# Darbo Fixed Point Criterion on Solutions of a Hadamard Nonlinear Variable Order Problem and Ulam-Hyers-Rassias Stability

Authors: Shahram Rezapour; Zoubida Bouazza; Mohammed Said Souid; Sina Etemad; Mohammed K. A. Kaabar
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The existence aspects along with the stability of solutions to a Hadamard variable order fractional boundary value problem are investigated in this research study. Our results are obtained via generalized intervals and piecewise constant functions and the relevant Green function, by converting the existing Hadamard variable order fractional boundary value problem to an equivalent standard Hadamard fractional boundary problem of the fractional constant order. Further, Darbo’s fixed point criterion along with Kuratowski’s measure of noncompactness is used in this direction. As well as, the Ulam-Hyers-Rassias stability of the proposed Hadamard variable order fractional boundary value problem is established. A numerical example is presented to express our results’ validity.


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## Body

## 1. Introduction

Fractional calculus is fundamentally established by having arbitrary numbers in the order of derivation operators instead of natural numbers. This idea is considered preliminary and simple. However, it involves remarkable effects and outcomes which describe some physical processes, dynamics, mathematical modelings, control theory, bioengineering, biomedical applications, etc. [1, 2]. The main effectiveness of this field can be found in recent studies. For example, Thabet et al. in [3] simulated a fractional model of pantograph in the Caputo conformable settings. In [4], Khan et al. designed a model of p-Laplacian FBVP in the form of a singular problem, and Matar et al. derived similar results for a new p-Laplacian model via generalized fractional derivative [5]. The fractional Langevin impulsive equations are studied by Rizwan et al. regarding existence property of solutions in [6], and Zada et al. [7] analyzed the Ulam-Hyers stability for an impulsive integro-differential equations. Etemad et al. used a new property entitled approximate endpoint for studying a novel fractional problem via the Caputo-Hadamard operators [8].Thabet et al. also modeled COVID-19 transmission by Caputo-Fabrizio operators and analyzed its dynamical behavior [9]. In [10], Shah et al. compared the results of two classical and fractional models of COVID-19 and showed the accuracy of fractional operators in simulation of processes. Pratap et al. [11] studied finite-time Mittag-Leffler stability criteria for fractional quaternion-valued memristive neural networks. Along with these, Boulares et al. [12] conducted a theoretical research on the generalized weakly singular integral inequalities and their applications to generalized FBVPs. Naifar et al. [13] studied a global Mittag-Leffler stabilization by output feedback in relation to a class of nonlinear systems of FBVPs.It is notable that in recent advanced mathematical models, constant fractional orders have not needed effectiveness for describing the specifications of some processes and phenomena, and consequently, some researchers had to model their boundary problems via the fractional operators equipped with orders as a real-valued functions. These operators are known as variable order ones [14, 15]. The investigation of the variable order fractional boundary problems (VOFBVPs) in the field of existence theory is considered as a new and important branch of fractional calculus, which are published limited research works in this regard. In 2018, Yang et al. presented a numerical scheme for a VOFBVP and analyzed their system from numerical point of view [16]. Zhang studied the solutions of a singular two-point VOFBVP for the first time in [17]. In recent years, Zhang et al. [18, 19] derived approximate solutions of two different VOFIVP on the half-axis in 2018 and 2019. In addition, a multiterm FBVP involving the nonlinear fractional differential equation (NFDE) of the variable order type was investigated in detail by Bouazza et al. in [20]. In this paper, motivated by other related works in this regard, we investigate the solutions’ existence of the Hadamard nonlinear VOFBVP as follows:
(1)HD1+ϑtrt+m1t,rt=0,t∈U≔1,T,via boundary conditions r1=rT=0, where 1<T<+∞, 1<ϑt≤2, and m1:U×S⟶S.S is a continuous function (S is a real (or complex) space) and HD1+ϑt specifies the Hadamard derivative of variable order ψt. For the first time, as the novelty of this research, we here consider a FBVP in the variable order Hadamard settings and establish the existence specifications of solutions to mentioned system on the generalized subintervals by combining the existing notions in relation to the Kuratowski’s measure of noncompactness (KMNCS) in the context of Darbo fixed point criterion. The piecewise constant functions will play a vital role in our study for converting the Hadamard VOFBVP (1) to the standard Hadamard FBVP. Lastly, another criterion of the behavior of solutions like the Ulam-Hyers-Rassias stability (UHRS) is analyzed, and a numerical illustrative example will complete the consistency of our findings.


