# Some Novel Generalized Strong Coupled Fixed Point Findings in Cone Metric Spaces with Application to Integral Equations

Authors: Saif Ur Rehman; Sami Ullah Khan; Abdul Ghaffar; Shao-Wen Yao; Mustafa Inc
Journal: Journal of Function Spaces
 (2021)

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## 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.


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## Body

## 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 CMSU,dc and prove that a cyclic mapping Γ:U×U⟶U has a strong coupled FP theorem in U,dc. 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 subsetP of a real Banach space E is called a cone if
(i)
P≠∅, closed, and P≠θ, where θ is the zero elements of E(ii)
Ifa,b∈0,∞ and any μ,ν∈P, then aμ+bν∈P(iii)
If both−u,μ∈P, then μ=θ
A partial ordering≼ on a given cone P⊆E is defined as μ≼ν iff μ−ν∈P and μ≺v and if μ≼v and μ≠v while μ≪v iff μ−v∈intP. A cone P is known as a normal cone if ∃κ>0 such that ∀μ,ν∈E:
(1)θ≼μ≼ν⟹μ≤κν,∀μ,ν∈E.
Then,κ is known as a normal constant of P.Definition 2 (see [8]).
LetU≠∅ be the set and a mapping dc:U×U⟶E is known as a cone metric, if dc satisfies the following:
(i)
θ≼dcμ1,μ2 and dcμ1,μ2=θ⟺μ1=μ2(2)dcμ1,μ2=dcμ2,μ1,dcμ1,μ3≼dcμ1,μ2+dcμ2,μ3,
∀μ1,μ2,μ3∈U. Then, a pair U,dc is called a CMS.Definition 3 (see [8]).
LetU,dc be a CMS, and let μ∈U and μℓ be a sequence in U. Then,
(i)
μℓ is said to be convergent and converges to μ if for any c≫θ in E and N is a positive integer such that dcμℓ,μ≪c, for ℓ≥N. We denote this by limℓ⟶∞μℓ=μ or μℓ⟶μ, as ℓ⟶∞(ii)
μℓ is said to be a Cauchy sequence if for any c≫θ in E and N is a positive integer such that dcμℓ,μk≪c, for ℓ,k≥N(iii)
U,dc is known as complete if every Cauchy sequence is convergent in U
Throughout this paper,E is a real Banach space, P may be a normal cone in E with normal constant κ, and intP≠∅, and “≼” is a partial ordering w.r.t. P.Lemma 4 (see [8]).
Letμℓ and vℓ be two sequences in U,dc. Then, the following statements hold:
(i)
limℓ⟶∞μℓ⟶μ iff limℓ⟶∞dcμℓ,μ⟶θ, and the limit of a convergent sequence is unique(ii)
μℓ is a Cauchy sequence iff limℓ,k⟶∞dcμℓ,μk⟶θ(iii)
limℓ⟶∞μℓ⟶μ and limℓ⟶∞νℓ⟶ν imply that limℓ⟶∞dcμℓ,νℓ⟶dcμ,ν
For more details, we shall refer the readers to study [8].Definition 5 (see [7]).
LetG and H be two nonempty subsets of a given set U. We call that a function Γ:U×U⟶U such that Γμ,ν∈G if μ∈H and ν∈G and Γμ,ν∈H if μ∈G and ν∈H is a cyclic mapping w.r.t. G and H.Definition 6 (see [7]).
LetU≠∅ be the set, and an element μ,ν∈U×U is called a coupled FP of a mapping Γ:U×U⟶U if Γμ,ν=μ and Γν,μ=ν. Moreover, μ is called a strong coupled FP of Γ, if μ=ν, that is, Γμ,μ=μ.Definition 7 (see [7]).
LetG and H be two nonempty subsets of a metric space U. Then, a mapping Γ:U×U⟶U is known as a cyclic coupled Kannan-type contraction w.r.t. G and H if Γ is cyclic w.r.t. G and H and satisfies the inequality for some a∈0,1/2, such that
(3)dcΓx,y,Γμ,ν≤adcx,Γx,y+dcμ,Γμ,ν,where x,ν∈G and y,μ∈H.
