# Langevin Equations with Generalized Proportional Hadamard–Caputo Fractional Derivative

Authors: M. A. Barakat; Ahmed H. Soliman; Abd-Allah Hyder
Journal: Computational Intelligence and Neuroscience
 (2021)

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## Abstract

We look at fractional Langevin equations (FLEs) with generalized proportional Hadamard–Caputo derivative of different orders. Moreover, nonlocal integrals and nonperiodic boundary conditions are considered in this paper. For the proposed equations, the Hyres–Ulam (HU) stability, existence, and uniqueness (EU) of the solution are defined and investigated. In implementing our results, we rely on two important theories that are Krasnoselskii fixed point theorem and Banach contraction principle. Also, an application example is given to bolster the accuracy of the acquired results.


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## Body

## 1. Introduction

In recent years, fractional calculus has gained great importance by numerous renowned mathematicians. The most essential feature of this topic is that it allows us to execute integrations and differentiations in any order, not necessarily integer ones Such a benefit has been encouraged by the applications in various areas, conceivably including fractal phenomena which appear in many sciences such as physics and engineering. Also, discovering the fractional derivatives and their generalizations was done by well-known mathematicians such as Riemann, Caputo, Hadamard, Euler, Liouville, Laplace, Laurent, and Fourier and was constantly the major path of research in the fractional calculus area. Moreover, these derivatives will provide us new chances to get generalized solutions of fractional differential equations [1–6]. It is worth to mention that the most important comprehensive treatments of differential equations with fractional order were initiated by Caputo [7] in 1969, and it was later called Caputo’s derivative. There are many generalizations and modifications of this derivative [8–12], for example, the authors in [9] used Caputo’s derivative to modify the Hadamard derivatives to a more beneficial concept that called Hadamard–Caputo derivatives (HCD). In 2019, Rahman et al. [12] introduced an integral form of Hadamard fractional derivatives which give generalized forms than the HCD and have shorted as GHCD. Many authors used HCD and their generalizations to study the existence, uniqueness, and stability of fractional differential equations [1, 6, 13, 14]. The researchers in [14] utilized the HCDs to present some results on the existence and stability for solutions of fractional Langevin equations with some conditions related to nonperiodic type boundary and nonlocal integral. Recently, Devi et al. [1] used two fixed point theorems due to Krasnoselskii and Banach as well as the HCDs of distinctive orders connected with nonlocal integral to establish the existence, uniqueness, and HU stability of solutions for fractional Langevin equations with nonperiodic boundary conditions.According to varied literature, like [15–17], it has verified that the FLEs are perfectly mathematical models for single-file prevalence and the conduct of unshackled particles driven by internal noises. This motivates us to examine the solutions of the FLEs and their features. So, in the present work, we use the generalized proportional Hadamard–Caputo derivative to investigate the FLEs with nonlocal integrals and nonperiodic boundary conditions. For the proposed equations, the HU stability and EU of the solution are defined and analyzed. In implementing our results, we relied on two main fixed point theorems, namely, Krasnoselskii’s theorem and Banach contraction operator. Furthermore, to reinforce the accuracy of the gained results, an application example is presented with adequate values for the parameters.Our paper arranged as follows. In Section2, we present all basic concepts related to the HCDs and their generalizations with some previous results which serve our results. Section 3 devoted to present our main results which in turn was divided into two subsections, namely, 3.1, introduced to the existence and uniqueness of the solutions of the FLEs, and 3.2, prepared to study the stability results for the solutions of the FLEs.


## 2. Preliminaries

We use the setAℂr1,e to refer to all absolutely continuous functions ξ such that its derivative of order r−1 is absolutely continuous on [1, e], where r∈ℕ and e is the renowned Euler’s number.Definition 1 (see [18]).
Ifξ:1,∞⟶ℝ is both integrable and continuous, the one side fractional integral of Hadamard of qth-order is given by(1)ℷqξϖ=1Γq∫1ϖlnϖ−lnuq−1ξuudu,q∈ℝ,q>0.Definition 2 (see [18]).
Theqth-order Hadamard fractional derivative is given by(2)Dqξϖ=ϖd/dϖΓm−q∫1ϖlnϖ−lnum−q−1ξuudu,where m=q+1 and q>0.Definition 3 (see [18]).
Theqth-order fractional HCD is given by(3)Dℭℌqξϖ=1Γm−q∫1ϖlnϖ−lnum−q−1udduξuduu.Rahman et al. [12] recently defined a generalized proportional Hadamard fractional integral of order q.Definition 4 (see [12]).
Theqth-order generalized proportional Hadamard fractional integral is given by(4)ℷq,ϱξϖ=1ϱqΓq∫1ϖeϱ−1/ϱlnϖ/ulnϖ−lnuq−1ξuudu,where q>1 and ϱ∈0,1.Finally, Jarad et al. [10] presented a broader range of fractional proportional integrals and derivatives.Definition 5 (see [10]).
Theqth-order of a broader range of the one side fractional proportional integrals is given by(5)ℷq,ϱ,χξϖ=1ϱqΓq∫1ϖeϱ−1/ϱχϖ−χuχϖ−χuq−1ξuudu,where q>1, ϱ∈0,1, and χ∈C1,e.Definition 6 (see [10]).
Theqth-order of the one side of a broader range of fractional proportional derivative is given by(6)Dq,ϱ,χξϖ=Dm,ϱ,χΓm−q∫1ϖeϱ−1/ϱχϖ−χuχϖ−χum−q−1ξuudu,where m=q+1, ϱ∈0,1, and q>0.Definition 7 (see [10]).
Theqth-order of the one side of Caputo-broader range of fractional proportional derivative is given by(7)Dℭq,ϱ,χξϖ=1Γm−q∫1ϖeϱ−1/ϱχϖ−χuχϖ−χum−q−1Dm,ϱ,χξuχ′udu,where m=q+1, ϱ∈0,1, and q>0.Remark 1.
Obviously, if we putχt=lnt in Definitions 5 and 7, we obtain the one side generalized proportional Caputo–Hadamard fractional integral and derivative, respectively.By using Proposition 3.1, Proposition 4.1, and Remark 1.1 in [10], we obtain the following lemma.Lemma 1 (see [10]).
Letq,p∈ℂ such that Rep>0 and Req>0. Then, for any ϱ∈0,1 and m=Req+1, we have(i)
ℷq,ϱeϱ−1/ϱlntlntp−1ϖ=Γp/ϱqΓp+qeϱ−1/ϱlnϖlnϖq+p−1(ii)
DCHPq,ϱ,χeϱ−1/ϱlntlntp−1ϖ=ϱqΓp/Γp−qeϱ−1/ϱlnϖlnϖp−q−1The following theorem is essential in implementing our results.Theorem 1 (see [8]).
LetK be a nonempty convex, bounded, and closed subset of a Banach space W. Consider J1andJ2 are two operators from K to W such that(i)
J1x+J2y∈K for each x,y∈K(ii)
J1 is continuous and compact(iii)
J2 satisfies a contraction condition
Then,∃u∈K such that u=J1u+J2u.


