# Common Fixed Point Theorems for Generalized Contractive Pair of Mappings in a Metric Space and Their Application to Fractional Calculus

Authors: Priya Goel; Manoj Kumar; Dimple Singh; Kamal Kumar
Journal: Journal of Applied Mathematics
 (2022)

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## Abstract

In this manuscript, we have established relation-theoretic version of some common fixed point results in metric space for generalizedβ−ϕ−Z-contractive pair of mappings furnished with an arbitrary binary relation R. Recently, the concept of binary relation is well known leading trend in fixed point theory. Our results extend and unify several fixed point theorems present in the literature. An illustrative example is given to support our main theorem. Finally, we exploit our main result for proving existence and uniqueness results to established the solution of a fractional differential equation of Caputo type.


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## Body

## 1. Introduction

Fixed point theory is a very extensive and active area of research which promises the existence and uniqueness of solution to many mathematical problems in various field of sciences. It begins with fixed point and proceeds through the Banach contraction principle [1]. Banach contraction principle is the most conspicuous tool in the field of fixed point theory which states that every contraction defined on a complete metric space possesses a unique fixed point that is a self- mapping U defined on a complete metric space S,σ has a unique fixed point if there exists h∈0,1 such that
(1)σUr,Us<hσr,s,∀r,s∈S.Furthermore,limϱUϱr=t for all r in S, where Uϱϱ≥1 is the ϱth iteration of U. It ensures us about the existence and uniqueness of solutions to substantial problems in various directions of mathematics. Banach contraction principle is employed in various field of mathematics as well as other domain of sciences and proved many new fixed point results and their uniqueness related to contraction type of mappings. In 1967, the first coincidence fixed point result is proved by Machuca [2], which was later improved by Goebel [3]. Further, in 1976, Jungck [4] has established the first common fixed point theorem. The principle of Banach contraction is generalized by many authors in numerous ways. Ran and Reurings [5] and Nieto and Rodriguez-Lopez [6] extended the Banach contraction principle in a new way by showing that if the metric space is endowed with an ordered binary relation, then it is sufficient to assume that the contraction condition holds only for those comparable elements. In 2006, T. Bhaskar and Lakshmikantham [7] established a fixed point theorem for a mixed monotone mapping in a metric space endowed with partial order. In 2007, Ben-El-Mechaiekh [8] generalized Ran–Reurings’ fixed point theorem using the concept of transitive binary relation in a complete metric space. Alam and Imdad [9] have presented a new variant of Banach contraction principle in a complete metric space under an universal binary relation and also utilized their result for transitive, strict order, preorder binary relation, etc. in 2015.In2012, Wardowski [10] introduced the concept of F- contraction and stated a fixed point theorem for F-contraction in complete metric spaces. In 2014, Wardowski and Van Dung [11] introduced the notion of F-weak contraction and stated a fixed point theorem for F-weak contraction in metric space. Jleli and Samet [12] gave a generalization of Banach contraction principle in complete generalized metric space in 2014. Later, in 2015, Aggarwal et al. [13] discussed the existence and uniqueness of the common fixed point of expansive mappings in the complete G-metric space. In 2018, Imdad et al. [14] demonstrated the idea of weak θ-contraction on metric space by generalizing the idea of θ-contraction introduced by JLeli and Samet and proved some relation-theoretical fixed point results for weak θ- contraction in a generalized metric space without completeness property with application in fractional calculus. In 2019, Alfaqih et al. [15] gave the notion of F,Rg- contraction and proved some common and coincident fixed point results in metric space endowed with binary relations.In2015, Khojasteh et al. [16] initiated the idea of Z-contraction by introducing a new function, known as simulation function. They generalized the Banach contraction principle by originating the idea of Z-contraction and proved some fixed point results in metric spaces. Moreover, many other researchers proved many coincidence and common fixed point results in various metric spaces using Z-contraction. Shahzad and Karapinar [17] demonstrated some coincidence fixed point results in metric spaces using the concept of Z-contractions. In b-metric spaces, existence and the uniqueness of few operators are introduced by Rashid et al. [18]. They established a new type of contractive condition by combining the idea of simulation functions with admissible functions. Argoubi et al. [19] slightly modified the definition of simulation function by withdrawing a condition that is ξ0,0=0 and proved some fixed point results for a pair of nonlinear operators satisfying nonlinear contraction results in partial ordered metric spaces based on Z-contraction.In2012, Samet et al. [20] initiated the idea of α-admissible functions and proposed a new category of contractive type mapping, known as α−ϕ-contractive type mapping. With the help of α- admissibility, they established some fixed point results for α−ϕ- contractive type mappings. Motivated Samet et al. [20], Durmaz et al. [21] attained the existence and uniqueness of the solution of a fourth order two-point boundary value problem. Further, Wardowski and Van Dung [11] gave a generalized form of α−ϕ-contractive type mappings and obtained various fixed point results. Recently, Sarwar et al. [22] have proposed some fixed point results in metric interval and normed interval spaces using the concept of simulation functions and α-admissibility in 2021. Also, in 2021, Kumar and Sharma [23] illustrated the idea of generalized α−ϕ−Z-contractive type mapping and proved some fixed point results in metric spaces. Here, the results are proved in terms of α-admissibility condition and a condition of existence a subsequence for some convergent sequence and for the validity of their results; they have also demonstrated some fixed point results in metric spaces endowed with a partial order.Fixed point theory also plays a vital role in establishing the existence and the uniqueness of the solution of a fractional differential equation. In 1996, Delbosco and Rodino [24] established the existence and the uniqueness of the solution of a nonlinear fractional differential equation. In 2009, Belmekki et al. [25], attained the existence of periodic solution of a nonlinear fractional differential equation. In [26], Zada et al. proved a coupled fixed point theorem and attained the solution of fractional variable order hybrid differential equations in 2021. In addition, Abdo et al. [27], in 2021, developed the theory of nonlinear system of pantograph-type fractional differential equations and proved fixed point result with the use of Banach contraction principle and Krasnoselskii’s fixed point theorem. In 2021, Aydi et al. established some fixed point results in extended b-metric space and existence and the uniqueness of a system of nonlinear fractional differential equation is obtained. For instance, the Caputo nonlinear fractional differential equation proposed by Kilbas, is given as
(2)CDpγrt=ht,rt;t∈0,1,1<γ≤2,r0=0,r1=∫0wrpdp,0<w<1,where CDpγ represents the Caputo fractional derivative of order γ, and h:0,1⟶R is a continuous function. This nonlinear fractional differential equation can also be written as
(3)rt=1Γγ∫0tt−pγ−1hp,rpdp−2t2−w2Γγ∫011−pγ−1hp,rpdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,rzdzdp.In this paper, we have extended and generalized the results proved by Kumar and Sharma in [23] in 2021. Our current study provides a new approach for proving fixed point results in terms of binary relation by relaxing the condition of existing a subsequence for some convergent sequence and modifying the condition of β-admissibility. Our results show some fixed point results for generalized β−ϕ−Z- contractive pair of mappings in a new manner using the conditions in terms of R-continuity and relation R in terms of closeness of functions. We also explore an example (Example 21) which shows that our results are better in comparison to the existing result [23]. In addition, we have attained the existence and the uniqueness of the solution of fractional differential equations in the context of our main result, and hence our study is also proved to be useful in finding the existence and the uniqueness of solutions of integral equations, boundary value problems, and fractional differential equations.In this paper, we begin with introduction in Section1. In Section 2, we have stated some fundamental definitions related to our work. In Section 3, we have demonstrated a relation-theoretical common fixed point result and its uniqueness for generalized β−ϕ−Z-contractive pair of mappings with the help of binary relation. In Section 4, we have proved some more fixed point results with the help of our main result that is proved in Section 3. In the support of our main theorem, we have also provided an example in Section 5, for which our main result is applicable in finding the existence and the uniqueness of the solution for a given mathematical problem. In Section 6, we have applied our main result in finding the existence and the uniqueness of the solutions of fractional differential equation in Caputo sense. We have also given an example with graphical representation for the reliability of our attained result in fractional calculus.