The following “cyclic coupled Kannan-type contraction theorem” was obtained in [7].Theorem 8 (see [7]).
Assume thatG and H are two nonempty closed subsets of a complete metric space U,dc, and let Γ:U×U⟶U be a cyclic coupled Kannan-type contraction w.r.t. G and H and G∩H≠∅. Then, Γ has a strong coupled FP in G∩H.


## 3. Main Result

Now, we are in the position to present our main results.LetΓ:U×U⟶U be a cyclic mapping w.r.t. G and H, where G and H are subsets of a CMS U,dc, under the generalized coupled cone contractive-type condition:
(4)dcΓx,y,Γμ,ν≤αdcx,Γx,y+dcμ,Γμ,ν+βdcx,Γμ,ν+dcμ,Γx,y,where x,ν∈G, y,μ∈H, and α,β∈0,∞. 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.
LetG and H be two nonempty closed subsets of a complete CMS U,dc, and let Γ:U×U⟶U be a cyclic mapping w.r.t. G and H. Assume that Γ satisfies (4) with 2α+β<1. Then, G∩H≠∅ and Γ has a strong coupled FP in G∩H.Proof.
Fixν0∈G and μ0∈H, and let νℓ and μℓ be two sequences defined as
(5)νℓ+1=Γμℓ,νℓ,μℓ+1=Γνℓ,μℓ,∀ℓ≥0.
Then,νℓ⊂G and μℓ⊂H, since Γ is cyclic mapping w.r.t. G and H. Denote
(6)λ=α+β1−α+β.
Then,λ∈0,1 for α+β<1/2. We claim that
(7)dcνℓ,μℓ+1+dcμℓ,νℓ+1≼λℓdcν0,μ1+dcμ0,ν1,ℓ≥0.
It is clear that (7) holds for ℓ=0. Assume that (7) holds for ℓ=k; then, by (4),
(8)dcνk+1,μk+2=dcΓμk,νk,Γνk+1,μk+1≼αdcμk,Γμk,νk+dcνk+1,Γνk+1,μk+1+βdcμk,Γνk+1,μk+1+dcνk+1,Γμk,νk=αdcμk,νk+1+dcνk+1,μk+2+βdcμk,μk+2+dcνk+1,νk+1≼αdcμk,νk+1+dcνk+1,μk+2+βdcμk,νk+1+dcνk+1,μk+2,which implies that
(9)dcνk+1,μk+2≼λdcμk,νk+1.
Similarly, we can get(10)dcμk+1,νk+2≼λdcνk,μk+1.
Thus, by the induction hypothesis, i.e., (7) with ℓ=k, we have
(11)dcνk+1,μk+2+dcμk+1,νk+2≼λdcνk,μk+1+dcμk,νk+1≼λ2dcνk−1,μk+dcμk−1,νk≼⋯≼λk+1dcν0,μ1+dcμ0,ν1.
That is, (7) holds for ℓ=k+1. Therefore, we have proven that (7) holds for all ℓ≥0 by induction. Meanwhile, by (4), for ℓ≥0,
(12)dcνℓ,νℓ+1+dcμℓ,μℓ+1≼dcνℓ,μℓ+1+dcμℓ+1,νℓ+1+dcμℓ,νℓ+1+dcνℓ+1,μℓ+1=dcνℓ,μℓ+1+dcμℓ,νℓ+1+2dcΓμℓ,νℓ,Γνℓ,μℓ≼dcνℓ,μℓ+1+dcμℓ,νℓ+1+2αdcμℓ,νℓ+1+dcνℓ,μℓ+1+βdcμℓ,μℓ+1+dcνℓ,νℓ+1=1+2αdcνℓ,μℓ+1+dcμℓ,νℓ+1+2βdcμℓ,μℓ+1+dcνℓ,νℓ+1.
