Unique and Non-Unique Fixed Points and their ApplicationsView this Special Issue
Rational Fuzzy Cone Contractions on Fuzzy Cone Metric Spaces with an Application to Fredholm Integral Equations
This paper is aimed at proving some common fixed point theorems for mappings involving generalized rational-type fuzzy cone-contraction conditions in fuzzy cone metric spaces. Some illustrative examples are presented to support our work. Moreover, as an application, we ensure the existence of a common solution of the Fredholm integral equations: and , for all , , and , where is the space of all -valued continuous functions on the interval and .
In 1922, Banach  proved a “Banach contraction principle,” which is stated as follows: “A self-mapping on a complete metric space verifying the contraction condition has a unique fixed point.” This principle plays a very important role in the fixed point theory. A number of researches have generalized it in many directions for single-valued and multivalued mappings in the context of metric spaces. Some of the findings can be found in [2–13] and the references therein. Currently, the fixed point theory is one of the most interested research areas in the field of mathematics. In the last decades, it has been investigated in many fields, such as game theory, graph theory, economics, computer sciences, and engineering.
The theory of fuzzy sets was introduced by Zadeh , while the concept of a fuzzy metric space (FM space) was given by Kramosil and Michalek . After that, the stronger form of the metric fuzziness was presented by George and Veeramani in . Later on, in , Gregori and Sapena proved some contractive-type fixed point results in complete FM spaces. Some more fixed point results in FM spaces can be found in [18–27] and the references therein.
Initially, in 2007, the concept of a cone metric space was reintroduced by Huang and Zhang . They proved some nonlinear contractive-type fixed point results in cone metric spaces. After the publication of this article, a number of researchers have contributed their ideas in cone metric spaces. Some of such works can be found in [29–34] and the references therein.
In 2015, the basic concept of a fuzzy cone metric space (FCM space) was given by Öner et al. . They presented some key attributes and a “fuzzy cone Banach contraction theorem” in FCM spaces. Later, Rehman and Li  extended and improved a “fuzzy cone Banach contraction theorem” and proved some generalized fixed point theorems in FCM spaces. Some more properties and related fixed point results can be found in [37–47].
The aim of this research work is to establish some rational-type fuzzy cone-contraction results in FCM spaces. We use the concept of [36, 39] and prove some common fixed theorems under generalized rational-type fuzzy cone-contraction conditions in FCM spaces. Some illustrative examples are presented. In the last section, we give an application of two Fredholm integral equations (FIEs).
Definition 1 . An operation is called a continuous -norm if (i) is commutative, associative, and continuous(ii) and , whenever and , for all The basic -norms: the minimum, the product, and the Lukasiewicz continuous -norms are defined by 
Definition 2 . A 3-tuple is said to be a FCM space if is a cone of , is an arbitrary set, is a continuous -norm, and is a fuzzy set on satisfying the following conditions: (1) and (2)(3)(4) is continuousfor all .
Definition 3 . Let be a FCM space and and be a sequence in . (i) converges to if for and there is such that , for . We may write this or as (ii) is Cauchy if for and there is such that , for (iii) is complete if every Cauchy sequence is convergent in (iv) is fuzzy cone contractive if there is so that
Lemma 4 . Let be a FCM space and let be sequence in converging to a point iff as for each .
Definition 5 . Let be a FCM space. The fuzzy cone metric is triangular if
Definition 6 . Let be a FCM space and . Then, is said to be fuzzy cone contractive if there is such that A “fuzzy cone Banach contraction theorem”  is stated as follows: “Let be a complete FCM space in which fuzzy cone contractive sequences are Cauchy and be a fuzzy cone contractive mapping. Then, has a unique fixed point.”
In this paper, we present some rational-type fuzzy cone-contraction theorems in FCM spaces by using the concept of [36, 39]. Namely, we prove some common fixed theorems under generalized rational-type fuzzy cone-contraction conditions in FCM spaces without the assumption that the fuzzy cone contractive sequences are Cauchy. We use “the triangular property of the fuzzy cone metric.” We also present some illustrative examples to support our work. In the last section, an application of Fredholm integral equations is provided.
3. Main Results
In this section, we prove some common fixed point theorems via generalized rational-type fuzzy cone-contraction conditions in FCM spaces.
Theorem 7. Let be a complete FCM space in which is triangular. Let be a pair of self-mappings so that for all , , , and with . Then, and have a common fixed point in .
Proof. Fix and construct a sequence of points in such that Then, by (5), for ,
By Definition 2 (3), , for . One writes
After simplification, we get that
where since . Similarly,
Again, by Definition 2 (3), , for . We have
After simplification, we have
which shows that is a fuzzy cone-contractive sequence in , and we get that
Note that is triangular; then, for all ,
which yields that is a Cauchy sequence in . Since is complete, there is such that
Now, we prove that . Since is triangular,
Again, by Definition 2 (3), , for . It follows that
Note that because . Then, , that is, . Similarly, we can show that because is triangular. Therefore,
Again, by Definition 2 (3), , for It follows that
Note that since . Then, , that is, .
Hence, is a common fixed point of and .
Example 1. Let , be a continuous -norm and be written as Then, easily one can verify that is triangular and is a complete FCM space. Now, we define by Then, for , we have Hence, the pair of self-mapping is a fuzzy cone-contraction. Now, from Definition 2 (3), and , for . We get the following: Hence, from the above, we conclude that all the conditions of Theorem 7 are satisfied with , , and . The mappings and have a common fixed point, i.e., .
Putting in Theorem 7, we get the following corollary.
Corollary 8. Let be a complete FCM space in which is triangular. Let be a pair of self-mappings so that for all , , , and with . Then, and have a common fixed point in .
In the following corollary, we prove that the mappings and have a unique common fixed point in by using the constant in Theorem 7.
Corollary 9. Let be a complete FCM space in which is triangular. Let be a pair of self-mappings so that for all , , , and with . Hence, and have a unique common fixed point in .
Proof. It follows from the proof of Theorem 7 that is a common fixed point of and in . For uniqueness, let be another common fixed point of and in such that and . Then, by view of (32), By Definition 2 (3), It follows that Since , one writes