## 3. Main Results

This section comprises our new results for the fractional Langevin equations with generalized proportional Hadamard–Caputo derivative. By using Krasnoselskii and Banach fixed point theorems, we investigate the existence, uniqueness, and HU stability for these FLEs. Before we can show our results, we have to prove the following helpful lemmas.Lemma 2.
Letq∈ℂ such that Req>0, ϱ∈0,1, m=Req+1,ξ∈L11,e, and ℷq,ϱξϖ∈Aℂm1,e. Then,(8)ℷq,ϱDCHPq,ϱξϖ=ξϖ−∑r=1mDq−r,ϱξ1Γq−r+1ϱq−r⋅eϱ−1/ϱlnϖlnϖq−r.Also, the fractional differential equation(9)DCHPq,ϱξϖ=0has a solution(10)ξϖ=eϱ−1/ϱlnϖ∑l=0m−1Mllnϖl,where Ml=Dl,ϱξ1/Γl+1ϱl.Proof.
(11)ℷq,ϱCHPDq,ϱξϖ=1ϱqΓq∫1ϖeϱ−1/ϱlnϖ−lnulnϖ−lnuq−1Dm,ϱℷm−q,ϱξuduu=D1,ϱϱq+1Γq∫1ϖeϱ−1/ϱlnϖ−lnulnϖ−lnuqDm,ϱℷm−q,ϱξuduu=D1,ϱϱmϱq+1Γq−m+1∫1ϖeϱ−1/ϱlnϖ−lnulnϖ−lnuq−mℷm−q,ϱξuduu−∑r=1mDq−r,ϱξ1Γq−r+1ϱq−r⋅eϱ−1/ϱlnϖlnϖq−r=D1,ϱℷq−m+1,ϱℷm−q,ϱξϖ−∑r=1mDq−r,ϱξ1Γq−r+2ϱq−r+1⋅eϱ−1/ϱlnϖlnϖq−r+1=D1,ϱℷ1,ϱξϖ−∑r=1mDq−r,ϱξ1Γq−r+2ϱq−r+1⋅eϱ−1/ϱlnϖlnϖq−r+1=ξu−∑r=1mDq−r,ϱξ1Γq−r+1ϱq−r⋅eϱ−1/ϱlnϖlnϖq−r.
Also,(12)ℷq,ϱCHPDq,ϱξϖ=ℷq,ϱℷm−q,ϱCHPDm,ϱξϖ=ℷm,ϱCHPDm,ϱξϖ.
By applying equation (11) in (12), we have(13)ℷq,ϱCHPDq,ϱξϖ=ξϖ−∑l=0m−1Dl,ϱξ1Γl+1ϱl⋅eϱ−1/ϱlnϖlnϖl,where the replacing of the integer m−r by l has been modified. Hence, the solution of the FDE (9) is given by(14)ξϖ=∑l=0m−1Dl,ϱξ1Γl+1ϱl⋅eϱ−1/ϱlnϖlnϖl.Lemma 3.
The solution of the following fractional Langevin equation in the integral form(15)CHPDq,ϱCHPDq,ϱ+λzt=ξt,zt,t∈1,e,z1=0,z′1=0,z″1=0,CHPDq,ϱz1=ℷϑ,ϱzη,1<η<e,CHPDq,ϱze+lze=0,l∈ℝ,is given by(16)zt=1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1ξu,zuudu−λϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1zuudu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,zuudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,zuudu−λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1zuudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu.Proof.
Applying the generalized proportional Hadamard fractional integralℷq,ϱ to equation (15), we get(17)DCHPp,ϱzt+λzt=ℷq,ϱξt,zt+eϱ−1/ϱlntM0+M1lnt.
Repeating integration by using generalized proportional Hadamard fractional integral operator of orderp, we obtain(18)zt=ℷq+p,ϱξt,zt−λℷp,ϱzt+M0ℷp,ϱeϱ−1/ϱlnt+M1ℷp,ϱeϱ−1/ϱlntlnt.
Since(19)ℷp,ϱeϱ−1/ϱlnt=eϱ−1/ϱlntlntpϱpΓp+1,(20)ℷp,ϱeϱ−1/ϱlntlnt=eϱ−1/ϱlntlntp+1ϱpΓp+2.
Then, by substituting in (18) from (19) and (20), we get(21)zt=ℷq+p,ϱξt,zt−λℷp,ϱzt+M0eϱ−1/ϱlntlntpϱpΓp+1+M1eϱ−1/ϱlntlntp+1ϱpΓp+2+M2eϱ−1/ϱlntlnt2+M3eϱ−1/ϱlntlnt+M4eϱ−1/ϱlnt,where M0,M1,M2,M3, and M4 are real constants.
By using the boundary condition, we get the following.
Fromz1=0⇒M4=0, z′1=0⇒M3=0, and z″1=0⇒M2=0.
Therefore,(22)zt=ℷq+p,ϱξt,zt−λℷp,ϱzt+M0eϱ−1/ϱlntlntpϱpΓp+1+M1eϱ−1/ϱlntlntp+1ϱpΓp+2.
From the conditions,(23)DCHPp,ϱz1=ℷϑ,ϱzη,1<η<e,DCHPp,ϱze+lze=0,l∈ℝ.
By puttingt=1 in (17), we get(24)M0=ℷϑ,ϱzη.
Also, by puttingt=e in (17) and (22) and from equation (24), we have(25)M1=Γp+2λ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,zuudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,zuudu−λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1zuudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu.
Substituting in (22) from equations (24) and (25), we obtain(26)zt=1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1ξu,zuudu−λϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1zuudu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,zuudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,zuudu−λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1zuudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu.Suppose that the spaceC1,e,. is a Bananch space and its norm is defined by f=supt∈1,eft such that, for each f∈C1,e, it implies that f:1,e⟶ℝ is continuous.Let us start by defining an operatorJ:C1,e⟶C1,e in order for(27)Jzt=1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1ξu,zuudu−λϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1zuudu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,zuudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,zuudu−λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1zuudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu.It should be noted that if the operatorJ has a fixed point, the solution of equation (15) exists.