This together with (7) implies that
(13)dcνℓ,νℓ+1+dcμℓ,μℓ+1≼1+2α1−2βλℓdcν0,μ1+dcμ0,ν1,ℓ≥0.
Then, forℓ,j≥0, without loss of generality, we assume that j>ℓ:
(14)dcνℓ,νj≼∑k=ℓj−1dcνk,νk+1≼∑k=ℓj−11+2α1−2βλkdcν0,μ1+dcμ0,ν1=1+2α1−2β1−λλℓdcν0,μ1+dcμ0,ν1⟶θ,asℓ⟶∞.
Hence,νℓ is a Cauchy sequence by Lemma 4 (ii), and the completeness of U yields that there exists ν∈U such that
(15)limℓ⟶∞νℓ=ν∈G.
Similarly, we can get that(16)limℓ⟶∞μℓ=μ∈H.
Then, from Lemma4 (iii), we have
(17)limℓ⟶∞dcνℓ,μℓ=dcν,μ.
On the other hand, by (7) and (13),
(18)dcνℓ,μℓ≼dcνℓ,νℓ+1+dcνℓ+1,μℓ+1≼1+2α1−2β+1λℓdcν0,μ1+dcμ0,ν1⟶θ,asℓ⟶∞.
Therefore,dcν,μ=θ by Lemma 4 (i), and ν=μ∈G∩H.
Now, we have to prove thatν is a strong coupled FP of Γ. Indeed, by (4), (15), and (16), we have
(19)dcν,Γν,μ≤dcν,νℓ+1+dcνℓ+1,Γν,μ=dcν,νℓ+1+dcΓμℓ,νℓ,Γν,μ≼dcν,νℓ+1+αdcμℓ,Γμℓ,νℓ+dcν,Γν,μ+βdcμℓ,Γν,μ+dcx,Γμℓ,νℓ≼1+βdcν,νℓ+1+αdcμℓ,νℓ+1+dcν,Γν,μ+βdcμℓ,Γν,μ⟶α+βdcν,Γν,μ,asℓ⟶∞.
This implies thatdcν,Γν,μ=θ since α+β<1/2. Then, Γν,μ=ν=μ, and ν is a strong coupled FP of Γ.Corollary 10.
LetG and H be two nonempty closed subsets of a complete CMS U,dc, and let Γ:U×U⟶U be a cyclic coupled Kannan-type cone contraction w.r.t. G and H. That is,
(20)dcΓx,y,Γμ,ν≤αdcx,Γx,y+dcμ,Γμ,ν,where α∈0,1/2, x,ν∈G, and y,μ∈H. Then, Γ has a strong coupled FP in G∩H.Corollary 11.
LetG and H be two nonempty closed subsets of a complete CMS U,dc, and let Γ:U×U⟶U be a cyclic coupled Chatterjea-type cone contraction w.r.t. G and H. That is,
(21)dcΓx,y,Γμ,ν≤αdcμ,Γx,y+dcx,Γμ,ν,where α∈0,1/2, x,ν∈G, and y,μ∈H. Then, Γ has a strong coupled FP in G∩H.Remark 12.
In the special case, whenU,dc is a complete metric space, Corollary 10 is the same as Theorem 8 (Theorem 5 in [7]), if β=0 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.
LetU=ℝ in CMS which is defined as dcμ,ν=μ−ν, ∀μ,ν∈U, and let G=−1,0 and H=0,1. Then, G and H are two nonempty closed subsets of a set U and dG,H=0. A mapping Γ:U×U⟶U can be defined as
(22)Γx,y=−2x3.
Then, easily one can prove thatΓ is a cyclic mapping w.r.t. G and H, for any x,ν∈G and y,μ∈H. A mapping Γ is not a cyclic coupled Kannan-type contraction, since
(23)dcΓx,y,Γμ,ν=Γx,y−Γμ,ν=2x−μ3,where a=2/3∉0,1/2. Now, from (4), we have
(24)dcΓx,y,Γμ,ν=2x−μ3≤2x+μ3≤17x+μ21=10x+μ21+x+μ3=275x+5μ3+155x+5μ3=27x+2x3+μ+2μ3+15x+2μ3+μ+2x3≤27x+2x3+μ+2μ3+15x+2μ3+μ+2x3=27dcx,Γx,y+dcμ,Γμ,ν+15dcx,Γμ,ν+dcμ,Γx,y.
