# On the Stability of Fractional Differential Equations Involving Generalized Caputo Fractional Derivative

Authors: Minh Duc Tran; Vu Ho; Hoa Ngo Van
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This work presents the results of the global existence for fractional differential equations involving generalized Caputo derivative with the case of the fractional order derivativeα∈1,2. In addition, the Ulam–Hyers–Mittag-Leffler stability of the given problems is also established.


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## Body

## 1. Introduction

In recent years, there are a vast number of various concepts for fractional integrals and derivatives, such as Riemann–Liouville, Riesz, Grünwald–Letnikov, Hadamard, and Caputo derivatives and/or integrals. One can notice that most of the research results on the topics of fractional differential equations involving Riemann–Liouville and Hadamard fractional derivatives have been paid more and more attention by a large number of mathematicians because of the interesting and their applications. For more details on fractional calculus theory and interesting applications, one can see the monographs and the interesting papers in [1–6] and the references cited therein. However, both of the definitions of Hadamard and Riemann–Liouville fractional derivatives have their own disadvantages as well; one of which is that the derivative of a constant is not equal to zero. Then, to overcome the disadvantage of two types of these fractional derivatives, the Caputo and Caputo–Hadamard fractional derivatives were proposed. In the past decade, in [7, 8], Katagampola has proposed a new generalized concept of the fractional derivative, the so-called Caputo–Katugampola, that unifies the definitions of Caputo and Caputo-Hadamard fractional derivatives into a single form. The parameter family ρ of Caputo–Katugampola fractional derivative, Da+α,ρC, of the noninteger order α allows one to interpolate two types of the Caputo and Caputo–Hadamard fractional derivatives. Other approaches of fractional operators based on using very general kernel functions have been also proposed in [1, 9]. These approaches relate to the various real data corresponding to different complex systems requiring different kernel functions. For more details, on Caputo–Katugampola fractional derivative and interesting applications, one can see the papers [1, 2, 10–15] and the references cited therein. Very recently, the motivation behind the approach of Caputo–Katugampola fractional operator relates to the chaos problems in fractional dynamical systems suggested in the security of image encryption [16, 17] and in quantum mechanics [12].During the past two decades, a large number of mathematicians have paid great attention to the studies of the concepts of Ulam’s stability because of its usefulness in many applications such as numerical analysis and optimization, where finding the exact solutions is quite difficult. In fact, it is not easy to get exact solutions to most of the problems of fractional differential equations. Therefore, it is vital to develop the concepts of Ulam’s stability for these problems because we need not obtain the exact solutions of the given problems when we study the properties of Ulam’s stability. This theory helps us getting an efficient and reliable technique for approximately solving fractional differential equations because there exists a close exact solution when the given problem is Ulam stable. More details from historical point of view and recent developments of such stabilities are reported in [9, 17–30] and the references cited therein. So, the motivation for the elaboration of this paper is the investigation of some kinds of the Ulam–Hyers stability for the following problem involving the concept of Caputo–Katugampola fractional derivative with the case of the α∈1,2:(1)Da+α,ρCψt=ft,ψt,ψa=ψ1,ψ′a=ψ2,t∈a,b,where α∈1,2,ρ>0 is a real parameter, f:a,b×ℝd⟶ℝd is a nonlinear continuous function, and Da+α,ρC is the Caputo–Katugampola fractional derivative. Based on (1), the parameter ρ allows one to get the initial value problem involving the Caputo fractional derivative if ρ tends to 1, and the initial value problem with the concept of Caputo–Hadamard fractional derivative if ρ tends to 0+. Our aim in this paper is to discuss the global existence of solutions of problem (1) by using Schauder’s and Weissinger’s fixed point theorem. In addition, some kinds of the Ulam–Hyers stability of problem (1) are also established. The rest of this paper is arranged as follows: some fundamental theories of Caputo–Katugampola fractional calculus are introduced in Section 2. Section 3 is devoted to discuss the global existence of solutions of problem (1), and the stability of problem (1) is presented in Section 4.