### 3.1. Existence Result

We now introduce the following conditions to confirm our existence result with the support of fixed point procedure.Theorem 2.
Let us imagine that the continuous functionξ:1,e×ℝ⟶ℝ satisfying the following inferences:(I)
ξt,zt≤kt∀t,zt∈1,e×ℝandkt∈C1,e,ℝ+,withsuptkt=k(II)
ξt,zt−ξt,z1t≤χzt−z1t∀t∈1,e,z,z1∈C1,eandχ>0
Then, the FLEs (15) has a minimum of one solution on 1,e if(28)ϒ3<1,where(29)ϒ3=l−λχϱp+qλ−l−ϱpΓp+2Γp+q+1+χϱqΓq+1λ−l−ϱpΓp+2+λλ−lϱpλ−l−ϱpΓp+2Γp+1+lnηϑl−λ+ϱpΓp+1ϱp+ϑΓp+1Γϑ+1λ−l−ϱpΓp+2+lnηϑϱp+ϑΓp+1Γϑ+1.Proof.
Chooser=ϒ1k/1−ϒ2, where(30)ϒ1=1ϱp+qΓp+q+1+l−λϱp+qλ−l−ϱpΓp+2Γp+q+1+1ϱqΓq+1λ−l−ϱpΓp+2,ϒ2=λϱpΓp+1+λλ−lϱpλ−l−ϱpΓp+2Γp+1+lnηϑl−λ+ϱpΓp+1ϱp+ϑΓp+1Γϑ+1λ−l−ϱpΓp+2+lnηϑϱp+ϑΓp+1Γϑ+1.
Define the closed ballB¯r=w∈C1,e:w≤r.
By combining the operatorsJ1,J2∈B¯r which are described as follows:(31)J1zt=1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1ξu,zuudu−λϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1zuudu,J2zt=eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,zuudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,zuudu−λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1zuudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu.
Keep this in mindJzt=J1zt+J2zt∈B¯r on 1,e.
First, we will show thatJzt=J1z1t+J2z2t∈B¯r, for each z1,z2∈B¯r.(32)J1z1+J2z2=supt∈1,eJ1z1t+J2z2t=supt∈1,e|1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1ξu,zuudu−λϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1zuudu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,zuudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,zuudu−λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1zuudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu|≤supt∈1,e1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1ξu,zuudu−λϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1zuudu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,zuudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,zuudu−λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1zuudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu.
Taking advantage of theeϱ−1/ϱlnt≤eϱ−1/ϱlnt−lnu≤1, for 0≤lnu<lnη<lnt≤1, and from I1, we get(33)J1z1+J2z2≤ksupt∈1,e1ϱp+qΓp+q∫1tlnt−lnup+q−1duu+l−λlntp+1ϱp+qλ−l−ϱpΓp+2Γp+1∫1e1−lnup+q−1duu+lntp+1ϱqΓqλ−l−ϱpΓp+2∫1e1−lnuq−1duu+z1λϱpΓpsupt∈1,e∫1tlnt−lnup−1duu+z2supt∈1,eλλ−llntp+1ϱpλ−l−ϱpΓp+2Γp∫1e1−lnup−1duu+lnηϑl−λ+ϱpΓp+1ϱp+ϑΓp+1Γϑ+1λ−l−ϱpΓp+2+lntpϱp+ϑΓp+1Γϑ∫1ηlnη−lnuϑ−1duu≤k1ϱp+qΓp+q+1+l−λϱp+qλ−l−ϱpΓp+2Γp+q+1+1ϱqΓq+1λ−l−ϱpΓp+2+rλϱpΓp+1+λλ−lϱpλ−l−ϱpΓp+2Γp+1+lnηϑl−λ+ϱpΓp+1ϱp+ϑΓp+1Γϑ+1λ−l−ϱpΓp+2+lnηϑϱp+ϑΓp+1Γϑ+1≤kϒ1+rϒ2≤r.
Hence,Jzt=J1z1t+J2z2t∈B¯r.
Second, we will demonstrate thatJ2 is a contraction operator. Let α1andα2 be two elements in Banach space C1,e,. such that(34)J2α1−J2α2=supt∈1,eJ1α1t−J2α2t≤supt∈1,eeϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,α1u−ξu,α2uudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,α1u−ξu,α2uudu+λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1α1u−α2uudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1α1u−α2uudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1α1u−α2uudu.
Taking advantage of theeϱ−1/ϱlnt≤eϱ−1/ϱlnt−lnu≤1 for 0≤lnu<lnη<lnt≤1, and from I2, we get(35)J2α1−J2α2≤l−λχϱp+qλ−l−ϱpΓp+2Γp+q+1+χϱqΓq+1λ−l−ϱpΓp+2+λλ−lϱpλ−l−ϱpΓp+2Γp+1+lnηϑl−λ+ϱpΓp+1ϱp+ϑΓp+1Γϑ+1λ−l−ϱpΓp+2+lnηϑϱp+ϑΓp+1Γϑ+1α1−α2≤ϒ3α1−α2.
Hence,J2 is a contraction map, as implied by assumption that ϒ3<1.
Finally, we shall demonstrate thatJ1 is continuous and compact.
To begin with, sinceξ is a continuous function on t∈1,e, operator J1 is also continuous.
J1 must be uniformly bounded and equicontinuous on B¯r in order to be seen to be compact.(36)J1z=supt∈1,eJ1zt≤supt∈1,e1ϱp+qΓp+q∫1teϱ−1/ϱlnη−lnulnη−lnup+q−1ξu,zuudu+λϱpΓp∫1teϱ−1/ϱlnη−lnulnη−lnup−1zuudu≤ksupt∈1,e1ϱp+qΓp+q∫1tlnt−lnup+q−1duu+rsupt∈1,eλϱpΓp∫1tlnη−lnup−1duu≤kϱp+qΓp+q+1+λrϱpΓp<∞.
This proves thatJ1 is uniformly bounded.
The compactness of operatorJ1 is then demonstrated. For each 1<t1<t2<e, we get(37)J1zt1−J1zt2=|1ϱp+qΓp+q∫1t2eϱ−1/ϱlnt2−lnulnt2−lnup+q−1ξu,zuudu−λϱpΓp∫1t2eϱ−1/ϱlnt2−lnulnt2−lnup−1zuudu−1ϱp+qΓp+q∫1t1eϱ−1/ϱlnt1−lnulnt1−lnup+q−1ξu,zuudu+λϱpΓp∫1t1eϱ−1/ϱlnt1−lnulnt1−lnup−1zuudu|,J1zt1−J1zt2=|1ϱp+qΓp+q∫1t1eϱ−1/ϱlnt2/ulnt2up+q−1−eϱ−1/ϱlnt1/ulnt1up+q−1ξu,zuudu+1ϱp+qΓp+q∫t1t2eϱ−1/ϱlnt2/ulnt2up+q−1ξu,zuudu−λϱpΓp∫1t1eϱ−1/ϱlnt2/ulnt2up−1−eϱ−1/ϱlnt1/ulnt1up−1zuudu+λϱpΓp∫t1t2eϱ−1/ϱlnt2−lnulnt2−lnup−1zuudu|≤kϱp+qΓp+q∫1t1eϱ−1/ϱlnt1/ueϱ−1/ϱlnt2/t1lnt2up+q−1−lnt1up+q−1duu+∫t1t2eϱ−1/ϱlnt2/ulnt2up+q−1duu+λrϱpΓp∫1t1eϱ−1/ϱlnt1/ueϱ−1/ϱlnt2/ulnt2up−1−lnt1up−1duu+∫t1t2eϱ−1/ϱlnt2−lnulnt2−lnup−1duu.
Taking advantage of theeϱ−1/ϱlnt2/u<eϱ−1/ϱlnt1/u≤1, eϱ−1/ϱlnt2/t1≤1, for each 1≤u<t1<t2≤e, and from the first mean value theorem, we get(38)J1zt1−J1zt2≤kϱp+qΓp+q∫1t1lnt2up+q−1−lnt1up+q−1duu+∫t1t2lnt2up+q−1duu+λrϱpΓp∫1t1lnt2up−1−lnt1up−1duu+∫t1t2lnt2−lnup−1duu.
Taking the limit ast2⟶t1, we have J1zt1−J1zt2⟶0.
J1 is an equicontinuous as a result of this. We may conclude that J1 compacts on B¯r based on the Arzela Ascoli theorem. As a result, in B¯r, there is a point z so that z=J1z. As a consequence, equation (1) has at least one solution on 1,e. □