## 2. Preliminaries

In this part, we have stated a few definitions associated with our work.Definition 1 (see [28]).
A subset ofS×S is termed as binary relation R where S is any nonempty set.Definition 2 (see [29]).
A sequencesϱ∈S is called R-preserving if sϱ,sϱ+1∈R for all ϱ∈N0, where R is a binary relation defined on a nonempty set S.Definition 3 (see [29]).
LetS,σ be any metric space and a binary relation R defined on set S. Thus, if each R-preserving cauchy sequence in S converges to a point in S, then S,σ is called R-complete.Definition 4 (see [29]).
Let us consider a metric spaceS,σ and a binary relation R defined on set S. Then, a self-mapping U:S⟶S is known as R-continuous at s∗ whenever sϱ⟶s∗ for any R-preserving sequence sϱ, then Usϱ⟶Us∗ and if U is continuous at every point of S, then it is called R-continuous.Definition 5 (see [16]).
A simulation function is a mappingξ:0,∞×0,∞⟶R requiring the following assumptions:
(1)
ξ0,0=0;(2)
ξr,s<s−r, for each r,s>0;(3)
If we have two sequencesUϱ and Vϱ∈0,∞ in such a manner that limϱ⟶∞Uϱ=limϱ⟶∞Vϱ=l∈0,∞, then(4)limϱ⟶∞Uϱ,Vϱ<0.Note:Z denotes the classes of all the simulation functions ξ:0,∞×0,∞⟶R.Definition 6 (see [20]).
If we consider a self-mappingU:S⟶S and a function β:S×S⟶0,+∞, then U is called β-admissible if
(5)βr,s≥1⇒βUr,Us≥1,∀r,s∈S.Definition 7 (see [30]).
LetU,V:S×S⟶0,∞ be two self-mappings, and β:S×S⟶0,+∞ is a function. Then, U is called β-admissible with respect to V if
(6)βVr,Vs≥1⇒βUr,Us≥1,∀r,s∈S.Definition 8 (see [20]).
Φ, is the collection of functions ϕ:1,∞⟶1,∞ where ϕ is required to hold:
(1)
ϕ:1,∞⟶1,∞ is nondecreasing(2)
∑ϱ=1+∞ϕϱs<∞ for all s>0, where ϕϱs is the ϱth iterate of ϕ.Definition 10 (see [20]).
Consider a metric spaceS,σ and a self-mapping U:S⟶S. Then, the self-mapping U is known as β−ϕ-contractive mapping if there exist two functions ϕ∈Φ and β:S×S⟶0,+∞ in such a manner that
(7)βr,sσUr,Us≤ϕσr,s,∀r,s∈S.Definition 11 (see [30]).
Consider a metric spaceS,σ and two self-mappings U,V:S⟶S. Then, the pair U,V is said to be generalized β−ϕ-contractive pair of mappings if there exist two functions ϕ∈Φ and β:S×S⟶0,+∞ such that
(8)βVr,VsσUr,Us≤ϕMVr,Vs≥0,∀r,s∈S,where
(9)MVr,Vs=maxσVr,Vs,σVr,Ur+σVs,Us2,σVr,Us+σVs,Ur2.Definition 12 (see [23]).
Consider a metric spaceS,σ and two self-mappings U,V:S⟶S. Then, the pair U,V is said to be generalized β−ϕ−Z-contractive pair of mappings with respect to ξ if
(10)ξβVr,VsσUr,Uv,ϕMVr,Vs≥0,∀r,s∈S,
whereβ:S×S⟶0,∞ and ϕ∈Φ and
(11)MVr,Vs=maxσVr,Vs,σVr,Ur+σVs,Us2,σVr,Us+σVs,Ur2.


