# On Solutions of Hybrid–Sturm-Liouville–Langevin Equations with Generalized Versions of Caputo Fractional Derivatives

Authors: Abdellatif Boutiara; Hanan A. Wahash; Heba Y. Zahran; Emad E. Mahmoud; Abdel-Haleem Abdel-Aty; El Sayed Yousef
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The main intention of this research article is to introduce a new class of generalized fractional differential equations that fall into the categories of Sturm-Liouville’s, Langevin’s, and hybrid’s problems involvingY-Caputo fractional derivatives. The existence of the solutions of the proposed equations is discussed by using the technique of the measure of noncompactness related to the fixed point theorem, which is a generalization of Darbo’s fixed point theorem. Additionally, pertinent examples are provided along with the different values of the function Y to confirm the validity of the reported results.


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## Body

## 1. Introduction

Fractional differential equations (FDEs) with their various branches such as Hybrid Equation (HE), Langevin Equation (LE), and Sturm-Liouville Equation (SLE) are currently well established, due to the number of papers and books edited worldwide. These types of equations have been applied in many applications in different fields, such as engineering and science. Since in recent years, it has achieved a great deal of development and interest by many researchers, for some of these developments in the theory of fractional differential equations, one can look at the monographs of Kilbas et al. [1] and Podlubny [2], where they presented some properties and applications appropriate for various types of fractional operators. Dhage and Lakshmikantham [3] and Dhage et al. [4] made excellent results on hybrid problems, as did Zhao et al. [5] and Ahmad and Ntouyas [6]. The LE [7] is formulated to be a powerful tool for describing the evolution of physical phenomena in volatile environments. Some of recent Langevin’s problem is studied through [8–10]. However, SLE has many applications in distinct areas of technical knowledge and engineering [11, 12]. The mix of both fractional SLE and fractional LE might give an adequate description of the dynamic processes described in a fractal medium where fractal and memory properties are inserted with a scattered memory kernel. Recently, the authors in [13] suggested an approach to the fractional model of the SLE and LE. Indeed, they discussed the existence of solutions to the considered systems through fixed point techniques and mathematical inequalities. Muensawat et al. [14] studied antiperiodic BVPs for fractional systems of generalized SL and LE. Boutiara et al. [15] considered fractional LE under Caputo function-dependent kernel fractional derivatives. Existence theorem for psi-fractional HEs has been proven by Suwan et al. [16]. Some qualitative analyses for multiterm LEs with generalized Caputo FDs and diffusion FDE with ABC operators can be found in [17, 18]. The authors in [19] considered a hybrid LE involving Caputo FD and Riemann-Liouville (RL) fractional integral (FI) as follows:
(1)cDζcDξϑσMσ,ϑνσ−λϑσ=Nσ,ϑμσ,Iγϑμσ,σ∈0,1,ϑ0=0,cDξϑσMσ,ϑνσσ=0=0,ϑ1=ζϑκ,0<κ<1,ζ,λ∈ℝ.Motivated by the above works aforesaid and inspired by [19, 21], in this paper, we deal with the existence of solutions for the following BVP to the nonlinear fractional hybrid–Sturm-Liouville–Langevin differential equation:
(2)cDζ,YpσcDξ,YϑσMσ,ϑνσ−qσϑσ=Nσ,ϑμσ,σ∈Π=a,b,ϑa=0,pbcDξ,YϑσMσ,ϑνσσ=b+qbϑb=0,where cDr,Y denotes the Y-Caputo FD of order r∈ζ,ξ, 0<ζ,ξ≤1. Here, M∈CΠ×ℝ,ℝ\0, N∈CΠ×ℝ,ℝ,ν,μ:Π⟶Π are given functions, p∈CΠ,ℝ\0, and q∈CΠ,ℝ. As in Banach spaces, a closed and bounded set is not generally a compact set; just continuity of the function M does not ensure the existence of a solution to differential equations. Our arguments are principally founded on Darbo’s fixed point technique mixed with the technique of measures of noncompactness to set up the existence of solutions for (2). In particular, problem (2) is formed as an overarching structure comprising both fractional SLE, LE, and HE, subjected to boundary conditions involving Y-Caputo FDs. In fact, choosing qσ≡0 on the one hand and pσ≡1, qσ=λ, and λ∈ℝ, on the other hand, reduces the problem (2) into the fractional Sturm-Liouville problem and the fractional Langevin problem, respectively. Besides, if we set pσ≡1 and qσ≡0, the problem (2) reduces to the fractional sequential hybrid problem.Observe also that the current results are consistent with some of the literature results whenYσ=σ, and they are new even for the special case: Yσ=logσ and Yσ=σρ.Here is a brief outline of the paper. In Section2, we provide some preliminary facts. Sections 3 and 4 handle the formulation of solutions and the existence of solutions for (2) by using the generalized Darbo’s fixed point theorem (D’sFPT) along with the approach of measures of noncompactness in the Banach algebras. Lastly, we give pertinent examples.