### 3.2. Uniqueness Result

Theorem 3.
Let us imagine that the continuous functionξ:1,e×ℝ⟶ℝ satisfying the inferences I,II and(I3)ℵ=supt∈1,eξt,0<∞
If(39)Ω=χϒ1+ϒ2≤1,then the FLEs (15) have a unique solution on 1,e.Proof.
Chooser1>ϒ1ℵ/1−ϒ1−ϒ2, to define the closed ball B¯r1=w∈C1,e:w≤r1.
JB¯r1 must be bounded. So, we must demonstrate that JB¯r1⊆B¯r1 and J is contractive.
For eachu∈1,e and w∈B¯r1, we get(40)ξu,wu=ξu,wu−ξu,0+ξu,0≤ξu,wu−ξu,0+ξu,0≤χwu+ℵ≤χr1+ℵ.
Therefore,(41)Jwt≤supt∈1,e1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1ξu,wuudu+λϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1wuudu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,wuudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,wuudu+λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1wuudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1wuudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1wuudu|≤χr1+ℵsupt∈1,e1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1duu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1duu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+21ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1duu+r1supt∈1,eλϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1duu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1duueϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1duu+eϱ−1/ϱlnt|lnt|pϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1duu.
Taking advantage of theeϱ−1/ϱlnt2/u<eϱ−1/ϱlnt1/u≤1, eϱ−1/ϱlnt2/t1≤1, for each 1≤u<t1<t2≤e,(42)Jwt≤χr1+ℵsupt∈1,e1ϱp+qΓp+q∫1tlnt−lnup+q−1duu+l−λlntp+1ϱp+qλ−l−ϱpΓp+2Γp+1∫1e1−lnup+q−1duu+lntp+1ϱqΓqλ−l−ϱpΓp+2∫1e1−lnuq−1duu+r1supt∈1,eλϱpΓp∫1tlnt−lnup−1duu+λλ−llntp+1ϱpλ−l−ϱpΓp+2Γp∫1e1−lnup−1duul−λ+ϱpΓp+1lntp+1ϱϑ+pΓp+1Γϑλ−l−ϱpΓp+2∫1ηlnη−lnuϑ−1duu+lntpϱp+ϑΓp+1Γϑ∫1ηlnη−lnuϑ−1duu≤χr1+ℵsupt∈1,e1ϱp+qΓp+q+1+l−λϱp+qλ−l−ϱpΓp+2Γp+q+1+1ϱqλ−l−ϱpΓp+2Γq+1+r1λϱpΓp+1+λλ−lϱpλ−l−ϱpΓp+2Γp+1l−λ+ϱpΓp+1lnηϑϱp+ϑΓp+1Γϑ+1λ−l−ϱpΓp+2+lnηϑϱp+ϑΓp+1Γϑ+1≤χr1+ℵϒ1+r1ϒ2≤r1,which suggests that Jw≤r1. Hence, JB¯r1⊆B¯r1.
Now, we will show thatJ is contractive. So, for all w1,w2∈C1,e, we get(43)Jw1t−Jw2t≤supt∈1,e1ϱp+qΓp+q∫1teϱ−1/ϱlnt/ulntup+q−1ξu,w1u−ξu,w2uudu+λϱpΓp∫1teϱ−1/ϱlnt/ulntup−1w1u−w2uudu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,w1u−ξu,w2uudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,w1u−ξu,w2uudu+λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1w1u−w2uudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1w1u−w2uudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1w1u−w2uudu≤χw1−w2supt∈1,e1ϱp+qΓp+q∫1tlnt−lnup+q−1duu+l−λlntp+1ϱp+qλ−l−ϱpΓp+2Γp+1∫1e1−lnup+q−1duu+lntp+1ϱqΓqλ−l−ϱpΓp+2∫1e1−lnuq−1duu+w1−w2supt∈1,eλϱpΓp∫1tlnt−lnup−1duu+λλ−llntp+1ϱpλ−l−ϱpΓp+2Γp∫1e1−lnup−1duul−λ+ϱpΓp+1lntp+1ϱϑ+pΓp+1Γϑλ−l−ϱpΓp+2∫1ηlnη−lnuϑ−1duu+lntpϱp+ϑΓp+1Γϑ∫1ηlnη−lnuϑ−1duu≤χw1−w2supt∈1,e1ϱp+qΓp+q+1+l−λϱp+qλ−l−ϱpΓp+2Γp+q+1+1ϱqλ−l−ϱpΓp+2Γq+1+w1−w2λϱpΓp+1+λλ−lϱpλ−l−ϱpΓp+2Γp+1l−λ+ϱpΓp+1lnηϑϱp+ϑΓp+1Γϑ+1λ−l−ϱpΓp+2+lnηϑϱp+ϑΓp+1Γϑ+1≤χϒ1+ϒ2w1−w2,since χϒ1+ϒ2<1. Then, J is a contractive operator. Hence, J has a unique fixed point. As a result, there is only one solution for FLEs (15).