## 2. Fundamental Theorems of Fractional Analysis

In this section, some definitions and basic results will be briefly presented which will be used throughout the paper. LetCa,b,Rd be the space of vector-valued continuous functions ψ from a,b⟶Rd endowed with the norm ψ0=supt∈a.bψt, where ⋅ is the vector norm in the d-dimensional Euclidean space. Denote by Cγa,b,ℝd the weighted space of continuous functions given by(2)Cγa,b,ℝd=ψ:a,b⟶ℝd:tρ−aρργψt∈Ca,b,ℝd,where γ∈0,1.Letα>0, then the Riemann–Liouville generalized fractional integral of ψ is defined by (see [7])(3)ψα,ρt≔Ia+α,ρψt=ρ1−αΓα∫atsρ−1tρ−sρα−1ψsds.Letα>0, then the Riemann–Liouville generalized derivative of ψ is defined by (see [7])(4)Da+α,ρRLψt=t1−pddtnIa+n−α,ρψt=ρα−n+1Γn−αt1−pddtn∫atsρ−1tρ−sρn−α−1ψsds,t>a,where n=α.Letα>0, then the Caputo-generalized fractional derivative of ψ denoted by Da+α,ρCψ is defined by(5)Da+α,ρCψt=Da+α,ρRLψt−∑k=0n−1tρ−aρkρkk!t1−pddtkψtt=a.By puttingψp,kt≔t1−pd/dtkψt, we have that(6)Da+α,ρCψt=Da+α,ρRLψt−∑k=0n−1ψp,kaρkk!Da+α,ρRLtρ−aρk=Da+α,ρRLψt−∑k=0n−1ψp,kaΓk−α+1tρ−aρρk−α,where ψρ,ka≔t1−pd/dtkψtt=a. If ψ∈Cna,b,ℝd, then we have that (see [7])(7)Da+α,ρCψt=ρα−n+1Γn−α∫atsρ−11−ntρ−sρn−α−1dnψdsnsds,t≥a.We observe that(8)ψα,ρ0≔supt∈a,bψα,ρt≤ψ0ρ1−αΓα∫atsρ−1tρ−sρα−1ds≤ψ0ραΓα+1bρ−aρα.Remark 1 (see [7]).
Letα,β,ρ>0, then the following properties are satisfied:(i)
Ia+α,ρψ+ξt=Ia+α,ρψt+Ia+α,ρξt.(ii)
Ia+α,ρIa+β,ρψt=Ia+α+β,ρψt.(iii)
For0<α<β, we have(9)Da+α,ρRLIa+α,ρψt=ψt,t∈a,b,Da+α,ρRLIa+β,ρψt=Ia+β−α,ρψt,t∈a,b.Remark 2.
Letn−1<α≤n∈ℕ and ψ1−α,ρ∈Cna,b,ℝd, then we have, for t∈a,b,(10)Ia+α,ρDa+α,ρRLψt=ψt−∑k=1nψn−α,ρn−kaΓα−k+1tρ−aρρα−k.Remark 3.
Letn−1<α≤n∈ℕ and ψ∈Cna,b,ℝd, then we have that, for t∈a,b,(11)Ia+α,ρDa+α,ρCψt=ψt−∑k=0n−1ψp,kak!tρ−aρρk,Da+α,ρCIa+α,ρψt=ψt,where n=α.Theorem 1 (see Theorem8 in [10]).
Letpt,qt be two integrable functions and rt be a continuous function on a,b. Assume that pandq are nonnegative, and r is nonnegative and nondecreasing. If(12)pt≤qt+rtρ1−α∫atsρ−1tρ−sρα−1psds,∀t∈a,b,then(13)pt≤qt+∫at∑k=1∞ρ1−kαrtΓαkΓkαsρ−1tρ−sρkα−1qsds,∀t∈a,b.Furthermore, if the functionq is nondecreasing, then(14)pt≤qtEα,1rtΓαtρ−aρρα,∀t∈a,b.The existence and uniqueness results are proved according to the following Schauder’s and Weissinger’s fixed point theorem [31].Theorem 2.
Assume thatX,d is a complete metric space, and let S be a closed convex subset of X. Furthermore, let ℙ:S⟶S be the map such that the set ℙx:x∈S is relatively compact in X. Then, the operator P has at least one fixed point x∗∈S such that ℙx∗=x∗.Theorem 3.
Assume thatX,d is a nonempty complete metric space and let λn≥0 for every n∈ℕ0 such that the series ∑n=0∞λn converges. Furthermore, let the mapping ℙ:X⟶X satisfy the inequality(15)dℙnx,ℙny≤λndx,y,for every n∈ℕ and for any x,y∈X. Then, the operator P has a unique fixed point x0. Moreover, for any x∗∈X, the sequence ℙnx∗n≥1 converges to the fixed point x0.


## 3. The Existence and Uniqueness of the Solution

In this section, we reconsider the following fractional differential equations:(16)Da+α,ρCψt=ft,ψt,ψa=ψ1,ψ′a=ψ2,t∈a,b.A functionψ:a,b⟶ℝd is said to be a solution of problem (16); if ψ is continuous, ψa=ψ1, ψ′a=ψ2, and Da+α,ρCψt=ft,ψ.Theorem 4.
Let the functionft,ψ belong to Cγa,b,ℝd, where γ∈0,1. Then, problem (16) is equivalent to the fractional integral equation:(17)ψt=ψ0+tρ−aρρψρ,1a+1Γα∫atsρ−1tρ−sρρα−1fs,ψsds,t∈a,b,where ψρ,1a≔t1−pd/dtψtt=a.Proof.