## 2. Preliminaries

Let us start this section with some auxiliary results used in the forthcoming analysis.Definition 1 (see [1]).
TheY-RL FI of order ζ>0 for an integrable function ϑ:Π⟶ℝ is given by
(3)Ia+ζ;Yϑσ=1Γζ∫aσY′ςYσ−Yςζ−1ϑςdς,where Γ is the gamma function. One can deduce that
(4)DσIa+ζ;Yϑσ=Y′σIa+ζ−1;Yϑσ,ζ>1,where Dσ=d/dt.Definition 2 (see [20]).
Forn−1<ζ<nn∈ℕ and ϑ,Y∈CnΠ,ℝ, the Y-Caputo FD of a function ϑ of order ζ is given by
(5)cDa+ζ;Yϑσ=Ia+n−ζ;YDσY′σnϑσ,where n=ζ+1 for ζ∉ℕ and n=ζ for ζ∈ℕ.
Also, we can expressY-Caputo FD by
(6)cDa+ζ;Yϑσ=∫aσY′ςYσ−Yςn−ζ−1Γn−ζDσY′ςnϑςdς,ifζ∉ℕ,DσY′σnϑσ,ifζ∈ℕ.Lemma 3 (see [1, 20]).
Letζ,ξ>0 and ϑ∈L1Π,ℝ. Then,
(7)Ia+ζ;YIa+ξ;Yϑσ=Ia+ζ+ξ;Yϑσ,a.e.σ∈Π.
In particular, ifϑ∈CΠ,ℝ, then Ia+ζ;YIa+ξ;Yϑσ=Ia+ζ+ξ;Yϑσ, σ∈Π.Lemma 4 (see [20]).
Letζ>0. Then, the following holds:
Ifϑ∈CΠ,ℝ, then
(8)cDa+ζ;YIa+ζ;Yϑσ=ϑσ,σ∈Π.
If∈CnΠ,ℝ,n−1<ζ<n. Then,
(9)Ia+ζ;YcDa+ζ;Yϑσ=ϑσ−∑k=0n−1ϑYkak!Yσ−Yak,σ∈Π.Lemma 5 (see [1, 20]).
Letσ>a,ζ≥0, and ξ>0. Then,
(i)
Ia+ζ;YYσ−Yaξ−1=Γξ/Γξ+ζYσ−Yaξ+ζ−1(ii)
cDa+ζ;YYσ−Yaξ−1=Γξ/Γξ−ζYσ−Yaξ−ζ−1(iii)
cDa+ζ;YYσ−Yak=0, for k<n,n∈ℕ
LetBυ,r~ be the closed ball in the Banach space E; if υ=0, then Br≡B0,r~. Let X~⊂E, such that X~ and Conv X~ are a closure and a convex closure of X~, respectively. And let ME be the family of the nonempty and bounded subsets of E, while PE denotes the subfamily of all relatively compact subsets of ME.Definition 6 (see [22]).
We say thatχ~:ME⟶0,∞ is a noncompactness measure in E if all the assumptions below hold:
(i)
kerχ~=X~∈ME:χ~X~=0 is nonempty and kerχ~⊂PE(ii)
Y~⊂X~, then χ~Y~≤χ~X~(iii)
χ~Y~=χ~Y~=χ~ConvY~(iv)
χ~λ1Y~+λ2X~≤λ1χ~Y~+λ2χ~X~,λ1+λ2=1(v)
In the case ofY~n being a sequence of closed subsets of ME with Y~n+1⊂Y~nn≥1 and limn⟶∞χ~Y~n=0, then ∩n=1∞Y~n≠∅Definition 7 (see [22]).
LetY~ be a nonempty bounded set and ⊂C~ be a Banach space. We say that M∈Y~ is a modulus of continuous function, denoted by ωM,ε; if ∀M∈Y~ and ∀e>0, we have
(10)ωM,ε=supMσ−Mς:σ,ς∈Π,σ−ς≤ε.
Moreover,(11)ωY~,ε=supωM,ε:M∈Y~,ω0Y~=limε⟶0ωY~,e.Definition 8 (see [23]).
A noncompactness measureχ~ in C~ satisfies the condition m if
(12)χ~MN≤Mχ~N+Nχ~M,for all M,N∈MCΠ, where C~≔CΠ is the Banach algebra.Lemma 9 (see [24]).
The condition (m) may be grasped by the noncompactness measure ϑ0 on C~.
Set(13)S=Y:0,∞⟶b,∞:∀υn⊂0,∞,limn⟶∞Yυn=b⟺limn⟶∞υn=0.
Now, we present D’sFPT and generalized D’sFPT to prove that there exists at least one fixed point.Theorem 10 (see [25, 26]).
LetC~ be a Banach space and Ξ⊂C~ be a nonempty, bounded, convex, and closed set. Let K:Ξ⟶Ξ be continuous. Assume that there is 0≤θ<1 with υ as a noncompactness measure in C~ meeting the following requirements:
(14)υKY~≤θχ~Y~,Θ≠Y~⊆Ξ.
Then,K has a fixed point in Ξ.Theorem 11 (see [26]).
LetC~ be a Banach space and V⊂C~ be a nonempty, bounded, convex, and closed set, and let K:V⟶V be continuous. Assume there exist Θ∈S and 0≤θ<1 such that for each nonempty subset D of V with χ~KD>0,(15)Θχ~KD≤Θχ~Dθ,where χ~ is a noncompactness measure in C~. Then, K has a fixed point in V.


