# On Solution of Nonlinear Integral and Fractional Differential Equations via Discontinuous Nonlinear Contractions

Authors: Santosh Kumar; Anita Tomar; Gopi Prasad
Journal: Journal of Mathematics
 (2022)

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## Abstract

The aim of this research work is to demonstrate the survival of exactly one common fixed point for nonlinear contractions without assuming containment of range spaces of an involved pair of mappings, commutativity, and continuity (or any of their weaker forms). It is interesting to note that these findings involve different techniques of proof and comparatively fewerassumptions related to underlying mappings to find the existence of common fixed points of certain mappings. In this way, such investigations are generalizations and improvements over the several prominent recent results of the existing literature. For the application part, we solve the system of nonlinear integral equations of Fredholm and Volterra type and fractional differential equation of Caputo type to validate our results. Moreover, we validate the main conclusion with the help of an example.


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## Body

## 1. Introduction and Preliminaries

The Banach contraction principle [1] provides a significant technique to guarantee the existence of the fixed point. In the last 100 years, it has been explored by several researchers in diverse directions (see, [2, 3]). Wongyat and Sintunavarat [4] applied fixed point results to find the solution of nonlinear Fredholm integral equations and Volterra integral equations together with nonlinear fractional differential equations of Caputo type. Boyd and Wong [5] utilized a function that is described on the closure of the range of a metrically convex metric space, to generalize the conclusion of Banach [1]. However, the involved mapping must be continuous in Banach [1] as well as Boyd and Wong's [5] results.In 1962, a note on contractive mappings was presented by Rakotch [6]. Recently Nziku and Kumar [7] presented the fixed point results via contractive mappings in partial metric spaces. The respective authors also presented the Boyd and Wong variant of the fixed point theorem in the same metric settings (see [8]). Imdad and Kumar [9] extended the existing results by relaxing “continuity” and lightening the “commutativity” requirement besides increasing the number of involved maps from “two” to “four.” Several other researchers extended these results in different directions. One can see the results proved in ([10–14]) and the references therein. In this continuation, most recently some interesting generalizations of these results were obtained by Karapinar et al. [15] and by Prasad ([16, 17]). In fact, mappings that shrink the distances have been of concern for several years.Inspired by the fact that the majority of the phenomena evolving in the physical world are discontinuous, we extend the result of Boyd and Wong [5] to a pair of discontinuous nonlinear mappings without utilizing the containment of range spaces of involved pair of mappings, a weaker form of commutativity or continuity (or any of its variants) via the technique of Akkouchi [18]. Next, we utilize our conclusions to solve Fredholm and Volterra type nonlinear integral equations and an application for the existence and uniqueness of a solution for the nonlinear fractional differential equation of Caputo type too. Our conclusions generalize, improve and extend two celebrated and renowned results due to (Banach [1], Boyd, and Wong [5]) besides comparable other ones. It is interesting to mention here that the containment of range spaces of mappings under consideration, commutativity or its weaker form as well as continuity or its variants are essential for the survival of a common fixed point (see, Singh and Tomar [19] and Tomar and Karapinar [20]).In this way, these investigations provide a different perspective to determine a common fixed point for nonlinear contraction without exploiting the continuity, commutativity (or any of its weaker forms), and containment of range spaces of an involved pair of discontinuous mappings. Furthermore, inspired by the reality that the nonlinear integral equations of Fredholm and Volterra and the fractional differential equation of Caputo types appear in numerous real-life problems. We resolve them to verify the genuineness and efficacy of the established results.Definition 1 (see [21]).
Letμn be a sequence in a metric space X,ϱ. Then,(i)
μn is convergent to a point μ∈X if and only if limn⟶∞ϱμn,μ=0(ii)
μn is a Cauchy sequence if there exists ε>0 so that ϱμn,μm<ϵ, n,m>N for some integers N≥0, that is, limn,m⟶+∞ϱμn,μm=0(iii)
X,ϱ is complete if every Cauchy sequence μn converges to a point μ∈XDefinition 2 (see [18]).
LetΦ symbolize the set of continuous functions ϕ:0,∞⟶0,∞, so that(i)
ϕt=0 if and only if t=0 and(ii)
ifϕtn is a decreasing sequence, then all the sequences tn of elements in 0,∞ are bounded, that is, supntn<∞.Noticeably, if either of the subsequent conditions holds for a continuous functionϕ:0,∞⟶0,∞ then it belongs to Φ;(iii)
ϕ is nondecreasing in 0,∞,(iv)
ϕt≥Mta, t>0, and M,a>0.
Let us use the following notations in the subsequent sections:(i)
P:=ϱμ,ν:μ,ν∈X,(ii)
Letχ:ϕP¯⟶0,∞ be an upper semicontinuous from the right on ϕP¯ (the closure of ϕP) so that χt<t, t∈ϕP¯\0.
Akkouchi [18] proved the following theorem by generalizing the contraction condition due to Boyd and Wong [5] and applying the control function χ on the right-hand side of the inequality as follows:Theorem 1 (see [18]).
LetM,ϱ be a complete metric space and let P:=ϱμ,ν:μ,ν∈M. Let ϕ∈Φ and χ:ϕP¯⟶0,∞ be an upper semicontinuous from the right on ϕP¯ and satisfies χt<t for all t∈ϕP¯\0 where ϕP¯ denotes the closure of ϕP. Let T be the mapping of M satisfying the following contractive condition:(1)ϕϱTμ,Tν≤χϕϱμ,ν,∀μ,ν∈M.Then,T has a unique fixed point z and ϱTnμ,z⟶0 for all μ∈M as n⟶∞.


