# Existence and Stability of a Caputo Variable-Order Boundary Value Problem

Authors: Amar Benkerrouche; Mohammed Said Souid; Sumit Chandok; Ali Hakem
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this study, we investigate the existence of a solution to the boundary value problem (BVP) of variable-order Caputo-type fractional differential equation by converting it into an equivalent standard Caputo (BVP) of the fractional constant order with the help of the generalized intervals and the piecewise constant functions. All our results in this study are proved by using Darbo’s fixed-point theorem and the Ulam–Hyers (UH) stability definition. A numerical example is given at the end to support and validate the potentiality of our obtained results.


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## Body

## 1. Introduction

Fractional differential equations of a constant order or fractional calculus, in general, have been studied by many researchers for more than three centuries compared to integer differential equations. In recent years, the notion of a variable-order operator is a much more recent improvement. Different authors have presented different definitions of variable-order differential; we refer to [1–6].Several investigators have studied boundary value problems (BVPs) for different types of fractional differential equations (FDEs), for example, Adiguzel et al. [7] obtain a solution for a nonlinear (FDEs) of order α∈2,3, Benchora and Souid [8] obtain a solution for implicit fractional-order differential equations, and Zhang [9] discuss the existence of solutions for two point (BVPs) with singular (FDEs) of variable order.Bai and Kong [10] studied the following problem:(1)Dc0+ωyt=ft,yt,I0+ωyt,t∈b1,b2,ω∈0,1,0<b1<b2<∞yb1=yb1,where D0+ωc and I0+ω are the Caputo–Hadamard derivative and Hadamard integral operators of constant order ω, respectively, f is a given continuous function, and yb1∈ℝ.Some existence and Ulam stability properties for FDEs are studied by many authors (see [11, 12] and references cited therein).Motivated by the above studies, we deal with the existence of solutions and the stability of the obtained solution in the sense of Ulam–Hyers (UH) to the following BVP of Caputo variable-order type:(2)D0+ωtcyt=f1t,yt,I0+ωtyt,t∈J,y0=0,yT=0,where J=0,T,0<T<∞, ωt:J⟶1,2 is the variable order of the fractional derivatives, f1:J×ℝ×ℝ⟶ℝ is a continuous function, and the left Riemann–Liouville fractional integral (RLFI) of variable-order ωt for function yt is (see, for example, [13–15])(3)I0+ωtyt=∫0tt−sωt−1Γωtysds,t∈J,and the left Caputo fractional derivative (CFD) of variable-order ωt for function yt is (see, for example, [13–15])(4)D0+ωtcyt=∫0tt−s1−ωtΓ2−ωty2sds,t∈J.