## 3. Solution Formulation

This section presents a formulation of the solution to problem (2) along with the assumptions required in the forthcoming analysis. Foremost, we denote by C~,. the space of real valued continuous functions defined on a unit interval Π. It is clearly the Banach space with the norm:
(16)ϑ=supσ∈Πϑσ,forϑ∈C~.Multiplication is defined as the usual product of real functions.To prove the existence of solutions to (2), we need the following lemma:Lemma 12.
The problem (2) is equivalent to the following fractional integral equation:
(17)ϑσ=Mσ,ϑνσIξ,Y1pIζ,YNσ,ϑμσ−Iξ,Yqpϑσ−Yσ−YaξpσΓξ+1Iζ,YNb,ϑμb.Proof.
Applying theζth-Y-RL integral on (2), we obtain
(18)cDξ,YϑσMσ,ϑνσ=Iζ,YNσ,ϑμσ−qσϑσ+k1pσ,where k1∈ℝ. From the BCs of (2), we get
(19)k1=−Iζ,YNb,ϑμb.
Taking theξth-Y-RL integral of (18), one has
(20)ϑσMσ,ϑνσ=Iξ,Y1pIζ,YNσ,ϑμσ−Iξ,Yqpϑσ−Yσ−YaξpσΓξ+1Iζ,YNb,ϑμb+k2,where k2∈ℝ. The BCs of (2) give k2=0. In this regard, if we apply the ξth-Y-Caputo FD and ζth-Y-Caputo FD to both sides of (17) and use Lemma 5, then the problem (2) immediately is established.Before giving the essential result, we shall investigate formula (17) under the following assumptions:
(i)
(AS1) Both functionsv,μ:Π⟶Π are continuous(ii)
(AS2)M∈CΠ×ℝ,ℝ\0, and N∈CΠ×ℝ2,ℝ(iii)
(AS3) There exists a real numberρ∈a,b with
(21)Mσ,v−Mσ,ϑ≤v−ϑ+dρ−dρ,∀σ∈Π,ϑ,v∈ℝ,d∈ℝ+.(iv)
(AS4) There exists a continuous nondecreasing functionφ:ℝ+⟶ℝ+ with φ0=0 such that
(22)Nσ,ϑ≤φϑ,σ∈Π,ϑ∈ℝ.(v)
(AS5) There existsr0>0 such that
(23)r0+dρ−dρ+Nφr0p~Yb−Yaζ+ξΓζ+ξ+1+q~p~r0Yb−YaζΓζ+1+1p~φr0Yb−Yaζ+ξΓζ+1Γξ+1≤r0,where
(24)Λ≔φr0p~Yb−Yaζ+ξΓζ+ξ+1+q~p~r0Yb−YaζΓζ+1+1p~φr0Yb−Yaζ+ξΓζ+1Γξ+1≤1,N=supMσ,0:σ∈Π,p~=supσ∈Πpσ,q~=supσ∈Πqσ.


