#### Abstract

This paper is aimed at proving some unique common fixed point theorems by using the compatible and weakly-compatible four self-mappings in fuzzy cone metric (FCM) space. We prove the results under the generalized rational contraction conditions in FCM spaces with the help of one self-map are continuous. Furthermore, we prove some rational contraction results with the weaker condition of the self-mapping continuity. Ultimately, our theoretical work has been utilized to prove the existence solution of the two nonlinear integral equations. This is an illustrative application of how FCM spaces can be used in other integral type operators.

#### 1. Introduction

The theory of fixed-point theory was introduced by Banach [1]. He 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 (FP).” Later on, many researchers have been generalized this principle in many directions and proved different contractive type FP and common fixed point (CFP) for single-valued and multivalued mappings in the context of metric spaces. Chatterjea [2], Chatterjea [3], and Kannan [4] proved some single-valued contractive type FP theorems. While Ali et al. [5], Covitz and Nadler [6], Daker and Kaneko [7, 8], Khan [9], and Patle et al. [10] proved multivalued contractive type FP and CFP results by using different types of spaces.

Zadeh [11], in 1965, introduced the concept of fuzzy sets. Later on, this concept was used in topology and functional analysis by many researchers. Kramosil and Michalek [12] introduced the notion of fuzzy metric FM space, and they established some basic properties. After that, George and Veeramani [13] presented the stronger form of the FM. Grabiec [14] proved two FP theorems by using the concept of complete and compact FM spaces. Gregori and Sapena [15] established some FP contraction results in the sense of [13, 15]. Hadzic and Pap [16] proved a FP theorem for multivalued mappings in probabilistic metric spaces and presented applications in FM spaces. Imdad and Ali [17] and Rehman et al. [18] proved some FP theorems in complete FM spaces. Pant and Chauhan [19] established some CFP theorems by using weakly-compatible mappings in menger spaces and FM spaces. Kiyani et al. [20] and Sadeghi et al. [21] proved some results for set-valued contractive type mappings in FM spaces.

The concept of cone metric space (CMS) was proposed by many researchers but it became popular after being rediscovered by Huang and Zhang [22]. They proved the convergence properties and FP theorems for nonlinear contractive type mappings. By using the concept of Huang and Zhang [22], many authors have contributed their work to the problems on CMSs. Some of such works can be found in ([23–28]).

In 2015, the notion of fuzzy cone metric space (FCM space) was introduced by Oner et al. [29]. They proved the key attributes of FCM space and a “fuzzy cone Banach contraction theorem for FP” in FCM space. In [30], Rehman and Li extended and improved a “fuzzy cone Banach contraction theorem” and established some generalized-contraction results for FP in FCM spaces. Rehman et al. [31, 32] proved different contractive type CFP-theorems in FCM spaces. Recently, the concept of weakly compatible self-mappings in FCM spaces was given by Jabeen et al. [33].

This paper is aimed at proving some unique CFP-theorems under the generalized rational contraction conditions in FCM spaces by using compatibility and weak-compatibility of four self-mappings. We prove our results by using the one self-map are continuous. Furthermore, we prove some results without the continuity of self-mappings with supportive examples. In addition, we present an application of two nonlinear integral equations (NIEs) for the existence of a common solution to support our main work. This paper is managed as follows: in Section 2, we present the basic preliminary concept. While in Section 3, we prove our main results for unique CFP-theorems under the generalized rational contraction conditions in FCM spaces by using compatibility and weakly-compatibility of four self-mappings. In Section 4, we present NIEs as an application to support our main work.

#### 2. Preliminaries

In this section, we recall some basic definitions and lemmas.

*Definition 1 (see [34]). *An operation is called a continuous -norm if:
(i) is associative, commutative, and continuous(ii) and , whenever and , for all

Schweizer are Sklar [34] define the following basic continuous -norms are (i)The minimum; (ii)The product; (iii)The Lukasiewicz;

For detail study (see [34]).

*Definition 2 (see [29]). *A 3-tuple is called 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 continuous and .

*Definition 3 (see [29]). *Let be a FCM space, and be any sequence in .
(i) converges to if for any , , and such that , for . This can be written as , or as (ii) is Cauchy if for any , , and such that , for (iii) is complete if every Cauchy sequence is convergent in (iv) is FC contractive if so that

Lemma 4 (see [29]). *“Let be a FCM space and a sequence iff as for each ”.*

*Definition 5 (see [30]). *Let be a FCM space. The FCM is triangular if

*Definition 6 (see [29]). *Let be a FCM space and . Then, is said to be FC contractive if there is so that
and .

*Definition 7 (see [23]). *Let set and let be the self-mappings on . If there exists such that for some . Then, is called a coincidence point of and , and is known as a point of coincidence of the mappings . A pair of self-mappings is known to be weakly-compatible if the self-mappings commute at their coincidence point, i.e., for .

