# A Coupled System of Caputo–Hadamard Fractional Hybrid Differential Equations with Three-Point Boundary Conditions

Authors: Meraa Arab; Muath Awadalla
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This article presents a study of the existence and uniqueness of solutions for a system of hybrid fractional differential equations involving fractional derivatives of the Caputo-Hadamard type with three-point hybrid boundary conditions. In addition to this, the “Hyres–Ulam” stability of the solutions for this type of equation is verified, and finally a numerical example was presented to support our theoretical results.


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## Body

## 1. Introduction

Fractional calculus has sprouted as a remarkable domain of realization given its extensive applications in the mathematical modeling of numerous complex and nonlocal nonlinear systems. Fractional differential equations were applied in several fields. In the recent past, fractional differentials have drawn the attention of researchers in various fields of research of engineering, bioengineering, mathematics, physics, viscosity, electrochemistry, and other physical processes (see [1–3]).The difference between the ordinary differential equation and the fractional differential equation is that the latter is an equation that contains fractional derivatives and also comes in a relationship so that the definition of the fractional derivative is an integral equation on the other side of this equation. The Hadamard fractional derivative contains a logarithmic function of an arbitrary exponent in the kernel of the integral appearing in its definition.For the details of Hadamard fractional calculus, we mention the reader to the articles [4–6]. Fractional differential equations involving Hadamard derivatives attracted remarkable interest in the latest years; for example, see [7–20] and [21, 22].Freshly, some authors have studied different characteristics of Hybrid FDEs including the existence of solutions (see [23–37]), and some go further and studied Hyers–Ulam stability for FDEs by different mathematical theories; see [28–30] for some details.In 2019, the authors [12] published a study investigating the existence results for the following boundary value problem:(1)DHpzω=fω,zω,ω∈1,T,0<p≤1,c1z1+c2zT=c3HIqzη+φ,0<q≤1.where DHp is the Caputo–Hadamard fractional derivative; let B denote Banach space. f∈1,T×B⟶B is a continuous function, ci∈ℝ,i=1,2,3, and η∈1,T,φ∈B.In 2022, the authors [38] published a study investigating the existence results for the following FDE:(2)DHα,β;ψzϖwϖ,zϖ=χϖ,yϖ,ϖ∈0,T,0<α<1,0≤β≤1,limϖ⟶0+ψϖ−ψ01−ξzϖ=z0∈ℝ,where DHα,β;ψ denoted Psi-Hilfer fractional derivative of order α and type β, w∈C0,T×ℝ,ℝ\0,, χ∈C0,T×ℝ,ℝ., ξ=μ+v1−μ.Some researchers focus on having solutions to a system of equations only (see [8, 25, 26, 29, 31]). However, others went deeper in their research beyond the issue of verifying the issue of the existence of a solution to such equations and studied the issue of the stability of these solutions (see [21, 35–37]).This paper aims to investigate the existence of solutions for the following nonlinear sequential hybrid fractional differential equation given by(3)DCHputz1t,ut,vt=w1t,ut,vt,t∈1,e,p∈1,2,DCHqvtz2t,ut,vt=w2t,ut,vt,q∈1,2.Subject to the following boundary conditions:(4)utz1t,ut,vtt=1=0,DCHutz1t,ut,vtt=e=λ1DCHutz1t,ut,vtt=η1,vtz2t,ut,vtt=1=0,DCHvtz2t,ut,vtt=e=λ2DCHvtz2t,ut,vtt=η2,where DCHγ,γ=p,q is the Caputo–Hadamard fractional derivative of order 1<γ≤2,λ1,λ2∈0,1, and. η1,η2∈1,e.Recently, interest in fractional differential equations has become so considerable that we can say that this field is an independent science in its own right. In fact, the follower of this topic in the literature notices immediately that a good number of researchers interested in the field did not pay much attention to the hybrid type, even the stability is not always investigated.The findings of this paper must be novel and generalize several earlier findings that are important to the research. To the best of our knowledge, there are no articles that discuss boundary value problems for systems of fractional differential equations withy-Caputo and no articles that investigate Ulam–Hyers stability for differential equations that contain y-Caputo derivatives.This work is organized as follows. The second section is devoted to illuminating the fundamental principles of fractional calculus and the associated definitions and lemmas. Leray-Schauder, Krasnoselskii, and Banach fixed point theorems are applied in Section3 to show their existence and uniqueness results. In Section 4, the stability of Hyers–Ulam solutions is examined, and a set of requirements are established that ensure the stability of these solutions. Section 5 provides some examples provided to sustain the theoretical results. In Section 6, a conclusion with future work is also introduced.