## 4. Existence Result

The aim of this section is to discuss the existence of solutions to the problem (2). For this end, we apply Theorems 10 and 11.Theorem 13 Under hypotheses (AS1)–(AS5).
Then, the problem (2) has a least one solution in the Banach algebra C~.Proof.
Consider the operatorK:C~⟶C~ on the Banach algebra C~ as
(25)Kϑσ=FϑσGϑσ,where
(26)Fϑσ=Mσ,ϑvσ,Gϑσ=Iξ,Y1pIζ,YNσ,ϑμσ−Iξ,Yqpϑσ−Yσ−YaξpσΓξ+1GN,GN=Iζ,YNb,ϑμb.
From (AS4), we have(27)GN≤Yb−YaζΓζ+1Yϑ.
For the sake of simplicity, we put(28)QYχσ,ς=Y′ςYσ−Yςχ−1Γχ,χ>0.Now, we divide the proof into several steps.Step 1.
K transforms C~ into itself.
At first, we show that∀ϑ∈C~ implies that Kϑ∈C~, i.e., FϑGϑ∈C~ for all ϑ∈C~. Certainly, (AS1) and (AS2) guarantee that if ϑ∈C~, then Fϑ∈C~. It remains to prove if ϑ∈C~, then Gϑ∈C~. Let ϑ∈C~ and σ2,σ1∈Π with σ2>σ1. By hypothesis (AS4), we get
(29)Gϑσ1−Gϑσ2=1p~∫aσ1QYζ+ξσ1,ςMς,ϑςdς−∫aσ2QYζ+ξσ2,ςMς,ϑςdς+q~p~∫aσ1QYζσ1,ςϑςdς−∫aσ2QYζ+ξσ2,ςϑςdς+1p~GNΓξ+1Yσ2−Yaξ−Yσ1−Yaξ≤1p~∫aσ2QYζ+ξσ1,ς−QYζ+ξσ2,ςMς,ϑςdς+1p~∫σ2σ1QYζ+ξσ1,ςMς,ϑςdς+q~p~∫aσ2QYζσ1,ς−QYζσ2,ςMς,ϑςdς+q~p~∫σ2σ1QYζσ1,ςMς,ϑςdς+1p~GNΓξ+1Yσ2−Yaξ−Yσ1−Yaξ≤1p~φϑΓζ+ξ+1Yσ1−Yaζ+ξ−Yσ2−Yaζ+ξ−Yσ1−Yσ2ζ+ξ+Yσ1−Yσ2ζ+ξ+q~p~ϑΓζ+1Yσ1−Yaζ−Yσ2−Yaζ−Yσ1−Yσ2ζ+Yσ1−Yσ2ζ+1p~φϑYb−YaζΓζ+1Γξ+1Yσ2−Yaξ−Yσ1−Yaξ,which tends to be zero uniformly once σ2⟶σ1. It is clear that Gϑ∈C~ for all ϑ∈C~.Step 2.
An estimate ofKϑ for ϑ∈C~.
Letϑ∈C~ and σ∈Π. Then, by using our hypothesis, we have
(30)Kϑσ=FϑσGϑσ≤Mσ,ϑvσ−Mσ,0+Mσ,0×1p~∫aσQYζ+ξσ,ςNς,ϑμςdς+q~p~∫aσQYζσ,ςϑςdς+1p~GNΓξ+1Yσ−Yaξ≤ϑ+dρ−dρ+N1p~∫aσQYζ+ξσ,ςYϑdς+q~p~∫aσQYζσ,ςϑdς+1p~φϑYb−Yaζ+ξΓζ+1Γξ+1.
Therefore,(31)Kϑ≤ϑ+dρ−dρ+Nφϑp~Yb−Yςζ+ξΓζ+ξ+1+q~p~ϑYb−YςζΓζ+1+1p~φϑYb−Yaζ+ξΓζ+1Γξ+1.Step 3.