Hence, all the axioms of Theorem9 are fulfilled at α=2/7 and β=1/5. A mapping Γ has a strong coupled FP which is 0, that is, Γ0,0=0.Theorem 14.
Assume thatG and H are two nonempty closed subsets of a complete CMS U,dc and Γ:U×U⟶U is a cyclic coupled contractive-type mapping w.r.t. G and H which satisfies
(25)dcΓx,y,Γμ,ν≤amaxdcx,Γx,y,dcμ,Γμ,ν,dcμ,Γx,y,dcx,Γμ,ν,where x,ν∈G, y,μ∈H, and a∈0,1. Then, G∩H≠∅ and Γ has a strong coupled FP in G∩H.Proof.
Fixν0∈G and μ0∈H, and let νℓ and μℓ be two sequences defined as
(26)νℓ+1=Γμℓ,νℓ,μℓ+1=Γνℓ,μℓ,forallℓ≥0.
Then,νℓ⊂G and μℓ⊂H, since Γ is a cyclic mapping w.r.t. G and H. Now, we shall show that νℓ is a Cauchy sequence. We claim that, for all ℓ≥0,
(27)dcνℓ+1,μℓ+2+dcμℓ+1,νℓ+2≤λℓ+1dcν0,μ1+dcμ0,ν1,whereλ=a1−a<1.
First, we have to prove that(28)dcνℓ+1,μℓ+2≤λdcμℓ,νℓ+1,whereλ=a1−a<1.
Then, by (25), we have
(29)dcνℓ+1,μℓ+2=dcΓμℓ,νℓ,Γνℓ+1,μℓ+1≤amaxdcμℓ,Γμℓ,νℓ,dcνℓ+1,Γνℓ+1,μℓ+1,dcνℓ+1,Γμℓ,νℓ,dcμℓ,Γνℓ+1,μℓ+1=amaxdcμℓ,νℓ+1,dcνℓ+1,μℓ+2,dcνℓ+1,νℓ+1,dcμℓ,μℓ+2=amaxdcμℓ,νℓ+1,dcνℓ+1,μℓ+2,dcμℓ,μℓ+2.
Now, we have three cases:(i)
Ifdcμℓ,νℓ+1 is maximum, then from (29), we have(30)dcνℓ+1,μℓ+2≤adcμℓ,νℓ+1≤a1−adcμℓ,νℓ+1.
It holds (28), as a<a/1−a, since a∈0,1.
(ii)
Ifdcνℓ+1,μℓ+2 is maximum, then from (29), we have(31)dcνℓ+1,μℓ+2≤adcνℓ+1,μℓ+2,which is not possible.
(iii)
Ifdcμℓ,μℓ+2 is maximum, then from (29), we have(32)dcνℓ+1,μℓ+2≤adcμℓ,μℓ+2≤adcμℓ,νℓ+1+dcνℓ+1,μℓ+2≤a1−adcμℓ,νℓ+1.It follows that (28) holds.
Hence, from all cases, we get that(33)dcνℓ+1,μℓ+2≤λdcμℓ,νℓ+1,whereλ=a1−a<1.
Similarly, we can prove(34)dcμℓ+1,νℓ+2≤λdcνℓ,μℓ+1,whereλ=a1−a<1.
Then, again from (25), we have
(35)dcμℓ+1,νℓ+2=dcΓνℓ,μℓ,Γμℓ+1,νℓ+1≤amaxdcνℓ,Γνℓ,μℓ,dcμℓ+1,Γμℓ+1,νℓ+1,dcμℓ+1,Γνℓ,μℓ,dcxn,Γμℓ+1,νℓ+1=amaxdcνℓ,μℓ+1,dcμℓ+1,νℓ+2,dcμℓ+1,μℓ+1,dcνℓ,νℓ+2,t=amaxdcνℓ,μℓ+1,dcμℓ+1,νℓ+2,dcνℓ,νℓ+2.