## 2. Preliminaries

This section is dedicated to presenting some definitions, Lemmas, and theorems related to the fixed point concept of solutions of differential equations, which will be used to verify the existence of a solution to the system of equations given by (3)Definition 1.
(see [12]). The Hadamard fractional integral of order ν for a continuous function φ is defined as(5)IHνφω=1Γν∫aωlnωτq−11τφτdτ,ν>0.Definition 2.
(see [12]). The Hadamard fractional derivative of order ν>0 for a continuous function φ:a,∞⟶ℝ is defined as(6)DHνφω=δnIHνφω,n−1<ν<n,n=ν+1,where δ=ωd/dω,ν denotes the integer part of the real number ν.Definition 3.
(see [12]). The Caputo–Hadamard fractional derivative of order ν for at least n−times differentiable function φ:a,∞⟶ℝ is defined as(7)DCHνφω=1Γn−ν∫aωlnωτn−ν−1δngττdτ.Remark 1.
For properties and more notes on the Caputo-Hadamard derivative, we ask the reader to refer to reference [8].Lemma 1 (see [12]).
Letu∈Cδna,T,ℝ,whereCδna,T=u:a,T⟶ℝ:δn−1u∈Ca,T. Then, IHνDHνuω=uω−∑k=1ncklnω/aν−k, and(8)IHνDCHνuω=uω−∑k=0n−1cklnωak.Theorem 1 (Banach’s contraction mapping principle [23]).
LetS,d be a complete metric space, H:S⟶S is a contraction, then(i)
H has a unique fixed point s∈S, that is, Hs=s(ii)
∀s0∈S, we have limn⟶∞Hnu0=uTheorem 2 (nonlinear alternative of Leray–Schauder type [23]).
Assume thatV is an open subset of a Banach space U, 0∈V and F:V¯⟶U be a contraction such that FV¯ is bounded, then either(i)
F has a fixed point in V¯ or(ii)
∃μ∈0,1 and v∈∂V such that v=μFv holdsTheorem 3 (Arzela–Ascoli theorem [23]).
F⊂CU,ℝ is compact if and only if it is closed, bounded, and equicontinuous.Theorem 4 (Krasnoselskii’s theorem [23]).
LetE,⋅ be Banach space, B be closed convex subset of E, A be an open subset of B, and p∈A. Assume that G:A¯⟶B can be written as G=G1+G2. In addition, GA¯ is bounded set in B satisfying the following:(i)
It is continuous and completely continuous(ii)
It is a contraction, that is, there is a continuous nondecreasing function
ϕ:0,∞⟶0,∞ with ϕx>x,x>0, such that G2x−G2y≤ϕx−y, for any x,y∈A¯.
Then, one of the following holds:(i)
G has a fixed point in A¯(ii)
There area∈∂A and λ∈0,1 with a=λGa+1−λp
In the next section, we will present the most important results of this study, as we will review finding a solution to (3). Using this result, we obtain the operator and then we develop theories that study the existence and uniqueness of solutions for such a system of equations.