## 2. Essential Preliminaries

Basic definitions are discussed in this section to be used later. Throughout the paper, the setS stands for the real numbers.The symbolCU,S denotes a set that contains all continuous functions f:U⟶S. It is a Banach space by defining
(2)f=supft:t∈U,U≔1,T.Definition 1 (see [21, 22]).
For1≤a1<a2<+∞, we consider the mappings h1t:a1,a2⟶0,+∞ and qt:a1,a2⟶n−1,n. The Hadamard ϑtth variable order integral of h1 is
(3)HIa1+ϑth1t=1Γϑt∫a1tlogtsϑt−1h1ssds,t>a1,and the Hadamard qtth variable order derivative of h1 is
(4)HDa1+qth1t=1Γn−qttddtn∫a1tlogtsn−qt−1h1ssds,t>a1.Obviously, in case ofϑt and qt are constant, then both above Hadamard variable order operators are in coincidence with the usual Hadamard constant order operators (refer to [1, 21, 22]).Lemma 2 (see [1]).
Assume thata1>1, γ1,γ2>0, h1∈La1,a2, and HDa1+γ1h1∈La1,a2. Then, the homogeneous differential equation
(5)HDa1+γ1h1=0admits the unique solution
(6)h1t=ω1logta1γ1−1+ω2logta1γ1−2+⋯+ωnlogta1γ1−n,HIa1+γ1HDa1+γ1h1t=h1t+ω1logta1γ1−1+ω2logta1γ1−2+⋯+ωnlogta1γ1−n,with n=γ1+1, ωȷ∈ℝ, and j=1,2,⋯,n.Moreover, for constantsαj>0,j=1,2,
(7)HDa1+α1HIa1+α1h1t=h1t,HIa1+α1HIa1+α2h1t=HIa1+α2HIa1+α1h1t=HIa1+α1+α1h1t.Remark 3.
The semigroup property is not fulfilled for the functionsϑt and qt, i.e.,
(8)HIa1+ϑtHIa1+qth1t≠HIa1+ϑt+qth1t.Example 1.
Let(9)ϑt=1,t∈1,2,2,t∈2,4,qt=3,t∈1,2,4,t∈2,4,h1t=2t2,t∈1,4.
We obtain(10)HI1+ϑtHI1+qth1t=1Γϑt∫1t1slogtsΨt−11Γqs∫1slogsτqs−1h1ττdτds=1ΓΨt∫121slogtsΨt−11Γqs∫1slogsτqs−1h1ττdτds+1Γϑt∫2t1slogtsϑt−11Γqs∫1slogsτqs−1h1ττdτds=1Γ1∫121slogts0∫1s1Γ3logsτ22τdτds+1Γ2∫2t1slogts1Γ3∫12logsτ22τdτ+1Γ4∫2slogsτ32τdτds,=∫12s4−12slogs2−12slogs−14sds+∫2t1slogts−23logs23+logs22+logs2−12logs2−12logs+18s2+14ds,(11)HI1+ϑt+qth1t=1Γϑt+qt∫1tlogtsϑt+qt−1h1ssds.
So,(12)HI1+ϑtHI1+qth1tt=3=−130log325+124log324+112log323+18log322−14log32−16log22log322−16log2log323−log236−log224−log24−14log2log322+1732≃0.0522.
On the other side,(13)HI1+ϑt+qth1tt=3=∫121Γ4log3s32sds+∫231Γ6log3s52sds=−130log325−112log324+13log323+34log322+34log32−16log33−14log32−14log3+1732≃0.1809.
Therefore, we obtain(14)HI1+ϑtHI1+qth1tt=3≠HI1+ϑt+qth1tt=3.Lemma 4 (see [23, 24]).
Ifϑ:U⟶1,2 has the continuity property, then for
(15)h1∈CβU,S=h1t∈CU,S,logtβh1t∈CU,S,0≤β≤1,the integral HI1+ϑth1t admits a finite value ∀t∈U.Lemma 5 (see [23, 24]).
Assume thatϑ:U⟶1,2 has the continuity property. Then,
(16)HI1+ϑth1t∈CU,Sforh1∈CU,S.Definition 6 (see [25–27]).
I⊆ℝ is termed as a generalized interval if I is either an interval, or a1, or ∅. A finite set F is defined to be a partition of I if every x∈I belongs to exactly one and one generalized interval I in F. Finally, w:I⟶S is piecewise constant w.r.t F as a partition of I; if ∀I∈F, w is constant on I.