## 3. Main Results

In this part, we have proved few common fixed point results and their uniqueness for generalizedβ−ϕ−Z-contractive pair of mappings by using the concept of binary relation and the simulation function.Theorem 13.
If we assume thatS,σ be any R-complete metric space equipped with a binary relation R defined on S and two self-mappings U,V:S⟶S are such that US⊆VS, and the pair U,V is a generalized β−ϕ−Z-contractive pair of mappings. Then, U and V have a unique common fixed point if we have the following assumptions:
(1)
SU,R is nonempty(2)
R is U-closed and V-closed(3)
βr,s≥1,∀r,s∈S;(4)
U and V are R-continuousProof.
In condition (1) we have given thatSU,R is nonempty i.e., SU,R≠ϕ. So, let r0∈SU,R i.e., r0,Ur0∈R and construct a Picard sequence rϱ,ϱ=0,1,2,⋯⋯ such that rϱ+1=Urϱ;∀ϱ∈N, and also it is given in condition (3) that R is U-closed, so we get
(12)r0,ur0,Ur0,U2r0,U2r0,U3r0,⋯.,Uϱr0,Uϱ+1r0=rϱ,rϱ+1∈R;∀ϱ∈N.
Thus,rϱ is a R-preserving sequence. Now, from the given condition i, we have US⊆VS, therefore, we can choose a point r1∈S such that Ur0=Vr1. Hence, by continuing the same process and selecting the points r2,r3,⋯⋯.,rϱ,⋯⋯∈S, we get
(13)Urϱ=Vrϱ+1,forallϱ=0,1,2,⋯⋯.Here, we have two cases.Case I: IfUrϱ+1=Urϱ for some ϱ, then from (13) equation, we have Urϱ+1=Vrϱ+1 for some ϱ. Thus, U and V have a common fixed point r=rϱ+1.Case II: IfUrϱ+1≠Urϱ and it is given that pair U,V is a generalized β−ϕ−Z- contractive pair of mappings, then
(14)ξβVr,VsσUr,Us,ϕMVr,Vs≥0,(15)ϕMVr,Vs−βVr,VsσUr,Uv≥0,(16)βVr,VsσUr,Us≤ϕMVr,Vs.Forr=rϱ and s=rϱ+1, the given inequality becomes
(17)βVrϱ,Vrϱ+1σUrϱ,Urϱ+1≤ϕMVrϱ,Vrϱ+1Now sinceU,V:S⟶S are two self-mappings, so
(18)Ur,Vr∈S;∀r∈Sand from given condition (3), we have
(19)βr,s≥1,∀r,s∈S.Hence, by comparing Equations (18) and (19), we get
(20)βVr,Vs≥1,∀r,s∈S.Using Condition (20) in Equation (14), we attain
(21)σUrϱ,Urϱ+1≤ϕMVrϱ,Vrϱ+1;where
(22)MVrϱ,Vrϱ+1=maxσVrϱ,Vrϱ+1,σVrϱ,Urϱ+σVrϱ+1,Urϱ+12+σVrϱ,Urϱ+1+σVrϱ+1,Urϱ2,=maxσUrϱ−1,Urϱ,σUrϱ−1,Urϱ+σUrϱ,Urϱ+12+σVrϱ,Urϱ+1+σVrϱ+1,Urϱ2,≤maxσUrϱ−1,Urϱ,σUrϱ,Urϱ+1.Now ifmaxσUrϱ−1,Urϱ,σUrϱ,Urϱ+1=σUrϱ,Urϱ+1; then we get,
(23)σUrϱ,Urϱ+1≤ϕσUrϱ,Urϱ+1<σUrϱ,Urϱ+1;∀ϱ≥1,which is a contradiction.
(24)Hence,maxσUrϱ−1,Urϱ,σUrϱ,Urϱ+1=σUrϱ−1,Urϱ.Thus, we have(25)σUrϱ,Urϱ+1≤ϕσUrϱ−1,Urϱ;∀ϱ≥1.By repeating the same process forr=rϱ−1,s=rϱ, and so on, we get
(26)σUrϱ−1,Urϱ≤ϕσUrϱ−2,Urϱ−1,σUrϱ−2,Urϱ−1≤ϕσUrϱ−3,Urϱ−2,⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯.σUr2,Ur1≤ϕσUr1,Ur0.Thus, by substituting all these inequalities in Equation (25), we get
(27)σUrϱ,Urϱ+1≤ϕϱσUr0,Ur1;∀ϱ≥1.Using triangular inequality and above inequality, we get:(28)σUrϱ,Urϱ+k≤∑p=ϱ∞ϕpσUr1,Ur0;∀k≥1.By letting,p⟶∞, we get Urϱ which is a cauchy sequence in S,σ, and since Urϱ=rϱ+1, so rϱ+1 i.e., rϱ is a cauchy sequence, and sequencerϱ is also the R-preserving. So, rm is the R-preserving cauchy sequence in S,σ, and S,σ is the R-complete. So, rϱ is the convergent sequence in S, that is, limϱ⟶∞rϱ⟶p, and p∈S. Since U is the R-continuous from the given condition (5), we have
(29)p=limϱ⟶∞rϱ=limϱ⟶∞rϱ+1=limϱ⟶∞Ur0=Ulimϱ⟶∞rϱ=Up.Therefore,p is a fixed point of U. Now, since Urϱ=Vrϱ+1;∀m=0,1,2,⋯⋯ and Urϱ is a cauchy sequence in S. Hence, Vrϱ+1 i.e. Vrϱ is also a cauchy sequence in S, and V is the R-continuous from the given condition (4), so
(30)p=limϱ⟶∞rϱ=limϱ⟶∞rϱ+1=limϱ⟶∞Vrϱ+1=Vlimϱ⟶∞rϱ+1=Vp.Thus,p is also a fixed point of V. Hence, p is the common fixed point of U and V.Uniqueness of common fixed point. Letq∈S be any other common fixed point point of U and V other than p. Then, by definition of common fixed point, we have
(31)Uq=Vq=qandalsoUp=Vp=pSo, we have to provep=q i.e., σp,q=0. By choosing r=p and s=rm, we attain
(32)σp,rϱ+1=σUp,Urϱ≤ϕMvp,Vrϱ,whereMVp,Vrϱ=maxσVp,Vrϱ,σVp,Up+σVrϱ,Urϱ2,σVp,Urϱ+σVrϱ,Up2,=maxσp,rϱ,σrϱ,rϱ+12.Thus, ifmaxσp,rϱ,1/2σrϱ,rϱ+1=1/2σrϱ,rϱ+1, then
(33)σp,rϱ+1≤ϕ12σrϱ,rϱ+1<σrϱ,rϱ+1.By lettingϱ⟶∞, we get 0<0, which is a contradiction. Hence,
(34)maxσp,rϱ,12σrϱ,rϱ+1=σp,rϱ.Hence,σp,rϱ+1≤ϕσp,rϱ.By continuing the same process, we get(35)σp,rϱ+1≤ϕϱσp,r0;∀ϱ∈N⇒σp,rϱ+1⟶0asϱ⟶∞.Similarly, we can prove thatσq,rϱ+1⟶0 as ϱ⟶∞.σp,q≤σp,rϱ+1+σq,rϱ+1 [By triangular inequality].σp,q⟶0 as n⟶∞ i.e., σp,q=0 i.e. p=q.Thus,U and V have a unique common fixed point.