## 2. Preliminaries

This section introduces some important fundamental definitions and results that will be needed in this paper.Denote byCJ,ℝ the Banach space of continuous functions y:J⟶ℝ, with the norm(5)y=supyt:t∈J.Remark 1.
In (2), the variable order ωt:J⟶1,2, but the RLFI could be defined for any ωt:J⟶0,∞.Remark 2.
In the case of a constant orderω in equations (2) and (3), the RLFI and CFD coincide with the standard Riemann–Liouville fractional integral and Caputo fractional derivative, respectively (see [13, 14, 16]).
Recall the following properties of fractional derivatives and integrals [16].Lemma 1.
Assume thatβ1>0, b1,b2>0, f2∈L1b1,b2, and Db1+β1cf2∈L1b1,b2. Then, the differential equation,(6)Db1+β1cf2=0,has a unique solution:(7)f2t=λ0+λ1t−b1+λ2t−b12+⋯+λn−1t−b1n−1,where n−1<β1≤n and λJ∈ℝ, j=0,1,…,n−1.Lemma 2.
Letβ1>0, b1,b2>0, f2∈L1b1,b2, and Db1+β1cf2∈L1b1,b2. Then,(8)Ib1+β1cDb1+β1f2t=f2t+λ0+λ1t−b1+λ2t−b12+⋯+λn−1t−b1n−1,where n−1<β1≤n and λj∈ℝ, j=0,1,…,n−1.Lemma 3.
Letβ1>0, b1,b2>0, and f2∈L1b1,b2. Then,(9)Db1+β1cIb1+β1f2t=f2t.Lemma 4.
Letβ1>0, b1,b2>0, and f2∈L1b1,b2. Then,(10)Ib1+β1Ib1+β2f2t=Ib1+β2Ib1+β1f2t=Ib1+β1+β2f2t.Remark 3.
It is to be hereby note that the semigroup property is not satisfied for general functionsωt and ψt (see [9, 17, 18]), that is,(11)Ib1+ωtIb1+ψtf2t≠Ib1+ωt+ψtf2t.Example 1.
Assume that(12)ωt=t,t∈0,4,ψt=2,t∈0,13,t∈1,4.f2t=2,t∈0,4,I0+ωtI0+ψtf2t=∫ott−sωt−1Γωt∫0ss−τvs−1Γvsf2τdτds,=∫ott−st−1Γt∫01s−τΓ22dτ+∫1ss−τ2Γ32dτds,=∫ott−st−1Γt2s−1+s−133ds,and(13)I0+ωt+ψtf2t|=∫0tt−sωt+ψt−1Γωt+ψtf2sds.
So, we obtain(14)I0+ωtI0+ψtf2t|t=3=∫033−s2Γ32s−1+s−133ds=2110,I0+ωt+ψtf2t|t=3=∫033−sωt+ψt−1Γωt+ψtf2sds,=∫013−s4Γ52ds+∫133−s5Γ62ds,=12∫01s4−12s3+54s2−108s+81ds,+160∫13−s5+15s4−90s3+270s2−405s+243ds,=665180.
Therefore, we obtain(15)I0+ωtI0+ψtf2t|t=3≠I0+ωt+ψtf2t|t=3.Definition 1 (see [19–21]).
LetA⊂ℝ, where A is named a generalized interval if it is either an interval, or b1 or ∅.
A finite setP is named a partition of A if each x in A lies in just one among the generalized intervals E in P.
A functiong:A⟶ℝ is defined to be piecewise constant with respect to partition P of A if g admits constant values on E, for any E∈P.
Zhang et al. [22] gave very interesting result.Lemma 5.
Ifu∈CJ,1,2, then both of the following holds:(a)
Forf2∈CJ,ℝ, I0+ωtf2t∈CJ,ℝ(b)
Forf2∈CκJ,ℝ=f2t∈CJ,ℝ,tκf2t∈CJ,ℝ,0≤κ≤1, the variable-order fractional integral I0+ωtf2t exists for any points on JDefinition 2 (see [23]).
LetΩ be a bounded subset of the Banach space X. The Kuratowski measure of noncompactness (KMNC) is a mapping ξ:Ω⟶0,∞ which is defined as follows:(16)ξD=infε>0:∃DJJ=1,2,…,n⊂X,D⊆∪J=1nDJ,diamDJ≤ε,where(17)diamDJ=supx−y:x,y∈DJ.
The KMNC satisfies the following properties.Proposition 1 (see [23, 24]).
LetX be a Banach space and D,D1, and D2 be bounded subsets of X. Then,(1)
ξD=0 if and only if D¯ is compact(2)
ξϕ=0(3)
ξD=ξD¯=ξconvD(4)
D1⊂D2 implies ξD1≤ξD2(5)
ξD1+D2≤ξD1+ξD2(6)
ξαD=αξD,α∈IR(7)
ξD1∪D2=maxξD1,ξD2(8)
ξD1∩D2=minξD1,ξD2(9)
ξD+x0=ξD for any x0∈XLemma 6 (see [25]).
LetB⊂CJ,X be a bounded and equicontinuous set; then,(i)
The functionξBt is continuous for t∈J, and(18)ξ^B=supt∈JξBt.(ii)
ξ∫0Txρdρ:x∈B≤∫0TξBρdρ, where(19)Bρ=xρ:x∈B,ρ∈J.Theorem 1 (DFPT, see [23]).
LetX be a Banach space and F be a nonempty, bounded, closed, and convex subset of X and Υ:F⟶F is a continuous operator satisfying(20)ξϒG≤kξG,k∈0,1,for anyG≠∅⊂F,i.e.,ϒisk−set contractions.Then,ϒ has at least one fixed point in F.Definition 3 (see [11]).
The BVP (2) is (UH) stable if ∃λf1>0,∀ε>0,∀z∈CJ,ℝ satisfies the following inequality:(21)|cD0+ωtzt−f1t,zt,I0+ωtzt|≤ε,t∈J,where ∃y∈CJ,ℝ solution of BVP (2) with(22)zt−yt≤λf1ϵ,t∈J.