The operatorK is continuous on Br0. Here, Br0 is a subset of C~ defined by
(32)Br0=ϑσ∈C~:ϑ≤r0:σ∈Π,with a fixed radius r0, which satisfies the inequality (AS5).
We shall need to show the continuity ofF and G on Br0, separately. For any ε>0 and ϑ,v∈Br0, there exists 0<δ<ε+dρ1/ρ−d,∋ϑ−v≤δ; it follows for σ∈Π that
(33)Fϑσ−Fvσ=Mσ,ϑνσ−Mσ,vνσ≤ϑνσ−νσ+dρ−dρ≤ϑ−v+dρ−dρ≤δ+dρ−dρ<ε.
Therefore,F is continuous on Br0. The continuity of the operator G is obtained by Lebesgue dominated convergence (LDC) theorem. Indeed, let ϑn be a sequence such that ϑn⟶ϑ in Br0 with ϑn−ϑ⟶0 as n⟶0. As μ:Π⟶Π is continuous, we obtain
(34)ϑnμσ≤r0,∀n∈ℕ,∀σ∈Π.
SinceN is continuous on Π×−r0,r0, it is uniformly continuous on Π×−r0,r0. Now, we set
(35)G0=maxσ,ϑ∈Π×−r0,r0Gϑσ,(36)κ0=G0Yb−YaζΓζ+1.
Applying the LDC theorem, we get(37)limn⟶∞Gϑnσ=limn⟶∞1p~∫aσQYζ+ξσ,ςNς,ϑnμςdς+q~p~∫aσQYζσ,ςϑnςdς+GϑnΓξ+1Yσ−Yaξ=1p~∫aσQYζ+ξσ,ςNς,ϑμςdς+q~p~∫aσQYζσ,ςϑςdς+1p~GϑΓξ+1Yσ−Yaξ=Gϑσ.
Thus,G is continuous in Br0.
Due to the continuity ofF and G, the operator K is continuous in Br0.Step 4.
We estimateϑ0FΞ and ϑ0GΞ for ∅≠Ξ⊂Br0.
At first, we estimateϑ0FΞ. Since ν:Π⟶Π is uniformly continuous, we obtain for any ε>0,∃δ>0 with δ<ε,∀σ1,σ2∈Π with σ2−σ1<δ, which implies νσ2−νσ1<ε. Taking ϑ∈Ξ and σ1,σ2∈Π with σ2−σ1<δ, under hypothesis (AS5), we get
(38)Fϑσ2−Fϑσ1=Mσ2,ϑνσ2−Mσ1,ϑνσ1≤Mσ2,ϑνσ2−Mσ2,ϑνσ1+Mσ2,ϑνσ1−Mσ1,ϑνσ1≤ϑνσ2−ϑνσ1+dρ−dρ+ωM,ε≤ωΞ,ε+dρ−dρ+ωM,ε.
Considering(39)ωM,ε=supMσ2,ϑ−Mσ1,ϑ:σ1,σ2∈Π,σ2−σ1<ε,ϑ∈−r0,r0,then we can write (38) as
(40)ωFΞ,ε≤ωΞ,ε+bp−bp+ωM,ε.
Obviously,Mσ,ϑ is uniformly continuous on Π×−r0,r0, and ωM,ε⟶0 once ε⟶0. Hence, (40) becomes as follows:
