Research Article | Open Access

Ing-Jer Lin, Chi-Ming Chen, Mirko Jovanović, Tzi-Huei Wu, "Fixed Points and Endpoints of Set-Valued Contractions in Cone Metric Spaces", *Abstract and Applied Analysis*, vol. 2012, Article ID 632628, 14 pages, 2012. https://doi.org/10.1155/2012/632628

# Fixed Points and Endpoints of Set-Valued Contractions in Cone Metric Spaces

**Academic Editor:**Gabriel Turinici

#### Abstract

The purpose of this paper is to present the fixed points and endpoints of set-valued contractions concerning with the stronger Meir-Keeler cone-type mappings in cone metric spcaes. Our results generalize the recent results of Kadelburg and Radenović, 2011; Wardowski, 2009.

#### 1. Introduction and Preliminaries

Throughout this paper, by , we denote the sets of all nonnegative real numbers and all real numbers, respectively, while is the set of all natural numbers. Let be a metric space, a subset of , and a map. We say is contractive if there exists such that, for all , The well-known Banach fixed-point theorem asserts that if , is contractive and is complete, then has a unique fixed-point in . It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, Kannan [2] and Chatterjea [3] introduced two conditions that can replace (1.1) in Banach's theorem as follows:

Kannan [2]: There exists such that, for all ,

Chatterjea [3]: There exists such that for all ,

After these three conditions, many papers have been written generalizing some of the conditions (1.1), (1.2), and (1.3). In 1969, Boyd and Wong [4] showed the following fixed-point theorem.

Theorem 1.1 (see [4]). *Let be a complete metric space and a map. Suppose there exists a function satisfying , for all and is right upper semicontinuous such that
**
Then has a unique fixed-point in .*

Later, Meir-Keeler [5], using a result of Chu and Diaz [6], extended Boyd-Wong's result to mappings satisfying the following more general condition: and Meir-Keeler proved the following very interesting fixed-point theorem which is a generalization of the Banach contraction principle.

Theorem 1.2 (Meir-Keeler [5]). *Let be a complete metric space and let be a Meir-Keeler contraction; that is, for every , there exists such that implies for all . Then has a unique fixed-point.*

Subsequently, some authors worked on this notion of Meir-Keeler contraction (e.g., [7–10]). In this paper, we introduce a new notion of the stronger Meir-Keeler-type mapping , , as follows.

*Definition 1.3. *A mapping , is called a stronger Meir-Keeler-type mapping in a metric space , if for each , there exists such that for with , there exists such that .

*Example 1.4. *Let be a metric space, and define , by
Then is a stronger Meir-Keeler-type mapping.

The existence of fixed-points for various multi valued contractive mappings had been studied by many authors under different conditions. In 1969, Nadler [11] extended the famous Banach contraction principle from single-valued mapping to multi valued mapping and proved the following fixed-point theorem for multi valued contraction. Let denote the collection of all nonempty subsets of , the collection of all nonempty bounded subsets of , the collection of all nonempty closed subsets of , the collection of all nonempty closed and bounded subsets of , and the collection of all nonempty sequentially compact subsets of .

Theorem 1.5 (see [11]). *Let be a complete metric space, and, be a mapping from into . Assume that there exists such that
**
Then has a fixed-point in .*

In 1989, Mizoguchi-Takahashi [12] proved the following fixed-point theorem.

Theorem 1.6 (see [12]). *Let be a complete metric space and a map from into . Assume that
**
for all , where satisfies for all . Then has a fixed-point in .*

*Remark 1.7. * It is clear that if the function , satisfies
then is also a stronger Meir-Keeler-type function.

In 2003, Rus [13] introduced the notion of endpoints (strict fixed-points) and proved some results on strict fixed-point theory on multi valued operator. An element is said to be a fixed-point of , if . If , then is called an endpoint of . We will denote the sets of all fixed-points and endpoints of by and , respectively. The following theorems were the main results of Rus [13].

