#### Abstract

Fixed point (FP) has been the heart of several areas of mathematics and other sciences. FP is a beautiful mixture of analysis (pure and applied), topology, and geometry. To construct the link between FP and applied mathematics, this paper will present some generalized strong coupled FP theorems in cone metric spaces. Our consequences give the generalization of “cyclic coupled Kannan-type contraction” given by Choudhury and Maity. We present illustrative examples in support of our results. This new concept will play an important role in the theory of fixed point results and can be generalized for different contractive-type mappings in the context of metric spaces. In addition, we also establish an application in integral equations for the existence of a common solution to support our work.

#### 1. Introduction

In a wide range of engineering and mathematical problems, the occurrence of a solution is identical to the existence of a FP for a suitable mapping. The occurrence of a FP is therefore of greatest importance in many diverse fields of engineering and applied mathematics. FP consequences make provision to provide many conditions under which maps have solutions. In the last five decades or so, the theory of FP has been disclosed as an important tool in the field of nonlinear phenomena. Particularly, FP methods have been used in several areas such as engineering, physics, chemistry, economics, biology, and game theory. The objective of this paper is to find new FP consequences and their use in which the indispensability of the FP results is mentioned.

In 2003, Kirk et al. [1] proved the result for FP on an exceptional variety of maps which are familiar as cyclic contractive maps. Some of these works are in [2–5] (and the references therein). Later, Lakshmikantham and Ćirić [6] established a new idea of coupled FP, which is a beautiful mixture of analysis (pure and applied), topology, and geometry. It has also a lot of applications in a wide range of mathematical problems. In 2014, Choudhury and Maity [7] proved a strong coupled FP result for cyclic coupled Kannan-type mapping.

Hang and Zhang [8] in 2007 discovered the idea of cone metric space (CMS). Moreover, they have presented some preliminary concepts and proved a cone Banach contraction theorem. After the publication of this article, many researchers, i.e., Abbas and Jungck [9], Ilić and Rakočević [10], and P. Vetro [11] generalize their results for the FP, common FP, and coincidence points in CMS by using the contraction conditions. Some other related results can be found in (see [12–23] and the references are therein).

In this manuscript, we study the more general cyclic coupled cone contraction results in complete CMS and prove that a cyclic mapping has a strong coupled FP theorem in . Moreover, we present an integral-type application by using the concept of Chen et al. [24] and Jabeen et al. [25] to support our work. We shall present the illustrative examples and the two Urysohn integral equations (UIEs) for finding the solution of problems to hold up our consequences. This paper is organized as follows: Section 2 consists of preliminary concepts. In Section 3, we present some strong coupled FP results by using cyclic-type mappings in complete cone metric spaces with illustrative examples. In Section 4, we present an application of Voltera integral equations for the existence of a solution to support our main work. In the last section, which is Section 5, we discuss the conclusion.

#### 2. Basic Definitions

*Definition 1 (see [18]). *A subset of a real Banach space is called a cone if
(i), closed, and , where is the zero elements of (ii)If and any , then (iii)If both , then A partial ordering on a given cone is defined as iff and and if and while iff . A cone is known as a normal cone if such that :
Then, is known as a normal constant of .

*Definition 2 (see [8]). *Let be the set and a mapping is known as a cone metric, if satisfies the following:
(i) and . Then, a pair is called a CMS.

*Definition 3 (see [8]). *Let be a CMS, and let and be a sequence in . Then,
(i) is said to be convergent and converges to if for any in and is a positive integer such that , for . We denote this by or as (ii) is said to be a Cauchy sequence if for any in and is a positive integer such that , for (iii) is known as complete if every Cauchy sequence is convergent in Throughout this paper, is a real Banach space, may be a normal cone in with normal constant , and , and “” is a partial ordering w.r.t. .

Lemma 4 (see [8]). *Let and be two sequences in . Then, the following statements hold:
*(i)* iff , and the limit of a convergent sequence is unique*(ii)* is a Cauchy sequence iff *(iii)* and imply that **For more details, we shall refer the readers to study [8].*

*Definition 5 (see [7]). *Let and be two nonempty subsets of a given set . We call that a function such that if and and if and is a cyclic mapping w.r.t. and .

*Definition 6 (see [7]). *Let be the set, and an element is called a coupled FP of a mapping if and . Moreover, is called a strong coupled FP of , if , that is, .

*Definition 7 (see [7]). *Let and be two nonempty subsets of a metric space . Then, a mapping is known as a cyclic coupled Kannan-type contraction w.r.t. and if is cyclic w.r.t. and and satisfies the inequality for some , such that
where and .

The following “cyclic coupled Kannan-type contraction theorem” was obtained in [7].

Theorem 8 (see [7]). *Assume that and are two nonempty closed subsets of a complete metric space , and let be a cyclic coupled Kannan-type contraction w.r.t. and and . Then, has a strong coupled FP in .*

#### 3. Main Result

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

Let be a cyclic mapping w.r.t. and , where and are subsets of a CMS , under the generalized coupled cone contractive-type condition: where , , and . Our results generalize and improve Theorem 8 (that is, Theorem 5 in [7]) (see Remark 12). Moreover, some illustrate examples, and the integral-type application is given in the paper to support our work.

