;(3) is -cone complete if every -cone Cauchy sequence in is -cone convergent in .

*Remark 1.5. *Clearly, a cone metric space in the sense of Huang and Zhang [1] is a special case of -cone metric spaces when is a -cone metric space with respect to a normal cone .

*Remark 1.6 (see [19, 22, 23]). *Let be a -cone metric space with a solid cone . The following properties are often used, particularly in the case when the underlying cone is nonnormal.(p1) If and , then ,(p2) If and , then ,(p3) If and , then ,(p4) If for each , then ,(p5) If for each , then ,(p6) If is with a cone , and if where and , then ,(p7) If , , and in locally convex , then there exists such that for all .

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

*Definition 1.7 (see [25]). *Let be a nonempty set, and let be a function satisfying the following axioms:(G1),(G2),(G3),(G4) (symmetric in all three variables),(G5). Then the function is called a generalized metric, or, more specifically a -metric on , and the pair is called a -metric space.

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

*Definition 1.8. *Let be a nonempty set and an ordered , and let be a function satisfying the following axioms:(G1) if and only if ,(G2),(G3),(G4) (symmetric in all three variables),(G5). Then the function is called a -generalized-cone metric, or, more specifically, a --cone metric on , and the pair is called a --cone metric space.

*Definition 1.9. *Let be a --cone metric space, , and a sequence in .(1)--cone converges to whenever, for every with , there exists such that for all . Here is called the limit of the sequence and is denoted by -cone-;(2) is a --cone Cauchy sequence whenever, for every with , there exists such that for all ;(3) is --cone complete if every --cone Cauchy sequence in is --cone convergent in .

Proposition 1.10. *Let be a --cone metric space, , and a sequence in . The following are equivalent:*(i)*--cone converges to ,*(ii)* as ,*(iii)* as ,*(iv)* as .*

In this paper, we also introduce the below concept of the ---cone-type function.

*Definition 1.11. *
One calls a ---cone-type function if the function satisfies the following condition for all and ; for all .

In this paeper, for a --cone metric space and for the family of subsets of , we introduce a new notion of the --cone metric with respect to , and we get a fixed result for the ---cone-type function in a complete --cone metric space . Our results generalize some recent results in the literature.

#### 2. Main Results

Let be a locally convex Hausdorff with its zero vector , a proper, closed and convex pointed cone in with , and a partial ordering with respect to . We introduce the below notion of the --cone metric with respect to --cone metric .

*Definition 2.1. *Let be a --cone metric space with a solid cone , and let be a collection of nonempty subsets of . A map is called a --cone metric with respect to if for any the following conditions hold:(),()(symmetry in all variables),,(),() one of the following is satisfied:(i),(ii),(iii).

We will prove that a --cone metric satisfies the conditions of .

Lemma 2.2. *Let be a --cone metric space with a solid cone , and let be a collection of nonempty subsets of , . If is a --cone metric with respect to , then pair is a --cone metric space.*

*Proof. * Let be a sequence such that for all and --. Take any and , . From , for each , there exists such that
Therefore, for each . By the closedness of , we conclude that .

Assume that . From H_{5}, we obtain for any . So .

Let . Assume that satisfy condition . Then, for each , there exists such that for all and . From , there exists a sequence satisfying for every . Obviously, for any and any and , we have
Now for each , there exists , such that . Consequently, we obtain that for each
Therefore,
In the case when or holds, we use the analogous method.

In the sequel, we denote by the class of functions satisfying the following conditions: is a ---cone-type-function; is subadditive, that is, for all .

Our main result is the following.

Theorem 2.3. *Let be a --cone complete metric space with a solid cone , let be a collection of nonempty closed subsets of , , and let be a --cone metric with respect to . If the mapping satisfies the condition that exists a such that for all holds
**
then has a fixed point in .*

*Proof. * Let us choose arbitrarily, and let be a sequence such that and . Let us choose arbitrarily and . If , then , and we are done. Assume that . Taking into account (2.5) and , there exists such that
Taking into account (2.5), (2.6), and and since , there exists such that
We continue in this manner. In general, for , , is chosen such that and
Since is arbitrary, letting and by the definition of the ---cone-type function, we obtain that

Next, we let , and we claim that the following result holds: for each ,ββthere is such that for all m, ,
We will prove (2.10) by contradiction. Suppose that (2.10) is false. Then there exists some such that for all , there are with satisfying(i) is even and is odd,(ii), and(iii) is the smallest even number such that conditions (i), (ii) hold.Since , by (ii), we have that and
It follows from ; let us choose arbitrarily such that
Taking into account (2.5), (2.11), and (2.12), we have that
Since is arbitrarily, letting and by letting , we have
a contradiction. So is a --cone Cauchy sequence. Since is a --cone complete metric space, is --cone convergent in and --. Thus, for every and sufficiently large , we have that
Since for , , by we obtain that for all there exists such that
Since , then for sufficiently large , we obtain that
which implies --. Since is closed, we obtain that .

For the case , , then and it is easy to get the following corollary.