### 2.1. Some Properties regarding KMNCS

Here, we regardS as a Banach space.Definition 7 (see [28]).
Suppose thatωS is a bounded set in S. The KMNCS is the function Φ:ωS⟶0,∞ as
(17)ΦP=infδ>0:P⊆∪=1nPȷ,DiamPȷ≤δ,P∈ωS,in which
(18)DiamPȷ=supx−r:x,r∈Pȷ.The symbolDiam denotes the diameter of the given set. Some valid properties KMNCS are as follows.Proposition 8 (see [28, 29]).
LetP,P1,P2 be bounded in S. Then,
(a)
P is relatively compact if ΦP=0(b)
Φ∅=0(c)
ΦP=ΦP¯(d)
P1⊂P2⇒ΦP1≤ΦP2(e)
ΦP1+P2≤ΦP1+ΦP2(f)
ΦλP=λΦP,λ∈ℝ(g)
ΦP1∪P2=maxΦP1,ΦP2(h)
ΦP1∩P2=minΦP1,ΦP2(i)
ΦP+x0=ΦP for any x0∈ℝLemma 9 (see [30]).
If the bounded setW⊂CU,S is equicontinuous, then
(i)
ΦW has continuity, and(19)Φ^W=supt∈UΦWt(ii)
Φ∫0Trθdθ:r∈W≤∫0TΦWθdθ,where
(20)Wθ=rθ:r∈W,θ∈UIn the next theorem, we point out the Darbo’s fixed point criterion.Theorem 10 (see [28]).
Consider the closed, convex, and bounded setΛ≠∅ in S and the continuous map Ϝ:Λ⟶Λ satisfying (k-set contractive property for Ϝ).
(21)ΦϜV≤kΦV,∀∅≠V⊂Λ,k∈0,1.
Then,Ϝ admits at least a fixed point belonging to Λ.


## 2.1. Some Properties regarding KMNCS

Here, we regardS as a Banach space.Definition 7 (see [28]).
Suppose thatωS is a bounded set in S. The KMNCS is the function Φ:ωS⟶0,∞ as
(17)ΦP=infδ>0:P⊆∪=1nPȷ,DiamPȷ≤δ,P∈ωS,in which
(18)DiamPȷ=supx−r:x,r∈Pȷ.The symbolDiam denotes the diameter of the given set. Some valid properties KMNCS are as follows.Proposition 8 (see [28, 29]).
LetP,P1,P2 be bounded in S. Then,
(a)
P is relatively compact if ΦP=0(b)
Φ∅=0(c)
ΦP=ΦP¯(d)
P1⊂P2⇒ΦP1≤ΦP2(e)
ΦP1+P2≤ΦP1+ΦP2(f)
ΦλP=λΦP,λ∈ℝ(g)
ΦP1∪P2=maxΦP1,ΦP2(h)
ΦP1∩P2=minΦP1,ΦP2(i)
ΦP+x0=ΦP for any x0∈ℝLemma 9 (see [30]).
If the bounded setW⊂CU,S is equicontinuous, then
(i)
ΦW has continuity, and(19)Φ^W=supt∈UΦWt(ii)
Φ∫0Trθdθ:r∈W≤∫0TΦWθdθ,where
(20)Wθ=rθ:r∈W,θ∈UIn the next theorem, we point out the Darbo’s fixed point criterion.Theorem 10 (see [28]).
Consider the closed, convex, and bounded setΛ≠∅ in S and the continuous map Ϝ:Λ⟶Λ satisfying (k-set contractive property for Ϝ).
(21)ΦϜV≤kΦV,∀∅≠V⊂Λ,k∈0,1.
Then,Ϝ admits at least a fixed point belonging to Λ.