## 4. Consequences

Here, we have stated some other results with the help of Theorem (13).Theorem 14.
Consider aR-complete metric space S,σ where R is a binary relation defined on set S and U,V:S⟶S are two self-mappings with ϕ∈Φ in such a manner that σUr,Us≤λϕMVr,Vs. Then, U and V have a unique common fixed point if we have the following assumptions:
(1)
SU,R is nonempty(2)
US⊆VS(3)
R is U-closed and V-closed(4)
U and V are R-continuousProof.
The result will hold directly from the given Theorem (13) by taking ξr,s=λs−r for all r,t>0 and λ∈0,1 and βr,s=1 for all r,s∈S.Theorem 15.
LetS,σ be any R-complete metric space where R is a binary relation defined on set S, and U:S⟶S is a self- mapping with ϕ∈Φ such that σUr,Us≤λϕMr,s. Then, U has a unique fixed point if the following propositions hold:
(1)
SU,R is nonempty(2)
R is U-closed(3)
U is R-continuousProof.
The result will hold directly from the given Theorem (13) by taking ξr,s=λs−t for all r,s>0 and λ∈0,1, βr,s=1 for all r,s∈S and Tr=r,∀r∈S.Theorem 16.
Suppose thatS,σ be an R-complete metric space where R is a binary relation defined on set S, and U,V:S⟶S are two mappings satisfying σUr,Us≤ϕσVr,Vs with ϕ∈Φ. Now if we have the following conditions:
(1)
SU,R is nonempty(2)
US⊆VS(3)
R is U-closed and V-closed(4)
U and V are R-continuousthen, U and V have a unique common fixed point.Proof.
The result can be hold directly from the given Theorem (13) by taking βr,s=1 for each r,s∈S and MVr,Vs=σVr,Vs.Theorem 17.
Assume thatS,σ be any R-complete metric space equipped with a binary relation R defined on set S, and U:S⟶S is a self-mapping satisfying the condition σUr,Us≤ϕσr,s where ϕ∈Φ. Thus, if we have the following assumptions:
(1)
SU,R is nonempty(2)
R is U-closed(3)
U is R-continuousthen, U has a unique fixed point.Proof.
The result can be attained directly from the given Theorem (13) for βr,s=1 for each r,s∈λ,MVr,Vs=σVr,Vs and the self-map V as Vr=r.Theorem 18.
Consider aR-complete metric space S,σ equipped with a binary relation R defined on set S, and U:S⟶S be a self-mappings in such a manner that σUr,Us≤σr,s, Then, U has a unique fixed point if we have the following propositions:
(1)
SU,R is nonempty(2)
R is U-closed(3)
U is R-continuousProof.
The result will hold directly from the given Theorem (13) by taking βr,s=1,MVr,Vs=σVr,Vs,ϕ=1, and Vr=r∀r,s∈U.Theorem 19.
Let us suppose thatS,σ be any R-complete metric space where R is a binary relation defined on set S, and U:S⟶S is a self-mapping with ϕ∈Φ such that σUr,Us≤λσr,Ur+σs,Us. Then, U has a unique fixed point if we have the following conditions:
(1)
SU,R is nonempty(2)
R is U-closed(3)
U is R-continuousProof.
The result can be attained directly from the given Theorem (13) by taking ξs,r=2λr−s,λ∈0,1/2,βr,s=1,ϕ=1 and Vr=r for every r,s∈U.Theorem 20.
Let us suppose thatS,σ be any R-complete metric space equipped with a binary relation R defined on set S, and U:S⟶S is a self-mapping such that σUr,Us≤λσr,Us+σs,Ur. Then, U has a unique fixed point if we have the following conditions:
(1)
SU,R is nonempty(2)
R is U-closed(3)
U is R-continuousProof.
The result will hold directly from the given Theorem (13) by taking ξr,s=2λs−r for all r,s>0 and λ∈0,1/2, βr,s=1,ϕ=1 and Vr=r for all r,s∈S.


## 5. Example

Here, in this section, we have demonstrated an example in support of Theorem (13) which shows that our main Theorem (13) is applicable to determine the existence of a unique common fixed point for the given problem, but the existing result in [31] is not applicable.Example 21.
Consider the spaceS=0,∞ with usual metric σ and a binary relation R defined on S. Then, a pair of mappings U,V:S⟶S defined as
(36)Ur=0,ifr∈0,1,2,ifr∈1,∞.Vr=0,ifr∈0,1,1,ifr∈1,2,2,ifr∈2,3,4,ifr∈3,∞,have a unique common fixed point with βs,r=1.Proof.
Forr=1.5 and s=2, we have Ur=2, Vr=1, Us=2, and Vs=1. Hence,
(37)βVr,Vs=β1,1=1,βUr,Us=β2,2=16.
Thus,βVr,Vs≥1⇏βUr,Us≥1 and hence, U is not β- admissible with respect to V. Hence, solution of this mathematical problem can not be find out with the help of existing result Theorem (2.2) in [32].
(1)
Clearly,SU,R is nonempty(2)
SinceUS=0,2,VS=0,1,2,4 and hence clearly, we have US⊆VS.(3)
Since for allr∈R, we have r,Ur,r,Vr∈R. Thus, R is U−closed and V-closed(4)
Also,U and V are R-continuous, as for any sequence rn with rn⟶r, we have Urn⟶Ur and Vrn⟶Vr.(5)
MVr,Vs=4 and maxσUr,Us=2.
βVr,VsσUr,Us≤2<4=MVr,Vs.
βVr,VsσUr,Us<MVr,Vs.
Now by choosing an monotonically increasing functionϕt=et, we obtain
βVr,VsσUr,Us<ϕMVr,Vs.
Thus,ξβVr,VsσUr,Us,ϕMVr,Vs≥0.
Hence, pairU,V is generalized β−ϕ−Z-contractive pair of mappings. Hence, all the conditions are satisfied of the above Theorem (13) and thus U and V have a unique common fixed point. Here, in our example, U and V have a unique common fixed point 0.Graphical representation of solution: the graphical solution of Example21 is given in Figure 1, which clearly shows that U and V have a unique common fixed point 0.Figure 1
Graphical solution of the Example21.