## 3.1. Existence Result

We now introduce the following conditions to confirm our existence result with the support of fixed point procedure.Theorem 2.
Let us imagine that the continuous functionξ:1,e×ℝ⟶ℝ satisfying the following inferences:(I)
ξt,zt≤kt∀t,zt∈1,e×ℝandkt∈C1,e,ℝ+,withsuptkt=k(II)
ξt,zt−ξt,z1t≤χzt−z1t∀t∈1,e,z,z1∈C1,eandχ>0
Then, the FLEs (15) has a minimum of one solution on 1,e if(28)ϒ3<1,where(29)ϒ3=l−λχϱp+qλ−l−ϱpΓp+2Γp+q+1+χϱqΓq+1λ−l−ϱpΓp+2+λλ−lϱpλ−l−ϱpΓp+2Γp+1+lnηϑl−λ+ϱpΓp+1ϱp+ϑΓp+1Γϑ+1λ−l−ϱpΓp+2+lnηϑϱp+ϑΓp+1Γϑ+1.Proof.
Chooser=ϒ1k/1−ϒ2, where(30)ϒ1=1ϱp+qΓp+q+1+l−λϱp+qλ−l−ϱpΓp+2Γp+q+1+1ϱqΓq+1λ−l−ϱpΓp+2,ϒ2=λϱpΓp+1+λλ−lϱpλ−l−ϱpΓp+2Γp+1+lnηϑl−λ+ϱpΓp+1ϱp+ϑΓp+1Γϑ+1λ−l−ϱpΓp+2+lnηϑϱp+ϑΓp+1Γϑ+1.
Define the closed ballB¯r=w∈C1,e:w≤r.
By combining the operatorsJ1,J2∈B¯r which are described as follows:(31)J1zt=1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1ξu,zuudu−λϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1zuudu,J2zt=eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,zuudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,zuudu−λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1zuudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu.
Keep this in mindJzt=J1zt+J2zt∈B¯r on 1,e.
First, we will show thatJzt=J1z1t+J2z2t∈B¯r, for each z1,z2∈B¯r.(32)J1z1+J2z2=supt∈1,eJ1z1t+J2z2t=supt∈1,e|1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1ξu,zuudu−λϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1zuudu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,zuudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,zuudu−λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1zuudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu|≤supt∈1,e1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1ξu,zuudu−λϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1zuudu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,zuudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,zuudu−λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1zuudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu.
Taking advantage of theeϱ−1/ϱlnt≤eϱ−1/ϱlnt−lnu≤1, for 0≤lnu<lnη<lnt≤1, and from I1, we get(33)J1z1+J2z2≤ksupt∈1,e1ϱp+qΓp+q∫1tlnt−lnup+q−1duu+l−λlntp+1ϱp+qλ−l−ϱpΓp+2Γp+1∫1e1−lnup+q−1duu+lntp+1ϱqΓqλ−l−ϱpΓp+2∫1e1−lnuq−1duu+z1λϱpΓpsupt∈1,e∫1tlnt−lnup−1duu+z2supt∈1,eλλ−llntp+1ϱpλ−l−ϱpΓp+2Γp∫1e1−lnup−1duu+lnηϑl−λ+ϱpΓp+1ϱp+ϑΓp+1Γϑ+1λ−l−ϱpΓp+2+lntpϱp+ϑΓp+1Γϑ∫1ηlnη−lnuϑ−1duu≤k1ϱp+qΓp+q+1+l−λϱp+qλ−l−ϱpΓp+2Γp+q+1+1ϱqΓq+1λ−l−ϱpΓp+2+rλϱpΓp+1+λλ−lϱpλ−l−ϱpΓp+2Γp+1+lnηϑl−λ+ϱpΓp+1ϱp+ϑΓp+1Γϑ+1λ−l−ϱpΓp+2+lnηϑϱp+ϑΓp+1Γϑ+1≤kϒ1+rϒ2≤r.