## 3. Existence Criterion of Solutions

To achieve the main purpose of this section, some assumptions are proposed as:(H1) ConsiderF=U1≔1,T1,U2≔T1,T2,U3≔T2,T3,⋯,Un≔Tn−1,T as a partition for the interval U and ϑt:U⟶1,2 as a piecewise constant function w.r.t F, i.e.,
(22)ϑt=∑i=1nϑȷJȷt=ϑ1,ift∈U1,1<ϑ1≤2,ϑ2,ift∈U2,1<ϑ2≤2,⋅⋅⋅ϑn,ift∈Un,1<ϑn≤2,in which Jȷ interprets the indicator of Jȷ interprets the indicator of Uȷ≔Tȷ−1,Tȷ,∈ℕ1n, so that T0=1 and Tn=T, and
(23)Jȷt=1,fort∈Uȷ,0,forelsewhere.(H2) Letlogtβm1:U×S⟶S be continuous, 0≤β≤1, and ∃K>0, such that logtβm1t,r−m1t,r¯≤Kr−r¯, for any r,r¯∈S and t∈U.Remark 11 (see [31]).
Note that the inequality(24)Φlogtβm1t,B1≤KΦB1is equivalent to (H2) for each B1⊂S and t∈U, where B1 is bounded.Further, for a supposed setW of all mappings w:U⟶S, define
(25)Wt=wt,w∈W,t∈U,Wt=wt:w∈W,t∈U.Let us now establish the solutions’ existence for the Hadamard VOFBVP (1) via KMNCS and Darbo’s criterion (Theorem 10).Here,∀ȷ∈1,2,⋯,n, the symbol Eȷ=CUȷ,r, indicated as Banach spaces of continuous mappings r:Uȷ⟶S is furnished with the norm
(26)rEȷ=supt∈Uȷrt,where ȷ∈1,2,⋯,n.First, we analyze the Hadamard VOFBVP defined in (1). In the light of (4), the Hadamard VOFBVP (1) can be rewritten by
(27)1Γ2−ϑttddt2∫1tlogts1−ϑtrssds+m1t,rt=0,t∈U.From (H1), equation (27) on the interval Uȷ,∈ℕ1n, can be expressed as
(28)tddt21Γ2−ϑ1∫1T1logts1−ϑ1rssds+⋯+1Γ2−ϑȷ∫Tȷ−1tlogts1−ϑȷrssds+m1t,rt=0,t∈Uȷ.Definition 12.
The Hadamard VOFBVP (1) admits a solution like functions rȷ,ȷ=1,2,⋯,n, if rȷ∈C1,Tȷ,S satisfies equation (28) and rȷ1=0=rȷTȷ.From the above, the Hadamard VOFBVP (1) written in (27) can be given as (28) on Uȷ,∈ℕ1n. For 1≤t≤Tȷ−1, put rt≡0. Then, (28) is formulated by
(29)DTȷ−1+ϑȷrt+m1t,rt=0,t∈Uȷ.In this case, we follow our study by considering the standard Hadamard constant-order FBVP (COFBVP) as follows:(30)DHTȷ−1+ϑȷrt+m1t,rt,t∈Uȷ,rTȷ−1=0,rTȷ=0.The fundamental part of our analysis regarding solutions of the Hadamard COFBVP (30) is discussed below.Lemma 13.
A functionr∈Eȷ is a solution of the Hadamard COFBVP (30) if r fulfills the integral equation
(31)rt=∫Tȷ−1Tȷ1sGȷt,sm1s,rsds,t∈Uȷ,where Gȷt,s stands for the Green function formulated by
(32)Gȷt,s=1ΓϑȷlogTȷTȷ−11−ϑȷlogtTȷ−1logTȷsϑȷ−1−logtsϑȷ−1,T−1≤s≤t≤Tȷ,logTȷTȷ−11−ϑȷlogtTȷ−1logTȷsϑȷ−1,T−1≤t≤s≤Tȷ,where ȷ∈ℕ1n.Proof.
Suppose thatr∈Eȷ satisfies the Hadamard COFBVP (30). Let us employ the operator IHTȷ−1+ψȷ on both sides (30) and using Lemma 2, we get
(33)rt=ω1logtτj−1ϑj−1+ω2logtτj−1ϑj−2_HIτj−1+ϑjm1t,rt,t∈uj,j∈ℕ1n.
From definition ofm1 along with rTȷ−1=0, we get ω2=0.
Suppose thatr satisfies rTȷ=0. Hence,
(34)ω1=logTȷTȷ−11−ϑȷIȷHTȷ−1+ϑȷm1Tȷ,rTȷ.
Thus,(35)rt=logTȷTȷ−11−ϑȷlogtTȷ−1ϑȷ−1HITȷ−1+ϑȷm1Tȷ,rTȷ−HITȷ−1+ϑȷm1t,rt,t∈Uȷ.
