Abstract and Applied Analysis

Volume 2014 (2014), Article ID 653841, 8 pages

http://dx.doi.org/10.1155/2014/653841

Research Article

Common Fixed Point Theorems of Contractions in Partial Cone Metric Spaces over Nonnormal Cones

Zhilong Li1,2 and Shujun Jiang³

¹School of Statistics, Jiangxi University of Finance and Economics, Nanchang 330013, China

²Research Center of Applied Statistics, Jiangxi University of Finance and Economics, Nanchang 330013, China

³Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang 330013, China

Received 21 November 2013; Accepted 18 March 2014; Published 10 April 2014

Academic Editor: Sehie Park

Copyright © 2014 Zhilong Li and Shujun Jiang. 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.

We prove some common fixed point theorems of contractions restricted with variable positive linear bounded mappings in -complete partial cone metric spaces over nonnormal cones and present some examples to support the usability of our results.

1. Introduction

In 2007, Huang and Zhang [1] introduced cone metric spaces, being unaware that they already existed under the name -metric and -normed spaces that were introduced and used in the middle of the 20th century in [2–9]. In both cases, the set of real numbers was replaced by an ordered Banach space . However, Huang and Zhang went further and defined the convergence via interior points of the cone by which the order in is defined. This approach allows the investigation of cone spaces in the case that the cone is not necessarily normal. Since then, there were many references concerned with fixed point results and common fixed point results in cone metric spaces over a nonnormal cone (see [10–18]). In 2012, based on the definition of cone metric spaces and partial metric spaces introduced by Matthews [19], Sonmez [20, 21] defined a partial cone metric space and proved some fixed point theorems of contractions restricted with constants in complete partial cone metric spaces over normal cones. Recently, without using the normality of the cone, Malhotra et al. [22] and Jiang and Li [23] extended the results of [20, 21] to -complete partial cone metric spaces. In addition, the contractions considered in [23] are not necessarily restricted with constants but restricted with positive linear bounded mappings.

In this paper, we prove some common fixed point theorems of contractions restricted with variable positive linear bounded mappings in -complete partial cone metric spaces over nonnormal cones, which improve the recent results of [22, 23].

2. Preliminaries

Let be a topological vector space. A cone of is a nonempty closed subset of such that for each and each , and , where is the zero element of . A cone of determines a partial order on by for each . In this case, is called an ordered topological vector space.

A cone of a topological vector space is solid if int , where int  is the interior of . For each with int , we write . Let be a solid cone of a topological vector space . A sequence of weakly converges [22] to (denote ) if, for each int , there exists a positive integer such that for all .

A subset of a topological vector space is order-convex if for each with , where . An ordered topological vector space is order-convex if it has a base of neighborhoods of consisting of order-convex subsets. In this case, the cone is said to be normal. In the case of a normed vector space, this condition means that the unit ball is order-convex, which is equivalent to the condition that there is some positive number such that and implies that , and the minimal is called a normal constant of . Another equivalent condition is that

It is not hard to conclude from (1) that is a nonnormal cone of a normed vector space if and only if there exist sequences such that

which implies that the Sandwich theorem does not hold. However, the Sandwich theorem holds in the sense of weak convergence even if is a nonnormal cone.

Lemma 1 (Sandwich theorem). Let be a solid cone of a topological vector space and . If

and there exists some such that and , then .

Proof. By and , for each int , there exists some positive integer such that, for all ,

Thus, by (3) and (4), we have for all ; that is, . The proof is completed.

The following lemma is needed in further arguments, which directly follows from Lemma 1 and Remark 1 of [23].

Lemma 2. Let be a solid cone of a normed vector space . Then, for each sequence , implies . Moreover, if is normal, then implies .

Let be a cone of a normed vector space and . The mapping is said to be a positive linear bounded mapping if , for each , and there exists some positive real number such that . In the sequel, and will denote the family of all positive linear bounded mappings and the identity mapping, respectively.

Lemma 3. Let be a solid cone of a normed vector space , and . If and , then .

Proof. Let , for all , where

It is clear that for all , and hence, for all , the inverse of exists (denoted by ). It follows from that for all for all , and then for all . By Lemma 2 and , for each int, there exists some positive integer such that for all . Note that for all implies that for all and each ; then, for all ; that is, . The proof is completed.

Let be a nonempty set and let be a cone of a topological vector space . A partial cone metric on is a mapping such that, for each ,

(p1);(p2);(p3);(p4).

The pair is called a partial cone metric space over . A partial cone metric on over a solid cone generates a topology on which has a base of the family of open -balls , where for each and each int .

Let be a partial cone metric space over a solid cone of a topological vector space . A sequence of converges to (denoted by ) if . A sequence of is -Cauchy, if . The partial cone metric space is -complete, if each -Cauchy sequence of converges to a point such that . Every complete partial cone metric space is -complete, but the converse may not be true (see [23]).