Letψ∈C1a,b,ℝd be a solution of (16), then from (16) and Remark 3 we have that, for α∈1,2,(18)Ia+α,ρDa+α,ρCψt=ψt−ψa−tρ−aρρψρ,1a,for t∈a,b. Because of the continuous hypotheses of the function f and from (16), it yields that(19)Ia+α,ρDa+α,ρCψt=Ia+α,ρft,ψt=1Γα∫atsρ−1tρ−sρρα−1fs,ψsds,t∈a,b.
Consequently, by (18) and (3) we get the necessity condition. Conversely, let ψ∈C1a,b,ℝd satisfy the integral equation (17). Using the continuity of f yields that t⟼fα,ρt,u≔Ia+α,ρft,u is continuous on a,b and fα,ρa,ua=limt⟶a+fα,ρt,u=0. Indeed, it follows from the hypothesis f∈Cγa,b,ℝd that tρ−aρ/ργft,u∈Ca,b,ℝd and, for all t∈a,b, there exists a positive constant C such that(20)tρ−aρργft,u≤C,orft,u≤Ctρ−aρρ−γ.
This infers that(21)Ia+α,ρft,u≤Ia+α,ρft,u≤CIa+α,ρtρ−aρρ−γ=CΓ1−γΓα−γ+1tρ−aρρα−γ.
Taking the limit whent⟶a+, we observe that the right-hand side of (21) tends to 0. Furthermore, ψa=ψ1 and ψ′a=ψ2. Next, by taking the Caputo–Katugampola fractional derivative, Da+α,ρRL, on the two sides of (17) and by Remark 1 one has that(22)Da+α,ρCψt=ft,ψt,t∈a,b.In the below theorems, we will present the existence and uniqueness of the local solution to problem (16) by using the Schauder fixed point theorem. We set It≔ψ0+ψρ,1atρ−aρ/ρ. Let r>0 be a given constant, and define. Ω≔t,ψ∈a,b×Ca,b,ℝd:supt∈a,bψt−It≤r.Theorem 5.
Letf:a,b×ℝd⟶ℝd be a continuous function and supt,ψ∈Ωft,ψ≤M. Then, problem (16) has at least one solution on a,b. Furthermore, if the following Lipschitz condition is held,(23)ft,ψ−ft,ξ≤Lψ−ξ,where L is a positive constant, then problem (16) has a unique solution on a,b.Proof.
Let(24)ΩT=ψ:ψ∈Ca,T,ℝd,supt∈a,Tψt−It≤r,where(25)T=minb,rραΓ1+αM1/α+aρ1/ρ.
We observe thatΩT⊆Ca,T,ℝd is nonempty, bounded, closed, and convex subset. Define the operator T:ΩT⟶ψψ∈Ca,T,ℝd by(26)Tψt=It+1Γα∫atsρ−1tρ−sρρα−1fs,ψsds.
The proof of this theorem is divided into two steps. In the first step, we shall prove that the operatorT has at least one fixed point by using Schauder’s fixed point principle and in the second one we also verify that the operator T has a unique solution by using Weissinger’s fixed point theorem.
Step 1. We shall show that the conditions of Theorem 2 are satisfied.
We show thatTΩT⊂ΩT. For any ψ∈ΩT, we obtain(27)Tψt−It≤1Γα∫atsρ−1tρ−sρρα−1fs,ψsds≤MΓα+1tρ−aρρα≤r.
On the other hand, letψ,ψn∈ΩT such that ψn⟶ψ in Ca,T,ℝd as n⟶∞. By the continuity of the function f one has ft,ψn⟶ft,ψ in Ca,T,ℝd as n⟶∞. Since(28)Tψnt−Tψt≤1Γα∫atsρ−1tρ−sρρα−1sups∈a,tfs,ψns−fs,ψsds,we have supt∈a,TTψnt−Tψt≤ρ−α/Γα+1tρ−aραsups∈a,tfs,ψns−fs,ψs. Then, Tψnt tends to Tψt in ΩT as n⟶∞. This yields T is continuous. Thus, we infer that Tψ∈ΩT, for ψ∈ΩT, i.e., TΩT⊆ΩT.
Next, we show thatTΩT=Tψt:ψ∈ΩT is a relatively compact set. For Tψ∈TΩT and t∈a,T, since(29)Tψt≤It+MΓα+1tρ−aρρα≤supt∈a,TIt+r,we conclude that the set TΩT is uniformly boundedness. Furthermore, for a≤t1≤t2≤T(30)Tψt1−Tψt2+It2−It1≤Mρ−αΓα+1t2ρ−t1ρα+Mρ−αΓα+1t2pα−t1pα−t2ρ−t1ρα≤Mρ−αΓα+1t2ρ−t1ρα+t2pα−t1pα.