## 6. Application

### 6.1. Existence and Uniqueness of a Common Solution of Nonlinear Fractional Differential Equation of Caputo Type

The aim of this part is to introduce an application of Theorem (13) to obtain a common solution of nonlinear fractional differential equation of Caputo type for a pair of generalized β−ϕ−z-contractive pair of mapping in metric space. The Caputo nonlinear fractional differential equation proposed by Kilbas, is given as follows:
(38)CDpαrt=ht,rt,t∈0,1,1<γ≤2,r0=0,r1=∫0wrpdp,0<w<1,where CDpγ represents the Caputo fractional derivative of order γ and h:0,1⟶R is a continuous function. This nonlinear fractional differential equation can also be represented in the form as follows:
(39)rt=1Γγ∫0tt−pγ−1hp,rpdp−2t2−w2Γγ∫011−pγ−1hp,rpdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,rzdzdp.A functionrt is a solution of above defined nonlinear fractional differential equation whenever it is the solution of the fractional integral Equation (39) and vice-versa.Theorem 22.
Consider the space of all continuous functionsS=C0,1 constructed on closed interval 0,1, equipped with a binary relation R=C0,1×C0,1. Let C0,1,S be the Banach space of all continuous functions from 0,1 into S with norm r=supt∈0,1rt;∀rt∈C0,1. This space defines the metric as follows:
(40)σr,s=supt∈0,1rt−st;∀r,s∈S.
Now if we construct two self-mapsU,V:S⟶S as
(41)Urt=1Γγ∫0tt−pγ−1hp,rpdp−2t2−w2Γγ∫011−pγ−1hp,rpdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,rzdzdp,(42)andVrt=1Γγ∫0tt−pγ−1hp,rpdp,
satisfying thatUS⊆VS. Then, U and V have a unique common fixed point if we have the following assumptions:
(1)
βr,s=1,∀r,s∈S,(2)
there exist a continuous functionh:0,1×R⟶R satisfying that ht,rt−ht,st=krt−st, where k is some constant(3)
tγ2−w2γ+1+2t1+γ+γ2/γ+12−w2≤1.Proof.
Obviously,C0,1,S is a complete metric space.
(1)
sinceU:S⟶S is a self map, and R=S×S, then for any r0∈S, we have Ur0∈S. Thus, r0,Ur0∈R, and hence SU,R is nonempty(2)
for anyr0∈S, Ur0,U2r0,⋯..,Uϱr0,Uϱ+1r0,⋯∈S as U:S⟶S is a self-map. Therefore, r0,Ur0,Ur0,U2r0,⋯⋯⋯,Uϱr0,Uϱ+1r0∈R and hence R is U-closed. In the same pattern, we can easily prove that R is V-closed.Thus, R is U-closed as well as V-closed(3)
from given condition (1), we haveβr,s=1, i.e. βr,s≥1, for each r,s∈S.(4)
it is only required to prove that pairU,V is generalized β−ϕ−z-contractive pair of mappings(43)σVrt,Vst=supt∈0,1Vrt−Vst,=supt∈0,11Γγ∫0tt−pγ−1hp,rpdp−1Γγ∫0tt−pγ−1hp,spdp,=1Γγ∫0tt−pγ−1supt∈0,1hp,rpdp−hp,spdp,=kΓγ∫0tt−pγ−1supt∈0,1rpdp−spdp,=kΓγr−s∞∫0tt−pγ−1dp=kr−s∞pγγΓγ2.σUrt,Ust=supt∈0,1Urt−Ust,=supt∈0,11Γγ∫0tt−pγ−1hp,rpdp−2t2−w2Γγ∫011−pγ−1hp,rpdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,rzdzdp−1Γγ∫0tt−pγ−1hp,spdp+2t2−w2Γγ∫011−pγ−1hp,spdp−2t2−w2Γγ∫0w∫0pp−zγ−1hz,szdzdp.σUrt,Ust=supt∈0,11Γγ∫0tt−pγ−1hp,rp−hp,spdp−2t2−w2Γγ∫011−pγ−1hp,rp−hp,spdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,rz−hz,szdzdp,≤supt∈0,11Γγ∫0tt−pγ−1hp,rp−hp,spdp+2t2−w2Γγ∫011−pγ−1hp,rp−hp,spdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,rz−hz,szdzdp,≤kΓγ∫0tt−pγ−1supt∈0,1rp−spdp+2t2−w2Γγ∫011−pγ−1supt∈0,1rp−spdp+2t2−w2Γγ∫0w∫0pp−zγ−1supt∈0,1rz−szdzdp.σUrt,Ust≤kr−s∞Γγ∫0tt−pγ−1dp−2t2−w2∫011−pγ−1dp+2t2−w2Γγ∫0w∫0pp−zγ−1dzdp,=kr−s∞γΓγ2tγ2−w2γ+1+2t1+γ+γ2γ+12−w2.
By using given conditionc, we get
(44)σUrt,Ust≤kr−s∞γΓγ2,=σVrt,Vst,≤MσVrt,Vst.
Now if we assume thatϕt=et, then by assuming MσVrt,Vst=t, we get
(45)eMσVrt,Vst>MσVrt,Vst,i.e.,ϕMσVrt,Vst>MσVrt,Vst.