3. Common Fixed Point Theorems

Let be a partial cone metric space. The mappings are called contractions restricted with variable positive linear bounded mappings if there exist such that

In particular, if (6) holds with and , then and are called contractions restricted with positive linear bounded mappings.

We first present a common fixed point theorem of contractions restricted with variable positive linear bounded mappings in a partial cone metric space over a nonnormal cone. In the sequel, will denote the set of all nonnegative integer numbers.

Theorem 4. Let be a -complete partial cone metric space over a solid cone of a normed vector space , and let be contractions restricted with variable positive linear bounded mappings. If

and , where denotes the spectral radius of linear bounded mappings,

where and denote the inverses of and , respectively. Then, and have a common fixed point in . Moreover, if

then and have a unique common fixed point such that, for each , , where is defined by

Proof. For each , by (7), the inverses of and exist. Then, it is clear that and are meaningful, and so are well defined. Moreover, by Neumann’s formula,

which together with implies that , and hence . By (6), (11), (p4), and ,

and so

Act the above inequality with ; then, by ,

Similarly, by (6), (p3), (p4), and ,

and so

Act the above inequality with ; then, by ,

Moreover, by (15), (18), and ,

In the following, we will prove that

For all , we have four cases: (i) ; (ii) ; (iii) ; and (iv) , where and are two nonnegative integers such that . We only show that (20) holds for case (i); the proofs of the other three cases are similar.

It follows from (p4) and (19) that

By ,

which implies that , and hence by Lemma 2. Thus, by (21) and Lemma 1, ; that is, (20) holds. It is proved that is a -Cauchy sequence in , and so by the -completeness of , there exists such that and ; that is,

For all , by (6) and (p4),

and so

Act the above inequality with ; then, by ,

where and . It is clear that and by and . Then, it follows from Lemma 3 and (23) that

which together with Lemma 1 and (26) implies that . Therefore, by (p1) and (p3). Similarly, we can show that . Hence, is a common fixed point of and .

Now, we show the uniqueness of fixed point. Let and be two common fixed points of and . Then, by (6), (p3), and ,

and so

It follows from (9) that the inverse of exists (denoted by ), and by Neumann’s formula. Act (29) with ; then, , and hence by (p1) and (p3). The proof is completed.

Remark 5. Theorem 3 of [23] is a special case of Theorem 4 with and , where are nonnegative numbers such that .

Note that Theorem 4 is still valid if are replaced with nonnegative bounded real functions; then, we have the following corollary for which is not necessarily confined to a normed vector space.

Corollary 6. Let be a -complete partial cone metric space over a solid cone of a topological vector space and . Assume that there exist four nonnegative bounded functions such that

If

and , where

Then, and have a unique common fixed point such that, for each , , where is defined by (11).

Corollary 7. Let be a -complete partial cone metric space over a solid cone of a normed vector space , and let be contractions restricted with positive linear bounded mappings. If

then and have a unique common fixed point such that, for each , , where is defined by (11).

Proof. Let . It is easy to check that (6) holds with , , where , , and . By (33) and Neumann’s formula,

for each ; that is, and . Note that both (7) and (10) hold with by (33); then, the conclusion directly follows from Theorem 4. The proof is completed.

Note that (33) hold naturally if . In this case, Corollary 7 holds true.

The following common fixed point theorem improves Theorem 2 of [23].

Theorem 8. Let be a -complete partial cone metric space over a solid cone of a normed vector space and . Assume that there exists such that

If , then and have a unique common fixed point such that, for each , there exists some positive integer such that , where is defined by

Proof. By and Gelfand’s formula, there exists such that , which implies that there exists a positive integer such that

By (35), (36), and (p2),

and so

which together with implies that

Thus, by (p4),

It follows from (37) that

which implies , and hence by Lemma 2. Therefore, by Lemma 1 and (41), we get is a -Cauchy sequence in . Then, by analogy with the proof of Theorem 4, by , and Lemma 3, we can prove that there exists some with such that , and is the unique common fixed point of and . For this , we have and ; that is, and are fixed points of and , respectively. It follows from (35) and that , and hence by (p1) and (p3). This shows that is a common fixed point of and . Note that is the unique common fixed point of and ; then, ; that is, is a common fixed point of and . Moreover, it is easy to show that is the unique common fixed point of and by and . The proof is completed.

Example 9. Let with the norm , and , which is nonnormal solid cone [24]. Define a mapping by

It follows from Example 2 of [22] that is a partial cone metric space. Let for each and , and for each . Clearly, is the unique common fixed point of and .

By the definitions of , , , and ,

that is, (35) is satisfied. It is clear that for each , and hence . Note that , and then

which implies that . Therefore, by Gelfand’s formula, since , and hence and have a unique common fixed point by Theorem 8.

Finally, we present a fixed point theorem of contractions restricted with positive linear bounded mappings, which generalizes Theorem 3.1 of [22].