Applying the mean value theorem, one obtains(31)Tψt1−Tψt2≤t2−t1τρ−1+Mρ−αΓα+1ταρ−1t2−t1ρα+ατρα−1t2−t1ρ,for some τ∈t1,t2⊆a,T. Therefore, if t2−t1<δ, we have(32)Tψt1−Tψt2≤Mδτρ−1Γα+1ταρ−1δα+ρ1−αατρα−1δ≤maxaρ−1,Tρ−1δ+MΓα+1maxaαρ−1,Tαρ−1δα+ρ1−ααmaxaρα−1,Tρα−1δ,where a>0, ρ>0 is fixed, and α∈1,2. This shows that TΩT is equicontinuous. Hence, by Arzela–Ascoli theorem (see Theorem 1.8 in [5]), this yields that TΩT is relatively compact. So, according to the conditions of Theorem 2, we can conclude that the operator T has a fixed point.
Step 2. For the uniqueness of solution, we suppose that ξ⋅:a,T⟶Ca,T,ℝd is another solution for problem (16) on a,T and ξa=ψa,ξ′a=ψ′a. Then, we obtain(33)Tψt−Tξt≤LΓα∫atsρ−1tρ−sρρα−1ψs−ξsds.
Thus, it follows that(34)supt∈a,TTψt−Tξt≤LΓα+1tρ−aρραsupt∈a,Tψs−ξs.
By the induction method, we will verify that(35)supt∈a,TTnψt−Tnξt≤LnΓnα+1tρ−aρρnαsupt∈a,Tψt−ξt,where(36)Tnψt=It+1Γα∫atsρ−1tρ−sρρα−1fs,Tn−1ψsds,for n∈ℕ. Indeed, we assume that (35) is satisfied for the case of n−1∈ℕ, and for n∈ℕ, one has that(37)supt∈a,TTnψt−Tnξt=supt∈a,TTTn−1ψt−TTn−1ξt≤LΓα+1∫atsρ−1tρ−sρρα−1sups∈a,TTn−1ψs−Tn−1ξsds≤ρ1−nαLnΓn−1α+1supt∈a,Tψt−ξt1Γα+1∫atsρ+ραn−1−1tρ−sρα−1ds=LnΓnα+1tρ−aρρnαsupt∈a,Tψt−ξt.
SettingKn≔Ln/Γnα+1tρ−aρ/ρnα, we observe that the series ∑n=0∞Kn converges to the Mittag-Leffler function Eα,1Ltρ−aρ/ρα. So, we can conclude that the series Kn is convergent. Based on Theorem 3, the operator T has a unique fixed point according to the conditions of Weissinger’s fixed point theorem.Theorem 6.
Assume that the functionf:a,b×ℝd⟶ℝd is a continuous function and supt,ψ∈Ωft,ψ≤M. Let ψt be a solution of problem (16) on a,β. If ψt is bounded on η,β for some η>a, then β=+∞.Proof.
This assertion will be divided into two steps. First of all, we verifylimt⟶β−ψt exists. It means there exists η∈a,β such that, for t∈η,β, ψt−ψ∗<ε, for any given ε>0, i.e.,(38)limt⟶β−ψt=ψ∗.
Sinceψt is bounded on η,β, there exists a sequence tk and a positive constant K>0 such that tk<tk+1,k∈ℕ, limk⟶∞tk=β, and ψtk≤K. In addition, as ψtk is a bounded convergent subsequence, it follows that limk⟶∞ψtk=ψ∗. This implies that there exists k0 such that tk0>η, and for k≥k0, we have(39)ψtk−ψ∗≤ε2.
If (38) is not true, then for k≥k0, there exists τk∈tk,β such that ψτk−ψ∗≥ε and ψt−ψ∗<ε,t∈tk,τk. Then, from (39) and triangle inequality one has(40)ε≤ψtk−ψ∗+ψτk−ψtk≤ε2+ψρ,1aρτkρ−tkρ+1Γα∫atksρ−1τkρ−sρρα−1−tkρ−sρρα−1fs,ψsds+1Γα∫tkτksρ−1tkρ−sρρα−1fs,ψsds≤ε2+ψρ,1aρτkρ−tkρ+MραΓα+1τkρ−tkρα+τkpα−tkpα.
Then, for sufficiently largek≥k0, we have(41)ε≤ψτk−ψ∗<ε/2+ε/2=ε.
