- Open Access
Multivalued fixed point theorems in tvs-cone metric spaces
© Azam and Mehmood; licensee Springer 2013
- Received: 5 April 2013
- Accepted: 28 June 2013
- Published: 12 July 2013
In this paper we extend the Kannan, Chatterjea and Zamfirescu theorems for multivalued mappings in a tvs-cone metric space without the assumption of normality on cones and generalize many results in literature.
- tvs-cone metric space
- non-normal cones
- multivalued contraction
- fixed points
The notion of cone metric space was introduced by Huang and Zhang in . They replaced the set of real numbers by an ordered Banach space and defined a cone metric space. They extended Banach fixed point theorems for contractive type mappings. Many authors [2–25] studied the properties of cone metric spaces and generalized important fixed point results of complete metric spaces. The concept of cone metric space in the sense of Huang-Zhang was characterized by Al-Rawashdeh et al. in . Indeed, is a cone metric space if and only if is an E-metric space, where E is a normed ordered space, with (, Theorem 3.8).
It is well known that the Banach contractions, Kannan mappings and Chatterjea mappings are independent in general. Zamfirescu  proved a remarkable fixed point theorem by combining the results of Banach, Kannan and Chatterjea. Afterwards, some authors investigated these results in many directions [29–34].
In the papers [35–39], the authors studied fixed point theorems for multivalued mappings in cone metric spaces. Seong and Jong  invented the generalized Hausdorff distance in a cone metric space and proved multivalued results in cone metric spaces. Shatanawi et al.  generalized it in the case of tvs-cone metric spaces. However, all these results presented in the literature [35–39] for the case of multivalued mappings in cone metric spaces are restricted to Banach contraction. In this paper, we have achieved the results for Kannan and Chatterjea contraction for multivalued mappings in tvs-cone metric spaces. We also extend the Zamfirescu theorem to multivalued mappings in tvs-cone metric spaces.
Let be a topological vector space with its zero vector θ. A nonempty subset P of is called a convex cone if and for . A convex cone P is said to be pointed (or proper) if ; and P is normal (or saturated) if has a base of neighborhoods of zero consisting of order-convex subsets. For a given cone , we define a partial ordering ≼ with respect to P by if and only if ; stands for and , while stands for , where intP denotes the interior of P. The cone P is said to be solid if it has a nonempty interior.
Now let us recall the following definitions and remarks.
Definition 2.1 
for all and if and only if ;
for all ;
for all .
The pair is then called a tvs-cone metric space.
Remark 2.1 
The concept of cone metric space is more general than that of metric space, because each metric space is a cone metric space, and a cone metric space in the sense of Huang and Zhang is a special case of tvs-cone metric spaces when is a tvs-cone metric space with respect to a normal cone P.
Definition 2.2 
tvs-cone converges to x if for every with there is a natural number such that for all . We denote this by ;
is a tvs-cone Cauchy sequence if for every with there is a natural number such that for all ;
is tvs-cone complete if every tvs-cone Cauchy sequence in X is tvs-cone convergent.
Remark 2.2 
The results concerning fixed points and other results, in the case of cone spaces with non-normal solid cones, cannot be provided by reducing to metric spaces, because in this case neither of the conditions of Lemmas 1-4 in  hold. Further, the vector cone metric is not continuous in the general case, i.e., from , it need not follow that .
If and , then .
If and , then .
If and , then .
If for each , then .
If for each , then .
In the sequel, denotes a locally convex Hausdorff topological vector space with its zero vector θ, P is a proper, closed and convex pointed cone in with , and ≼ denotes the induced partial ordering with respect to P.
Definition 3.1 
Definition 3.2 
Let be a cone metric space with a solid cone P. The cone P is complete if for every bounded above nonempty subset A of , supA exists in . Equivalently, the cone P is complete if for every bounded below nonempty subset A of , infA exists in .
An is called lower bound of T associated with . By we denote the set of all lower bounds T associated with . Moreover, is denoted by .
Definition 3.4 Let be a tvs-cone metric space with a solid cone P. The multivalued mapping is said to have greatest lower bound property (g.l.b. property) on X if the greatest lower bound of exists in for all .
Let us recall the following lemma, which will be used to prove our main Theorem 3.1.
Lemma 3.1 
Let . If , then .
Let and . If , then .
Let and let and . If , then .
For all and . Then if and only if there exist and such that .