## 2. Main Results

Firstly, we state and prove the main result and then validate it with an illustrative example.Theorem 2.
LetS and T be self-mappings of a complete metric space X,ϱ so that:(2)ϕϱSμ,Tν≤χϕMμ,ν.
μ,ν∈X, where Mμ,ν=maxϱμ,ν,ϱμ,Sμ,ϱν,Tν,ϱμ,Tν+ϱν,Sμ/2. Then, S and T have a unique common fixed point. Further, any fixed point of S is also a fixed point of T and vice-versa.Proof.
Supposeμ0∈X. For a positive integer n, define a sequence μn by μn+1=Sμn and μn+2=Tμn+1. If μn=μn+1, then μn is a fixed point of S. Similarly, if μn+1=μn+2, then μn+1 is a fixed point of T.
Letμn≠μn+1 and an=ϱμn,μn+1, μn∈X, that is, ϕan+1=ϕϱμn+1,μn+2. Utilizing the nonlinear contraction (2), we get(3)ϕϱμn+1,μn+2=ϕϱSμn,Tμn+1≤χϕMμn,μn+1=χϕmaxϱμn,μn+1,ϱμn,Sμn,ϱμn+1,Tμn+1,ϱμn,Tμn+1+ϱμn+1,Sμn2=χϕmaxϱμn,μn+1,ϱμn,μn+1,ϱμn+1,μn+2,ϱμn,μn+2+ϱμn+1,μn+12.
Since,(4)ϱμn,μn+2+ϱμn+1,μn+12≤ϱμn,μn+1+ϱμn+1,μn+2+ϱμn+1,μn+12=ϱμn,μn+1+ϱμn+1,μn+22,ϕϱμn+1,μn+2=ϕϱSμn,Tμn+1≤χϕmaxϱμn,μn+1,ϱμn+1,μn+2.
Ifmaxϱμn,μn+1,ϱμn+1,μn+2=ϱμn+1,μn+2. Then,(5)ϕϱμn+1,μn+2=ϕϱSμn,Tμn+1≤χϕϱμn+1,μn+2<ϕϱμn+1,μn+2,which is a contradiction.
Somaxϱμn,μn+1,ϱμn+1,μn+2=ϱμn,μn+1. Hence,(6)ϕϱμn+1,μn+2=ϕϱSμn,Tμn+1≤χϕϱμn,μn+1<ϕϱμn,μn+1,that is, the sequence ϕan is decreasing and bounded below in 0,∞ for n≥0.
Leta be the limit point of ϕan. Obviously a∈0,∞. We assert that a=0. Instead, if a>0, taking limsupn⟶∞ in inequality (6) we get,(7)0<a≤limsupn⟶∞χϕan≤χa<a,which is a contradiction of the presumption on the mapping χ, that is, a=0. Hence, limn⟶∞ϕan=limn⟶∞ϕϱμn,μn+1=0.
Clearly,limϕan=ϕliman=0. Utilizing the definition of ϕ,(8)limn⟶∞an=limn⟶∞ϱμn,μn+1=0.
Next, we assert that a sequenceμn is Cauchy. Instead, if μn is not a Cauchy sequence in X, then there exists ϵ>0 and a sequence of integers mk,nk so that,(9)ϱμnk,μmk≥ϵ,nk>mk≥k,k≥0.
Ifmk is the smallest integer, so that inequality (9) holds,(10)ϱμk,μmk−1<ϵ.
Fornk>mk we obtain,(11)ϱμnk,μmk≤ϱμnk,μmk−1+ϱμmk−1,μmk,≤ϱμnk,μmk−1+ank.
Ask⟶∞, considering (8) we get,(12)ϱμnk,μmk⟶ϵ.
By elementary computations we have,(13)ϱμnk−1,μmk−1⟶ϵ.