Then, the solution of the Hadamard COFBVP (30) is given by
(36)rt=logTȷTȷ−11−ϑȷlogtTȷ−1ϑȷ−11Γϑȷ∫Tȷ−1TȷlogTȷsϑȷ−1m1s,rssds−1Γϑȷ∫Tȷ−1tlogtsϑȷ−1m1s,rssds=1Γϑȷ∫Tȷ−1tlogTȷTȷ−11−ϑȷlogtTȷ−1ϑi−1logTȷsϑȷ−1−logtsϑȷ−1m1s,rssds+∫tTȷlogTȷTȷ−11−ϑȷlogtTȷ−1ϑȷ−1logTȷsϑȷ−1m1s,rssds,and the continuity property of the Green function gives
(37)rt=∫Tȷ−1Tȷ1sGȷt,sm1s,rsds,t∈Uȷ.
Conversely, letr∈Eȷ be the integral equation’s (31) solution. Because of the continuity of logtβm1 and by Lemma 2, it is simply verified that r satisfies the Hadamard COFBVP (30) solution.Proposition 14.
Assume thatlogtβm1,0≤β≤1 belongs to CU×S,S and ϑt:U⟶1,2 satisfies (H1). Then, Gȷt,s given by (32) satisfy the following: (ȷ∈ℕ1n)
(1)
0≤Gȷt,s,∀Tȷ−1≤t,s≤Tȷ(2)
maxt∈UȷGȷt,s=Gȷs,s,s∈Uȷ(3)
Gȷs,s has a maximum value uniquely given by(38)maxs∈UȷGȷs,s=1ΓϑȷlogTȷ−logT−14ϑȷ−1,where ȷ=1,2,⋯,nProof.
Letφt,s=logTȷ/Tȷ−11−ϑȷlogt/Tȷ−1logTȷ/sϑȷ−1−logt/sϑȷ−1. We see that
(39)φtt,s=ϑȷ−1tlogTȷTȷ−11−ϑȷlogTȷsϑȷ−1logtTȷ−1ϑȷ−2−ϑȷ−1tlogtsϑȷ−2≤ϑȷ−1tlogTȷs1−ϑȷlogTȷsϑȷ−1logtsϑȷ−2−ϑȷ−1tlogtsϑȷ−2=0,which means that φt,s is nonincreasing w.r.t t, so φt,s≥φTȷ,s=0, for Tȷ−1≤s≤t≤Tȷ. Thus, 0≤Gȷt,s for any Tȷ−1≤t,s≤Tȷ,ȷ=1,⋯,n..
Sinceφt,s is nonincreasing w.r.t t, then φt,s≤φs,s for Tȷ−1≤s≤t≤Tȷ.
On the other hand, forTȷ−1≤t≤s≤Tȷ, we get
(40)logTȷTȷ−11−ϑȷlogtTȷ−1logTȷsϑȷ−1≤logTȷTȷ−11−ϑȷlogsTȷ−1logTȷsϑȷ−1.
These confirm thatmaxt∈Tȷ−1,TȷGȷt,s=Gȷs,s,s∈Tȷ−1,Tȷ,ȷ=1,⋯,n..
Further, we verify (3) of Proposition14. Clearly, the maximum points of Gȷs,s are not Tȷ−1 and Tȷ, ȷ=∈ℕ1n. For s∈Tȷ−1,Tȷ,ȷ=1,⋯,n., we have
(41)dGȷs,sds=ϑȷ−1slogTȷTȷ−11−ϑȷlogsTȷ−1ϑȷ−2logTȷsϑȷ−2logTȷs−logsTȷ−1,=ϑȷ−1slogTȷTȷ−11−ϑȷlogsTȷ−1ϑȷ−2logTȷsϑȷ−2logTȷTȷ−1−logs2,
which indicates that the maximum points ofGȷs,s is s=TȷTȷ−1,ȷ=1,⋯,n.
Hence, forȷ=1,⋯,n,(42)maxs∈Tȷ−1,TȷGȷs,s=GȷTȷTȷ−1,TȷTȷ−1=1Γϑȷ14logTȷTȷ−1ϑȷ−1=1ΓϑȷlogTȷ−logTȷ−14ϑȷ−1.
This shows the completion of the proof.The existence criterion of solutions for the Hadamard VOFBVP (1) in this work depends on the hypotheses of Theorem 10 which we investigate them in this position.Theorem 15.
Suppose that both (H1) and (H2) hold, and(43)KlogTȷ1−β−logTȷ−11−βlogTȷ−logTȷ−1ϑȷ−14ϑȷ−11−βΓϑȷ<1.
Then, there is a solution to the Hadamard VOFBVP (1) on U.Proof.
We construct the operator(44)Z:Eȷ⟶Eȷby
(45)Zrt=∫Tȷ−1Tȷ1sGȷt,sms,rsds,t∈Uȷ.
Some properties of fractional integrals along with the continuity for the functionlogβm1 imply that the operator Z:Eȷ⟶Eȷ defined in (45) is well-defined.
Let∃Rȷ>0 so that
(46)Rȷ≥m⋆logTȷ−logTȷ−1ϑȷ/4ϑȷ−1Γϑȷ1−KlogTȷ1−β−logTȷ−11−βlogTȷ−logTȷ−1ϑȷ−1/4ϑȷ−11−βΓϑȷ,with
(47)m⋆=supt∈Uȷm1t,0.
Let us consider the following set:(48)BRȷ=r∈Eȷ,rEȷ≤Rȷ.
ClearlyBRȷ≠∅ contains all three properties of the convexity, boundedness, and closedness.