Then again, we have the further three cases:(i)
Ifdcνℓ,μℓ+1 is maximum, then from (35), we have(36)dcμℓ+1,νℓ+2≤adcνℓ,μℓ+1≤a1−adcνℓ,μℓ+1.
It holds (34), as a<a/1−a, since a∈0,1.
(ii)
Ifdcμℓ+1,νℓ+2 is maximum, then from (35), we have(37)dcμℓ+1,νℓ+2≤adcμℓ+1,νℓ+2,which is not possible.
(iii)
Ifdcνℓ,νℓ+2 is maximum, then from (35), we have(38)dcμℓ+1,νℓ+2,t≤adcνℓ,νℓ+2≤adcνℓ,μℓ+1+dcμℓ+1,νℓ+2≤a1−adcνℓ,μℓ+1.It follows that (34) holds.
Hence, from all cases, we get that(39)dcμℓ+1,νℓ+2≤λdcνℓ,μℓ+1,whereλ=a1−a<1.
Now, by adding (33) and (39), we have that
(40)dcνℓ+1,μℓ+2+dcμℓ+1,νℓ+2≤λdcμℓ,νℓ+1+dcνℓ,μℓ+1.
Now, again by (25) and similar as above, we can get
(41)dcμℓ,νℓ+1≤λdcνℓ−1,μℓ,whereλ=a1−a<1,(42)dcνℓ,μℓ+1≤λdcμℓ−1,νℓ,whereλ=a1−a<1.
Now, again by adding (41) and (42) and then putting in (40), we have that
(43)dcνℓ+1,μℓ+2+dcμℓ+1,νℓ+2≤λ2dcνℓ−1,μℓ+dcμℓ−1,νℓ.
Continuing this process, we have that(44)dcνℓ+1,μℓ+2+dcμℓ+1,νℓ+2≤λℓ+1dcν0,μ1+dcμ0,ν1.
Hence, it is proven that (27) is exact ∀ℓ≥0. Now, for integer k, we have
(45)dcνk+1,μk+1=dcΓμk,νk,Γνk,μk≤amaxdcμk,Γμk,νk,dcνk,Γνk,μk,dcνk,Γμk,νk,dcμk,Γνk,μk≤amaxdcμk,νk+1,dcνk,μk+1,dcνk,νk+1,dcμk,μk+1.
Then, we have the following four cases:(a)
Ifdcμk,νk+1 is maximum, then from (45), we have(46)dcvk+1,μk+1≤adcμk,νk+1≤λdcμk,νk+1,where a<λ=a/1−a<1, since a∈0,1.
(b)
Ifdcνk,μk+1 is maximum, then from (45), we have(47)dcνk+1,μk+1≤adcνk,μk+1≤λdcνk,μk+1,where a<λ=a/1−a<1, since a∈0,1.
(c)
Ifdcνk,νk+1 is maximum, then from (45), we have(48)dcνk+1,μk+1≤adcνk,νk+1≤adcνk,μk+1+dcμk+1,νk+1≤λdcνk,μk+1,where a<λ=a/1−a<1.
(d)
Ifdcμk,μk+1,t is maximum, then from (45), we have(49)dcνk+1,μk+1≤adcμk,μk+1≤adcμk,νk+1+dcνk+1,μk+1≤λdcμk,νk+1,where a<λ=a/1−a<1.
Hence, from (a) and (d), we have(50)dcνk+1,μk+1≤λdcμk,νk+1,whereλ=a1−a<1.
And from (b) and (c), we have(51)dcνk+1,μk+1≤λdcνk,μk+1,whereλ=a1−a<1.