Theorem 1.8 (see [13]). *Let be a complete metric space and a multi valued operator. Suppose that *(i)* for all *(ii)*there exist a comparison function (see [14]) and a Picard sequence , such that
where . ** Then as and is a unique strict fixed-point of . *

Recently, Wardowski [15] proved the following theorems concerning fixed-points and endpoints of set-valued contractions in complete cone metric spaces.

*Definition 1.9. *A function is called lower semicontinuous, if, for any sequence and ,

If is a cone metric space and , then, for , we denote

Theorem 1.10 (see [15]). *Let be a complete cone metric space, let be a normal cone with normal constant , and let . Assume that a function defined by , is lower semicontinuous. If there exist , such that
**
then .*

Theorem 1.11 (see [15]). * Let be a complete cone metric space, let be a normal cone with normal constant , and let . Assume that a function defined by , is lower semicontinuous. The following hold*(a)* If there exist , such that
then .*(b)* If there exist , such that
then .*

In 1997, Zabrejko [16] introduced the -metric and -normed linear spaces and showed the existence and uniqueness of fixed-points for operators which act in -metric or -normed linear spaces. Huang and Zhang [17] 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 like Abbas and Jungck [18] have generalized the results of Huang and Zhang [17] and studied the existence of common fixed-points of a pair of self-mappings satisfying a contractive type condition in the framework of normal cone metric spaces. However, authors like Rezapour and Hamlbarani [19] studied the existence of common fixed-points of a pair of self- and non-self-mappings satisfying a contractive type condition in the situation in which the cone does not need to be normal. Many authors studied this subject and many results on fixed-point theory are proved (see, e.g., [12, 19–39]).

*Definition 1.12 (see [17]). *Let be a real Banach space and a nonempty subset of . , where denotes the zero element of , is called a cone if and only if: (i) is closed;(ii), , ;(iii) and .

For given a cone , we define a partial ordering with respect to by or if and only if for all . The real Banach space equipped with the partial order induced by is denoted by . We shall write to indicate that but , while or will stand for , where denotes the interior of .

Proposition 1.13 (see [40]). *Suppose is a cone in a real Banach space . Then one has the following. *(i)If and , then .(ii)If and , then .(iii)If and , then . (iv)If and for each , then .

Proposition 1.14 (see [41]). *Suppose , and . Then there exists such that for all .*

The cone is called normal if there exists a real number such that, for all ,

The least positive number , satisfying the above is called the normal constant of .

The cone is called regular if every increasing sequence which is bounded from above is convergent, that is, if is a sequence such that for some , then there is such that as . Equivalently, the cone is regular if and only if every decreasing sequence which is bounded from below is convergent. It is well known that a regular cone is a normal cone.

*Definition 1.15 (see [17]). *Let be a nonempty set, a real Banach space, and a cone in . Suppose that the mapping satisfies(i), for all ; (ii) if and only if ; (iii), for all ; (iv), for all . Then is called a cone metric on , and is called a cone metric space.

*Definition 1.16 (see [17]). * Let be a cone metric space, and let be a sequence in and . If for every with , there is such that
then is said to be convergent and converges to .

*Definition 1.17 (see [17]). *Let be a cone metric space, and let be a sequence in . We say that is a Cauchy sequence if, for any with , there is such that

*Definition 1.18 (see [17]). *Let be a cone metric space. If every Cauchy sequence is convergent in , then is called a complete cone metric space.

*Remark 1.19 (see [17]). * If is a normal cone, then converges to if and only if as . Further, in the case is a Cauchy sequence if and only if as .

The purpose of this paper is to present the fixed-points and endpoints of set-valued contractions concerning with the stronger Meir-Keeler cone-type mappings in cone metric spcaes. Our results generalize the recent results of Kadelburg and Radenović [42–44] and Wardowski [15].

#### 2. Main Results

In this section, we first introduce the following notion of stronger Meir-Keeler cone-type mapping.

*Definition 2.1. *Let be a cone metric space with cone , and let
Then the function is called a stronger Meir-Keeler cone-type mapping, if, for each , there exists such that, for with , there exists such that .

We now are in a position to present the following fixed-point theorem.