From the nonlinear contraction (2) we obtain,(14)ϕϱμnk,μmk=ϕϱSμnk−1,Tμmk−1≤χϕMμnk−1,μmk−1,where (15)Mμnk−1,μmk−1=maxϱμnk−1,μmk−1,ϱμnk−1,Sμnk−1,ϱμmk−1,Tμmk−1,ϱμnk−1,Tμmk−1+ϱμmk−1,Sμnk−12=maxϱμnk−1,μmk−1,ϱμnk−1,μnk,ϱμmk−1,μmk,ϱμnk−1,μmk+ϱμmk−1,μnk2.
Ask⟶∞, utilizing (12) and (13), inequality (14) transforms to,(16)0<ϕε≤χϕε<ϕε.
Sinceχt<t, w∈Xt>0 and ϕε∈ϕP¯\0 then inequality (16) is not valid. As a result, the sequence μn is a Cauchy sequence and limn,m⟶∞ϱμn,μm=0.
SinceX is complete, we select a point w∈X so that limn⟶∞ϱμn,w=0. Next, we assert that w is a fixed point of T. Instead, if w≠Tw.
From inequality (2), we attain(17)ϕϱμn+1,Tw=ϕϱSμn,Tw≤χϕMμn,w,where(18)Mμn,w=maxϱμn,w,ϱμn,Sμn,ϱw,Tw,ϱμn,Tw+ϱw,Sμn2=maxϱμn,w,ϱμn,μn+1,ϱw,Tw,ϱμn,Tw+ϱw,μn+12.
Taking the limit asn⟶∞ we attain,(19)Mμn,w⟶ϱw,Tw.
Taking the limit asn⟶∞ in the inequality (17), we obtain(20)ϕϱw,Tw≤χϕϱw,Tw<ϕϱw,Tw,which is a contradiction. Thus w=Tw.
Now, we assert thatw is a unique common fixed point of S and T.
Instead, ifS and T have two common fixed points μ0 and ν0 in X and μ0≠ν0. Then, from inequality (2), for all μ0,ν0∈X we obtain,(21)ϕϱSμ0,Tν0=ϕϱμ0,ν0≤χϕMμ0,ν0,where(22)Mμ0,ν0=maxϱμ0,ν0,ϱμ0,Sμ0,ϱν0,Tν0,ϱμ0,Tν0+ϱν0,Sμ02=maxϱμ0,ν0,ϱμ0,μ0,ϱν0,ν0,ϱμ0,ν0+ϱν0,μ02=ϱμ0,ν0.
Thus, inequality (21) transforms to,(23)ϕϱSμ0,Tν0=ϕϱμ0,ν0≤χϕϱμ0,ν0<ϕϱμ0,ν0,which is a contradiction and hence μ0=ν0. As a result, S and T have a unique common fixed point.
On contrary, to the hypothesis, ifμ0 is a fixed point of S and μ0≠Tμ0, utilizing inequality (2), for μ0∈X we obtain,(24)ϕϱSμ0,Tμ0=ϕϱμ0,Tμ0≤χϕMμ0,Tμ0,where (25)Mμ0,Tμ0=maxϱμ0,Tμ0,ϱμ0,Sμ0,ϱTμ0,T2μ0,ϱμ0,T2μ0+ϱTμ0,Sμ02=maxϱμ0,Tμ0,ϱμ0,μ0,ϱTμ0,T2μ0,ϱμ0,T2μ0+ϱTμ0,μ02=ϱμ0,Tμ0.
Thus, inequality (24) transforms to,(26)ϕϱSμ0,Tμ0=ϕϱμ0,Tμ0≤χϕϱμ0,Tμ0<ϕϱμ0,Tμ0,which is a contradiction, that is, μ0=Tμ0. Similarly, we may demonstrate that a fixed point of T is also a fixed point of S.Remark 1.
By settingT=S, one gets the following corollary as an extended variant of Theorem 1.2 [18].Corollary 1.
LetS be a self-mapping of a complete metric space X,ϱ so that:(27)ϕϱSμ,Sν≤χϕMμ,ν.μ,ν∈X, where Mμ,ν=maxϱμ,ν,ϱμ,Sμ,ϱν,Sν,ϱμ,Sν+ϱν,Sμ/2. Then, S has a unique fixed point.Remark 2.
By settingT=I, the identity map, one gets the subsequent corollary:Corollary 2.