Then, by comparing Equations (45) and (6.6), we get
(46)σUrt,Ust≤ϕMσVrt,Vst,βVrt,VstσUrt,Ust≤ϕMσVrt,Vst,βVrt,Vst=1,ξβVrt,VstσUrt,Ust,ϕMσVrt,Vst≥0.
This shows that pairU,V is generalized β−ϕ−z-contractive pair of mapping. Therefore, all the hpothesis of Theorem (13) are satisfied, and hence, U and V have a unique fixed point.Example 23.
Consider the space of all continuous functionsS=C0,1, defined on closed interval 0,1 endowed with a metric σ defined as σr,s=supt∈0,1rt−st, and a binary relation R=C0,1×C0,1. Then, the two self-maps U,V:S⟶S are constructed as
(47)Vrt=1Γγ∫0tt−pγ−1hp,rpdp,andUrt=1Γγ∫0tt−pγ−1hp,rpdp−2t2−w2Γγ∫011−pγ−1hp,rpdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,rzdzdp,with v=1/2,γ=3/2 and t∈0,1, have a unique common fixed point for rt=2t+1/t,st=4+6t+8t2+1/t and ht,zt=1/2rt−1/t.Proof.
Sincert=2t+1/t,st=4+6t+8t2+1/t and ht,zt=1/2zt−1/t. Then, ht,rt=t,ht,st=2+3t+4t2.
(48)rt−st=41+t+2t2,ht,rt−ht,st=21+t+2t2.Hence,ht,rt−ht,st≤rt−stVrt=1Γγ∫0tt−pγ−1hp,rpdp=1Γ3/2∫0tt−p1/2pdp,=1Γ3/2t1/2∫0p1−pt1/2pdp,Vst=1Γγ∫0tt−pγ−1hp,spdp,=1Γ3/2∫0pt−p1/22+3p+4p2dp,=1Γ3/2t1/22∫0p1−pt1/2dp+3∫0p1−pt1/2pdp+4∫0p1−pt1/2p2dp.
By puttingp/t=l, we get p=tl,dp=tdl, at p=0,l=0 and at p=t,l=1, we get
(49)Vrt=1Γ3/2t5/2∫011−l1/2ldl;=2t5/2Γ3/2Γ2πΓ7/2=8t5/215π,Vst=1Γ3/2t1/22∫011−l1/2tdl+3∫011−l1/2t2ldl+4∫011−l1/2t3l2dl,Vst=4t3/2πΓ3/2Γ1Γ5/2+6t5/2πΓ3/2Γ2Γ7/2+8t7/2πΓ3/2Γ3Γ9/2;=8t3/2π13+t5+16t2105.Urt=1Γγ∫0tt−pγ−1hp,rpdp−2t2−w2Γγ∫011−pγ−1hp,rpdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,rzdzdp,=1Γ3/2∫0tt−p1/2pdp−2t2−w2Γ3/2∫011−p1/2pdp+2t2−w2Γ3/2∫0w∫0pp−z1/2zdzdp,=1Γ3/2t1/2∫0t1−pt1/2pdp−2t2−w2Γ3/2∫011−p1/2pdp+2t2−w2Γ3/2∫0wp1/2∫0p1−zp1/2zdzdp.Ust=1Γγ∫0tt−pγ−1hp,spdp−2t2−w2Γγ∫011−pγ−1hp,spdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,szdzdp,=1Γ3/2∫0tt−p1/22+3p+4p2dp−2t2−w2Γ3/2∫011−p1/22+3p+4p2ds+2t2−w2Γ3/2∫0w∫0pp−z1/22+3p+4p2dzdp,=1Γ3/2t12∫0t1−pt1/22+3p+4p2dp−2t2−w2Γ3/2∫011−p1/22+3p+4p2ds+2t2−w2Γ3/2∫0wp1/2∫0p1−zp1/22+3z+4z2dzds.
By substitutingp/t=ρ, we get p=tϱ,dp=tdϱ, and at p=0,ϱ=0, at p=t,ϱ=1 and z/p=l, we get z=pl,dz=pdl, and at z=0,l=0, at z=p,l=1, we get
(50)Urt=2πt5/2∫011−ϱ1/2ϱdϱ−2t2−w2∫011−p1/2pdp+2t2−w2∫0wp5/2∫0p1−l1/2ldldp,=2πt5/2Γ3/2Γ2Γ7/2−2t2−w2Γ3/2Γ2Γ7/2+2t2−w2Γ3/2∫0wp5/2Γ3/2Γ2Γ7/2dp,=8t15πt3/2−22−w2+4w7/272−w2.Ust=1Γ3/2t1/2∫0t1−pt1/22+3p+4p2dp−2t2−w2∫011−p1/22+3p+4p2dp+2t2−w2∫0wp1/2∫0p1−zp1/22+3z+4z2dzdp,=2πt1/2∫011−ϱ1/22+3tϱ+4tϱ2tdϱ−2t2−w2∫011−p1/22+3p+4p2dp+2t2−w2∫0wp1/2∫011−l1/22+3lp+4lp2pdldp.Ust=2π2t3/2Γ3/2Γ1Γ5/2+3t5/2Γ3/2Γ2Γ7/2+4t7/2Γ3/2Γ3Γ9/2−2t2−w22Γ3/2Γ1Γ5/2+3Γ3/2Γ2Γ7/2+4Γ3/2Γ3Γ9/2+2t2−w2∫0wp3/2Γ3/2Γ1Γ5/2+3p5/2Γ3/2Γ2Γ7/2+4p7/2Γ3/2Γ3Γ9/2,=2π4t3/23+4t5/25+64t7/2105−2t2−w243+45+64105+2t2−w2∫0w4p3/23+4p5/25+64p7/2105=8tπt1/213+15+16t2105−48t352−w2+48t352−w24w5/2115+w35+16w2105.
Thus, all the hypotheses of Theorem (5.1) are fulfilled, and hence,U and V have a unique common fixed point. Here, clearly, U and V have a unique fixed point t=0.
Graphical representation of solution: Figure2 represents the solution of Example 23 in graphical form for the function rt=2t+1/t, which shows that 0 is the unique common fixed point of U and V.Figure 2
Graphical solution of the Example23 for the function rt=2t+1/t.Figure3 represents the solution of Example 23 in graphical form for the function st=4+6t+8t2+1/t, which shows that 0 is the unique common fixed point of U and V, and in this graph, s is represented by r.Figure 3
Graphical solution of the Example23 for the function st=4+6t+8t2+1/t.