We shall show thatZ satisfies Theorem 10 in four stages.
Step A. Claim:ZBRȷ⊆BRȷ. For r∈BRȷ, by Proposition 14 and (H2), we get
(49)Zrt=∫Tȷ−1Tȷ1sGȷt,xm1s,rsds≤∫Tȷ−1Tȷ1sGȷt,sm1s,rsds≤1ΓϑȷlogTȷ−logTȷ−14ϑȷ−1∫Tȷ−1Tȷ1sm1s,rsds≤1ΓϑȷlogTȷ−logTȷ−14ϑȷ−1∫Tȷ−1Tȷ1sm1s,rs−m1s,0ds+1ΓϑȷlogTȷ−logTȷ−14ϑȷ−1∫Tȷ−1Tȷ1sm1s,0ds≤1ΓϑȷlogTȷ−logTȷ−14ϑȷ−1∫T−1Tȷ1slogs−βKrsds+m⋆logTȷ−logTȷ−1ϑȷ4ϑȷ−1Γϑȷ≤KΓϑȷlogTȷ−logTȷ−14ϑȷ−1rGȷ∫Tȷ−1Tȷ1slogs−βds+m⋆logTȷ−logTȷ−1ϑȷ4ϑȷ−1Γϑȷ≤KΓϑȷlogTȷ−logTȷ−14ϑȷ−1RȷlogTȷ1−β−logTȷ−11−β1−β+m⋆logTȷ−logTȷ−1ϑȷ4ϑȷ−1Γϑȷ≤KlogTȷ1−β−logTȷ−11−βlogTȷ−logTȷ−1ϑȷ−14ϑȷ−11−βΓϑȷRȷ+m⋆logTȷ−logTȷ−1ϑȷ4ϑȷ−1Γϑȷ≤Rȷ,which means that ZBRȷ⊆BRȷ.
Step B. Claim:Z is continuous.
The sequencern is supposed to be convergent to r in Eȷ and t∈Uȷ. Then,
(50)Zrnt−Zrt≤∫Tȷ−1Tȷ1sGȷt,sm1s,rns−m1s,rsds≤1ΓϑȷlogTȷ−logTȷ−14ϑȷ−1∫Tȷ−1Tȷ1sm1s,rns−m1s,rsds≤1ΓϑȷlogTȷ−logTȷ−14ϑȷ−1∫Tȷ−1Tȷ1slogs−βKrns−rsds≤1ΓϑȷlogTȷ−logTȷ−14ϑȷ−1Krn−yGȷ∫Tȷ−1Tȷ1slogs−βds≤KlogTȷ1−β−logTȷ−11−βlogTȷ−logTȷ−1ϑȷ−14ϑȷ−11−βΓϑȷrn−rGȷ,
i.e., we get(51)Zrn−ZrGȷ⟶0asn⟶∞,and the correctness of the claim in this step is confirmed for Z.
Step C. Claim:Z is bounded and equicontinuous.
From A,ZBRȷ=Zr:r∈BRȷ⊂BRȷ; thus, for each r∈BRȷ, we get ZrEȷ≤Rȷ; in other ways, it means that ZBRȷ is bounded. It remains to check the equicontinuity of ZBRȷ.
Now,∀t1<t2∈Uȷ,t1<t2 and r∈BRȷ, we write
(52)Zrt2−Zrt1=∫Tȷ−1Tȷ1sGȷt2,sm1s,rsds−∫Tȷ−1Tȷ1sGȷt1,sm1s,rsds≤∫Tȷ−1Tȷ1sGȷt2,s−Gȷt1,sm1s,rsds≤∫Tȷ−1Tȷ1sGȷt2,s−Gȷt1,sm1s,rsds≤∫Tȷ−1Tȷ1sGȷt2,s−Git1,sm1s,rs−m1s,0+m1s,0ds≤∫Tȷ−1Tȷ1sGȷt2,s−Gȷt1,slogs−βKrs+m⋆ds≤∫Tȷ−1TȷGȷt2,s−Gȷt1,s1slogs−βKrEȷ+1sm⋆ds≤KlogTȷ−1−βTȷ−1rEȷ∫Tȷ−1TȷGȷt2,s−Gȷt1,sds+m⋆Tȷ−1∫Tȷ−1TȷGȷt2,s−Gȷt1,sds,using the continuity of G. Hence, Zrt2−Zrt1Eȷ⟶0 as t2−t1⟶0. It yields that ZBRȷ is equicontinuous.
Step D. Claim:Z is k-set contraction.
This time, letW∈BRȷ and t∈Uȷ. So,
(53)ΦZWt=ΦZrt,r∈W≤∫Tȷ−1Tȷ1sGȷt,sΦm1s,rsdsr∈W.
Remark11 indicates that
(54)ΦZWt≤∫Tȷ−1Tȷ1sGȷt,sKΦrs,r∈W≤1ΓϑȷlogTȷ−logTȷ−14ϑȷ−1KΦ^W∫Tȷ−1Tȷ1slogs−βds,r∈W≤KlogTȷ1−β−logTȷ−11−βlogTȷ−logTȷ−1ϑȷ−14ϑȷ−11−βΓϑȷΦ^W,
for anys∈Uȷ.
Therefore,(55)Φ^ZW≤KlogTȷ1−β−logTȷ−11−βlogTȷ−logTȷ−1ϑȷ−14ϑȷ−11−βΓϑȷΦ^W.
Consequently by (43), we deduce that Z admits a set contraction.
The conclusion of Theorem10 gives this result that the Hadamard COFBVP (30) involves at least a solution rȷ~ in BRȷ.
Assume that(56)rȷ=0,t∈1,Tȷ−1,r~ȷ,t∈Uȷ.
We know thatrȷ∈C1,Tȷ,S defined by (56) satisfies equation
(57)d2dt2∫1T1t−s1−ϑ1Γ2−ϑ1rȷsds+⋯+∫Tȷ−1tt−s1−ϑȷΓ2−ϑȷrȷsds+m1s,rȷs=0,for t∈Uȷ, which implies that rȷ is regarded as a solution for (28) along with rȷ1=0 and rȷTȷ=r~ȷTȷ=0.
Then,(58)rtr1t,t∈U1,r2t=0,t∈U1,x~2,t∈U2,...rȷt=0,t∈1,Tȷ−1,r~ȷ,t∈Uȷgives the solution for the Hadamard VOFBVP (1) and this completes the argument.