## 3. Existence Result

Lemma 2.
Givenh∈C1,e,ℝ, the integral solution to the problem(9)DCHputz1t,ut,vt=ht,t∈1,e,p∈1,2,utz1t,ut,vtt=1=0,DCHutz1t,ut,vtt=e=λ1DCHutz1t,ut,vtt=η1,is given by(10)ut=z1t,ut,vt×−1Γp∫1tlogtrp−1hrdrr+logt1−λ1Γp−1∫1elogerp−2hrdrr−λ1∫1η1logη1rp−2hrdrr.Proof.
First, we applyIH1p to the equation DCHput/z1t,ut,vt=ht; using the properties of Caputo–Hadamard fractional derivatives, we get(11)utz1t,ut,vt=−IH1pht+a0+a1logt.
Observe thatDCHut/z1t,ut,vt=−IH1p−1ht+a1.
The first boundary conditionut/z1t,ut,vtt=1=0, yields a0=0; thus,(12)utz1t,ut,vt=−IH1pht+a1logt.
Using the second boundary conditionDCHut/z1t,ut,vtt=e=λ1DCHut/z1t,ut,vtt=η1 yields(13)a1=11−λ1IH1p−1he−λ1IH1p−1hη1,which on substituting in (11) gives(14)utz1t,ut,vt=−IH1pht+11−λ1IH1p−1he−λ1IH1p−1hη1logt.
Alternatively, we have(15)ut=z1t,ut,vt×−1Γp∫1tlogtrp−1hrdrr+logt1−λ1Γp−1∫1elogerp−2hrdrr−λ1∫1η1logη1rp−2hrdrr,t∈1,e.
The proof is completed.
LetH=C1,e denote the Banach space of all continuous functions with the norm defined as h=sup0≤t≤1ht. The product space H×H,u,v with the norm u,v=u+v,, ∀u,v∈H×H is indeed a Banach space too. By lemma 2, we define an operator ƛ:H×H⟶H×H as(16)ƛu,vt=ƛ1u,vtƛ2u,vt,where(17)ƛ1u,vt=z1t,ut,vt×−1Γp∫1tlogtrp−1w1r,ur,vrdrr+logt1−λ1Γp−1∫1elogerp−2w1r,ur,vrdrr−λ1∫1η1logη1rp−2w1r,ur,vrdrr,ƛ2u,vt=z2t,ut,vt×−1Γq∫1tlogtrq−1w2r,ur,vrdrr+logt1−λ2Γq−1∫1elogerq−2w2r,ur,vrdrr−λ2∫1η2logη2rq−2w2r,ur,vrdrr.
To construct the necessary conditions for the results of uniqueness and existence of a problem (3), let us consider the following hypotheses:(1)
Ξ1 Let the functions z1,z2 be assumed to be continuous and bounded, that is, ∃λz1,λz2>0 such that(18)z1t,u,v≤λz1,z2t,u,v≤λz2,∀t,u,v∈1,e×ℝ2.(2)
Ξ2 Let the functions w1,w2 are assumed to be continuous, and ∃υi,τi>0,i=1,2 such that(19)w1t,u1,v1−w1t,u2,v2≤υ1u1−u2+υ2v1−v2,w2t,u1,v1−w2t,u2,v2≤τ1u1−u2+τ2v1−v2,∀t∈1,e,ui,vi∈ℝ,i=1,2.(3)
Ξ3 There are positive constants ω0,θ0 and ωi,θi≥0i=1,2 such that w1t,u,v≤ω0+ω1u+ω2v, and w2t,u,v≤θ0+θ1u+θ2v,∀t∈1,e,u,v∈ℝ,i=1,2.(4)
Ξ4 Let R⊂H×H be a bounded set, then ∃κi>0,i=1,2 such that w1t,ut,vt≤κ1, and w2t,ut,vt≤κ2, ∀u,v∈R.
To facilitate the calculations as follows, let us say(20)Λ1=sup1≤t≤e−1Γp∫1tlogtrp−1drr++logt1−λ1Γp−1∫1elogerp−2drr−λ1∫1η1logη1rp−2drr,≤1Γp+1+11−λ1Γp1+λ1logη1p−1,Λ2=sup1≤t≤e−1Γq∫1tlogtrq−1drr++logt1−λ2Γq−1∫1elogerq−2drr−λ2∫1η2logη2rq−2drr,≤1Γq+1+11−λ2Γq1+λ2logη2q−1.Theorem 5.
If bothΞ1 and Ξ2 are satisfied and assume that λz1Λ1υ1+υ2+λz2Λ2τ1+τ2<1, then the boundary value problem (3) has a unique solution.Proof.
LetNw1=sup1≤t≤ew1t,0,0,Nw2=sup1≤t≤ew2t,0,0, then we select ε≥λz1Λ1Nw1+λz2Λ2Nw2/1−λz1Λ1υ1+υ2+λz2Λ2τ1+τ2; next, we show ƛQ¯ε⊂Qε¯, where(21)Qε¯=u,v∈H×H:u,v≤ε.
Observe thatw1t,u,v=w1t,u,v−w1t,0,0+w1t,0,0≤υ1u+υ2v+Nw1≤υ1+υ2ε+Nw1.