(41)ω0FΞ≤ω0Ξ+bp−bp.
Next, sinceμ:Π⟶Π is uniformly continuous, we have ∀ε>0, ∃δ>0 with δ=δε, ∀σ1,σ2∈Π with σ2−σ1<δ, which implies μσ2−μσ1<ε. Take into account equations (32), (35), and (36) for each ε>0. Set
(42)δ=min12,Γξ+1εκ0,p∗Γζ+1εq∗r0,p∗Γζ+ξ+1ε4G0.
Choosingϑ∈Ξ and σ1,σ2∈Π with σ2−σ1≤δ yields
(43)Gϑσ1−Gϑσ2=1p~∫aσ1QYζ+ξσ1,ςMσ,ϑςdς−∫aσ2QYζ+ξσ2,ςMς,ϑςdς+q~p~∫aσ1QYζσ1,ςϑςdς−∫aσ2QYζσ2,ςϑςdς+1p~GNΓξ+1Yσ2−Yaξ−Yσ1−Yaξ≤1p~G0Γζ+ξ+1Yσ1−Yaζ+ξ−Yσ2−Yaζ+ξ−Yσ1−Yσ2ζ+ξ+Yσ1−Yσ2ζ+ξ+q~p~r0Γζ+1Yσ1−Yaζ−Yσ2−Yaζ−Yσ1−Yσ2ζ+Yσ1−Yσ2ζ+1p~κ0Γξ+1Yσ2−Yaξ−Yσ1−Yaξ.
For simplicity’s sake, we set(44)HYχσ=Yσ−Yaχ,χ>0.
The factorsHYζσ2−HYζσ1, HYξσ2−HYξσ1, and HYζ+ξσ2−HYζ+ξσ1 can be estimated as in the following cases:
Case 1. If0≤HYσ1<δ,HYσ2≤2δ, then
(45)HYζσ2−HYζσ1≤HYζσ2<2δζ≤2ζδ≤2δ,HYξσ2−HYξσ1≤HYξσ2<2δξ≤2ξδ≤2δ,HYζ+ξσ2−HYζ+ξσ1≤HYζ+ξσ2<2δζ+ξ≤2ζ+ξδ≤4δ.
Case 2. If0<HYσ1<HYσ2≤δ, then
(46)HYζσ2−HYζσ1≤HYζσ2<δζ≤ζδ<2δ,HYξσ2−HYξσ1≤HYξσ2<δξ≤ξδ<2δ,HYζ+ξσ2−HYζ+ξσ1≤HYζ+ξσ2<δζ+ξ≤ζ+ξδ<4δ.
Case 3. Ifδ≤HYσ1<HYσ2≤1, then
(47)HYζσ2−HYζσ1<ζδ<2δ,HYξσ2−HYξσ1<ξδ<2δandσ2a+ξ−σ1a+ξ<ζ+ξδ<4δ.
Accordingly, we obtainGϑσ2−Gϑσ1≤ε, which implies that ωGϑ,ε≤ε.
Letε⟶0. Then,
(48)ω0GΞ=0.Step 5.
We estimateω0KΞ for ∅=Ξ∈Br0.
By Lemma9 and equations (32), (41), and (48), we obtain
(49)ω0KΞ=ω0FΞ.GΞFΞω0GΞ+GΞω0FΞ≤FBr0ω0GΞ+GBr0ω0FΞ≤ω0Ξ+dρ−dρφr0p~Yσ−Yςζ+ξΓζ+ξ+1+q~p~r0Yσ−YςζΓζ+1+1p~φr0Yb−Yaζ+ξΓζ+1Γξ+1=ω0Ξ+dρ−dρΛ.
SinceΛ≤1, the assumption (AS5) gives
(50)ω0KΞ+dρ≤ω0Ξ+dρ.
Thanks to Theorem10, the contractive condition is fulfilled with φϑ=ϑ+b, where φ∈S. By applying Theorem 11, K has at least fixed point in Br0. Hence, the problem (2) has at least one solution in Br0.