Theorem 10. Let be a -complete partial cone metric space over a solid cone of a normed vector space and . Assume that there exist such that

If and

then and have a unique common fixed point such that, for each , , where for all .

Proof. Let . Then, by (47), and so the inverse of exists (denoted by ). It follows from Neumann’s formula that and

Let . Then, by and . Moreover, by (47) and (48),

By (46), (p4), and ,

Similarly, take and in (46), and we get

It follows from (50) and (51) that

Act the above inequality with ; then, by ,

and so, by ,

By (p4),

It follows from (49) that

which implies , and hence by Lemma 2. Therefore, by Lemma 1 and (55), ; that is, is a -Cauchy sequence in . By analogy with the proof of Theorem 4, by , , and Lemma 3, we can prove that there exists some with such that and is a fixed point of . Note that (47) implies ; then, similar to the proof of Theorem 4, we can show is the unique fixed point of . The proof is completed.

Remark 11. It is easy to check that all the conditions of Theorem 10 are satisfied if . Therefore, Theorem 10 is valid with , and hence Theorem 3.1 of [22] is a special case of Theorem 10 with , where are four nonnegative real numbers such that .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was supported by the Natural Science Foundation of China (11161022), Natural Science Foundation of Jiangxi Province (20114BAB211006, 20122BAB201015), Educational Department of Jiangxi Province (GJJ12280, GJJ13297), and Program for Excellent Youth Talents of JXUFE (201201).

References

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.

L. Kantorovitch, “The method of successive approximations for functional equations,” Acta Mathematica, vol. 71, pp. 63–97, 1939.

L. V. Kantorovič, “The principle of the majorant and Newton's method,” Doklady Akademii Nauk SSSR, vol. 76, pp. 17–20, 1951.

L. V. Kantorovitch, “On some further applications of the Newton approximation method,” Vestnik Leningradskogo Universiteta, vol. 12, no. 7, pp. 68–103, 1957.

W. A. Kirk and B. C. Kang, “A fixed point theorem revisited,” Journal of the Korean Mathematical Society, vol. 34, pp. 285–291, 1972.

K. J. Chung, “Nonlinear contractions in abstract spaces,” Kodai Mathematical Journal, vol. 4, no. 2, pp. 288–292, 1981.

M. A. Krasnoseljski and P. P. Zabreiko, Geometrical Methods in Nonlinear Analysis, Springer, 1984.

I. A. Rus, A. Petrusel, and G. Petrusel, Fixed Point Theory, Cluj University Press, Cluj-Napoca, Romania, 2008.

R. P. Agarwal, “Contraction and approximate contraction with an application to multipoint boundary value problems,” Journal of Computational and Applied Mathematics, vol. 9, no. 4, pp. 315–325, 1983.

N. Cakić, Z. Kadelburg, S. Radenović, and A. Razani, “Common fixed point results in cone metric spaces for a family of weakly compatible maps,” Advances and Applications in Mathematical Sciences, vol. 1, no. 1, pp. 183–207, 2009.

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 16, Article ID 761086, 2009.

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 13, Article ID 643840, 2009.

Z. Kadelburg, S. Radenović, and V. Rakočević, “Remarks on “Quasi-contraction on a cone metric space”,” Applied Mathematics Letters, vol. 22, no. 11, pp. 1674–1679, 2009.

Z. Kadelburg, S. Radenović, and B. Rosić, “Strict contractive conditions and common fixed point theorems in cone metric spaces,” Fixed Point Theory and Applications, vol. 2009, article 14, Article ID 173838, 2009.

S. Radenović and B. E. Rhoades, “Fixed point theorem for two non-self mappings in cone metric spaces,” Computers & Mathematics with Applications, vol. 57, no. 10, pp. 1701–1707, 2009.

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.

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.

W.-S. Du, “Nonlinear contractive conditions for coupled cone fixed point theorems,” Fixed Point Theory and Applications, vol. 2010, article 16, Article ID 190606, 2010.

S. G. Matthews, “Partial metric topology,” in Proceedings of the 8th Summer Conference on General Topology and Applications, vol. 728, pp. 183–197, Academy of Sciences, New York, NY, USA, 1994.

A. Sonmez, “Fixed point theorems in partial cone metric spaces, Arxiv,” 2011, http://arxiv.org/abs/1101.2741v1

A. Sonmez, “On partial cone metric space, Arxiv,” 2012, http://arxiv.org/abs/arXiv:1207.6766v1

S. K. Malhotra, S. Shukla, and R. Sen, “Some fixed point results in -complete partial cone metric spaces,” Journal of Advanced Mathematical Studies, vol. 6, no. 2, pp. 97–108, 2013.

S. Jiang and Z. Li, “Extensions of Banach contraction principle to partial cone metric spaces over a non-normal solid cone,” Fixed Point Theory and Applications, vol. 2013, article 250, 2013.

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.