By adding (50) and (51), we have
(52)dcνk+1,μk+1≤λdcνk,μk+1+dcμk,νk+1,where η=λ/2, and in view of (44), we have
(53)dcνk+1,μk+1≤ηλkdcν0,μ1+dcμ0,ν1,fork≥0.
Since, by triangular inequality (44) and (53), we have
(54)dcνℓ,νℓ+1+dcμℓ,μℓ+1≤dcνℓ,μℓ+dcμℓ,νℓ+1+dcμℓ,νℓ+dcνℓ,μℓ+1=dcνℓ,μℓ+dcμℓ,νℓ+dcμℓ,νℓ+1+dcνℓ,μℓ+1≤2ηλℓ−1dcν0,μ1+dcμ0,ν1+λℓdcν0,μ1,t+dcμ0,ν1=1+2ηλλℓdcν0,μ1+dcμ0,ν1,forℓ≥0.
Now, forℓ,j≥0 and j>ℓ, we have
(55)dcνℓ,νj≤∑k=ℓj−1dcνk,νk+1≤∑k=ℓj−11+2ηλλkdcν0,μ1+dcμ0,ν1≤1+2ηλλℓ1−λdcν0,μ1+dcμ0,ν1⟶θ,asℓ⟶∞.
Hence, it is proven thatνℓ is a Cauchy sequence and it is convergent in U. Since G is a closed subset of U, therefore
(56)νℓ⟶ν∈G,asℓ⟶∞.
Similarly,(57)μℓ⟶μ∈H,asℓ⟶∞.
Hence, from (56) and (57), we have
(58)limℓ⟶∞dcνℓ,μℓ=dcν,μ.
By triangular inequality (44) and (54), we have
(59)dcνℓ,μℓ≤dcνℓ,νℓ+1+dcνℓ+1,μℓ≤λ+2ηλ+1λℓdcν0,μ1+dcμ0,ν1⟶θ,asℓ⟶∞.
Therefore,dcν,μ=0, which implies that ν=μ∈G∩H.
Now, we prove a strong coupled FP ofΓ in U,dc; therefore,
(60)dcν,Γν,μ≤dcν,νℓ+1+dcνℓ+1,Γν,μ.
Then, by the view of (25), (56), and (57), we have
(61)≤amaxdcμℓ,Γμℓ,νℓ,dcν,Γν,μ,dcν,Γμℓ,νℓ,dcμj,Γν,μ≤amaxdcμℓ,νℓ+1,dcν,Γν,μ,dcν,νℓ+1,dcμℓ,Γν,μ⟶adcν,Γν,μ,asℓ⟶∞.
Hence, from (60),
(62)dcν,Γν,μ≤dcν,νℓ+1+dcνℓ+1,Γν,μ⟶adcν,Γν,μ,asℓ⟶∞,which implies that dcν,Γν,μ=θ, since 1−a≠0. Hence, Γν,μ=ν=μ, which shows that ν is a strong coupled FP of Γ.Corollary 15.
Assume thatG and H are two nonempty closed subsets of a complete CMS U,dc and Γ:U×U⟶U is a cyclic coupled contractive-type mapping w.r.t. G and H which satisfies:
(63)dcΓx,y,Γμ,ν≤amaxdcx,Γx,y,dcμ,Γμ,ν,where x,ν∈G, y,μ∈H, and a∈0,1. Then, G∩H≠∅ and Γ has a strong coupled FP in G∩H.Example 16.
From Example13, a mapping Γ:U×U⟶U can be defined as
(64)Γx,y=−3x7.
Then, easily one can verify thatΓ is a cyclic mapping w.r.t. G and H, for x,ν∈G and y,μ∈H. A mapping Γ is a cyclic coupled Kannan-type contraction, since
(65)dcΓx,y,Γμ,ν=Γx,y−Γμ,ν=37x−μwhere a=3/7∈0,1/2. Now, from (25), we have
(66)dcΓx,y,Γμ,ν=3x−μ7≤30x−μ49≤37·107maxx,μ,7μ+3x10,7x+3μ10=37max10x7,10μ7,7μ+3x7,7x+3μ7≤37maxx+3x7,μ+3μ7,7μ+3x7,7x+3μ7=37maxdcx,Γx,y,dcμ,Γμ,ν,dcμ,Γx,y,dcx,Γμ,ν.