## 3. Existence of Solution

In this section, we investigate the existence of solution for a BVP of a Caputo-type fractional differential equation using DFPT and KMNC.Let us introduce the following assumptions.(H1) Letn∈ℕ be an integer,(23)P=J1≔0,T1,J2≔T1,T2,J3≔T2,T3…Jn≔Tn−1,T,a partition of the interval J, and let ωt:J⟶1,2 be a piecewise constant function with respect to P, i.e.,(24)ωt=∑j=1nωjIjt=ω1,ift∈J1,ω2,ift∈J2,⋅⋅⋅ωn,ift∈Jn,where 1<ωj≤2 are constants, and Ij is the indicator of the interval Jj≔Tj−1,Tj,j=1,2,…,n (with T0=0andTn=T), such that(25)Ijt=1,fort∈Jj,0,for elsewhere.Remark 4.
According to remark of Benchohra (p.20 in [26]), it is not difficult to show that condition (H2) and the following inequality,(26)ξtκf1t,B1,B2≤KξB1+LξB2,are equivalent for any bounded sets B1,B2⊂X and for each t∈J.
Furthermore, for a given setB of functions v:J⟶X, we denote flushleft:(27)Bt=vt,v∈B,t∈J,and(28)BJ=vt:v∈B,t∈J.
The symbolEj=CJj,ℝ, which indicated the Banach space of continuous functions y:Jj⟶ℝ equipped with the norm(29)yEj=supt∈Jjyt,where j∈1,2,…,n,
Then, fort∈Jj,j=1,2,…,n, the left Caputo fractional derivative (CFD), defined by (4), could be presented as a sum of left Caputo fractional derivatives of constant orders ωl,l=1,2,…,j,(30)D0+ωtcyt=∫0tt−s1−ωtΓ2−ωty2sds,=∫0T1t−s1−ω1Γ2−ω1y2sds+∫T1T2t−s1−ω2Γ2−ω2y2sds+⋯+∫Tj−1tt−s1−ωjΓ2−ωjy2sds.
Thus, by (30), BVP (2) can be written, for any t∈Jj,j=1,2,…,n as the following form:(31)∫0T1t−s1−ω1Γ2−ω1y2sds+∫T1T2t−s1−ω2Γ2−ω2y2sds+⋯+∫Tj−1tt−s1−ωjΓ2−ωJy2sds=f1t,yt,I0+ωjyt.
Now, we will present the definition of the solution to BVP (2).Definition 4.
BVP (2) has a solution if there are functions yj,j=1,2,…,n, so that yj∈C0,Tj,ℝ fulfilling equation (31) and yj0=0=yjTj.
Let the functiony∈CJ,ℝ be a solution of integral (31), such that yt≡0 on t∈0,Tj−1. Then, (31) is reduced to(32)DTj−1+ωjcyt=f1t,yt,ITj−1+ωjyt,t∈Jj.
We consider the following auxiliary BVP:(33)DTj−1+ωjcyt=f1t,yt,ITj−1+ωjyt,t∈JjyTj−1=0,yTj=0.
The following lemma is necessary in our next analysis of BVP (33).Lemma 7.