Hence,Jzt=J1z1t+J2z2t∈B¯r.
Second, we will demonstrate thatJ2 is a contraction operator. Let α1andα2 be two elements in Banach space C1,e,. such that(34)J2α1−J2α2=supt∈1,eJ1α1t−J2α2t≤supt∈1,eeϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,α1u−ξu,α2uudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,α1u−ξu,α2uudu+λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1α1u−α2uudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1α1u−α2uudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1α1u−α2uudu.
Taking advantage of theeϱ−1/ϱlnt≤eϱ−1/ϱlnt−lnu≤1 for 0≤lnu<lnη<lnt≤1, and from I2, we get(35)J2α1−J2α2≤l−λχϱp+qλ−l−ϱpΓp+2Γp+q+1+χϱqΓq+1λ−l−ϱpΓp+2+λλ−lϱpλ−l−ϱpΓp+2Γp+1+lnηϑl−λ+ϱpΓp+1ϱp+ϑΓp+1Γϑ+1λ−l−ϱpΓp+2+lnηϑϱp+ϑΓp+1Γϑ+1α1−α2≤ϒ3α1−α2.
Hence,J2 is a contraction map, as implied by assumption that ϒ3<1.
Finally, we shall demonstrate thatJ1 is continuous and compact.
To begin with, sinceξ is a continuous function on t∈1,e, operator J1 is also continuous.
J1 must be uniformly bounded and equicontinuous on B¯r in order to be seen to be compact.(36)J1z=supt∈1,eJ1zt≤supt∈1,e1ϱp+qΓp+q∫1teϱ−1/ϱlnη−lnulnη−lnup+q−1ξu,zuudu+λϱpΓp∫1teϱ−1/ϱlnη−lnulnη−lnup−1zuudu≤ksupt∈1,e1ϱp+qΓp+q∫1tlnt−lnup+q−1duu+rsupt∈1,eλϱpΓp∫1tlnη−lnup−1duu≤kϱp+qΓp+q+1+λrϱpΓp<∞.
This proves thatJ1 is uniformly bounded.
The compactness of operatorJ1 is then demonstrated. For each 1<t1<t2<e, we get(37)J1zt1−J1zt2=|1ϱp+qΓp+q∫1t2eϱ−1/ϱlnt2−lnulnt2−lnup+q−1ξu,zuudu−λϱpΓp∫1t2eϱ−1/ϱlnt2−lnulnt2−lnup−1zuudu−1ϱp+qΓp+q∫1t1eϱ−1/ϱlnt1−lnulnt1−lnup+q−1ξu,zuudu+λϱpΓp∫1t1eϱ−1/ϱlnt1−lnulnt1−lnup−1zuudu|,J1zt1−J1zt2=|1ϱp+qΓp+q∫1t1eϱ−1/ϱlnt2/ulnt2up+q−1−eϱ−1/ϱlnt1/ulnt1up+q−1ξu,zuudu+1ϱp+qΓp+q∫t1t2eϱ−1/ϱlnt2/ulnt2up+q−1ξu,zuudu−λϱpΓp∫1t1eϱ−1/ϱlnt2/ulnt2up−1−eϱ−1/ϱlnt1/ulnt1up−1zuudu+λϱpΓp∫t1t2eϱ−1/ϱlnt2−lnulnt2−lnup−1zuudu|≤kϱp+qΓp+q∫1t1eϱ−1/ϱlnt1/ueϱ−1/ϱlnt2/t1lnt2up+q−1−lnt1up+q−1duu+∫t1t2eϱ−1/ϱlnt2/ulnt2up+q−1duu+λrϱpΓp∫1t1eϱ−1/ϱlnt1/ueϱ−1/ϱlnt2/ulnt2up−1−lnt1up−1duu+∫t1t2eϱ−1/ϱlnt2−lnulnt2−lnup−1duu.
Taking advantage of theeϱ−1/ϱlnt2/u<eϱ−1/ϱlnt1/u≤1, eϱ−1/ϱlnt2/t1≤1, for each 1≤u<t1<t2≤e, and from the first mean value theorem, we get(38)J1zt1−J1zt2≤kϱp+qΓp+q∫1t1lnt2up+q−1−lnt1up+q−1duu+∫t1t2lnt2up+q−1duu+λrϱpΓp∫1t1lnt2up−1−lnt1up−1duu+∫t1t2lnt2−lnup−1duu.
Taking the limit ast2⟶t1, we have J1zt1−J1zt2⟶0.
J1 is an equicontinuous as a result of this. We may conclude that J1 compacts on B¯r based on the Arzela Ascoli theorem. As a result, in B¯r, there is a point z so that z=J1z. As a consequence, equation (1) has at least one solution on 1,e. □


