Abstract and Applied Analysis
Volume 2019 (2019), Article ID 2841606, 10 pages
https://doi.org/10.1155/2019/2841606

Research Article

Common Fixed Point Theorems for a Pair of Self-Mappings in Fuzzy Cone Metric Spaces

Saif Ur Rehman,¹ Yongjin Li,² Shamoona Jabeen,³ and Tayyab Mahmood⁴

¹Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
²Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
³School of Mathematics and Systems Science, Beihang University, Beijing 100191, China
⁴Department of Mathematics, COMSATS University, Islamabad Wah Cantt. 47040, Pakistan

Correspondence should be addressed to Saif Ur Rehman; saif.urrehman27@yahoo.com

Received 1 November 2018; Revised 29 January 2019; Accepted 19 February 2019; Published 1 April 2019

Academic Editor: Douglas R. Anderson

Copyright © 2019 Saif Ur Rehman 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.

In this paper, we present some common fixed point theorems for a pair of self-mappings in fuzzy cone metric spaces under the generalized fuzzy cone contraction conditions. We extend and improve some recent results given in the literature.

1. Introduction

In 1965, Zadeh [1] came up with a fabulous idea. He introduced the theory of fuzzy set, which is the generalization of crisp sets. A mapping is from to ; then is known as a fuzzy set. Later on, the fuzzy metric space concept was given by Kramosil and Michalek [2], which is performing the probabilistic metric space and would approach the fuzzy set. In [3], George and Veeramani were given the stronger form of the fuzz metric. Some more set-valued mapping results for fixed point on fuzzy metric spaces can be seen, for example, in [4–6] and the references therein.

In 2007, Som [7] proved some continuous self-mapping results for common fixed point in fuzzy metric spaces. He generalized the results of Pant [8], Som [9], and Vasuki [10]. Some other common fixed point results in the fuzzy metric space can be found in [11–16] and the references therein.

Huang and Zhang [17] introduced the concept of cone metric space. They proved the convergent sequences, Cauchy sequences, and some fixed point theorems for contractive-type mappings in cone metric spaces. Later on, Abbas and Jungck [18] proved some noncommuting mapping results in cone metric spaces. After that, a series of authors proved some fixed point and common fixed point results for different contractive-type mappings in cone metric spaces (see, e.g., [19–25]).

Recently, the concept of fuzzy cone metric space was introduced by Oner et al. [26]. They proved some basic properties and a Banach contraction theorem for fixed point with the assumption of Cauchy sequences. Rehman and Li [27] generalized the result of Oner et. al. [26] and proved some fixed point theorems in fuzzy cone metric spaces without the assumption of Cauchy sequences. Some more fixed point and common fixed point results in fuzzy cone metric spaces can be found in [27–31].

In the demonstration of this research work, we generalize the results of Oner [26] and Rehman [27] for a pair of self-mappings in fuzzy cone metric spaces and prove some unique common fixed theorems with illustrative examples.

2. Preliminaries

Definition 1 ([32]). An operation is known as a continuous -norm if it holds the following:

(1) is commutative, associative, and continuous.(2), .(3), whenever and , for every .

Meanwhile, the basic -norm continuous conditions are as follows.

The minimum, product, and Lukasiewicz -norms are defined, respectively, as (see [32])

Definition 2 ([17]). A subset of a real Banach space is called a cone if

(1), closed and , where represents the zero element of ,(2), if and ,(3), if both .

All the cones have nonempty interior and the natural numbers set is denoted by .

Definition 3 ([26]). A 3-tuple is known as a fuzzy cone metric space, if is a continuous -norm, is an arbitrary set, is a cone of , and is a fuzzy set on if the following hold:

(i) and if ,(ii),(iii),(iv) is continuous,

for all and .

Remark 4 ([27]). If we suppose that , , and , then every fuzzy metric space becomes a fuzzy cone metric space.

Definition 5 ([26]). Let be a fuzzy cone metric space, , and a sequence in is

(i)converging to if and such that , . We can write this as or as .(ii)Cauchy sequence if and such that , .(iii) is complete if every Cauchy sequence is convergent in .(iv)fuzzy cone contractive if , satisfying

for all , .

Definition 6 ([27]). Let be a fuzzy cone metric space. A fuzzy cone metric is triangular if

and .

Lemma 7 ([26]). Let and let be a sequence in . Then in a fuzzy cone metric space if as , for each .

For more properties of fuzzy cone metric spaces, see [26].

Definition 8 ([26]). A mapping is known as fuzzy cone contractive in a fuzzy cone metric space , if such that

for all , , and is known as a contraction constant of .

Theorem 9 ([26]). A self-mapping in a complete fuzzy cone metric space, in which the fuzzy cone contractive sequences are Cauchy, has a unique fixed point.