Letf1∈CJj×ℝ×ℝ,ℝ. Suppose that there exists a number κ∈0,1 such that tκf1∈CJJ×ℝ×ℝ,ℝ, for any j∈1,2,…,n.Then, the solution of BVP (33) can be expressed by the integral equation:(34)yt=−Tj−Tj−1−1t−Tj−1ITj−1+ωjf1Tj,yTj,ITj−1+ωjyTj+ITj−1+ωjf1t,yt,ITj−1+ωjyt.Proof.
Lety∈Ej is a solution of BVP (33). Taking (RLFI) ITj−1+ωj to both sides of (33) and using Lemma 2, we find(35)yt=λ1+λ2t−Tj−1+1Γωj∫Tj−1tt−sωj−1f1s,ys,ITj−1+ωjysds,t∈Jj.
ByyTj−1=0, we get λ1=0.
ByyTj=0, we observe that(36)λ2=−Tj−Tj−1−1ITj−1+ωjf1Tj,yTj,ITj−1+ωjyTj.
Then,(37)yt=−Tj−Tj−1−1t−Tj−1ITj−1+ωjf1Tj,yTj,ITj−1+ωjyTj+ITj−1+ωjf1t,yt,ITj−1+ωjyt,t∈Jj.
Conversely, lety∈Ej be solution of integral equation (34). Employing the operator (CFD) cDTj−1+ωj to both sides of (34) and Lemma 3, we deduce that y is the solution of BVP (33).
Based on concept of MNCK and DFPT, we have the following theorem for the existence of a solution for BVP (33).Theorem 2.
In addition to the conditions of Lemma7, suppose that there exist constants K,L>0, such that, for any xl,zl∈ℝ, l=1,2, t∈Jj, and(38)tκf1t,x1,z1−f1t,x2,z2≤Kx1−x2+Lz1−z2,the following inequality holds:(39)2Tj−Tj−1ωj−1Tj1−κ−Tj−11−κ1−κΓωjK+LTj−Tj−1ωjΓωj+1<1.Then, BVP (33) has at least one solution Ej.Proof.
Consider the operatorW:Ej⟶Ej defined by(40)Wyt=−Tj−Tj−1−1t−Tj−1ITj−1+ωjf1Tj,yTj,ITj−1+ωjyTj+1Γωj∫Tj−1tt−sωj−1f1s,ys,ITj−1+ωjysds,t∈Jj.
From the properties of fractional integrals and the continuity of functiontκf1, then the operator W:Ej⟶Ej defined in (40) is well defined.
Let(41)Rj≥2f∗Tj−Tj−1ωj/Γωj1−2Tj−Tj−1ωj−1Tj1−κ−Tj−11−κ/1−κΓωjK+LTj−Tj−1ωj/Γωj+1,with(42)f∗=supt∈Jjf1t,0,0.
We consider the set(43)BRj=y∈Ej,yEj≤Rj.
Clearly,BRj is nonempty, bounded, closed, and convex.
Now, we prove thatW satisfies the assumption of Theorem 1.Step 1
:WBRj⊆BRj.
Fory∈BRj and by (H2), we obtain(44)Wyt≤Tj−Tj−1−1t−Tj−1Γωj∫Tj−1TjTj−sωj−1f1s,ys,ITj−1+ωjysds,+1Γωj∫Tj−1tt−sωj−1f1s,ys,ITj−1+ωjysds,≤2Γωj∫Tj−1TjTj−sωj−1f1s,ys,ITj−1+ωjysds,≤2Γωj∫Tj−1TjTj−sωj−1f1s,ys,ITj−1+ωjys−f1s,0,0ds,+2Γωj∫Tj−1TjTj−sωj−1f1s,0,0ds,≤2Γωj∫Tj−1TjTj−sωj−1s−κKys+LITj−1+ωjysds+2f∗Tj−Tj−1ωjΓωj,≤2Tj−Tj−1ωj−1Γωj∫Tj−1Tjs−κK+LTj−Tj−1ωjΓωj+1ysds+2f∗Tj−Tj−1ωjΓωj,≤2Tj−Tj−1ωj−1Tj1−κ−Tj−11−κ1−κΓωjK+LTj−Tj−1ωjΓωj+1Rj+2f∗Tj−Tj−1ωjΓωj,≤Rj,which means that WBRj⊆BRj.Step 2:
W is continuous.
Letyn be a sequence such that yn⟶y in Ej and t∈Jj. Then,(45)Wynt−Wyt≤Tj−Tj−1−1t−Tj−1Γωj∫Tj−1TjTj−sωj−1f1s,yns,ITj−1+ωjyns−f1s,ys,ITj−1+ωjysds+1Γωj∫Tj−1tt−sωj−1f1s,yns,ITj−1+ωjyns−f1s,ys,ITj−1+ωjysds≤2Γωj∫Tj−1TjTj−sωj−1f1s,yns,ITj−1+ωjyns−f1s,ys,ITj−1+ωjysds≤2Γωj∫Tj−1Tjs−κTj−sωj−1Kyns−ys+LITj−1+ωjyns−ysds≤2KΓωjyn−yEj∫Tj−1Tjs−κTj−sωj−1ds+2LΓωjITj−1+ωjyn−yEj∫Tj−1Tjs−κTj−sωj−1ds≤2KΓωjyn−yEj∫Tj−1Tjs−κTj−sωj−1ds+2LTj−Tj−1ωjΓωjΓωj+1yn−yEj∫Tj−1Tjs−κTj−sωj−1ds≤2KΓωj+2LTj−Tj−1ωjΓωjΓωj+1yn−yEj∫Tj−1Tjs−κTj−sωj−1ds≤2Tj−Tj−1ωj−1Tj1−κ−Tj−11−κ1−κΓωjK+LTj−Tj−1ωjΓωj+1yn−yEj.
That is, we obtain(46)Wyn−WyEj⟶0asn⟶∞.
Thus, the operatorW is a continuous on Ej.Step 3:
W is bounded and equicontinuous.
By Step 1, we haveWyEj≤Rj which means that WBRj is bounded. It remains to verify that WBRj is equicontinuous.
Fory∈BRJ and t1,t2∈Jj,t1<t2, we have(47)Wyt2−Wyt1=−Tj−Tj−1−1t2−Tj−1ΓωJ∫Tj−1TjTj−sωj−1f1s,ys,ITj−1+ωjysds+1Γωj∫Tj−1t2t2−sωj−1f1s,ys,ITj−1+ωjysds+Tj−Tj−1−1t1−Tj−1Γωj∫Tj−1TjTj−sωj−1f1s,ys,ITj−1+ωjysds−1Γωj∫Tj−1t1t1−sωj−1f1s,ys,ITj−1+ωjysds≤Tj−Tj−1−1Γωjt2−Tj−1−t1−Tj−1∫Tj−1TjTj−sωj−1f1s,ys,ITj−1+ωjysds+1Γωj∫Tj−1t1t2−sωj−1−t1−sωj−1f1s,ys,ITj−1+ωjysds+1Γωj∫t1t2t2−sωj−1f1s,ys,ITj−1+ωjysds≤Tj−Tj−1−1Γωjt2−Tj−1−t1−Tj−1∫Tj−1TjTj−sωj−1f1s,ys,ITj−1+ωjys−f1s,0,0ds+Tj−Tj−1−1Γωjt2−Tj−1−t1−Tj−1∫Tj−1TjTj−sωj−1f1s,0,0ds+1Γωj∫Tj−1t1t2−sωj−1−t1−sωj−1f1s,ys,ITj−1+ωjys−f1s,0,0ds+1Γωj∫Tj−1t1t2−sωj−1−t1−sωj−1f1s,0,0ds+1Γωj∫t1t2t2−sωj−1f1s,ys,ITj−1+ωjys−f1s,0,0ds+1Γωj∫t1t2t2−sωj−1f1s,0,0ds≤Tj−Tj−1−1Γωjt2−Tj−1−t1−Tj−1∫Tj−1TjTj−sωj−1s−κKys+LITj−1+ωjysds+f∗Tj−Tj−1−1Γωjt2−Tj−1−t1−Tj−1∫Tj−1TjTj−sωj−1ds+1Γωj∫Tj−1t1s−κt2−sωj−1−t1−sωj−1Kys+LITj−1+ωjysds+f∗Γωj∫Tj−1t1t2−sωj−1−t1−sωj−1ds+1Γωj∫t1t2s−κt2−sωj−1Kys+LITj−1+ωjysds+f∗Γωj∫t1t2t2−sωj−1ds≤Tj−Tj−1ωj−2ΓωJt2−Tj−1−t1−Tj−1KyEj+LITj−1+ωjyEj∫Tj−1Tjs−κds+f∗Tj−Tj−1ωj−1Γωj+1t2−Tj−1−t1−Tj−1+1ΓωjKyEj+LITj−1+ωjyEj∫Tj−1t1s−κt2−t1ωj−1ds+f∗Γωjt2−Tj−1ωjωj−t2−t1ωjωj−t1−Tj−1ωjωj+t2−t1ωj−1ΓωjKyEj+LITj−1+ωjyEj∫t1t2s−κds+f∗Γωjt2−t1ωjωj≤Tj−Tj−1ωj−2Tj1−κ−Tj−11−κ1−κΓωjt2−Tj−1