## 5. Examples

Here, we provide two examples to illustrate previous results.Example 14.
Consider the problem (2) with following specific data:
(51)pσ=1,qσ=λ=1100,Yσ=σ.Then, the problem (2) reduces to
(52)cD1/4cD1/23ϑeσ−1/2+42−1100ϑσ=σ10sinϑσ,σ∈Π=0,1,ϑ0=0,22ϑσϑeσ−1/2+42σ=1+1100ϑ1=0,where
(53)a=0,b=1,ζ=14,ξ=12,λ=1100,νσ=eσ−12,μσ=σ.Mσ,ϑ=ϑ+42,Nσ,ϑ=1/10sinϑ, d=16, and N=supσ∈0,1Mσ,0=4. Thus, (AS1) and (AS2) hold. For (AS3), we obtain ρ=1/2. Furthermore, let zϑ=ϑ+42−22. Then, z0=0, and it is a concave function. Since zσ is concave. As a result, the subadditive property of the concave function allows us to conclude
(54)Mσ,ϑ2−Mσ,ϑ1=zϑ2−zϑ1≤zϑ2−ϑ1=ϑ2−ϑ1+42−22.Thus, (AS3) holds, withρ=1/2. Moreover, for every σ∈Π and ϑ∈ℝ, we obtain
(55)Nσ,ϑ=σ10sinϑσ≤110ϑσ,∀σ∈Π.Hence, (AS4) holds withφϑ=1/10ϑ. Finally, (AS5) permitted to provide us the range of r0 which is obviously
(56)0<r0≤1.7924.Accordingly, (AS5) confirms that the illustrated example (52) has a solution in C~ due to
(57)Λ=0.544529299<1.Example 15.
Depending on the previous example, we present some special cases ofY with different values for some parameters as in Table 1.Table 1
Examples with some special cases of theY function.
Yσa,bζξp~q~r0Λσ0,11412111000<r0≤1.79240.5445<1eσ0,1133422250<r0≤0.3850.1613<1lnσ1,e2313351250<r0≤1.26030.3034<12σ1,21212321350<r0≤0.18070.0449<1


## 6. Conclusions

In this work, we have successfully studied some qualitative properties of the solution to a fractional problem that integrates three different types of BVP; more precisely, we have investigated the existence of the solutions of the Sturm-Liouville–Langevin–hybrid-type FDEs. Our analysis has been based on the technique of the measure of noncompactness along with the generalized Darbo’s fixed point theorem. The results were consistent with some of the literature results whenYσ=σ, and they are new even for the special case: Yσ=logσ and Yσ=σρ.The problem studied can be extended to a more general problem containingY-Hilfer FD, and this is what we are considering in future research.



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