LetS be a self-mapping of a complete metric space X,ϱ so that(28)ϕϱSμ,ν≤χϕMμ,ν.μ,ν∈X, where Mμ,ν=maxϱμ,ν,ϱμ,Sμ,ϱμ,ν+ϱν,Sμ/2. Then, S has a unique fixed point.Remark 3.
It is interesting to notice that, Theorem2, Corollaries 1 and 2 continue to be valid if we substitute, besides Mμ,ν=ϱμ,ν retaining the rest of the assumptions. In this transformation Theorem 2 and Corollary 2 are still generalizations and extensions over the existing results, however, Corollary 1 reduces to Theorem 1.4 [18] due to Akkouchi.Remark 4.
We also point out the fact that, if we setϕ=I, an identity map, χt=kt for k∈0,1 and t≥0, in light of the above remark in Corollary 1, Corollary 1 reduces to the Banach contraction principle [1].Remark 5.
If we compare Theorem2 with Theorem 2 in [6], one can see that our results are more generalized than that of Rakotch [6]. Theorem 2 is proved for two maps while Theorem 2 [6] is proved for one map. Also in our case, the continuity condition is relaxed.
We provide an example to authenticate the genuineness of the obtained conclusion.Example 1.
LetX = 0,∞ be equipped with the usual metric. Noticeably, X,ϱ is a complete metric space. Define two self-mappings S and T on X such that(29)Sμ=μ3,ifμ∈1,∞,0,ifμ∈0,1,Tν=ν2,ifν∈1,∞,0,ifν∈0,1.
Forμ,ν∈X and t∈a,b. Define a function ϕ:P⟶0,∞ as:(30)ϕϱμ,ν=μ+ν.
Clearly,ϕ is nondecreasing and ϕt=0 if and only if t=0 and so, ϕ∈Φ. Define a function χ:0,∞⟶0,∞ by χt=t/2, t≥0. Now, we shall show that S and T validate the nonlinear contractive assumption (2) of Theorem 2 in light of Remark 3. Consider the subsequent two cases:Case 1.
Ifμ∈0,∞ and ν∈1,∞, then we obtain(31)ϕϱSμ,Tν=Sμ+Tν=μ3+ν2≤μ+ν2=χϕϱμ,ν.Case 2.
Ifμ∈0,∞ and ν∈1,∞, then we obtain(32)ϕϱSμ,Tν=Sμ+Tν=μ3+0≤μ+ν2=χϕϱμ,ν.
Thus, forμ,ν∈X,(33)ϕϱSμ,Tν≤χϕϱμ,ν,that is, S and T satisfy the nonlinear contractive assumption (2). Therefore, all the assumptions of Theorem 2 are validated(taking Mμ,ν=ϱμ,ν and μ=0 is a unique common fixed point of S and T.
Noticeably, Example1 demonstrates that inequality (2) is not a linear contraction which marks the supremacy of Theorem 2 over analogous renowned conclusions.