This implies the contradiction thatlimt⟶β−ψt exists. For the next step, we show that ψt is continuable. Define ψβ=limt⟶β−ψt. Then, ψt is continuous on a,β. Consider the operator T as follows:(42)Tψt=It+1Γα∫aβsρ−1tρ−sρρα−1fs,ψsds+1Γα∫βtsρ−1tρ−sρρα−1fs,ψsds,where ψ∈Cβ,β+1,ℝd,t∈β,β+1. As a result of Theorem 5, since on the interval a,β, equation (42) possesses a solution, and we can rewrite equation (42) as follows:(43)Tψt=ψ1t+1Γα∫βtsρ−1tρ−sρρα−1fs,ψsds,where(44)ψ1t=It+1Γα∫aβsρ−1tρ−sρρα−1fs,ψsds,is the known function. Let Ω∗=t,ψ:β≤t≤β+1,ψ≤supt∈β,β+1ψ1t+ρ1. Because of the continuation of f on Ω∗, denote M∗=maxt,ψ∈Ω∗ft,ψ. Again let(45)Ωh=ψ:ψ∈Cβ,β+1,ℝd,supt∈β,β+hψt−ψ1t≤ρ1,ψβ=ψ1β,where(46)h=min1,ρ1ραΓ1+αM∗1/α+βρ1/p.
By the same argument as in Theorem5, we show that T has a fixed point. Let ψn⊆Cβ,β+h,ℝd, such that ψn⟶ψ in Cβ,β+h,ℝd as n⟶∞. By the continuity hypothesis of the function f we have ft,ψn⟶ft,ψ in Cβ,β+h,ℝd as n⟶∞. Since(47)Tψnt−Tψt≤hpαραΓα+1sups∈β,β+hfs,ψns−fs,ψs,we have supt∈β,β+hTψnt−Tψt⟶0 as n⟶∞, which yields that operator T is continuous. Next, it follows that TΩh=Tψt:ψ∈Ωh is relatively compact. For Tψ∈TΩh and t∈β,β+h, we have(48)Tψt−ψ1≤M1tρ−βραραΓα+1≤ρ1.
Thus,TΩh⊂Ωh. Furthermore, for ψ∈Ωh,β≤t1≤t2≤β+h, one has(49)Tψt1−Tψt2+ψ1t2−ψ1t1≤ρ1−αΓα∫βt1sρ−1t2ρ−sρα−1−t1ρ−sρα−1fs,ψsds+ρ1−αΓα∫t1t2sρ−1t2ρ−sρα−1fs,ψsds≤M∗ρ−αΓα+1t2ρ−t1ρα+t2pα−t1pα.With the same argument as in the proof of Theorem 5, since ψ1t is continuous on β,β+1, it follows that TΩh is equicontinuous. Hence, by Arzela–Ascoli theorem, we deduce that TΩh is relatively compact. Therefore, by the Schauder’s fixed point theorem, operator T has a fixed point ψ^t∈Ωh, i.e.,(50)ψ^t=ψ1t+1Γα∫βtsρ−1tρ−sρρα−1fs,ψ^sds=It+1Γα∫atsρ−1tρ−sρρα−1fs,ψ^sds,t∈β,β+h, where(51)ψ^t=ψt,t∈a,β,ψ^t,t∈β,β+h.
According to Theorem4, we conclude that ψ^t is a solution of problem (16) on a,β+h. By the assumption of this theorem again, it follows that ψt can be extended beyond β. So, the solution of problem (16) exists on a,∞ and so β=+∞.Theorem 7.
Letf:a,+∞ℝd⟶ℝd be a continuous function and there exists a continuous function qt:a,+∞⟶0,+∞ such that ft,ψ≤Mψ+qt, where M>0 is a constant. Then, problem (16) has at least one solution on a,+∞. Furthermore, if the following Lipschitz condition is satisfied,(52)ft,ψ−ft,ξ≤Lψ−ξ,∀t∈a,+∞,then problem (16) has a unique solution on a,+∞.Proof. It follows from the assertion of Theorem5 that there exists a solution ψt of problem (16). By Theorem 4, ψt satisfies the following integral equation(53)ψt=ψ0+tρ−aρρψρ,1a+1Γα∫atsρ−1tρ−sρρα−1fs,ψsds.Step 3 (the existence of solution).
Suppose that the solutionψ admits a maximal existence interval, denoted by a,β β<+∞. By the condition of the function f, one has the estimate(54)ψt≤ψ0+ψρ,1aρtρ−aρ+1Γα∫atsρ−1tρ−sρρα−1Mψs+qsds.
By puttingvt=ψ0+ψρ,1a/ρtρ−aρ+ρ1−α/Γα∫atsρ−1tρ−sρα−1qsds., then (54) leads to(55)ψt≤vt+Mρ1−αΓα∫atsρ−1tρ−sρα−1ψsds.
We now prove thatwt≔ψt is bounded on a,β. Define the functional(56)ℱwt=Mρ1−αΓα∫atsρ−1tρ−sρα−1wsds.