Proposition 8 (see [23]). *Let be weakly-compatible self-mappings on . If and have a unique point of coincidence, that is, , then, is a unique CFP of the mappings and .*

*Definition 9 (see [32]). *A self-mapping pair is said to be compatible on a FCM space , if, for , whenever is a sequence in so that for some .

#### 3. Main Results

Now, we are in the position to present our main results.

Theorem 10. *Let be the four self-mappings on a complete FCM space in which a FCM is triangular and satisfies
**, , and with . If , and consider that [(1)]
*(1)* is a continuous self-mapping*(2)*A pair is compatible, and*(3)*A pair is weakly-compatible*

Then, the mappings and have a unique CFP in .

*Proof. *Fix and by the hypothesis , , we define the iterative sequences in so that
Then, by (4),
By Definition 2 (3), , for . One writes
This implies that
where since . Similarly,
Again, by Definition 2 (3), . We have
This implies that
where the value of is the same as in (8). Now, from (3), (8), (11), and by induction, we have
Its prove that a sequence is a FC contractive, and we get that
Since is triangular, then ,
Hence, proved that is a Cauchy sequence. Now by the completeness of FCM space , so that as . Now for its subsequences, we have that
Since, a self-mapping is continuous, therefore
By hypothesis (2), a is compatible, therefore,
Next, we have to prove that , then, by Definition 2 (3),
Since, a pair is compatible, by using limit , and by the view of (15), (17), and (18), we have
Hence, for . Now, we prove that , then again by Definition 2 (3),
Again by using limit , and by the view of (15), (17), and (20), we have
Hence, for . Thus, we get that . Next, we have to prove that . Now by hypothesis (1), i.e., , and there exists such that . Then, by view of (4), for ,
Again, by Definition 2 (3), It follows that
Noticing that , therefore, for , hence, . Now by hypothesis (3), a pair is weakly compatible, therefore,
Next, we have to prove that , then again by view of (4) and by using Definition 2 (3), for ,
After simplification, we obtain
Since , therefore, , for , which further implies that . Hence, proved that , that is, is the CFP of the mappings , and .☐

Uniqueness: let be the other CFP of the mappings and in such that . Then by view of (4) and by using Definition 2 (3), for ,

After simplification, we obtain

Since , therefore, , for . This completes the proof.

Corollary 11. *Let be the four self-mappings on a complete FCM space in which a FCM is triangular and satisfies
**, , and with . If , and consider that
*(1)* is a continuous self-mapping*(2)*A pair is compatible, and*(3)*A pair is weakly-compatible*

Then, the mappings and have a unique CFP in .

Corollary 12. *Let be the four self-mappings on a complete FCM space in which a FCM is triangular and satisfies
**, , and with . If , and consider that
*(1)* is a continuous self-mapping*(2)*A pair is compatible, and*(3)*A pair is weakly-compatible**Then, the mappings and have a unique CFP in .*

Corollary 13. *Let be the four self-mappings on a complete FCM space in which a FCM is triangular and satisfies
**, , and with . If , and consider that
*(1)* is a continuous self-mapping*(2)*A pair is compatible, and*(3)*A pair is weakly-compatible*

Then, the mappings and have a CFP in .

*Example 14. *Assume that , is a continuous -norm and be written as

Then, it is easy to verify that FCM is triangular and is a complete FCM space. Now, the mappings, , be defined by, (for all );

and

Since, from the above equation, and , so that we conclude that or . Then,

and

Hence, the mappings , and on satisfying the fuzzy cone-contraction condition in FCM space. Now, from Definition 2 (3), and , for . Now, we calculate the following terms of (4), for ,

Next, we calculate

Similarly,

Thus, after routine calculation, all the conditions of Theorem 10 are satisfied with , , and , and the mappings and on have a unique CFP, i.e., .

If we choose the self-mappings and in Theorem 10, we obtain the following corollary.

Corollary 15. *Let be two self-mappings on a complete FCM space in which a FCM is triangular and satisfies
**, , and with . If , is a continuous self-mapping, and a pair is weakly-compatible. Then, the mappings and have a unique CFP in .*

Next result, we shall prove without the continuity of self-mapping, i.e., and we replaced the completeness of by the completeness of or .

Theorem 16. *Let be two self-mappings on a complete FCM space in which a FCM is triangular and satisfies
**, , and with . If , or is complete and a pair is weakly-compatible. Then, the mappings and have a unique CFP in .*

*Proof. *From the proof of Theorem 10, we assume that is a Cauchy sequence in , and the iterative sequences are earlier defined in the proof of Theorem 10, that are,
We know that is complete, and , so that , as . Therefore,
Since by triangular property,
Now by the view of (41), (43), and by using Definition 2 (3), for , we have that