Further, in this paper, we shall study some common fixed point results in . Let be two self-mappings satisfying the following more generalized fuzzy cone contraction condition:

where and the constants . It is noted that (5) is the same as (4) if , , and . On the other hand, the mappings and may not hold the fuzzy cone contraction condition if (5) is satisfied, which is shown in Example 14. Thus, in this research work, we generalize some recent results given in the literature (see Remark 13 and Example 14).

3. Main Result

Theorem 10. Let , be two self mappings and is triangular in a complete fuzzy cone metric space which satisfies (5) with . Then and have a unique common fixed point in .

Proof. Fix and we define the iterative sequences in as

By view of (5), for ,

Then

where , since .
Let us denote by ; then, from (8), we have

Similarly,

Then

where , since . Then (11) can be written as

Now, from (9) and (12), we can get the following inequalities:

Thus, we have

Hence, from the above, we conclude that a sequence is fuzzy cone contractive in ; that is,

Let and let be the above sequence; we assume that . Then, two cases arise.
Case (i). If is an even number,

Case (ii). If is an odd number,

Thus, the right-hand sides of (16) and (17) converge to zero as , which yields that is a Cauchy sequence. Since is complete, such that

Since is triangular,

By using (5), (15), and (18), for ,

Then

The above (21) together with (18) and (19) implies that

, since ; then ; that is, . Similarly, by triangular,

Again, by using (5), (15), and (18), similar to the above, after simplification, we can get

The above (24) together with (18) and (23) implies that

, since ; then ; that is, . Hence, the fact that is the common fixed point of and in is proven.
Uniqueness: let be the other common fixed point of and in . Then, again by view of (5), for ,

We note that , where . Therefore , implying that . Hence the fact that the common fixed point of and is unique is proven.

Corollary 11. Let , be two self-mappings and is triangular in the complete fuzzy cone metric space which satisfies

for all , , and such that . Then and have a unique common fixed point in .

Corollary 12. Let , be two self-mappings and is triangular in the complete fuzzy cone metric space which satisfies

for all , , and such that . Then and have a unique common fixed point in .

Remark 13. (i) In special case, Theorem 10, Corollaries 11 and 12, and [26, Theorem 3.3] (i.e., Theorem 9) all have the same results. In fact, if , and in (5).
(ii) Theorem 10 and [27, Theorem 3.1] both have similar proof. If , and in (5).

Example 14. Let ; is a continuous -norm and is defined as

and . Then, one can easily prove that is triangular and is a complete fuzzy cone metric space. Now we define as

And

Then and are not fuzzy cone contractive, since

In special case, if , then Theorem 9 does not hold. But it can be easily proven that all the conditions of Theorem 10 hold with , , . Thus, and have a unique common fixed point in , that is, .

Theorem 15. Let , be two self-mappings and is triangular in the complete fuzzy cone metric space which satisfies

for all , , and . Then and have a unique common fixed point in .

Proof. Fix and a point such that and such that . If , then we have that

which implies that if and only if . Then the proof is complete. Otherwise, we assume that and let us take . Now we define the iterative sequence in such as

By view of (33),

Now there are three possibilities.
(i) If is minimum, then will be the maximum in the above (36). Then, we have

(ii) If is minimum, then will be the maximum in the above (36). Then, we have

which is not possible.
(iii) If is minimum, then will be the maximum in the above (36). Then, we have

which implies that

where , since . Thus, . Now, from (i), (ii), and (iii), for all and ,

which shows that a sequence is fuzzy cone contractive. Thus,

Since is triangular, for all , we have

which shows that is a Cauchy sequence. Since is complete and , we have

Now we shall show that . By the triangular property of , we have

Now, by using (33), (42), and (44), for , we have

Thus,

The above (47) together with (44) and (45) implies that

and , since . This implies that ; that is, . Similarly, we can prove that . Thus, .
Uniqueness: let such that . Then, by using (33), for every , we have

This implies that

, since . This implies that ; that is, . Hence the fact that and have a unique common fixed point is proven. That is, .

Corollary 16. Let , be two self-mappings and is triangular in the complete fuzzy cone metric space which satisfies

for all , , and . Then and have a unique common fixed point in .

Example 17. From Example 14, we define as

Then, and are fuzzy cone contractive, since

for all . Then, all the conditions of Theorem 15 easily hold with , as well as Theorem 9, if . Thus, and have a unique common fixed point in , that is, .

4. Conclusion

We gave the concept of common fixed point for a pair of self-mappings in fuzzy cone metric spaces and proved some unique common fixed point results in fuzzy cone metric spaces. We also proved that a pair of self-mappings may not be a fuzzy cone contraction if it satisfies (5), which is shown in Example 14. According to this concept, one can study some more common fixed point results for two or more self-mappings in fuzzy cone metric spaces for different contractive-type mappings.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors share equal contributions to the final manuscript.

References

L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, pp. 338–353, 1965.

I. Kramosil and J. Michalek, “Fuzzy metrics and statistical metric spaces,” Kybernetika, vol. 11, no. 5, pp. 326–334, 1975.

A. George and P. Veeramani, “On some results in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 64, no. 3, pp. 395–399, 1994.