## 3. Applications

The aim of this section is to provide some applications validating our main results in the previous section. In this way, we demonstrate the existence and uniqueness of a solution for various problems.


### 3.1. Existence of a Solution for Nonlinear Integral Equations

The theory of systems of nonlinear integral equations is a substantial field of mathematics, having various applications in numerous branches of physics, biology, chemistry, engineering, and other fields related to real-life problems. Nowadays, several authors explored distinct results of this theory (see for instance, [4, 22, 23]) in the settings of metric spaces. Based on this fact, we present a unique solution for the systems of nonlinear Fredholm and Volterra integral equations by utilizing the main result in light of Remark 3.Theorem 3.
Consider the nonlinear Fredholm integral equation(34)μt=αt+∫abKt,s,μsds.Letμ,ν∈Ca,b, a,b∈ℝ with a<b,α:a,b⟶ℝ and K:a,b×a,b×ℝ⟶ℝ are continuous mappings. Assume that(35)Kt,s,μs+Kt,s,νs≤χsups∈a,bμs+sups∈a,bνs−2αtb−a,t,s∈a,b. Then, the integral equation (34) has a unique solution.Proof.
LetX=Ca,b. Define self-mappings S and T on X, such that(36)Sμt=αt+∫abKt,s,μsds,Tνt=αt+∫abKt,s,νsds,μ,ν∈X and s,t∈a,b. Let ϱ:X×X⟶0,∞ be the metric defined as ϱμ,ν=supt∈a,bμt−νt, μ,ν∈X. Then, X,ϱ is a complete metric space. For t∈a,b, define a function ϕ:P⟶0,∞ as follows:(37)ϕϱμ,ν=supt∈a,bμt+supt∈a,bνt.
Noticeably,ϕ is nondecreasing and ϕt=0 if and only if t=0 and so, ϕ∈Φ.
Now, we assert thatS and T verify the nonlinear contractive assumption (2). For μ,ν∈X and s,t∈a,b,(38)Sμt+Tνt=αt+∫abKt,s,μsds+αt+∫abKt,s,νsds≤αt+∫abKt,s,μsds+αt+∫abKt,s,νsds≤2αt+∫abKt,s,μsds+∫abKt,s,νsds=2αt+∫abKt,s,μs+Kt,s,νsds≤2αt+∫abχsups∈a,bμs+sups∈a,bνs−2αtb−ads=2αt+1b−a∫abχϕμ,ν−2αtds,=χϕμ,ν,that is,(39)supt∈a,bSμt+supt∈a,bTνt≤χϕμ,ν.
Hence, forμ,ν∈X, we obtain(40)ϕϱSμ,Tν≤χϕμ,ν.
In this way,S and T satisfy the contractive assumption (highlighted by Remark 3) on the setting Mμ,ν=ϱμ,ν. As a result, all the suppositions of Theorem 2 are validated. Hence S and T have a unique common fixed point which is a unique solution (34).
Utilizing the almost identical procedure as presented in Theorem3, we have the subsequent conclusion for the system of nonlinear Volterra integral equations.Theorem 4.
Consider the nonlinear Volterra integral equation(41)μt=αt+∫atKt,s,μsds.Letμ,ν∈a,b, a,b∈ℝ with a<b,α:a,b⟶ℝ and K:a,b×a,b×ℝ⟶ℝ are continuous mappings. Assume that(42)Kt,s,μs+Kt,s,νs≤χsups∈a,bμs+sups∈a,bνs−2αtb−a,t,s∈a,b. Then, the integral (41) has a unique solution.Proof.
The proof of this theorem follows as above.