Theorem 2.2. * Let be a complete cone metric space, and let be a normal cone in and . Assume that a function defined by , is lower semicontinuous. The following holds.*(*)* There exists a stronger Meir-Keeler cone-type mapping
such that
**Then .*

* Proof. *Given , we define the sequence recursively as follows. Take any . By , there exist and such that
Thus,
If , then , and we are done. Assume that . Put . By the definition of the stronger Meir-Keeler cone-type mapping , corresponding to use, there exists and with such that . Taking into account (2.5), we have that

By the same process, for , there exist and such that
Thus,
Now, put . By the definition of the stronger Meir-Keeler cone-type mapping , corresponding to use, there exists and with such that . And we put and . Taking into account (2.9), we have that

We continue in this manner. Inductively, for , there exist and such that
Thus,
Put . By the definition of the stronger Meir-Keeler cone-type mapping , corresponding to use, there exist and with such that . And we put and . Taking into account (2.14), we have that

Let such that . Taking into account (2.15) and (2.16), we obtain that
Let be given. Since and , we get
Thus, there exists such that, for all ,
and we also conclude that for all . This implies that is a Cauchy sequence. Since is complete, there exists such that

By the definition of , since , there exists a sequence such that , for all . From the convergence of the sequence and from the lower semicontinuity of the function , we obtain that
Thus,
We claim that . To prove this, on the contrary, assume that . Then by (2.22), there exists a sequence such that , and hence . Thus, for any ,
Let be given. Then there exists such that for all , and , and so
This implies that is a Cauchy sequence in . Since is complete, there exists such that
By the closedness of the , we deduce that . Then, for any ,
Let be given. Since and , we can deduce that there exists such that, for all , and , and hence
By Proposition 1.13, we obtain that that is, , which is a contradiction. Therefore, .

Applying Remark 1.7, it is easy to establish the following corollary.

Corollary 2.3. *Let be a complete cone metric space, a normal cone in , and . Assume that a function defined by :, is lower semicontinuous. The following holds.*(**)* There exists a mapping
such that
**Then .*

Corollary 2.4. * Let be a complete cone metric space, a normal cone in , and . Assume that a function defined by , is lower semicontinuous. The following holds.*(***)*There exists a stronger Meir-Keeler cone-type mapping
such that
** Then .*

*Example 2.5. *Let , , where , , , defined by , , , with for all and if and only if . Let be defined by , , and . Then , , . Take for all . Calculation shows that, for ,
that is, according to Corollary 2.4 we have that .

Theorem 2.6. * Let be a complete cone metric space, a normal cone in , and . Assume that a function defined by , , is lower semicontinuous. The following hold:*(A)* There exists a stronger Meir-Keeler cone-type mapping
such that
**Then .*(B)* There exists a stronger Meir-Keeler cone-type mapping
such that
**Then .*

*Proof. *(A) By the same proof of the part (i) of Theorem 3.2 in [15], we have that for all . The remainder proof is similar to Theorem 2.2, we can deduce that .

(B) By (A), we obtain that the existence of such that, . Taking any , we have that, for all , . Since , we get , and hence , which gives . Thus .

*Example 2.7. *Let , , . Let for all , and let , . Then , . Let for all . It is easy to check that all conditions of Theorem 2.6 are satisfied and that .

Corollary 2.8. *Let be a complete cone metric space, a normal cone in , and . Assume that a function defined by , is lower semicontinuous. The following hold:*(C)* There exists a mapping
such that
**Then .*(D)* There exists a mapping
such that
**Then .*