Then, one has thatwt≤vt+ℱwt, and by iterating consecutively, we also obtain for n∈ℕ,(57)wt≤∑i=0n−1ℱivt+ℱnwt.
In addition, by mathematical induction andvt>0 for t∈a,β, then (56) leads to the following estimation:(58)ℱivt≤MiΓiα∫atsρ−1tρ−sρρiα−1vsds,i∈ℕ.
Indeed, fori=1, (58) is obvious. If (58) is valid for i∈ℕ, then by using Dirichlet’s formula, we obtain(59)ℱi+1vt=ℱℱivt≤Mρ1−αΓα∫atsρ−1tρ−sρα−1MiΓiα∫asτp−1sρ−τρρiα−1vτdτds=Mi+1ρ2−i+1αΓαΓiα∫at∫assρ−1tρ−sρα−1τp−1sρ−τρiα−1vτdτds=Mi+1ρ2−i+1αΓαΓiα∫atsρ−1vs∫stτp−1tρ−τρα−1τρ−sρiα−1dτds.With the change of variables z=sρ−aρ/tρ−aρ, we obtain(60)ℱi+1vt≤Mi+1Γiα+α∫atsρ−1tρ−sρρi+1α−1vsds.
Thus, (58) is valid for i+1. Besides, we observe that ℱnwt⟶0 as n⟶∞. Hence, from (57), we have(61)wt≤∑i=0∞ℱivt≤vt+∫at∑i=1∞MiΓiαsρ−1tρ−sρρiα−1vsds.
Furthermore, if the functionv is nondecreasing, then for all s∈a,t, we have vs≤vt and so(62)wt≤vt1+∑i=1∞MiΓiα∫atsρ−1tρ−sρρiα−1ds=vtEα,1Mtρ−aρρα.
Therefore, we deduce thatψt is bounded on a,β. It follows from Theorem 6 that the solution ψt can be extended to the right side of β. This obviously contradicts the assumption that a,β is the maximal existence interval. This consequently implies that β=+∞.Step 4 (the uniqueness of solution).
By the Lipschitz condition and inequality (62) the uniqueness of solution can be proved. Indeed, let us assume that ξt is another solution for problem (16) and ξa=ψa,ξ′a=ψ′a. Then, one has that(63)ψt−ξt≤ψ0t−ξ0t+LΓα∫atsρ−1tρ−sρρα−1ψs−ξsds,where ψ0t≔ψa+ψρ,1atρ−aρ/ρ. Define kt=ψt−ξt, and note that since ξa=ψa,ξ′a=ψ′a, we have ψ0t−ξ0t=0. Applying Theorem 1, we get kt≤0. This implies that ψt=ξt, which completes the proof.


## 4. The Stability of Problem (16)

In the sequel, some kinds of the Ulam–Hyers–Mittag-Leffler stability for problem (16) will be investigated. Denote J≔a,b. Let α∈1,2,ρ>0, ε>0, and φ∈Ca,b,ℝ+. We consider the inequalities as follows:(64)Da+α,ρCξt−ft,ξt≤ε,(65)Da+α,ρCξt−ft,ξt≤εφt.Remark 4.
From inequalities (64) and (65), we observe that(i)
If a function ξ is a solution of (64), then there exists a function δ1∈CJ,ℝd such that δ1≤ε, for all t∈J, and Da+α,ρCξt=ft,ξt+δ1t(ii)
If a function ξ is a solution of (65), then there exists a function δ2∈CJ,ℝd such that δ2≤εφt, for all t∈J, and Da+α,ρCξt=ft,ξt+δ2t
Definition 1. Letβf≥0. roblem (16) is said to be(i)
Ulam–Hyers–Mittag-Leffler stable if there is a constantCf>0 such that, for each ε>0 and for each solution ξ of (64), there exists a solution ψ of problem (16) satisfying the estimate(66)ψt−ξt≤εCfEα,1βftρ−aρρα,t∈a,b.(ii)
Ulam–Hyers–Rassias–Mittag-Leffler stable with respect toφ,η if there is a constant Cφ>0 such that, for each ε>0 and for each solution ξ of (65), there exists a solution ψ of problem (16) satisfying the estimate(67)ψt−ξt≤εCf,φEα,1βftρ−aρραφt+η,t∈a,b.
In the following theorem, the Ulam–Hyers–Mittag-Leffler stability for problem (16) is presented.Theorem 8.