For anyu,v∈Qε¯,t∈1,e, we have(22)ƛ1u,vt=|z1t,ut,vt×−1Γp∫1tlogtrp−1w1r,ur,vrdrr+logt1−λ1Γp−1∫1elogerp−2w1r,ur,vrdrr−λ1∫1η1logη1rp−2w1r,ur,vrdrr|≤λz1sup1≤t≤e−1Γp∫1tlogtrp−1w1r,ur,vrdrr+logt1−λ1Γp−1∫1elogerp−2w1r,ur,vrdrr−λ1∫1η1logη1rp−2w1r,ur,vrdrr≤λz1υ1+υ2ε+Nw1sup1≤t≤e−1Γp∫1tlogtrp−1drr+logt1−λ1Γp−1∫1elogerp−2drr−λ1∫1η1logη1rp−2drr≤λz1υ1+υ2ε+Nw1Λ1.
Similar to what was done above, we get(23)ƛ2u,v≤λz2τ1+τ2ε+Nw2Λ2.
From (22) and (23), we deduce that ƛu,v≤ε.
Next, for anyu1,v1,u2,v2∈H×H,∀t∈1,e, we have(24)ƛ1u1,v1t−ƛ1u2,v2t≤λz1sup1≤t≤e−1Γp∫1tlogtrp−1w1r,u1r,v1r−w1r,u2r,v2rdrr++logt1−λ1Γp−1∫1elogerp−2w1r,u1r,v1r−w1r,u2r,v2rdrr−λ1∫1η1logη1rp−2w1r,u1r,v1r−w1r,u2r,v2rdrr≤λz1υ1u1−u2+υ2v1−v2sup1≤t≤e−1Γp∫1tlogtrp−1drr+logt1−λ1Γp−1∫1elogerp−2drr−λ1∫1η1logη1rp−2drr≤λz1Λ1υ1u1−u2+υ2v1−v2≤λz1Λ1υ1+υ2u1−u2+v1−v2.
Similarly, we can find(25)ƛ2u1,v1−ƛ2u2,v2≤λz2Λ2τ1+τ2u1−u2+v1−v2.
Combining (24) and (25) yields(26)ƛu1,v1−ƛu2,v2≤λz1Λ1υ1+υ2+λz2Λ2τ1+τ2×u1−u2+v1−v2.
Sinceλz1Λ1υ1+υ2+λz2Λ2τ1+τ2<1, then ƛu1,v1−ƛu2,v2≤u1−u2+v1−v2, that is, the defined operator ƛ is a contraction; consequently, Banach fixed point theorem applies; thus, problem (3) has a unique solution on 1,e.Theorem 6.
IfΞ1, Ξ3, and Ξ4 are satisfied and if λz1Λ1ω1+λz2Λ2θ1<1 and λz1Λ1ω2+λz2Λ2θ2<1, then the boundary value problem (3) has at least one solution.Proof.
First, we show that the operatorƛ:H×H⟶H×H is completely continuous; obviously, the operator is continuous as a result that z1,z2,w1, and w2 are all assumed to be continuous.
By the aid ofΞ4, ∀u,v∈R, we have(27)ƛ1u,vt=|z1t,ut,vt×−1Γp∫1tlogtrp−1w1r,ur,vrdrr+logt1−λ1Γp−1∫1elogerp−2w1r,ur,vrdrr−λ1∫1η1logη1rp−2w1r,ur,vrdrr|≤λz1sup1≤t≤e−1Γp∫1tlogtrp−1w1r,ur,vrdrr+logt1−λ1Γp−1∫1elogerp−2w1r,ur,vrdrr−λ1∫1η1logη1rp−2w1r,ur,vrdrr≤λz1κ1sup1≤t≤e−1Γp∫1tlogtrp−1drr+logt1−λ1Γp−1∫1elogerp−2drr−λ1∫1η1logη1rp−2drr≤λz1Λ1κ1.
Similarly,(28)ƛ2u,v≤λz2Λ2κ2.
Combining inequalities (27) and (28) yields ƛu,v≤λz1Λ1κ1+λz2Λ2κ2, that is, the operator ƛ is uniformly bounded.
Next, to verify the equicontinuity for the operatorƛ, we let t1,t2∈1,e,t1<t2 and then(29)ƛ1u,vt2−ƛ1u,vt1≤λz1κ1sup1≤t≤e−1Γp∫1t1logt2rp−1−logt1rp−1drr+−1Γp∫t1t2logt2rp−1drr+logt2−logt11−λ1Γp−1∫1elogerp−2drr−λ1∫1η1logη1rp−2drr.(30)ƛ2u,vt2−ƛ2u,vt1≤λz2κ2sup1≤t≤e−1Γq∫1t1logt2rq−1−logt1rq−1drr+−1Γq∫t1t2logt2rq−1drr+logt2−logt11−λ2Γq−1∫1elogerq−2drr−λ2∫1η2logη2rq−2drr.
The R.H.S for both (29) and (30) tends to zero as t1⟶t2, and they are both independent on u,v. This implies that the operator ƛu,v is equicontinuous and yields that the operator ƛu,v is completely continuous.
Finally, we establish the bounded set given byS=u,v∈H×H|u,v=βƛu,v,β∈0,1, then ∀t∈1,e; the equation u,v=βƛu,v gives(31)ut=βƛ1u,vt,vt=βƛ2u,vt.
Using the hypothesisΞ3, we get(32)u≤λz1Λ1ω0+ω1u+ω2v,v≤λz2Λ2θ0+θ1u+θ2v.
Consequently, we have(33)u+v≤λz1Λ1ω0+λz2Λ2θ0+λz1Λ1ω1+λz2Λ2θ1u+λz1Λ1ω2+λz2Λ2θ2v.
Inequality (33) can be written as follows:(34)u,v≤λz1Λ1ω0+λz2Λ2θ0Λ0.where Λ0=min1−λz1Λ1ω1+λz2Λ2θ1,1−λz1Λ1ω2+λz2Λ2θ2.
Inequality (34) shows that S is bounded. Hence, Leray–Schauder alternative applies, which implies that problem (3) has at least one solution.