#### References

- S. Banach, “Sur les operations dans les ensembles abstraits et leur application aux equations integerales,”
*Fundamenta Mathematicae*, vol. 3, pp. 133–181, 1922. View at: Google Scholar - R. Kannan, “Some results on fixed points,”
*Bulletin of the Calcutta Mathematical Society*, vol. 60, pp. 71–76, 1968. View at: Google Scholar | Zentralblatt MATH - S. K. Chatterjea, “Fixed-point theorems,”
*Comptes Rendus de l'Académie Bulgare des Sciences*, vol. 25, pp. 727–730, 1972. View at: Google Scholar | Zentralblatt MATH - D. W. Boyd and J. S. W. Wong, “On nonlinear contractions,”
*Proceedings of the American Mathematical Society*, vol. 20, pp. 458–464, 1969. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. Meir and E. Keeler, “A theorem on contraction mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 28, pp. 326–329, 1969. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. C. Chu and J. B. Diaz, “Remarks on a generalization of Banach's principle of contraction mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 11, pp. 440–446, 1965. View at: Publisher Site | Google Scholar | Zentralblatt MATH - C. Di Bari, T. Suzuki, and C. Vetro, “Best proximity points for cyclic Meir-Keeler contractions,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 69, no. 11, pp. 3790–3794, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Janković, Z. Kadelburg, S. Radenović, and B. E. Rhoades, “Assad-Kirk-type fixed point theorems for a pair of nonself mappings on cone metric spaces,”
*Fixed Point Theory and Applications*, vol. 2009, Article ID 761086, 16 pages, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - T. Suzuki, “Fixed-point theorem for asymptotic contractions of Meir-Keeler type in complete metric spaces,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 64, no. 5, pp. 971–978, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - T. Suzuki, “Moudafi's viscosity approximations with Meir-Keeler contractions,”
*Journal of Mathematical Analysis and Applications*, vol. 325, no. 1, pp. 342–352, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. B. Nadler Jr., “Multi-valued contraction mappings,”
*Pacific Journal of Mathematics*, vol. 30, pp. 475–488, 1969. View at: Google Scholar | Zentralblatt MATH - N. Mizoguchi and W. Takahashi, “Fixed point theorems for multivalued mappings on complete metric spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 141, no. 1, pp. 177–188, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH - I. A. Rus, “Strict fixed point theory,”
*Fixed Point Theory*, vol. 4, no. 2, pp. 177–183, 2003. View at: Google Scholar | Zentralblatt MATH - I. A. Rus,
*Generalized Contractions and Applications*, Cluj University Press, Cluj-Napoca Romania, 2001. - D. Wardowski, “Endpoints and fixed points of set-valued contractions in cone metric spaces,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 71, no. 1-2, pp. 512–516, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - P. P. Zabrejko, “$K$-metric and $K$-normed linear spaces: survey,”
*Collectanea Mathematica*, vol. 48, no. 4–6, pp. 825–859, 1997. View at: Google Scholar - 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. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Abbas and G. Jungck, “Common fixed point results for noncommuting mappings without continuity in cone metric spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 341, no. 1, pp. 416–420, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Rezapour and R. Hamlbarani, “Cone metric spaces and fixed point theorems of contractive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 345, no. 2, pp. 719–724, 2008. View at: Publisher Site | Google Scholar - A. Amini-Harandi and M. Fakhar, “Fixed point theory in cone metric spaces obtained via the scalarization method,”
*Computers & Mathematics with Applications*, vol. 59, no. 11, pp. 3529–3534, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Arshad, A. Azam, and P. Vetro, “Some common fixed point results in cone metric spaces,”
*Fixed Point Theory and Applications*, vol. 2009, Article ID 493965, 11 pages, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Asadi, H. Soleimani, and S. M. Vaezpour, “An order on subsets of cone metric spaces and fixed points of set-valued contractions,”
*Fixed Point Theory and Applications*, vol. 2009, Article ID 723203, 8 pages, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. Azam and M. Arshad, “Common fixed points of generalized contractive maps in cone metric spaces,”
*Bulletin of the Iranian Mathematical Society*, vol. 35, no. 2, pp. 255–264, 2009. View at: Google Scholar | Zentralblatt MATH - C. Di Bari and P. Vetro, “$\phi $-pairs and common fixed points in cone metric spaces,”