## 6.1. Existence and Uniqueness of a Common Solution of Nonlinear Fractional Differential Equation of Caputo Type

The aim of this part is to introduce an application of Theorem (13) to obtain a common solution of nonlinear fractional differential equation of Caputo type for a pair of generalized β−ϕ−z-contractive pair of mapping in metric space. The Caputo nonlinear fractional differential equation proposed by Kilbas, is given as follows:
(38)CDpαrt=ht,rt,t∈0,1,1<γ≤2,r0=0,r1=∫0wrpdp,0<w<1,where CDpγ represents the Caputo fractional derivative of order γ and h:0,1⟶R is a continuous function. This nonlinear fractional differential equation can also be represented in the form as follows:
(39)rt=1Γγ∫0tt−pγ−1hp,rpdp−2t2−w2Γγ∫011−pγ−1hp,rpdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,rzdzdp.A functionrt is a solution of above defined nonlinear fractional differential equation whenever it is the solution of the fractional integral Equation (39) and vice-versa.Theorem 22.
Consider the space of all continuous functionsS=C0,1 constructed on closed interval 0,1, equipped with a binary relation R=C0,1×C0,1. Let C0,1,S be the Banach space of all continuous functions from 0,1 into S with norm r=supt∈0,1rt;∀rt∈C0,1. This space defines the metric as follows:
(40)σr,s=supt∈0,1rt−st;∀r,s∈S.
Now if we construct two self-mapsU,V:S⟶S as
(41)Urt=1Γγ∫0tt−pγ−1hp,rpdp−2t2−w2Γγ∫011−pγ−1hp,rpdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,rzdzdp,(42)andVrt=1Γγ∫0tt−pγ−1hp,rpdp,
satisfying thatUS⊆VS. Then, U and V have a unique common fixed point if we have the following assumptions:
(1)
βr,s=1,∀r,s∈S,(2)
there exist a continuous functionh:0,1×R⟶R satisfying that ht,rt−ht,st=krt−st, where k is some constant(3)
tγ2−w2γ+1+2t1+γ+γ2/γ+12−w2≤1.Proof.
Obviously,C0,1,S is a complete metric space.
(1)
sinceU:S⟶S is a self map, and R=S×S, then for any r0∈S, we have Ur0∈S. Thus, r0,Ur0∈R, and hence SU,R is nonempty(2)
for anyr0∈S, Ur0,U2r0,⋯..,Uϱr0,Uϱ+1r0,⋯∈S as U:S⟶S is a self-map. Therefore, r0,Ur0,Ur0,U2r0,⋯⋯⋯,Uϱr0,Uϱ+1r0∈R and hence R is U-closed. In the same pattern, we can easily prove that R is V-closed.Thus, R is U-closed as well as V-closed(3)
from given condition (1), we haveβr,s=1, i.e. βr,s≥1, for each r,s∈S.(4)
it is only required to prove that pairU,V is generalized β−ϕ−z-contractive pair of mappings(43)σVrt,Vst=supt∈0,1Vrt−Vst,=supt∈0,11Γγ∫0tt−pγ−1hp,rpdp−1Γγ∫0tt−pγ−1hp,spdp,=1Γγ∫0tt−pγ−1supt∈0,1hp,rpdp−hp,spdp,=kΓγ∫0tt−pγ−1supt∈0,1rpdp−spdp,=kΓγr−s∞∫0tt−pγ−1dp=kr−s∞pγγΓγ2.σUrt,Ust=supt∈0,1Urt−Ust,=supt∈0,11Γγ∫0tt−pγ−1hp,rpdp−2t2−w2Γγ∫011−pγ−1hp,rpdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,rzdzdp−1Γγ∫0tt−pγ−1hp,spdp+2t2−w2Γγ∫011−pγ−1hp,spdp−2t2−w2Γγ∫0w∫0pp−zγ−1hz,szdzdp.σUrt,Ust=supt∈0,11Γγ∫0tt−pγ−1hp,rp−hp,spdp−2t2−w2Γγ∫011−pγ−1hp,rp−hp,spdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,rz−hz,szdzdp,≤supt∈0,11Γγ∫0tt−pγ−1hp,rp−hp,spdp+2t2−w2Γγ∫011−pγ−1hp,rp−hp,spdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,rz−hz,szdzdp,≤kΓγ∫0tt−pγ−1supt∈0,1rp−spdp+2t2−w2Γγ∫011−pγ−1supt∈0,1rp−spdp+2t2−w2Γγ∫0w∫0pp−zγ−1supt∈0,1rz−szdzdp.σUrt,Ust≤kr−s∞Γγ∫0tt−pγ−1dp−2t2−w2∫011−pγ−1dp+2t2−w2Γγ∫0w∫0pp−zγ−1dzdp,=kr−s∞γΓγ2tγ2−w2γ+1+2t1+γ+γ2γ+12−w2.
By using given conditionc, we get
(44)σUrt,Ust≤kr−s∞γΓγ2,=σVrt,Vst,≤MσVrt,Vst.
Now if we assume thatϕt=et, then by assuming MσVrt,Vst=t, we get
(45)eMσVrt,Vst>MσVrt,Vst,i.e.,ϕMσVrt,Vst>MσVrt,Vst.
Then, by comparing Equations (45) and (6.6), we get
(46)σUrt,Ust≤ϕMσVrt,Vst,βVrt,VstσUrt,Ust≤ϕMσVrt,Vst,βVrt,Vst=1,ξβVrt,VstσUrt,Ust,ϕMσVrt,Vst≥0.
This shows that pairU,V is generalized β−ϕ−z-contractive pair of mapping. Therefore, all the hpothesis of Theorem (13) are satisfied, and hence, U and V have a unique fixed point.Example 23.
Consider the space of all continuous functionsS=C0,1, defined on closed interval 0,1 endowed with a metric σ defined as σr,s=supt∈0,1rt−st, and a binary relation R=C0,1×C0,1. Then, the two self-maps U,V:S⟶S are constructed as