## 4. Stability

In the current section, we are interested in studying the U–H of the proposed system of the coupled sequential fractional differential BVPs (1).Definition 4.
The system of the coupled sequential fractional differential BVPs (1) is stable with in U–H sense if a real number c=maxc1,c2>0 exists so that for any ε=maxε1,ε2>0 and for any u¯,v¯∈H×H satisfying(35)DCHpu¯tz1t,u¯t,v¯t−w1t,u¯t,v¯t<ε1,t∈1,e,DCHqu¯tz2t,u¯t,v¯t−w2t,u¯t,v¯t<ε2.
There exists a unique solutionu,v∈H×H of (3) with(36)u,v−u¯,v¯<cε.
It is clear thatu¯,v¯∈H×H satisfies inequality (35) if there exists a function f1,f2∈H×H (which depends on u¯,v¯), such that(i)
f1t<ε1 and f2t<ε2, t∈1,e(ii)
fort∈1,e
And(37)DCHpu¯tz1t,u¯t,v¯t=w1t,u¯t,v¯t+f1t,DCHqu¯tz2t,u¯t,v¯t=w2t,u¯t,v¯t+f2t.Theorem 7.
Suppose thatΞ2 is fulfilled. Moreover,(38)λz1Λ1υ1+υ2<1,λz2Λ2τ1+τ2<1,Δ=1−λz1Λ1υ1+υ21−λz2Λ2τ1+τ2−λz1Λ1υ1+υ2λz2Λ2τ1+τ2>0.Then, the system of coupled sequential fractional differential BVPs (1) is U–H stable.Proof.
Assume that forε1,ε2>0 a couple u¯,v¯∈H×H satisfies the inequalities (35). Introduce the following operators.
Then,(39)u¯t=z1t,u¯t,v¯t×−1Γp∫1tlogtrp−1w1r,u¯r,v¯rdrr+logt1−λ1Γp−1∫1elogerp−2w1r,u¯r,v¯rdrr−λ1∫1η1logη1rp−2w1r,u¯r,v¯rdrr,+z1t,ut,vt×−1Γp∫1tlogtrp−1f1rdrr+logt1−λ1Γp−1∫1elogerp−2f1rdrr−λ1∫1η1logη1rp−2f1rdrr,(40)v¯t=z2t,u¯t,v¯t×−1Γq∫1tlogtrq−1w2r,u¯r,v¯rdrr+logt1−λ2Γq−1∫1elogerq−2w2r,u¯r,v¯rdrr−λ2∫1η2logη2rq−2w2r,u¯r,v¯rdrr,+z2t,ut,vt×−1Γq∫1tlogtrq−1f2rdrr+logt1−λ2Γq−1∫1elogerq−2f2rdrr−λ2∫1η2logη2rq−2f2rdrr.
From (39) and (40), we obtain(41)u¯t−z1t,u¯t,v¯t×−1Γp∫1tlogtrp−1w1r,u¯r,v¯rdrr+logt1−λ1Γp−1∫1elogerp−2w1r,u¯r,v¯rdrr−λ1∫1η1logη1rp−2w1r,u¯r,v¯rdrr≤z1t,u¯t,v¯t×−1Γp∫1tlogtrp−1f1rdrr+logt1−λ1Γp−1∫1elogerp−2f1rdrr−λ1∫1η1logη1rp−2f1rdrr.(42)v¯t−z2t,u¯t,v¯t×−1Γq∫1tlogtrq−1w2r,u¯r,v¯rdrr+logt1−λ2Γq−1∫1elogerq−2w2r,u¯r,v¯rdrr−λ2∫1η2logη2rq−2w2r,u¯r,v¯rdrr≤z2t,u¯t,v¯t×−1Γq∫1tlogtrq−1f2rdrr+logt1−λ2Γq−1∫1elogerq−2f2rdrr−λ2∫1η2logη2rq−2f2rdrr.