−t1−Tj−1KyEj+LTj−Tj−1ωjΓωj+1yEj+f∗Tj−Tj−1ωj−1Γωj+1t2−Tj−1−t1−Tj−1+t11−κ−Tj−11−κt2−t1ωj−11−κΓωjKyEj+LTj−Tj−1ωjΓωj+1yEj+f∗Γωj+1t2−Tj−1ωj−t2−t1ωj−t1−Tj−1ωj+t21−κ−t11−κt2−t1ωj−11−κΓωjKyEj+LTj−Tj−1ωjΓωj+1yEj+f∗t2−t1ωjΓωj+1≤Tj−Tj−1ωj−2Tj1−κ−Tj−11−κ1−κΓωjK+LTj−Tj−1ωjΓωj+1yEj+f∗Tj−Tj−1ωj−1Γωj+1t2−Tj−1−t1−Tj−1+t21−κ−Tj−11−κ1−κΓωjK+LTj−Tj−1ωjΓωj+1yEjt2−t1ωj−1+f∗Γωj+1t2−Tj−1ωj−t1−Tj−1ωj.
Hence,Wyt2−Wyt1Ej⟶0 as t2−t1⟶0. It implies that TBRj is equicontinuous.Step 4:
W is k-set contractions.
Lett∈Jj and B∈BRj; then,(48)ξWBt=ξWyt,y∈B≤Tj−Tj−1−1t−Tj−1Γωj∫Tj−1TjTj−sωj−1ξf1s,ys,ITj−1+ωjysds+1Γωj∫Tj−1tt−sωj−1ξf1s,ys,ITj−1+ωjysds,y∈B.
By Remark4, we have, for each s∈Jj,(49)ξWBt≤Tj−Tj−1−1t−Tj−1Γωj∫Tj−1TjTj−sωj−1Kξ^B∫Tj−1Tjs−κds+LTj−Tj−1ωjΓωj+1ξ^B∫Tj−1Tjs−κds+1Γωj∫Tj−1tt−sωj−1Kξ^B∫Tj−1ts−κds+LTj−Tj−1ωjΓωj+1ξ^B∫Tj−1ts−κds,y∈B≤Tj−Tj−1ωj−2t−Tj−1Γωj∫Tj−1TjKξ^B∫Tj−1Tjs−κds+LTj−Tj−1ωjΓωj+1ξ^B∫Tj−1Tjs−κds+t−Tj−1ωj−1Γωj∫Tj−1tKξ^B∫Tj−1ts−κds+LTj−Tj−1ωjΓωj+1ξ^B∫Tj−1ts−κds,y∈B≤Tj1−κ−Tj−11−κTj−Tj−1ωj−2t−Tj−11−κΓωjK+LTj−Tj−1ωjΓωj+1ξ^B+t1−κ−Tj−11−κt−Tj−1ωj−11−κΓωjK+LTj−Tj−1ωjΓωj+1ξ^B≤2Tj1−κ−Tj−11−κTj−Tj−1ωj−11−κΓωjK+LTj−Tj−1ωjΓωj+1ξ^B.
Thus,(50)ξ^WB≤2Tj1−κ−Tj−11−κTj−Tj−1ωj−11−κΓωjK+LTj−Tj−1ωjΓωj+1ξ^B.
So, BVP (33) has at least a solution yj˜∈BRj. Since BRj⊂Ej, we have completed the proof of Theorem 2.
Now, we will be interested in proving the existence of solution for BVP (2). We begin by presenting the following assumption.
(H2) Letf1:J×ℝ×ℝ⟶ℝ be a continuous function, and there exists κ∈0,1 such that tκf1∈CJ×ℝ×ℝ,ℝ, and there exist a constants K,L>0, such thattκf1t,x1,z1−f1t,x2,z2≤Kx1−x2+Lz1−z2, for any x1,x2,z1,z2∈ℝ and t∈J.Theorem 3.
Let (H1) and (H2) hold and inequality (39) be satisfied for any j∈1,2,…,n. Then, BVP (2) has at least one solution in CJ,ℝ.Proof.
By Theorem2, BVP (33) possesses a solution yj˜∈Ej, j∈1,2,…,n.
We define the function(51)yj=0,t∈0,Tj−1,y˜j,t∈Jj,,j∈1,2,…,n.
Thus, fort∈JJ, the integral equation (31) has the solution yJ∈C0,TJ,ℝ with yJ0=0andyJTJ=y˜JTJ=0.
Then, the function,(52)yt=y1t,t∈J1,y2t=0,t∈J1,y˜2,t∈J2⋅⋅⋅ynt=0,t∈0,Tj−1,y˜j,t∈Jjis a solution of BVP (2) in CJ,ℝ.