(47)Vrt=1Γγ∫0tt−pγ−1hp,rpdp,andUrt=1Γγ∫0tt−pγ−1hp,rpdp−2t2−w2Γγ∫011−pγ−1hp,rpdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,rzdzdp,with v=1/2,γ=3/2 and t∈0,1, have a unique common fixed point for rt=2t+1/t,st=4+6t+8t2+1/t and ht,zt=1/2rt−1/t.Proof.
Sincert=2t+1/t,st=4+6t+8t2+1/t and ht,zt=1/2zt−1/t. Then, ht,rt=t,ht,st=2+3t+4t2.
(48)rt−st=41+t+2t2,ht,rt−ht,st=21+t+2t2.Hence,ht,rt−ht,st≤rt−stVrt=1Γγ∫0tt−pγ−1hp,rpdp=1Γ3/2∫0tt−p1/2pdp,=1Γ3/2t1/2∫0p1−pt1/2pdp,Vst=1Γγ∫0tt−pγ−1hp,spdp,=1Γ3/2∫0pt−p1/22+3p+4p2dp,=1Γ3/2t1/22∫0p1−pt1/2dp+3∫0p1−pt1/2pdp+4∫0p1−pt1/2p2dp.
By puttingp/t=l, we get p=tl,dp=tdl, at p=0,l=0 and at p=t,l=1, we get
(49)Vrt=1Γ3/2t5/2∫011−l1/2ldl;=2t5/2Γ3/2Γ2πΓ7/2=8t5/215π,Vst=1Γ3/2t1/22∫011−l1/2tdl+3∫011−l1/2t2ldl+4∫011−l1/2t3l2dl,Vst=4t3/2πΓ3/2Γ1Γ5/2+6t5/2πΓ3/2Γ2Γ7/2+8t7/2πΓ3/2Γ3Γ9/2;=8t3/2π13+t5+16t2105.Urt=1Γγ∫0tt−pγ−1hp,rpdp−2t2−w2Γγ∫011−pγ−1hp,rpdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,rzdzdp,=1Γ3/2∫0tt−p1/2pdp−2t2−w2Γ3/2∫011−p1/2pdp+2t2−w2Γ3/2∫0w∫0pp−z1/2zdzdp,=1Γ3/2t1/2∫0t1−pt1/2pdp−2t2−w2Γ3/2∫011−p1/2pdp+2t2−w2Γ3/2∫0wp1/2∫0p1−zp1/2zdzdp.Ust=1Γγ∫0tt−pγ−1hp,spdp−2t2−w2Γγ∫011−pγ−1hp,spdp+2t2−w2Γγ∫0w∫0pp−zγ−1hz,szdzdp,=1Γ3/2∫0tt−p1/22+3p+4p2dp−2t2−w2Γ3/2∫011−p1/22+3p+4p2ds+2t2−w2Γ3/2∫0w∫0pp−z1/22+3p+4p2dzdp,=1Γ3/2t12∫0t1−pt1/22+3p+4p2dp−2t2−w2Γ3/2∫011−p1/22+3p+4p2ds+2t2−w2Γ3/2∫0wp1/2∫0p1−zp1/22+3z+4z2dzds.
By substitutingp/t=ρ, we get p=tϱ,dp=tdϱ, and at p=0,ϱ=0, at p=t,ϱ=1 and z/p=l, we get z=pl,dz=pdl, and at z=0,l=0, at z=p,l=1, we get
(50)Urt=2πt5/2∫011−ϱ1/2ϱdϱ−2t2−w2∫011−p1/2pdp+2t2−w2∫0wp5/2∫0p1−l1/2ldldp,=2πt5/2Γ3/2Γ2Γ7/2−2t2−w2Γ3/2Γ2Γ7/2+2t2−w2Γ3/2∫0wp5/2Γ3/2Γ2Γ7/2dp,=8t15πt3/2−22−w2+4w7/272−w2.Ust=1Γ3/2t1/2∫0t1−pt1/22+3p+4p2dp−2t2−w2∫011−p1/22+3p+4p2dp+2t2−w2∫0wp1/2∫0p1−zp1/22+3z+4z2dzdp,=2πt1/2∫011−ϱ1/22+3tϱ+4tϱ2tdϱ−2t2−w2∫011−p1/22+3p+4p2dp+2t2−w2∫0wp1/2∫011−l1/22+3lp+4lp2pdldp.Ust=2π2t3/2Γ3/2Γ1Γ5/2+3t5/2Γ3/2Γ2Γ7/2+4t7/2Γ3/2Γ3Γ9/2−2t2−w22Γ3/2Γ1Γ5/2+3Γ3/2Γ2Γ7/2+4Γ3/2Γ3Γ9/2+2t2−w2∫0wp3/2Γ3/2Γ1Γ5/2+3p5/2Γ3/2Γ2Γ7/2+4p7/2Γ3/2Γ3Γ9/2,=2π4t3/23+4t5/25+64t7/2105−2t2−w243+45+64105+2t2−w2∫0w4p3/23+4p5/25+64p7/2105=8tπt1/213+15+16t2105−48t352−w2+48t352−w24w5/2115+w35+16w2105.
Thus, all the hypotheses of Theorem (5.1) are fulfilled, and hence,U and V have a unique common fixed point. Here, clearly, U and V have a unique fixed point t=0.
Graphical representation of solution: Figure2 represents the solution of Example 23 in graphical form for the function rt=2t+1/t, which shows that 0 is the unique common fixed point of U and V.Figure 2
Graphical solution of the Example23 for the function rt=2t+1/t.Figure3 represents the solution of Example 23 in graphical form for the function st=4+6t+8t2+1/t, which shows that 0 is the unique common fixed point of U and V, and in this graph, s is represented by r.Figure 3
Graphical solution of the Example23 for the function st=4+6t+8t2+1/t.


## 7. Conclusion

The work presented here was carried out in the context of investigating some fixed point results forZ-contractive pair of mappings in a metric space under some binary relation assumed conditions. The notion of a binary relation is found to be more flexible in nature in the context of fixed point results. Due to the less restrictive nature of binary relations, the fixed point results attained in our paper have a much wider scope of applications. In the context of our main relation-theoretic common fixed point result, we also present an application in fractional calculus. In the future, we can try to prove the same results and many other results for some particular binary relations like transitive relations, symmetrical relations, reflexive relations, etc. The reader can also apply the concept of binary relation to other generalized metric spaces such as b-metric space, nontriangular metric space, cone metric space, and so on.



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