## 3.2. Uniqueness Result

Theorem 3.
Let us imagine that the continuous functionξ:1,e×ℝ⟶ℝ satisfying the inferences I,II and(I3)ℵ=supt∈1,eξt,0<∞
If(39)Ω=χϒ1+ϒ2≤1,then the FLEs (15) have a unique solution on 1,e.Proof.
Chooser1>ϒ1ℵ/1−ϒ1−ϒ2, to define the closed ball B¯r1=w∈C1,e:w≤r1.
JB¯r1 must be bounded. So, we must demonstrate that JB¯r1⊆B¯r1 and J is contractive.
For eachu∈1,e and w∈B¯r1, we get(40)ξu,wu=ξu,wu−ξu,0+ξu,0≤ξu,wu−ξu,0+ξu,0≤χwu+ℵ≤χr1+ℵ.
Therefore,(41)Jwt≤supt∈1,e1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1ξu,wuudu+λϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1wuudu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,wuudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,wuudu+λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1wuudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1wuudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1wuudu|≤χr1+ℵsupt∈1,e1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1duu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1duu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+21ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1duu+r1supt∈1,eλϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1duu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1duueϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1duu+eϱ−1/ϱlnt|lnt|pϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1duu.
Taking advantage of theeϱ−1/ϱlnt2/u<eϱ−1/ϱlnt1/u≤1, eϱ−1/ϱlnt2/t1≤1, for each 1≤u<t1<t2≤e,(42)Jwt≤χr1+ℵsupt∈1,e1ϱp+qΓp+q∫1tlnt−lnup+q−1duu+l−λlntp+1ϱp+qλ−l−ϱpΓp+2Γp+1∫1e1−lnup+q−1duu+lntp+1ϱqΓqλ−l−ϱpΓp+2∫1e1−lnuq−1duu+r1supt∈1,eλϱpΓp∫1tlnt−lnup−1duu+λλ−llntp+1ϱpλ−l−ϱpΓp+2Γp∫1e1−lnup−1duul−λ+ϱpΓp+1lntp+1ϱϑ+pΓp+1Γϑλ−l−ϱpΓp+2∫1ηlnη−lnuϑ−1duu+lntpϱp+ϑΓp+1Γϑ∫1ηlnη−lnuϑ−1duu≤χr1+ℵsupt∈1,e1ϱp+qΓp+q+1+l−λϱp+qλ−l−ϱpΓp+2Γp+q+1+1ϱqλ−l−ϱpΓp+2Γq+1+r1λϱpΓp+1+λλ−lϱpλ−l−ϱpΓp+2Γp+1l−λ+ϱpΓp+1lnηϑϱp+ϑΓp+1Γϑ+1λ−l−ϱpΓp+2+lnηϑϱp+ϑΓp+1Γϑ+1≤χr1+ℵϒ1+r1ϒ2≤r1,which suggests that Jw≤r1. Hence, JB¯r1⊆B¯r1.
Now, we will show thatJ is contractive. So, for all w1,w2∈C1,e, we get(43)Jw1t−Jw2t≤supt∈1,e1ϱp+qΓp+q∫1teϱ−1/ϱlnt/ulntup+q−1ξu,w1u−ξu,w2uudu+λϱpΓp∫1teϱ−1/ϱlnt/ulntup−1w1u−w2uudu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,w1u−ξu,w2uudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,w1u−ξu,w2uudu+λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1w1u−w2uudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1w1u−w2uudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1w1u−w2uudu≤χw1−w2supt∈1,e1ϱp+qΓp+q∫1tlnt−lnup+q−1duu+l−λlntp+1ϱp+qλ−l−ϱpΓp+2Γp+1∫1e1−lnup+q−1duu+lntp+1ϱqΓqλ−l−ϱpΓp+2∫1e1−lnuq−1duu+w1−w2supt∈1,eλϱpΓp∫1tlnt−lnup−1duu+λλ−llntp+1ϱpλ−l−ϱpΓp+2Γp∫1e1−lnup−1duul−λ+ϱpΓp+1lntp+1ϱϑ+pΓp+1Γϑλ−l−ϱpΓp+2∫1ηlnη−lnuϑ−1duu+lntpϱp+ϑΓp+1Γϑ∫1ηlnη−lnuϑ−1duu≤χw1−w2supt∈1,e1ϱp+qΓp+q+1+l−λϱp+qλ−l−ϱpΓp+2Γp+q+1+1ϱqλ−l−ϱpΓp+2Γq+1+w1−w2λϱpΓp+1+λλ−lϱpλ−l−ϱpΓp+2Γp+1l−λ+ϱpΓp+1lnηϑϱp+ϑΓp+1Γϑ+1λ−l−ϱpΓp+2+lnηϑϱp+ϑΓp+1Γϑ+1≤χϒ1+ϒ2w1−w2,since χϒ1+ϒ2<1. Then, J is a contractive operator. Hence, J has a unique fixed point. As a result, there is only one solution for FLEs (15).