Letf:J×ℝd⟶ℝd be a continuous function which satisfies the assumption as follows. There exists a positive constant L such that, for all w,z∈ℝd, one has(68)ft,w−ft,z≤Lw−z.Then, we have the following assertions:(A1) For everyε>0, if a function ξ satisfies (64) for all t∈J, then there exists a unique solution ψ of problem (16) with initial conditions ψa=ξa,ψ′a=ξ′a, which satisfies(69)ψt−ξt≤εLEα,1Ltρ−aρρα,∀t∈J.Furthermore, this yields that problem (16) is Ulam–Hyers–Mittag-Leffler stable.(A2) Assume that the following hypothesis holds: Letφ:J⟶ℝ+ in the inequality (65) be a nondecreasing function for all t∈J. We assume that there exists a positive constant Kφ which satisfies KφL<1, and for t∈J,(70)ρ1−αΓα∫atsρ−1tρ−sρα−1φsds≤Kφφt.Then, for everyε>0, if a function ξ satisfies (65), for all t∈J, there exists a unique solution ψ of problem (16) with initial conditions ψa=ξa,ψ′a=ξ′a, and problem (16) is Ulam–Hyers–Rassias–Mittag-Leffler stable.Proof.
In the view of Remark4 and Theorem 4, we notice that if functions ξandξ^ satisfy inequalities (64) and (65), respectively, then there exists δ1t,δ2t∈CJ,ℝd, respectively, such that ξa=ψa,ξ^a=ψa,ξ′a=ψ′a,andξ′^a=ψ′a, and for t∈a,b,(71)ξt−ξa−tρ−aρρξρ,1a=1Γα∫atsρ−1tρ−sρρα−1fs,ξs+δ1sds,(72)ξ^t−ξ^a−tρ−aρρξ^ρ,1a=1Γα∫atsρ−1tρ−sρρα−1fs,ξ^s+δ2sds,respectively, where δ1t≤ε and δ2t≤εφt. To show the results of this theorem, the method of successive approximations will be used.
Prove the assertion (A1): we defineψ0t=ξt,t∈a,b, and we consider the sequence ψnn≥1∈CJ,ℝd which is defined as follows:(73)ψnt−ξa−tρ−aρρξρ,1a=ρ1−αΓα∫atsρ−1tρ−sρρα−1fs,ψn−1sds,∀t∈J.
By the abovementioned definition of successive approximations, forn=1 and from (71), one has(74)ψ1t−ψ0t=ψ1t−ξa−tρ−aρρξρ,1a−ξt−ξa−tρ−aρρξρ,1a≤ρ1−αΓα∫atsρ−1tρ−sρρα−1δ1sds≤εtρ−aραραΓα+1.
By the Lipschitz condition of the functionf, for any t∈J and n=2,3,4,…, one has(75)ψnt−ψn−1t=ψnt−ξa−tρ−aρρξρ,1a−ψn−1t−ξa−tρ−aρρξρ,1a≤Lρ1−αΓα∫atsρ−1tρ−sρα−1ψn−1s−ψn−2sds.
So, from (74) and for n=2, one obtains(76)ψ2t−ψ1t≤εLL2tρ−aρ2αρ2αΓ2α+1,and for n=3, we also obtain(77)ψ3t−ψ2t≤εLL3tρ−aρ3αρ3αΓ3α+1.
By using the mathematical induction method, forn≥4, we have(78)ψnt−ψn−1t≤εLLntρ−aρnαρnαΓnα+1.
Now, if we assume that (78) holds for n=k, then by (4) one obtains(79)ψk+1t−ψkt≤LΓα∫atsρ−1tρ−sρρα−1ψks−ψk−1sds≤Lρ1−αΓα∫atsρ−1tρ−sρα−1εLLksρ−aρkαρkαΓkα+1ds≤εLLk+1Γk+1α+1tρ−aρρk+1α.which is the inequality (78) for n=k+1. This yields that the inequality (78) is satisfied for all n≥1. Then, one has(80)∑n=1∞ψnt−ψn−1t≤εL∑n=1∞Ltρ−aραnραnΓnα+1≤εLEα,1Ltρ−aρρα−1.
Since the series of the right-hand side of the above inequality is convergent to(81)εLEα,1Ltρ−aρρα−1,that is,(82)∑n=1∞ψnt−ψn−1t≤εLEα,1Ltρ−aρρα−1.
This yields that the seriesψ0t+∑n=1∞ψnt−ψn−1t is uniformly convergent on J with respect to the norm ⋅. Now, assume that(83)ψ¯t=ψ0t+∑n=1∞ψnt−ψn−1t.
Then, we takeψkt which is the kth partial of series (83) of the form(84)ψkt=ψ0t+∑n=1kψnt−ψn−1t,∀t∈J.
By (83) and (84), we notice(85)limk⟶∞ψkt−ψ¯t=0,∀t∈J.
Setψt=ψ¯t for t∈J. We prove that the limit function ψ is a solution of the following integral equation on J:(86)ψt−ξa−tρ−aρρξρ,1a=ρ1−αΓα∫atsρ−1tρ−sρα−1fs,ψsds.