## 4. Ulam-Hyers-Rassias Stability

The stability issue has gained substantially important attention in several research fields through applications. There are many kinds of stability; one of them is the stability introduced by Ulam in 1940. Since then, the problem is known as Ulam-Hyers stability or simply Ulam stability. Later, other generalizations of this notion were introduced by other researchers. Its applications for many types of equations have been investigated by many mathematicians. Ben Makhlouf [32] derived sufficient conditions of different types of stability such as uniform stability, Mittag-Leffler stability, and asymptotic uniform stability for a nonlinear Caputo fraction BVP via a method with respect to Lyapunov-like functions. Ahmad et al. [33] investigated the notion of stability for a nonlinear coupled implicit switched singular fractional differential system with p-Laplacian operator. Now, we aim to accomplish an argument regarding the UHRS stability of the given Hadamard VOFBVP (1) in the framework of Theorem 17.Definition 16 (see [34]).
Letϱ∈CU,S. Then, the Hadamard VOFBVP (1) is Ulam-Hyers-Rassias stable (UHRS) w.r.t ϱ if ∃cm>0, so that ∀ϵ>0 and for every solution ς∈CU,S of
(59)HD1+ϑtςt−−m1t,ςt≤ϵϱt,t∈U,
∃ a solution r∈CU,S of (1) with
(60)ςt−rt≤cmϵϱt.Theorem 17.
Let both (H1) and (H2) along with (43) hold. Also,
(H3) Letϱ∈CUȷ,S is a increasing function and ∃λϱ>0 provided
(61)IHTȷ−1+ϑȷϱt≤λϱtϱt,foranyt∈Uȷ.
Then, the Hadamard VOFBVP (1) is UHRS stable w.r.t the function ϱ.Proof.
Assume thatϵ>0 is chosen arbitrarily and ς from ς∈CUȷ,ℝ satisfies (59). Now, ∀ȷ∈ℕ1n, the following are defined: ς1t≡ςt,t∈1,T1 and for ȷ=2,3,⋯,n:(62)ςȷt=0,t∈0,Tȷ−1,ςt,t∈Uȷ.
TakingIHTȷ−1+ϑȷ on both sides (59), we get
(63)ςt−−∫TI−1TI1sGIt,sm1s,ςsds≤ϵΓϑI∫TI−1t1slogtsϑI−1ϱsds≤ϵλϱtϱt.
According to the argument above, the Hadamard VOFBVP (1) involves a solution r∈CU,ℝ formulated as rt=rItfort∈UI,∈ℕ1n, in which
(64)rj=0,t∈0,TJ−1,r~j,t∈UJ,and r~I∈EI is a solution of the Hadamard COFBVP (30). In accordance with Lemma 13, we have
(65)r~It=−TI−TI−1−1t−TI−1ΓϑI∫TI−1TITI−sϑI−1m1s,r~Isds+1ΓϑI∫TI−1tt−sϑI−1m1s,r~Isds.
Suppose thatt∈UI,=1,2,⋯,n. Then, by equations (64) and (65), we get
(66)ςt−rt=ςt−rIt=ςIt−r~It=ςIt−∫TI−1TI1sGIt,sm1s,r~Isds≤ςIt−∫TI−1TI1sGIt,sm1s,ςIsds+∫TI−1TI1sGIt,sm1s,ςIs−m1s,r~Isds≤ςIt+∫T−1TI1sGIt,sm1s,ςIsds+∫TI−1TI1sGIt,sm1s,ςIs−m1s,r~~Isds≤λϱtϵϱt+1ΓϑIlogTI−logTI−14ϑI−1∫TI−1TIlogs−βKςIs−r~Issds≤λϱtϵϱt+KΓϑIlogTI−logTI−14ϑI−1ςI−r~IGI∫TI−1TI1slogs−βds≤λϱtϵϱt+KlogTI1−β−logTI−11−βlogTI−logTI−1ϑI−11−β4ϑI−1ΓϑIςI−r~IGI≤λϱtϵϱt+μς−r,where
(67)μ=maxI=1,2,⋯,nKlogTI1−β−logTI−11−βlogTI−logTI−1ϑI−11−β4ϑI−1ΓϑI.
Then,(68)ς−r1−μ≤λϱtϵϱt,and so by assuming cm1≔λϱt/1−μ,
(69)ςt−rt≤λϱtϱt1−μϵ≔cm1ϵϱt.
Therefore, the Hadamard VOFBVP (1) is UHRS stable w.r.t ϱ. This result completes the proof.