## 4. Stability Results

The HU stability of equation (15) is discussed in this section. The following is the definition of HU stability:Definition 8.
It is said that the integral equation (26) is HU stable. If ∃ε∗≥0 for some fixed constant ζ∗>0 satisfying the following:(44)|zt−1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1ξu,zuudu−λϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1zuudu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,zuudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,zuudu−λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1zuudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zuudu|≤ζ∗,then there exists a continuous function z1t which satisfies(45)z1t=1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1ξu,z1uudu−λϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1z1uudu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,z1uudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,z1uudu−λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1z1uudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1z1uudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1z1uudu,such that(46)zt−z1t≤ε∗ζ∗.Theorem 4.
Let us imagine that the continuous functionξ:1,e×ℝ⟶ℝ satisfies inferences I and II, then equation (15) is HU stable.Proof.
(47)zt−z1t≤sup1ϱp+qΓp+q∫1teϱ−1/ϱlnt−lnulnt−lnup+q−1ξu,zu−ξu,z1uudu−λϱpΓp∫1teϱ−1/ϱlnt−lnulnt−lnup−1zu−z1uudu+eϱ−1/ϱlntlntp+1ϱpλ−l−ϱpΓp+2l−λϱqΓp+q∫1eeϱ−1/ϱlnu1−lnup+q−1ξu,zu−ξu,z1uudu+1ϱq−pΓq∫1eeϱ−1/ϱlnu1−lnuq−1ξu,zu−ξu,z1uudu−λλ−lΓp∫1eeϱ−1/ϱlnu1−lnup−1zu−z1uudu+l−λ+ϱpΓp+1ϱϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zu−z1uudu+eϱ−1/ϱlntlntpϱp+ϑΓp+1Γϑ∫1ηeϱ−1/ϱlnη−lnulnη−lnuϑ−1zu−z1uudu≤χz−z1supt∈1,e1ϱp+qΓp+q∫1tlnt−lnup+q−1duu+l−λlntp+1ϱp+qλ−l−ϱpΓp+2Γp+1∫1e1−lnup+q−1duu+lntp+1ϱqΓqλ−l−ϱpΓp+2∫1e1−lnuq−1duu+z−z1supt∈1,eλϱpΓp∫1tlnt−lnup−1duu+λλ−llntp+1ϱpλ−l−ϱpΓp+2Γp∫1e1−lnup−1duu+l−λ+ϱpΓp+1lntp+1ϱϑ+pΓp+1Γϑλ−l−ϱpΓp+2∫1ηlnη−lnuϑ−1duu+lntpϱp+ϑΓp+1Γϑ∫1ηlnη−lnuϑ−1duu≤χz−z1supt∈1,e1ϱp+qΓp+q+1+l−λϱp+qλ−l−ϱpΓp+2Γp+q+1+1ϱqλ−l−ϱpΓp+2Γq+1+z−z1λϱpΓp+1+λλ−lϱpλ−l−ϱpΓp+2Γp+1+l−λ+ϱpΓp+1lnηϑϱp+ϑΓp+1Γϑ+1λ−l−ϱpΓp+2+lnηϑϱp+ϑΓp+1Γϑ+1≤χϒ1+ϒ2z−z1.
Hence, by using equation (47), equation (26) is HU stable. Therefore, the FLEs (15) are HU stable.


## 5. An Example

Let us take a look at the following FLEs:(48)CHPD2.5,0.5CHPD1.5,0.5+150zt=ξt,zt,t∈1,e,z1=0,z′1=0,z″1=0,CHPD1.5,0.5z1=ℷ4,0.5ze2,CHPD1.5,0.5ze+14ze=0.Choose(49)ξt,zt=1t+25zz+10.It is obvious that the prerequisites (I1), (I2), and (I3) are fulfilled.By puttingχ=1/4, we get ϒ3=0.372486<1, ϒ2=0.0565441, and ϒ1=1.98731. Hence, all assumptions of Theorem 2 are met. As a result of Theorem 2, there must be at least one solution to equation (15).On the other hand,χϒ1+ϒ2=0.553373<1, this implies that Theorem 3 demonstrates that equation (15) has only one solution. The requirements for HU stability can also be found in a simple manner. Equation (15) is HU stable, so we conclude.


## 6. Conclusion

Fractional Langevin equations have a considerable role in modeling varied physical phenomena. For instance, they have been employed for describing single-file prevalence [19] and the conduct of unshackled particles driven by internal noises [16]. The present paper examined the fractional Langevin equations with generalized proportional Hadamard–Caputo derivative of different orders. Nonlocal integrals and nonperiodic boundary conditions have been deemed as well. With the assistance of Krasnoselskii and Banach fixed point theorems, existence, uniqueness, and Hyres–Ulam stability for the solution have been amply investigated. Moreover, an application example has been offered to reinforce the accuracy of the gained outcomes.



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*Source: 6316477-2021-12-08.xml*