### 3.2. Existence of a Solution for Nonlinear Fractional Differential Equations

Fractional order differential equations are generalized and noninteger order differential equations. The theory of nonlinear fractional differential equations nowadays has gotten extensive attention. The main reason is due to the rapid development of the subject fractional calculus itself and of having various applications in numerous branches of science and technology.The first definition of the fractional derivative was introduced at the end of the nineteenth century by Liouville and Riemann, but the concept of noninteger derivative and integral was mentioned already in 1695 by Leibniz and L’Hospital. In fact, first-order differential equations are considered an alternative model to integer differential equations. The definition of the Riemann–Liouville derivative was established by Riemann in 1876. Thereafter many applications of the fractional derivatives and integrals of this Riemann–Liouville type have been presented in various fields of science and technology. These include studies on controllability, thermoelasticity, vibration, diffusion processes, and other complex phenomena (see [24, 25]).Recently, many researchers are focused on understanding the various properties of fractional derivatives and their usefulness in certain complex systems. This research direction hinges on the results presented by various researchers on complex systems of nonlocal elements, that is, a local operator of ordinary derivative leads to a nonlocal fractional operator. Along these research lines, it is important to study fractional-order models in terms of global optimization. At the beginning of the twentieth century, another definition of fractional derivative was proposed by Caputo in regard to a Riemann–Liouville fractional integral (see for instance, [26–30]). In continuation of it, we present an application for the existence and uniqueness of a solution for the nonlinear fractional differential equation of Caputo type:Let us recall some basic definitions of fractional calculus (see [24, 25]). For a continuous function g:0,∞⟶ℝ, the Caputo derivative of g of order β>0 is defined as follows:(43)DCβgt=1Γn−β∫0tt−sn−β−1gnsds.Dβgt=1Γn−β∫0tt−sn−β−1gnsds,where n=β+1 with β denoting the integer part of a positive real number β, and Γ is the gamma function. Consider the nonlinear fractional differential equation(44)DCβμt=ft,μt,with integral boundary conditions(45)μ0=0,μ1=∫0ημsds,where 1<β≤2,0<η<1,μ∈C0,1, and f:0,1×ℝ⟶ℝ is a continuous function (see [21]). It is well known that if f is continuous, then (44) is immediately inverted as the very familiar integral equation(46)μt=1Γβ∫0tt−sβ−1fs,μsds−2t2−η2Γβ∫011−sβ−1fs,μsds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,μmdmds.Now, we prove the following existence theorem.Theorem 5.
Consider the nonlinear fractional differential (44). and for each μ,ν∈C0,1, we have(47)fs,μs+fs,νs≤Γβ+15supt∈0,1μt+supt∈0,1νt,for all s∈0,1. Then, the nonlinear fractional differential equation of Caputo type (44) has a unique solution. Moreover, for each μ0∈C0,1, the Picard iteration μn defined by(48)μnt=1Γβ∫0tt−sβ−1fs,μn−1sds−2t2−η2Γβ∫011−sβ−1fs,μn−1sds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,μn−1mdmds,for all n∈ℕ converges to a unique solution of the nonlinear fractional differential equation of Caputo type (44).Proof.
LetX=C0,1. Define self-mappings S and T on X, such that(49)Sμt=1Γβ∫0tt−sβ−1fs,μsds−2t2−η2Γβ∫011−sβ−1fs,μsds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,μmdmds,Tνt=1Γβ∫0tt−sβ−1fs,νsds−2t2−η2Γβ∫011−sβ−1fs,νsds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,νmdmds,for all μ,ν∈X and s,t∈0,1. Let ϱ:X×X⟶0,∞ be the metric defined as ϱμ,ν=supt∈a,bμt−vt, μ,ν∈X. Then, X,ϱ is a complete metric space. For t∈a,b, define a function ϕ:P⟶0,∞ as follows:(50)ϕϱμ,ν=supt∈a,bμt+supt∈a,bνt.
Noticeably,ϕ is nondecreasing and ϕt=0 if and only if t=0 and so, ϕ∈Φ. Now, we assert that S and T verify the nonlinear contractive assumption (2). For μ,ν∈X and s,t∈a,b(51)Sμt+Tνt=1Γβ∫0tt−sβ−1fs,μsds−2t2−η2Γβ∫011−sβ−1fs,μsds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,μmdmds+1Γβ∫0tt−sβ−1fs,νsds−2t2−η2Γβ∫011−sβ−1fs,νsds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,νmdmds≤1Γβ∫0tt−sβ−1fs,μs+fs,νsds−2t2−η2Γβ∫011−sβ−1fs,μs+fs,νsds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,μm+fm,νmdmds≤1Γβ∫0tt−sβ−1Γβ+15supt∈0,1μt+supt∈0,1νtds−2t2−η2Γβ∫011−sβ−1Γβ+15supt∈0,1μt+supt∈0,1νtds+2t2−η2Γβ∫0η∫0ss−mβ−1Γβ+15supt∈0,1μt+supt∈0,1νtdmds≤Γβ+15supt∈0,1μt+supt∈0,1νt×supt∈0,11Γβ∫0tt−sβ−1ds+2t2−η2Γβ∫011−sβ−1ds+2t2−η2Γβ∫0η∫0ss−mβ−1dmds≤χϕμ,ν.
This implies that(52)supt∈0,1Sμt+supt∈0,1Tνt≤χϕμ,ν,and so(53)ϕϱSμ,Tν≤χϕμ,ν,for all μ,ν∈X.
It follows thatS and T satisfies the contractive assumption on the setting Mμ,ν=ϱμ,ν. Therefore, all the suppositions of Theorem 2 are validated. Hence S and T have a unique solution of the nonlinear fractional differential equation of Caputo type (44). This completes the proof.