## 4. The Stability

In this section, we show that BVP (2) is UH stable.Theorem 4.
Let all the conditions of Theorem3 be satisfied. Then, BVP (2) is UH stable.Proof.
Let the functionzt from z∈CJj,ℝ satisfy inequality (21).
We define the functions:(53)zjt=0,t∈0,Tj−1,zt,t∈Jj,,j∈1,2,…,n.
By equality (30), for j∈1,2,…,n and t∈Jj, we obtain(54)DcTj−1+ωtzjt=∫Tj−1tt−s1−ωjΓ2−ωjz2sds.
Taking the RLFIITj−1+ωj of both sides of inequality (21), we obtain(55)zjt+Tj−Tj−1−1t−Tj−1Γωj∫Tj−1TjTj−sωj−1f1s,zjs,ITj−1+ωjzjsds,−1Γωj∫Tj−1tt−sωj−1f1s,zjs,ITj−1+ωjzjsds,≤ε∫Tj−1tt−sωj−1ΓωjdsεTj−Tj−1ωjΓωj+1,≤εTj−Tj−1ωjΓωj+1.
According to Theorem3, BVP (2) has a solution y∈CJ,ℝ that is given for any t∈Jj,j=1,2,…,n, as yt=yjt, where(56)yj=0,t∈0,Tj−1,y˜j,t∈Jj,and y˜j is a solution of (33), which is given according to Lemma 7 by(57)y˜jt=−Tj−Tj−1−1t−Tj−1Γωj∫Tj−1TjTj−sωj−1f1s,y˜js,ITj−1+ωjy˜jsds+1Γωj∫Tj−1tt−sωj−1f1s,y˜js,ITj−1+ωjy˜jsds.
Then, by equations (56) and (57), for t∈Jj,j=1,2,…,n, we obtain(58)zt−yt=zt−yjt=zjt−y˜jt,=|zjt+Tj−Tj−1−1t−Tj−1Γωj∫Tj−1TjTj−sωj−1f1s,y˜js,ITj−1+ωjy˜jsds,−1Γωj∫Tj−1tt−sωj−1f1s,y˜js,ITj−1+ωjy˜jsds|,≤|zjt+Tj−Tj−1−1t−Tj−1Γωj∫Tj−1TjTj−sωj−1f1s,zjs,ITj−1+ωjzjsds,−1Γωj∫Tj−1tt−sωj−1f1s,zjs,ITj−1+ωjzjsds|+Tj−Tj−1−1t−Tj−1Γωj,∫Tj−1TjTj−sωj−1f1s,zjs,ITj−1+ωjzj−f1s,y˜js,ITj−1+ωjy˜jds,+1Γωj∫Tj−1tt−sωj−1f1s,zjs,ITj−1+ωjzj−f1s,y˜js,ITj−1+ωjy˜jds,≤ϵTj−Tj−1ωjΓωj+1+Tj−Tj−1−1t−Tj−1Γωj∫Tj−1TjTj−sωj−1s−κKzjs−y˜js+LITj−1+ωjzjs−y˜jsds,+1Γωj∫Tj−1tt−sωj−1s−κKzjs−y˜js+LITj−1+ωjzjs−y˜jsds,≤ϵTj−Tj−1ωjΓωj+1+Tj−Tj−1ωj−1ΓωjKzj−y˜jEj+LITj−1+ωjzj−y˜jEj∫Tj−1Tjs−κds,+TJ−TJ−1ωJ−1ΓωJK‖zJ−y˜J‖EJ+L‖ITJ−1+ωJzJ−y˜J‖EJ∫TJ−1ts−κds≤ϵTJ−TJ−1ωJΓωJ+1+TJ−TJ−1ωJ−1TJ−11−κ−TJ−11−κ1−κΓωJK‖zJ−y˜J‖EJ+LTJ−TJ−1ωJΓωJ+1‖zJ−y˜J‖EJ+TJ−TJ−1ωJ−1t1−κ−TJ−11−κ1−κΓωJK‖zJ−y˜J‖EJ+LTJ−TJ−1ωJΓωJ+1‖zJ−y˜J‖EJ≤εTJ−TJ−1ωJΓωJ+1+2TJ−TJ−1ωJ−1TJ1−κ−TJ−11−κ1−κΓωJK+LTJ−TJ−1ωJΓωJ+1zJ−y˜JEJ≤εTJ−TJ−1ωJΓωJ+1+μz−y,where(59)μ=maxj=1,2,…,n2Tj−Tj−1ωj−1Tj1−κ−Tj−11−κ1−κΓωjK+LTj−Tj−1ωjΓωj+1.
Then,(60)z−y1−μ≤Tj−Tj−1ωjΓωj+1ε.
Thus, fort∈JJ, we have(61)zt−yt≤z−y≤Tj−Tj−1ωj1−μΓωj1ε≔λf1ε.
Therefore, BVP (2) is UH stable.