From (41) and (42), we obtain(43)u¯t−z1t,u¯t,v¯t×−1Γp∫1tlogtrp−1w1r,u¯r,v¯rdrr+logt1−λ1Γp−1∫1elogerp−2w1r,u¯r,v¯rdrr−λ1∫1η1logη1rp−2w1r,u¯r,v¯rdrr≤λz1Λ1f1≤λz1Λ1ε1.(44)v¯t−z2t,u¯t,v¯t×−1Γq∫1tlogtrq−1w2r,u¯r,v¯rdrr+logt1−λ2Γq−1∫1elogerq−2w2r,u¯r,v¯rdrr−λ2∫1η2logη2rq−2w2r,u¯r,v¯rdrr≤λz2Λ2f2≤λz2Λ2ε2.
Letu,v∈H×H be a solution of (3). Thanks to lemma 2, it is equivalent to the following integral equations:(45)ut=z1t,ut,vt×−1Γp∫1tlogtrp−1w1r,ur,vrdrr+logt1−λ1Γp−1∫1elogerp−2w1r,ur,vrdrr−λ1∫1η1logη1rp−2w1r,ur,vrdrr,vt=z2t,ut,vt×−1Γq∫1tlogtrq−1w2r,ur,vrdrr+logt1−λ2Γq−1∫1elogerq−2w2r,ur,vrdrr−λ2∫1η2logη2rq−2w2r,ur,vrdrr.
By the same arguments in theorem2, we get(46)ut−u¯t=ƛ1u,vt−ƛ1u¯,v¯t−z1t,u¯t,v¯t×−1Γp∫1tlogtrp−1f1rdrr+logt1−λ1Γp−1∫1elogerp−2f1rdrr−λ1∫1η1logη1rp−2f1rdrr≤ƛ1u,vt−ƛ1u¯,v¯t+z1t,u¯t,v¯t×−1Γp∫1tlogtrp−1f1rdrr+logt1−λ1Γp−1∫1elogerp−2f1rdrr−λ1∫1η1logη1rp−2f1rdrr≤λz1Λ1υ1+υ2u−u¯+v−v¯+λz1Λ1ε1.(47)vt−v¯t=ƛ2u,vt−ƛ2u¯,v¯t−z2t,u¯t,v¯t×−1Γq∫1tlogtrq−1f2rdrr+logt1−λ2Γq−1∫1elogerq−2f2rdrr−λ2∫1η2logη2rq−2f2rdrr≤ƛ2u,vt−ƛ2u¯,v¯t+z2t,u¯t,v¯t×−1Γq∫1tlogtrq−1f2rdrr+logt1−λ2Γq−1∫1elogerq−2f2rdrr−λ2∫1η2logη2rq−2f2rdrr≤λz2Λ2τ1+τ2u−u¯+v−v¯+λz2Λ2ε2.
It follows that(48)u−u¯−λz1Λ1υ1+υ2u−u¯+v−v¯≤λz1Λ1ε1,v−v¯−λz2Λ2τ1+τ2u−u¯+v−v¯≤λz2Λ2ε2.
Representing these inequalities as matrices, we get(49)1−λz1Λ1υ1+υ2−λz1Λ1υ1+υ21−λz2Λ2τ1+τ2−λz2Λ2τ1+τ2u−u¯v−v¯≤λz1Λ1ε1λz2Λ2ε2.
Solving the above inequality, we get(50)u−u¯≤1−λz1Λ1υ1+υ2Δλz1Λ1ε1+λz1Λ1υ1+υ2Δλz2Λ2ε2,v−v¯≤λz2Λ2τ1+τ2Δλz1Λ1ε1+1−λz2Λ2τ1+τ2Δλz2Λ2ε2,where Δ=1−λz1Λ1υ1+υ21−λz2Λ2τ1+τ2−λz1Λ1υ1+υ2λz2Λ2τ1+τ2≠0.
Thus,(51)u−u¯+v−v¯≤1−λz1Λ1υ1+υ2Δ+λz2Λ2τ1+τ2Δλz1Λ1ε1+1−λz2Λ2τ1+τ2Δ+λz1Λ1υ1+υ2Δλz2Λ2ε2.
Forε=maxε1,ε2 and(52)c=1−λz1Λ1υ1+υ2+λz2Λ2τ1+τ2λz1Λ1+1−λz2Λ2τ1+τ2+λz1Λ1υ1+υ2λz2Λ2Δ,we get(53)u,u¯−v,v¯≤u−u¯+v−v¯≤cε.
Therefore, by means of Definition4, the solution to problem (3) is U–H stable.