*Rendiconti del Circolo Matematico di Palermo. Second Series*, vol. 57, no. 2, pp. 279–285, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - C. Di Bari and P. Vetro, “Weakly $\phi $-pairs and common fixed points in cone metric spaces,”
*Rendiconti del Circolo Matematico di Palermo. Second Series*, vol. 58, no. 1, pp. 125–132, 2009. View at: Publisher Site | Google Scholar - C. Di Bari, R. Saadati, and P. Vetro, “Common fixed points in cone metric spaces for $CJM$-pairs,”
*Mathematical and Computer Modelling*, vol. 54, no. 9-10, pp. 2348–2354, 2011. View at: Publisher Site | Google Scholar - H. S. Ding, Z. Kadelburg, E. Karapinar, and S. Radenović, “Common fixed points of weak contractions in cone metric spaces,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 793862, 18 pages, 2012. View at: Publisher Site | Google Scholar - J. Harjani and K. Sadarangani, “Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 72, no. 3-4, pp. 1188–1197, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. H. Haghi and S. Rezapour, “Fixed points of multifunctions on regular cone metric spaces,”
*Expositiones Mathematicae*, vol. 28, no. 1, pp. 71–77, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. Huang, C. Zhu, and X. Wen, “A common fixed point theorem in cone metric spaces,”
*International Journal of Mathematical Analysis*, vol. 4, no. 13–16, pp. 721–726, 2010. View at: Google Scholar | Zentralblatt MATH - S. Janković, Z. Kadelburg, and S. Radenović, “On cone metric spaces: a survey,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 74, no. 7, pp. 2591–2601, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z. Kadelburg, S. Radenović, and V. Rakočević, “A note on the equivalence of some metric and cone metric fixed point results,”
*Applied Mathematics Letters*, vol. 24, no. 3, pp. 370–374, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - D. Klim and D. Wardowski, “Dynamic processes and fixed points of set-valued nonlinear contractions in cone metric spaces,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 71, no. 11, pp. 5170–5175, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - P. D. Proinov, “A unified theory of cone metric spaces and its applications to the fixed point theory,” submitted, http://arxiv.org/abs/1111.4920. View at: Google Scholar
- S. Rezapour, R. H. Haghi, and N. Shahzad, “Some notes on fixed points of quasi-contraction maps,”
*Applied Mathematics Letters*, vol. 23, no. 4, pp. 498–502, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Rezapour, H. Khandani, and S. M. Vaezpour, “Efficacy of cones on topological vector spaces and application to common fixed points of multifunctions,”
*Rendiconti del Circolo Matematico di Palermo. Second Series*, vol. 59, no. 2, pp. 185–197, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Rezapour and R. H. Haghi, “Fixed point of multifunctions on cone metric spaces,”
*Numerical Functional Analysis and Optimization*, vol. 30, no. 7-8, pp. 825–832, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - K. Włodarczyk, R. Plebaniak, and C. Obczyński, “Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 72, no. 2, pp. 794–805, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z. Zhao and X. Chen, “Fixed points of decreasing operators in ordered Banach spaces and applications to nonlinear second order elliptic equations,”
*Computers & Mathematics with Applications*, vol. 58, no. 6, pp. 1223–1229, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Rezapour, M. Drafshpour, and R. Hamlbarani, “A review on topological properties of cone metric spaces,” in
*Proceedings of the Analysis, Topology and Applications (ATA '08)*, Vrnjacka Banja, Serbia, 2008. View at: Google Scholar - G. Jungck, S. Radenović, S. Radojević, and V. Rakočević, “Common fixed point theorems for weakly compatible pairs on cone metric spaces,”
*Fixed Point Theory and Applications*, vol. 2009, Article ID 643840, 13 pages, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z. Kadelburg and S. Radenović, “Meir-Keeler-type conditions in abstract metric spaces,”
*Applied Mathematics Letters*, vol. 24, no. 8, pp. 1411–1414, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z. Kadelburg and S. Radenović, “Some results on set-valued contractions in abstract metric spaces,”
*Computers & Mathematics with Applications*, vol. 62, no. 1, pp. 342–350, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Radenović and Z. Kadelburg, “Some results on fixed points of multifunctions on abstract metric spaces,”
*Mathematical and Computer Modelling*, vol. 53, no. 5-6, pp. 746–754, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH

#### Copyright

Copyright © 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.