O. Hadzic and E. Pap, “A fixed point theorem for multivalued mappings in probabilistic metric spaces and an application in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 127, no. 3, pp. 333–344, 2002.

F. Kiany and A. Amini-Harandi, “Fixed point and endpoint theorems for set-valued fuzzy contraction maps in fuzzy metric spaces,” Fixed Point Theory and Applications, vol. 94, 2011.

Z. Sadeghi, S. M. Vaezpour, C. Park, R. Saadati, and C. Vetro, “Set-valued mappings in partially ordered fuzzy metric spaces,” Journal of Inequalities and Applications, vol. 2014, article 157, 17 pages, 2014.

T. Som, “Some results on common fixed point in fuzzy metric spaces,” Soochow Journal of Mathematics, vol. 33, no. 4, pp. 553–561, 2007.

R. P. Pant, “Common fixed points of noncommuting mappings,” Journal of Mathematical Analysis and Applications, vol. 188, no. 2, pp. 436–440, 1994.

T. Som, “Some fixed point theorems on metric and Banach spaces,” Indian Journal of Pure and Applied Mathematics, vol. 16, no. 6, pp. 575–585, 1985.

R. Vasuki, “Common fixed points for R-weakly commuting maps in fuzzy metric spaces,” Indian Journal of Pure and Applied Mathematics, vol. 30, no. 4, pp. 419–423, 1999.

M. Imdad and J. Ali, “Some common fixed point theorems in fuzzy metric spaces,” Mathematical Communications, vol. 11, no. 2, pp. 153–163, 2006.

S. Kutukcu, D. Turkoglu, and C. Yildiz, “Common fixed points of compatible maps of type (β) on fuzzy metric spaces, Communications of the Korean Society,” Communications of the Korean Mathematical Society, vol. 21, no. 1, pp. 89–100, 2006.

P. P. Murthy, S. Kumar, and K. Tas, “Common fixed points of self maps satisfying an integral type contractive condition in fuzzy metric spaces,” Mathematical Communications, vol. 15, no. 2, pp. 521–537, 2010.

B. D. Pant and S. Chauhan, “Common fixed point theorems for two pairs of weakly compatible mappings in Menger spaces and fuzzy metric spaces,” Scientific Studies and Research. Series Mathematics and Informatics, vol. 21, no. 2, pp. 81–96, 2011.

S. Sedghi, D. Turkoglu, and N. Shobe, “Common fixed point of compatible maps of type (γ) on complete fuzzy metric spaces,” Communications of the Korean Mathematical Society, vol. 24, no. 4, pp. 581–594, 2009.

P. V. Subrahmanyam, “A common fixed point theorem in fuzzy metric spaces,” Information Sciences, vol. 83, no. 3-4, pp. 109–112, 1995.

L. 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.

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.

M. Abbas, B. E. Rhoades, and T. Nazir, “Common fixed points for four maps in cone metric spaces,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 80–86, 2010.

I. Altun, B. Damjanovic, and D. Djoric, “Fixed point and common fixed point theorems on ordered cone metric spaces,” Applied Mathematics Letters, vol. 23, no. 3, pp. 310–316, 2010.

I. Altun and G. Durmaz, “Some fixed point theorems on ordered cone metric spaces,” Rendiconti del Circolo Matematico di Palermo , vol. 58, no. 2, pp. 319–325, 2009.

D. Ilić and V. Rakočević, “Common fixed points for maps on cone metric space,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 876–882, 2008.

D. Ilic and V. Rakocevic, “Quasi-contraction on a cone metric space,” Applied Mathematics Letters, vol. 22, no. 5, pp. 728–731, 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.

P. Vetro, “Common fixed points in cone metric spaces,” Rendiconti del Circolo Matematico di Palermo, Serie II, vol. 56, no. 3, pp. 464–468, 2007.

T. Oner, M. B. Kandemir, and B. Tanay, “Fuzzy cone metric spaces,” Journal of Nonlinear Sciences and Applications, vol. 8, no. 5, pp. 610–616, 2015.

S. U. Rehman and H.-X. Li, “Fixed point theorems in fuzzy cone metric spaces,” Journal of Nonlinear Sciences and Applications. JNSA, vol. 10, no. 11, pp. 5763–5769, 2017.

A. M. Ali and G. R. Kanna, “Intuitionistic fuzzy cone metric spaces and fixed point theorems,” International Journal of Applied Mathematics, vol. 5, pp. 25–36, 2017.

T. Oner, “On some results in fuzzy cone metric spaces,” International Journal of Computer Science and Network, vol. 4, pp. 37–39, 2016.

T. Oner, “On the metrizability of fuzzy cone metric spaces,” International Journal of Management and Applied Science, vol. 2, pp. 133–135, 2016.

T. Oner, “Some topological properties of fuzzy cone metric spaces,” Journal of Nonlinear Sciences and Applications, vol. 9, no. 3, pp. 799–805, 2016.

B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, pp. 313–334, 1960.