Theorem 9. *Let and be two nonempty closed subsets of a complete CMS , and let be a cyclic mapping w.r.t. and . Assume that satisfies (4) with . Then, and has a strong coupled FP in .*

*Proof. *Fix and , and let and be two sequences defined as
Then, and , since is cyclic mapping w.r.t. and . Denote
Then, for . We claim that
It is clear that (7) holds for . Assume that (7) holds for ; then, by (4),
which implies that
Similarly, we can get
Thus, by the induction hypothesis, i.e., (7) with , we have
That is, (7) holds for . Therefore, we have proven that (7) holds for all by induction. Meanwhile, by (4), for ,
This together with (7) implies that
Then, for , without loss of generality, we assume that :
Hence, is a Cauchy sequence by Lemma 4 (ii), and the completeness of yields that there exists such that
Similarly, we can get that
Then, from Lemma 4 (iii), we have
On the other hand, by (7) and (13),
Therefore, by Lemma 4 (i), and .

Now, we have to prove that is a strong coupled FP of . Indeed, by (4), (15), and (16), we have
This implies that since . Then, , and is a strong coupled FP of .

Corollary 10. *Let and be two nonempty closed subsets of a complete CMS , and let be a cyclic coupled Kannan-type cone contraction w.r.t. and . That is,
where , , and . Then, has a strong coupled FP in .*

Corollary 11. *Let and be two nonempty closed subsets of a complete CMS , and let be a cyclic coupled Chatterjea-type cone contraction w.r.t. and . That is,
where , , and . Then, has a strong coupled FP in .*

*Remark 12. *In the special case, when is a complete metric space, Corollary 10 is the same as Theorem 8 (Theorem 5 in [7]), if in (4). Therefore, our results generalize the result given in [7]. Moreover, the following example shows that Theorem 8 does not apply while Theorem 9 does.

*Example 13. *Let in CMS which is defined as , , and let and . Then, and are two nonempty closed subsets of a set and . A mapping can be defined as
Then, easily one can prove that is a cyclic mapping w.r.t. and , for any and . A mapping is not a cyclic coupled Kannan-type contraction, since
where . Now, from (4), we have
Hence, all the axioms of Theorem 9 are fulfilled at and . A mapping has a strong coupled FP which is 0, that is, .

Theorem 14. *Assume that and are two nonempty closed subsets of a complete CMS and is a cyclic coupled contractive-type mapping w.r.t. and which satisfies
where , , and . Then, and has a strong coupled FP in .*

*Proof. *Fix and , and let and be two sequences defined as
Then, and , since is a cyclic mapping w.r.t. and . Now, we shall show that is a Cauchy sequence. We claim that, for all ,
First, we have to prove that
Then, by (25), we have
Now, we have three cases:
(i)If is maximum, then from (29), we haveIt holds (28), as , since .
(ii)If is maximum, then from (29), we havewhich is not possible.
(iii)If is maximum, then from (29), we haveIt follows that (28) holds.

Hence, from all cases, we get that
Similarly, we can prove
Then, again from (25), we have
Then again, we have the further three cases:
(i)If is maximum, then from (35), we haveIt holds (34), as , since .
(ii)If is maximum, then from (35), we havewhich is not possible.
(iii)If is maximum, then from (35), we haveIt follows that (34) holds.

Hence, from all cases, we get that
Now, by adding (33) and (39), we have that
Now, again by (25) and similar as above, we can get
Now, again by adding (41) and (42) and then putting in (40), we have that
Continuing this process, we have that
Hence, it is proven that (27) is exact . Now, for integer , we have
Then, we have the following four cases:
(a)If is maximum, then from (45), we havewhere , since .
(b)If is maximum, then from (45), we havewhere , since .
(c)If is maximum, then from (45), we havewhere .
(d)If is maximum, then from (45), we havewhere .

Hence, from (a) and (d), we have
And from (b) and (c), we have
By adding (50) and (51), we have
where , and in view of (44), we have
Since, by triangular inequality (44) and (53), we have
Now, for and , we have
Hence, it is proven that is a Cauchy sequence and it is convergent in . Since is a closed subset of , therefore
Similarly,
Hence, from (56) and (57), we have
By triangular inequality (44) and (54), we have
Therefore, , which implies that .

Now, we prove a strong coupled FP of in ; therefore,
Then, by the view of (25), (56), and (57), we have
Hence, from (60),
which implies that , since . Hence, , which shows that is a strong coupled FP of .

Corollary 15. *Assume that and are two nonempty closed subsets of a complete CMS and is a cyclic coupled contractive-type mapping w.r.t. and which satisfies:
where , , and . Then, and has a strong coupled FP in .*

*Example 16. *From Example 13, a mapping can be defined as
Then, easily one can verify that is a cyclic mapping w.r.t. and , for and . A mapping is a cyclic coupled Kannan-type contraction, since
where . Now, from (25), we have
Hence, Example 16 is satisfying all conditions of Theorem 14 with , and , which is a strong coupled FP in .

#### 4. Integral-Type Application

This section is intended to present two UIEs for the existence of a common solution to support our main result. Let be the Banach space of all continuous functions defined on with the supremum norm and the induced metric can be defined as

Now, we are in the position to introduce two UIEs for finding the solution of problems to hold up our consequences.

Theorem 17. *The two UIEs are
where and .**Let such that , for , , and ; therefore,
for all . If there exists such that
where
**then the two UIEs (69) and (70) have a unique common solution.*

*Proof. *Define a mapping :
Then, we may have the following four cases:
(a)If is the maximum in (73), then we have