Hence, Example16 is satisfying all conditions of Theorem 14 with a=3/5, and Γ0,0=0, which is a strong coupled FP in U.


## 4. Integral-Type Application

This section is intended to present two UIEs for the existence of a common solution to support our main result. LetU=Ca0,b0,ℝ be the Banach space of all continuous functions defined on a0,b0 with the supremum norm
(67)μ=supr∈a0,b0μr,whereμ∈Ca0,b0,ℝ,and the induced metric dc can be defined as
(68)dcμ,ν=supr∈a0,b0μr−νr,whereμ,ν∈Ca0,b0,ℝ.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(69)μl=∫a0b0K1l,s,μsds+f1l,(70)μl=∫a0b0K2l,s,μsds+f2l,where l∈a0,b0⊂ℝ and μ,f1,f2∈U.
LetK1,K2:a0,b02×ℝ⟶ℝ such that Gx,y,Hμ,ν∈U, for x,ν∈G, y,μ∈H, and G,H⊆U; therefore,
(71)Gx,yl=∫a0b0K1l,s,x,ysds,Hμ,νl=∫a0bK2l,s,μ,νsds,for all l∈a0,b0. If there exists a∈0,1 such that
(72)Gx,y+f1−Hμ,ν+f2≤aNx,y,μ,ν,where
(73)Nx,y,μ,ν=maxGx,y+f1−x,Hμ,ν+f2−μ,Gx,y+f1−μ,Hμ,ν+f2−x,
then the two UIEs (69) and (70) have a unique common solution.Proof.
Define a mappingΓ:U×U⟶U:
(74)Γx,y=Gx,y+f1,Γμ,ν=Hμ,ν+f2.
Then, we may have the following four cases:(a)
If∥Gx,y+f1−x∥ is the maximum in (73), then we have(75)Nx,y,μ,ν=Gx,y+f1−x.
Hence, we get that(76)dcΓx,y,Γμ,ν=Γx,y−Γμ,ν≤aΓx,y−x=adcx,Γx,y.(b)
IfHμ,ν+f2−μ is the maximum in (73), then we have(77)Nx,y,μ,ν=Hμ,ν+f2−μ.
Hence, we get that(78)dcΓx,y,Γμ,ν=Γx,y−Γμ,ν≤aΓμ,ν−μ=adcμ,Γμ,ν.(c)
IfGx,y+f1−μ is the maximum in (73), then we have(79)Nx,y,μ,ν=Gx,y+f1−μ.
Hence, we get that(80)dcΓx,y,Γμ,ν=Γx,y−Γμ,ν≤aΓx,y−μ=adcμ,Γx,y.(d)
IfHμ,ν+f2−x is the maximum in (73), then we have(81)Nx,y,μ,ν=Hμ,ν+f2−x.
Hence, we get that(82)dcΓx,y,Γμ,ν=Γx,y−Γμ,ν≤aΓμ,ν−x=adcx,Γμ,ν.
Hence, from all cases, for everyx,ν∈G, y,μ∈H, and G,H⊆U with a∈0,1 in Theorem 14, the two UIEs (69) and (70) have a common solution in U.


## 5. Conclusions

It is well known that FP has been the heart of several areas of mathematics and other sciences. In this work, we have presented some generalized strong coupled FP theorems in cone metric spaces by using the cyclic-type mappings with illustrative examples. The result of our article gives the generalization of “cyclic coupled Kannan-type contraction” given by Choudhury and Maity in [7]. Finally, to support our main result, we have established an application of the two UIEs for the existence of a common solution to support our work. This concept will be very useful for finding different contractive-type strong coupled FP results in the context of metric spaces with different types of applications.



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*Source: 5541981-2021-05-17.xml*