## 5. Illustration

In this section, we construct an illustrative example to express the validity of the obtained results.Example 2.
Consider the following BVP:(62)D0+ωtcyt=t−1/3e−teet2/1+t+4e2t+11+yt+I0ωtyt,t∈J≔0,2,y0=0,y2=0.
Suppose that(63)f1t,x,z=t−1/3e−teet2/1+t+4e2t+11+y+z,t,x,z∈0,2×0,+∞×0,+∞.ωt=32,t∈J1≔0,1,95,t∈J2≔1,2.
Then, we have(64)t1/3f1t,x1,z1−f1t,x2,z2=e−teet2/1+t+4e2t+111+x1+z1−11+x2+z2=e−tx1−x2+z1−z2eet2/1+t+4e2t+11+x1+z11+x2+z2≤e−teet2/1+t+4e2t+1x1−x2+z1−z2≤1e+5x1−x2+1e+5z1−z2.
Hence, condition (H2) holds withκ=1/3 and K=L=1/e+5.
According to (33) and by (15), we have the following two auxiliary BVP:(65)cD0+3/2yt=t−1/3e−teet2/1+t+4e2t+11+yt+I03/2yt,t∈J1,y0=0,y1=0,and(66)D1+9/5cyt=t−1/3e−teet2/1+t+4e2t+11+yt+I09/5yt,t∈J2,y1=0,y2=0.
Next, we shall check that condition (39) is satisfied for j=1. Indeed,(67)T11−κ−T01−κT1−T0ω1−11−κΓω12K+2LT1−T0ω1Γω1+1=22/3e+5Γ3/21+1Γ5/2≃0.7685<1.
Accordingly, condition (39) is achieved. By Theorem 2, BVP (65) has a solution y˜1∈E1.
We shall check that condition (39) is satisfied for j=2. Indeed,(68)T21−κ−T11−κT2−T1ω2−11−κΓω22K+2LT2−T1ω2Γω2+1=22/3−12/3Γ9/52e+51+1Γ14/5≃0.3913<1.
Thus, condition (39) is satisfied.
By Theorem2, BVP (66) has a solution y˜2∈E2.
Thus, by Theorem3, BVP (62) possesses a solution:(69)yt=y˜1t,t∈J1,y2t,t∈J2,where(70)y2t=0,t∈J1,y˜2t,t∈J2.
According to Theorem4, BVP (62) is UH stable.


## 6. Conclusion

In this work, we presented results about the existence and uniqueness of solutions to the BVP of Caputo fractional differential equations of variable-orderωt, where ωt:0,T⟶1,2 is a piecewise constant function. All our results are based on Darbo’s fixed-point theorem, and we studied Ulam–Hyers (UH) stability of solutions to our problem. Finally, we illustrated the theoretical findings by a numerical example.The variable-order BVPs are important and interesting to all researchers. Therefore, all results in this paper show a great potential to be applied in various applications of multidisciplinary sciences.



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