## 5. Example

In this part, we present an applied example to support the theoretical results we reached in the previous part; consider the following system:(54)CHD43ut5/6sinut+7/6=7e5t+19t3+24v1+v+145tan−1u,t∈1,e,CHD54vt1/6cosvt=2e−3tt+120tan−1u+tan−1v,ut1/2sinut+7/5t=1=0,CHDut1/2sinut+7/5t=e=34CHDut1/2sinut+7/5t=2,vt1/3cosvtt=1=0,CHDvt1/3cosvtt=e=34CHDvt1/3cosvtt=52.Here,3/4=λ1=λ2, p=4/3,q=5/4,η1=2,η2=5/2,z1t,ut,vt=5/6sinut+7/6,z2t,ut,vt=1/6cosvt.(55)w1t,ut,vt=7e5t+19t3+24v1+v+145tan−1u,w2t,ut,vt=2e−3tt+117tan−1u+tan−1v.Observe that(56)w1t,u1,v1−w1t,u2,v2≤145u2−u1+145v2−v1,w2t,u1,v1−w2t,u2,v2≤117u2−u1+117v2−v1,λz1Λ1υ1+υ2+λz2Λ2τ1+τ2≤2×8.294×145+16×8.534×117=0.452288<1.Thus, the boundary value problem (54) satisfies all the conditions of Theorem 5; accordingly, we conclude that the BVP has a unique solution on 1,e.


## 6. Conclusion

In previous works, researchers investigated the existence and uniqueness of linear fractional differential equations involving Caputo–Hadamard. The legacy of this work lies in verifying the existence and uniqueness of solutions to a coupled system of Caputo–Hadamard hybrid fractional differential equations with Hybrid boundary conditions. Our major findings are demonstrated using the Banach fixed point theorem and the alternative of Leray–Schauder. The stability of the solutions involved in the Hyers–Ulam type was investigated. We provide an example to demonstrate the study results. Caputo–Hadamard calculus has its own prominence. For example, some researchers showed that by considering different different fractional derivative's order, a particular natural phenomenon can be remodeled with more accuracy when fractional derivative is replaced by the ordinary one.In future studies, researchers can verify the existence, uniqueness, and stability of the solutions for the system of equations given by (3) using the ψ-Hilfer fractional derivative or any other derivatives such as the fractional Katugambula derivative. In addition, this system can be used in practical applications of the subject by taking our results as proven facts [39, 40].



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*Source: 1500577-2022-07-11.xml*