Corollary 2.4. *Let be a --cone complete metric space with a solid cone , let be a collection of nonempty closed subsets of , , and let be a --cone metric with respect to . If the mapping satisfies the condition that exists such that for all holds
**
then has a fixed point in .*

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

#### Acknowledgment

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

#### References

- L.-G. Huang and X. Zhang, βCone metric spaces and fixed point theorems of contractive mappings,β
*Journal of Mathematical Analysis and Applications*, vol. 332, no. 2, pp. 1468β1476, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - M. Abbas and G. Jungck, βCommon fixed point results for noncommuting mappings without continuity in cone metric spaces,β
*Journal of Mathematical Analysis and Applications*, vol. 341, no. 1, pp. 416β420, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - S. Janković, Z. Kadelburg, S. Radonevic, and B. E. Rhoades, βAssad-Kirktype fixed point theorems for a pair of nonself mappings on cone metric
spaces,β
*Fixed Point Theory and Applications*, vol. 2009, Article ID 761086, 16 pages, 2009. View at Publisher Β· View at Google Scholar - Sh. Rezapour and R. Hamlbarani, βSome notes on the paper: ‘Cone metric spaces and fixed point theorems of contractive mappings’,β
*Journal of Mathematical Analysis and Applications*, vol. 345, no. 2, pp. 719β724, 2008. View at Publisher Β· View at Google Scholar - M. Arshad, A. Azam, and P. Vetro, βSome common fixed point results in cone metric spaces,β
*Fixed Point Theory and Applications*, vol. 2009, Article ID 493965, 11 pages, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Azam and M. Arshad, βCommon fixed points of generalized contractive maps in cone metric spaces,β
*Iranian Mathematical Society. Bulletin*, vol. 35, no. 2, pp. 255β264, 2009. View at Google Scholar Β· View at Zentralblatt MATH - C. Di Bari and P. Vetro, β
*ϕ*-pairs and common fixed points in cone metric spaces,β*Rendiconti del Circolo Matematico di Palermo*, vol. 57, no. 2, pp. 279β285, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - C. Di Bari and P. Vetro, βWeakly
*ϕ*-pairs and common fixed points in cone metric spaces,β*Rendiconti del Circolo Matematico di Palermo*, vol. 58, no. 1, pp. 125β132, 2009. View at Publisher Β· View at Google Scholar - J. Harjani and K. Sadarangani, βGeneralized contractions in partially ordered metric spaces and applications to ordinary differential equations,β
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 72, no. 3-4, pp. 1188β1197, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - R. H. Haghi and Sh. Rezapour, βFixed points of multifunctions on regular cone metric spaces,β
*Expositiones Mathematicae*, vol. 28, no. 1, pp. 71β77, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - D. Klim and D. Wardowski, βDynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces,β
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 71, no. 11, pp. 5170β5175, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Kadelburg, S. Radenović, and V. Rakočević, βRemarks on ‘Quasi-contraction on a cone metric space’,β
*Applied Mathematics Letters*, vol. 22, no. 11, pp. 1674β1679, 2009. View at Publisher Β· View at Google Scholar - Z. Kadelburg, M. Pavlović, and S. Radenović, βCommon fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces,β
*Computers & Mathematics with Applications*, vol. 59, no. 9, pp. 3148β3159, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - S. Rezapour, H. Khandani, and S. M. Vaezpour, βEfficacy of cones on topological vector spaces and application to common fixed points of multifunctions,β
*Rendiconti del Circolo Matematico di Palermo*, vol. 59, no. 2, pp. 185β197, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Sh. Rezapour and R. H. Haghi, βFixed point of multifunctions on cone metric spaces,β
*Numerical Functional Analysis and Optimization*, vol. 30, no. 7-8, pp. 825β832, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - W.-S. Du, βA note on cone metric fixed point theory and its equivalence,β
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 72, no. 5, pp. 2259β2261, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Azam, I. Beg, and M. Arshad, βFixed point in topological vector space-valued cone metric spaces,β
*Fixed Point Theory and Applications*, vol. 2010, Article ID 604084, 9 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - I. Beg, A. Azam, and M. Arshad, βCommon fixed points for maps on topological vector space valued cone metric spaces,β
*International Journal of Mathematics and Mathematical Sciences*, vol. 2009, Article ID 560264, 8 pages, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Kadelburg, S. Radenović, and V. Rakočević, βTopological vector space-valued cone metric spaces and fixed point theorems,β
*Fixed Point Theory and Applications*, vol. 2011, Article ID 170253, 17 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Kadelburg and S. Radenović, βCoupled fixed point results under
*tvs*–cone metric and*w*-cone-distance,β*Advanced in Fixed Point Theory*, vol. 2, no. 1, pp. 29β46, 2012. View at Google Scholar - L. J. Ćirić, H. Lakzian, and V. Rakoćević, βFixed point theorems for
*w*-cone distance contraction mappings in*tvs*-cone metric spaces,β*Fixed Point Theory and Applications*, vol. 2012, p. 3, 2012. View at Publisher Β· View at Google Scholar - S. Radenović, S. Simić, N. Cakić, and Z. Golubović, βA note on
*tvs*-cone metric fixed point theory,β*Mathematical and Computer Modelling*, vol. 54, no. 9-10, pp. 2418β2422, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Kadelburg, S. Radenović, and V. Rakočević, βA note on the equivalence of some metric and cone metric fixed point results,β
*Applied Mathematics Letters*, vol. 24, no. 3, pp. 370β374, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - B. C. Dhage, βGeneralized metric space and mapping with fixed point,β
*Bulletin of the Calcutta Mathematical Society*, vol. 84, pp. 329β336, 1992. View at Google Scholar - Z. Mustafa and B. Sims, βA new approach to generalized metric spaces,β
*Journal of Nonlinear and Convex Analysis*, vol. 7, no. 2, pp. 289β297, 2006. View at Google Scholar Β· View at Zentralblatt MATH