## 3.1. Existence of a Solution for Nonlinear Integral Equations

The theory of systems of nonlinear integral equations is a substantial field of mathematics, having various applications in numerous branches of physics, biology, chemistry, engineering, and other fields related to real-life problems. Nowadays, several authors explored distinct results of this theory (see for instance, [4, 22, 23]) in the settings of metric spaces. Based on this fact, we present a unique solution for the systems of nonlinear Fredholm and Volterra integral equations by utilizing the main result in light of Remark 3.Theorem 3.
Consider the nonlinear Fredholm integral equation(34)μt=αt+∫abKt,s,μsds.Letμ,ν∈Ca,b, a,b∈ℝ with a<b,α:a,b⟶ℝ and K:a,b×a,b×ℝ⟶ℝ are continuous mappings. Assume that(35)Kt,s,μs+Kt,s,νs≤χsups∈a,bμs+sups∈a,bνs−2αtb−a,t,s∈a,b. Then, the integral equation (34) has a unique solution.Proof.
LetX=Ca,b. Define self-mappings S and T on X, such that(36)Sμt=αt+∫abKt,s,μsds,Tνt=αt+∫abKt,s,νsds,μ,ν∈X and s,t∈a,b. Let ϱ:X×X⟶0,∞ be the metric defined as ϱμ,ν=supt∈a,bμt−νt, μ,ν∈X. Then, X,ϱ is a complete metric space. For t∈a,b, define a function ϕ:P⟶0,∞ as follows:(37)ϕϱμ,ν=supt∈a,bμt+supt∈a,bνt.
Noticeably,ϕ is nondecreasing and ϕt=0 if and only if t=0 and so, ϕ∈Φ.
Now, we assert thatS and T verify the nonlinear contractive assumption (2). For μ,ν∈X and s,t∈a,b,(38)Sμt+Tνt=αt+∫abKt,s,μsds+αt+∫abKt,s,νsds≤αt+∫abKt,s,μsds+αt+∫abKt,s,νsds≤2αt+∫abKt,s,μsds+∫abKt,s,νsds=2αt+∫abKt,s,μs+Kt,s,νsds≤2αt+∫abχsups∈a,bμs+sups∈a,bνs−2αtb−ads=2αt+1b−a∫abχϕμ,ν−2αtds,=χϕμ,ν,that is,(39)supt∈a,bSμt+supt∈a,bTνt≤χϕμ,ν.
Hence, forμ,ν∈X, we obtain(40)ϕϱSμ,Tν≤χϕμ,ν.
In this way,S and T satisfy the contractive assumption (highlighted by Remark 3) on the setting Mμ,ν=ϱμ,ν. As a result, all the suppositions of Theorem 2 are validated. Hence S and T have a unique common fixed point which is a unique solution (34).
Utilizing the almost identical procedure as presented in Theorem3, we have the subsequent conclusion for the system of nonlinear Volterra integral equations.Theorem 4.
Consider the nonlinear Volterra integral equation(41)μt=αt+∫atKt,s,μsds.Letμ,ν∈a,b, a,b∈ℝ with a<b,α:a,b⟶ℝ and K:a,b×a,b×ℝ⟶ℝ are continuous mappings. Assume that(42)Kt,s,μs+Kt,s,νs≤χsups∈a,bμs+sups∈a,bνs−2αtb−a,t,s∈a,b. Then, the integral (41) has a unique solution.Proof.
The proof of this theorem follows as above.