Remark 3.1 
Let be a tvs-cone metric space. If and , then is a metric space. Moreover, for , is the Hausdorff distance induced by d. Also, for all .
Now, let us prove the following result which is a Kannan-type multivalued theorem in tvs-cone metric spaces.
then T has a fixed point in X.
Hence, . Since Tv is closed, so . □
In the following we provide a Chatterjea-type multivalued theorem in a tvs-cone metric space.
then T has a fixed point in X.
Therefore . Since Tv is closed, so . □
In the following we establish a Zamfirescu-type result in a tvs-cone metric space.
then T has a fixed point in X.
Proof (i): A special case of  when .
(ii): As .
Now, by Theorem 3.1, T has a fixed point in X.
Now, by Theorem 3.2, T has a fixed point in X. □
Hence T satisfies all the conditions of Theorem 3.3 to obtain a fixed point of T.
The authors would like to thank the editor and the referees for their valuable comments and suggestions which improved greatly the quality of this paper.
- Huang L, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 2007, 332: 1468–1476. 10.1016/j.jmaa.2005.03.087MathSciNetView ArticleGoogle Scholar
- Abbas M, Jungck G: Common fixed point results for noncommuting mappings without continuity in cone metric spaces. J. Math. Anal. Appl. 2008, 341: 416–420. 10.1016/j.jmaa.2007.09.070MathSciNetView ArticleGoogle Scholar
- Abbas M, Vetro P, Khan SH: On fixed points of Berinde’s contractive mappings in cone metric spaces. Carpath. J. Math. 2010, 26(2):121–133.MathSciNetGoogle Scholar
- Agarwal RP, Meehan M, O’Regan D: Fixed Point Theory and Applications. Cambridge University Press, Cambridge; 2001.MATHView ArticleGoogle Scholar
- Arshad M, Azam A, Vetro P: Some common fixed point results in cone metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 493965Google Scholar
- Azam A, Arshad M, Beg I: Common fixed points of two maps in cone metric spaces. Rend. Circ. Mat. Palermo 2008, 57: 433–441. 10.1007/s12215-008-0032-5MathSciNetView ArticleGoogle Scholar
- Azam A, Beg I, Arshad M: Fixed point in topological vector space-valued cone metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 9Google Scholar
- Azam A, Arshad M, Beg I: Existence of fixed points in complete cone metric spaces. Int. J. Mod. Math. 2010, 5(1):91–99.MathSciNetGoogle Scholar
- Azam A: Fuzzy fixed points of fuzzy mappings via a rational inequality. Hacet. J. Math. Stat. 2011, 40(3):421–431.MathSciNetGoogle Scholar
- Ciric L, Samet B, Vetro C, Abbas M: Fixed point results for weak contractive mappings in ordered K -metric spaces. Fixed Point Theory 2012, 13(1):59–72.MathSciNetGoogle Scholar
- Cho SH, Bae JS: Fixed points and variational principle with applications to equilibrium problems. J. Korean Math. Soc. 2013, 50: 95–109. 10.4134/JKMS.2013.50.1.095MathSciNetView ArticleGoogle Scholar
- Cho SH, Bae JS: Variational principles on cone metric spaces. Int. J. Pure Appl. Math. 2012, 77: 709–718.Google Scholar
- Cho SH, Bae JS, Na KS: Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings in cone metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 133Google Scholar
- Di Bari C, Vetro P: Common fixed points in cone metric spaces for MK-pairs and L-pairs. Ars Comb. 2011, 99: 429–437.MathSciNetGoogle Scholar
- Haghi RH, Rezapour S, Shahzad N: Some fixed point generalizations are not real generalizations. Nonlinear Anal., Theory Methods Appl. 2011, 74(5):1799–1803. 10.1016/j.na.2010.10.052MathSciNetView ArticleGoogle Scholar
- Janković S, Kadelburg Z, Radenović S: On cone metric spaces. A survey. Nonlinear Anal. 2011, 74: 2591–2601. 10.1016/j.na.2010.12.014MathSciNetView ArticleGoogle Scholar
- Khani M, Pourmahdian M: On the metrizability of cone metric spaces. Topology Appl. 2011, 158(2):190–193. 10.1016/j.topol.2010.10.016MathSciNetView ArticleGoogle Scholar