## 5. Numerical Illustrative Example

Example 2.
Consider the Hadamard VOFBVP (based on the VOFBVP (1)) as follows:
(70)HD1+ϑtrt+logtϑtπ+14logt−1/4rt=0,t∈U≔1,e,r1=0,re=0.
Hence,T=e and
(71)m1t,r=logtϑtπ+14logt−1/4rt,t,r∈1,e×0,+∞,(72)ϑt1.2,t∈U1≔1,2,1.6,t∈U2≔2,e.
Then, we get(73)logt1/4mt,r−mt,r¯=logt1/4logtϑtπ+14rt−logt1/4logtϑtπ−14r¯t≤14rt−r¯t.
(H2) holds withβ=1/4 and K=1/4.
From (72), the Hadamard VOFBVP (70) is classified into the following:
(74)HD1+1.2rt+logt1.2π+14logt−1/4rt=0,t∈U1,HD2+1.6rt+logt1.6π+14logt−1/4rt=0,t∈U2.
Fort∈U1, the Hadamard VOFBVP (70) is equivalent to the Hadamard COFBVP
(75)HD1+1.2rt+logt1.2π+14logt−1/4rt=0,t∈U1,r1=0,r2=0.
Let us now show that condition (43) is satisfied. Clearly, the following value is obtained
(76)KlogT11−β−logT01−βlogT1−logT0ϑ1−14ϑ1−11−βΓϑ1=1/4log23/4log20.240.23/4Γ1.2≃0.1941<1.
On the other side, letϱt=logt1/2. In this case,
(77)HI1+ϑ1ϱt=1Γ1.2∫12logts1.2−1logt1/2sds≤logt1/2Γ1.2∫12log2s0.2dss≤log21.2Γ2.2logt1/2≔λϱtϱt.
As a result, (H3) is fulfilled withϱt=logt and λϱt=log21.2/Γ2.2∈ℝ.
Theorem15 guarantees the existence of a solution for the Hadamard COFBVP (75) like r1∈E1, and from Theorem 17, the Hadamard constant-order system (75) is UHRS stable. For t∈U2, the Hadamard VOFBVP (70) can be written as the following COFBVP, i.e.,
(78)HD2+1.6rt+logt1.6π+14logt−1/4rt=0,t∈U2,r2=0,re=0.
We see that(79)KlogT21−β−logT11−βlogT2−logT1ϑ2−14ϑ2−11−βΓϑ2=1/41−log23/41−log20.640.63/4Γ1.6≃0.0191<1.
Accordingly, condition (43) is achieved on the subinterval U2. Further,
(80)HI2+ϑ1ϱt=1Γ1.6∫2elogts1.6−1logt1/2sds≤logt1/2Γ1.6∫2eloges0.6dss≤loge/21.6Γ2.6logt1/2≔λϱtϱt.
As a result, (H3) is also valid withϱt=logt and λϱt=loge/21.6/Γ2.6∈ℝ.
On account of Theorem15, the Hadamard COFBVP (78) possesses a solution x~2∈E2. Further, Theorem 17 yields that the mentioned Hadamard system (78) is UHRS stable. It is known that
(81)r2t=0,t∈U1,r~2t,t∈U2.
Consequently, the Hadamard VOFBVP (70) has a solution
(82)rt=r1t,t∈U1,r2t=0,t∈U1,r~2t,t∈U2.
From Theorem17, the Hadamard VOFBVP given by (70) is UHRS stable.


## 6. Conclusions

In this paper, the nonlinear Hadamard VOFBVP (1) was considered in which we established some theorems regarding existence and stability of solutions of it by following a new method based on the generalized subintervals and piecewise constant functions. By applying such notions, we converted the given Hadamard VOFBVP (1) to the standard Hadamard COFBVP (30). After investigating some specifications of the Green function, we focused on the solutions’ existence via a combined technique in terms of KMNCS in the context of Darbo’s fixed point criterion. The UHRS stability of the proposed Hadamard VOFBVP was also studied. At the end, a numerical example has been discussed to validate the applicability of our results. In future works, our results can be extended to other fractional mathematical models equipped with variable orders such as studies on simulations and dynamical behaviors of COVID-19.



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