## 3.2. Existence of a Solution for Nonlinear Fractional Differential Equations

Fractional order differential equations are generalized and noninteger order differential equations. The theory of nonlinear fractional differential equations nowadays has gotten extensive attention. The main reason is due to the rapid development of the subject fractional calculus itself and of having various applications in numerous branches of science and technology.The first definition of the fractional derivative was introduced at the end of the nineteenth century by Liouville and Riemann, but the concept of noninteger derivative and integral was mentioned already in 1695 by Leibniz and L’Hospital. In fact, first-order differential equations are considered an alternative model to integer differential equations. The definition of the Riemann–Liouville derivative was established by Riemann in 1876. Thereafter many applications of the fractional derivatives and integrals of this Riemann–Liouville type have been presented in various fields of science and technology. These include studies on controllability, thermoelasticity, vibration, diffusion processes, and other complex phenomena (see [24, 25]).Recently, many researchers are focused on understanding the various properties of fractional derivatives and their usefulness in certain complex systems. This research direction hinges on the results presented by various researchers on complex systems of nonlocal elements, that is, a local operator of ordinary derivative leads to a nonlocal fractional operator. Along these research lines, it is important to study fractional-order models in terms of global optimization. At the beginning of the twentieth century, another definition of fractional derivative was proposed by Caputo in regard to a Riemann–Liouville fractional integral (see for instance, [26–30]). In continuation of it, we present an application for the existence and uniqueness of a solution for the nonlinear fractional differential equation of Caputo type:Let us recall some basic definitions of fractional calculus (see [24, 25]). For a continuous function g:0,∞⟶ℝ, the Caputo derivative of g of order β>0 is defined as follows:(43)DCβgt=1Γn−β∫0tt−sn−β−1gnsds.Dβgt=1Γn−β∫0tt−sn−β−1gnsds,where n=β+1 with β denoting the integer part of a positive real number β, and Γ is the gamma function. Consider the nonlinear fractional differential equation(44)DCβμt=ft,μt,with integral boundary conditions(45)μ0=0,μ1=∫0ημsds,where 1<β≤2,0<η<1,μ∈C0,1, and f:0,1×ℝ⟶ℝ is a continuous function (see [21]). It is well known that if f is continuous, then (44) is immediately inverted as the very familiar integral equation(46)μt=1Γβ∫0tt−sβ−1fs,μsds−2t2−η2Γβ∫011−sβ−1fs,μsds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,μmdmds.Now, we prove the following existence theorem.Theorem 5.
Consider the nonlinear fractional differential (44). and for each μ,ν∈C0,1, we have(47)fs,μs+fs,νs≤Γβ+15supt∈0,1μt+supt∈0,1νt,for all s∈0,1. Then, the nonlinear fractional differential equation of Caputo type (44) has a unique solution. Moreover, for each μ0∈C0,1, the Picard iteration μn defined by(48)μnt=1Γβ∫0tt−sβ−1fs,μn−1sds−2t2−η2Γβ∫011−sβ−1fs,μn−1sds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,μn−1mdmds,for all n∈ℕ converges to a unique solution of the nonlinear fractional differential equation of Caputo type (44).Proof.
LetX=C0,1. Define self-mappings S and T on X, such that(49)Sμt=1Γβ∫0tt−sβ−1fs,μsds−2t2−η2Γβ∫011−sβ−1fs,μsds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,μmdmds,Tνt=1Γβ∫0tt−sβ−1fs,νsds−2t2−η2Γβ∫011−sβ−1fs,νsds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,νmdmds,for all μ,ν∈X and s,t∈0,1. Let ϱ:X×X⟶0,∞ be the metric defined as ϱμ,ν=supt∈a,bμt−vt, μ,ν∈X. Then, X,ϱ is a complete metric space. For t∈a,b, define a function ϕ:P⟶0,∞ as follows:(50)ϕϱμ,ν=supt∈a,bμt+supt∈a,bνt.
Noticeably,ϕ is nondecreasing and ϕt=0 if and only if t=0 and so, ϕ∈Φ. Now, we assert that S and T verify the nonlinear contractive assumption (2). For μ,ν∈X and s,t∈a,b(51)Sμt+Tνt=1Γβ∫0tt−sβ−1fs,μsds−2t2−η2Γβ∫011−sβ−1fs,μsds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,μmdmds+1Γβ∫0tt−sβ−1fs,νsds−2t2−η2Γβ∫011−sβ−1fs,νsds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,νmdmds≤1Γβ∫0tt−sβ−1fs,μs+fs,νsds−2t2−η2Γβ∫011−sβ−1fs,μs+fs,νsds+2t2−η2Γβ∫0η∫0ss−mβ−1fm,μm+fm,νmdmds≤1Γβ∫0tt−sβ−1Γβ+15supt∈0,1μt+supt∈0,1νtds−2t2−η2Γβ∫011−sβ−1Γβ+15supt∈0,1μt+supt∈0,1νtds+2t2−η2Γβ∫0η∫0ss−mβ−1Γβ+15supt∈0,1μt+supt∈0,1νtdmds≤Γβ+15supt∈0,1μt+supt∈0,1νt×supt∈0,11Γβ∫0tt−sβ−1ds+2t2−η2Γβ∫011−sβ−1ds+2t2−η2Γβ∫0η∫0ss−mβ−1dmds≤χϕμ,ν.
This implies that(52)supt∈0,1Sμt+supt∈0,1Tνt≤χϕμ,ν,and so(53)ϕϱSμ,Tν≤χϕμ,ν,for all μ,ν∈X.
It follows thatS and T satisfies the contractive assumption on the setting Mμ,ν=ϱμ,ν. Therefore, all the suppositions of Theorem 2 are validated. Hence S and T have a unique solution of the nonlinear fractional differential equation of Caputo type (44). This completes the proof.


## 4. Conclusion

This paper provides yet another view to find a common fixed point for nonlinear contraction without exploiting the continuity, commutativity (or any of its weaker forms), and containment of range spaces of an involved pair of discontinuous mappings. Further, inspired by the reality that nonlinear integral equations of Fredholm and Volterra appear in numerous real-life problems, we resolve them to verify the genuineness and efficacy of the established conclusions. The theory of nonlinear fractional differential equations nowadays has gotten extensive attention due to its various applications in numerous branches of science and technology. We present an application for the existence and uniqueness of a solution for the nonlinear fractional differential equation of Caputo type too. In this way, these investigations provide yet another variant of common fixed point results and explored the possibility of their applications too.



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