- Li Z, Jiang S: On fixed point theory of monotone mappings with respect to a partial order introduced by a vector functional in cone metric spaces. Abstr. Appl. Anal. 2013., 2013: Article ID 349305Google Scholar
- Nieto JJ, Pouso RL, Rodríguez-López R: Fixed point theorems in ordered abstract spaces. Proc. Am. Math. Soc. 2007, 135: 2505–2517. 10.1090/S0002-9939-07-08729-1View ArticleGoogle Scholar
- Rezapour S: Best approximations in cone metric spaces. Math. Moravica 2007, 11: 85–88.MathSciNetGoogle Scholar
- Rezapour S, Hamlbarani R: Some notes on paper ‘Cone metric spaces and fixed point theorems of contractive mappings’. J. Math. Anal. Appl. 2008, 345: 719–724. 10.1016/j.jmaa.2008.04.049MathSciNetView ArticleGoogle Scholar
- Rezapour SH, Khandani H, Vaezpour SM: Efficacy of cones on topological vector spaces and application to common fixed points of multifunctions. Rend. Circ. Mat. Palermo 2010, 59: 185–197. 10.1007/s12215-010-0014-2MathSciNetView ArticleGoogle Scholar
- Simi S: A note on Stone’s, Baire’s, Ky Fan’s and Dugundji’s theorem in tvs-cone metric spaces. Appl. Math. Lett. 2011, 24: 999–1002. 10.1016/j.aml.2011.01.014MathSciNetView ArticleGoogle Scholar
- Vetro P, Azam A, Arshad M: Fixed point results in cone metric spaces for contractions of Zamfirescu type. Indian J. Math. 2010, 52(2):251–261.MathSciNetGoogle Scholar
- Wanga S, Guo B: Distance in cone metric spaces and common fixed point theorems. Appl. Math. Lett. 2011, 24: 1735–1739. 10.1016/j.aml.2011.04.031MathSciNetView ArticleGoogle Scholar
- Al-Rawashdeh A, Shatanawi W, Khandaqji M: Normed ordered and E -metric spaces. Int. J. Math. Math. Sci. 2012., 2012: Article ID 272137 10.1155/2012/272137Google Scholar
- Beg I, Azam A, Arshad M: Common fixed points for maps on topological vector space valued cone metric spaces. Int. J. Math. Math. Sci. 2009., 2009: Article ID 560264 10.1155/2009/560264Google Scholar
- Chatterjea SK: Fixed-point theorems. C. R. Acad. Bulgare Sci. 1972, 25: 727–730.MathSciNetGoogle Scholar
- Zamfirescu T: Fixed point theorems in metric spaces. Arch. Math. 1972, 23: 292–298. 10.1007/BF01304884MathSciNetView ArticleGoogle Scholar
- Beg I, Azam A: Fixed points of asymptotically regular multivalued mappings. J. Aust. Math. Soc. 1992, 53: 313–326. 10.1017/S1446788700036491MathSciNetView ArticleGoogle Scholar
- Berinde V, Berinde M: On Zamfirescu’s fixed point theorem. Rev. Roum. Math. Pures Appl. 2005, 50: 443–453.MathSciNetGoogle Scholar
- Ilić D, Pavlović V, Rakočević V: Extensions of the Zamfirescu theorem to partial metric spaces. Math. Comput. Model. 2012, 55: 801–809. 10.1016/j.mcm.2011.09.005View ArticleGoogle Scholar
- Neammanee K, Kaewkhao A: Fixed point theorems of multi-valued Zamfirescu mapping. J. Math. Res. 2010, 2: 150–156.View ArticleGoogle Scholar
- Raphaeli P, Pulickunnel S: Fixed point theorems for T. Zamfirescu operators. Kragujev. J. Math. 2012, 36: 199–206.Google Scholar
- Cho SH, Bae JS: Fixed point theorems for multivalued maps in cone metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 87Google Scholar
- Klim DW: Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces. Nonlinear Anal. 2009, 71: 5170–5175. 10.1016/j.na.2009.04.001MathSciNetView ArticleGoogle Scholar
- Latif A, Shaddad FY: Fixed point results for multivalued maps in cone metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 941371Google Scholar
- Rezapour S, Haghi RH: Fixed points of multifunctions on cone metric spaces. Numer. Funct. Anal. Optim. 2009, 30: 1–8. 10.1080/01630560802678549MathSciNetView ArticleGoogle Scholar
- Shatanawi W, Rajic VC, Radenovic S, Al-Rawashdeh A: Mizoguchi-Takahashi-type theorems in tvs-cone metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 106Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.