By using definition of successive approximation for anyt∈J and by the Lipschitz condition of f, we have(87)ψt−ξa−tρ−aρρξρ,1a−ρ1−αΓα∫atsρ−1tρ−sρα−1fs,ψsds≤ψ¯t−ψkt+ρ1−αΓα∫atsρ−1tρ−sρα−1fs,ψk−1sds−ρ1−αΓα∫atsρ−1tρ−sρα−1fs,ψsds≤ψ¯t−ψkt+Lρ1−αΓα∫atsρ−1tρ−sρα−1ψs−ψk−1sds.
On the other hand, (83) and (84) yield(88)ψ¯t−ψkt=∑n=k+1∞ψnt−ψn−1t≤∑n=k+1∞ψnt−ψn−1t.
Therefore, we obtain fort∈J(89)ψt−ψkt=ψ¯t−ψkt≤εL∑n=k+1∞Ltρ−aραnραnΓnα+1.
So, it follows from (87) that, for t∈J,(90)ψt−ξa−tρ−aρρξρ,1a−ρ1−αΓα∫atsρ−1tρ−sρρα−1fs,ψsds≤εL∑n=k+1∞Ltρ−aραnραnΓnα+1+εLLΓα∫atsρ−1tρ−sρα−1∑n=k+1∞Lsρ−aραnραnΓnα+1ds=εL∑n=k+1∞Ltρ−aραnραnΓnα+1+εL∑n=k+1∞Ltρ−aραn+1ραn+1Γn+1α+1.
Then, by taking limitk⟶∞, the right-hand side of (90) tends to 0. This yields (86) is a solution of problem (16) with initial conditions ψa=ξa,ψ′a=ξ′a. In addition, from (82) and (83), we also get the following estimation between the solution of problem (16) ψt and the solution of inequality (64):(91)ψt−ξt≤εLEα,1Ltρ−aρρα−1≤εLEα,1Ltρ−aρρα,∀t∈J.
Then, by Definition 1-(i) we can deduce problem (16) is Ulam–Hyers–Mittag-Leffler stable, where Cf≔1/L and βf≔L.
Prove the assertion (A2): similar to the proof of the assertion (I), we also consider the sequenceψnn≥0∈Ca,b,ℝd given by ψ0t=ξ^t,t∈a,b, and for n≥1,(92)ψnt−ξ^a−tρ−aρρξ^ρ,1a=ρ1−αΓα∫atsρ−1tρ−sρα−1fs,ψn−1sds,∀t∈a,b.
By similar processing as the case of (A1), we get forn≥1 and for t∈J,(93)ψnt−ψn−1t≤εLKφLnφt.
In order to check the validity of (93), for n=1, from (72) and by using the definition of successive approximations (92), one obtains(94)ψ1t−ψ0t=ψ1t−ξ^a−tρ−aρρξ^ρ,1a−ξ^t−ξ^a−tρ−aρρξ^ρ,1a≤ρ1−αΓα∫atsρ−1tρ−sρα−1δ2sds≤εKφφt.
Suppose that (93) is true for n=k∈ℕ, that is,(95)ψkt−ψk−1t≤εLKφLkφt.
Then, by the Lipschitz condition off and the hypothesis of the assertion (A2), we obtain(96)ψk+1t−ψkt=ψk+1t−ξ^a−tρ−aρρξ^ρ,1a−ψkt−ξ^a−tρ−aρρξ^ρ,1a≤Lρ1−αΓα∫atsρ−1tρ−sρα−1ψks−ψk−1sds≤Lρ1−αΓα∫atsρ−1tρ−spα−1εLKφLkφsds≤εLKφLk+1φt.
This proves inequality (93) is valid for all n≥1. On the other hand, by the hypothesis KφL∈0,1 and from (93) we have that, for t∈J,(97)∑n=1∞ψnt−ψn−1t≤εφt∑n=1∞KφLn≤εKφ1−KφLφt.
Since the functionφt is continuous on J, it is bounded. So, inequality (97) yields that the series ψ0t+∑n=1∞ψnt−ψn−1t is absolutely and uniformly convergent on J with respect to the distance ⋅. Therefore, we set, for t∈J,(98)ψt=ψ0t+∑n=0∞ψn+1t−ψnt.
Then, similar to the proof of (A1) we also get the estimation between the solution of problem (16) ψt and the solution of inequality (65) ξ^t as follows:(99)ψt−ξ^t≤εKφ1−KφLφt,t∈a,b.
Taking the integralIa+α,ρ on a,b of the order α∈1,2 on both sides of (99) and using hypothesis (H3), one obtains(100)p1−αΓα∫atsρ−1tρ−sρα−1ψs−ξsds≤εKφ1−KφLρ1−αΓα∫atsρ−1tρ−sρα−1φsds≤εKφ21−KφLφt.
This yields that(101)ψt−ξt+ρ1−αΓα∫atsρ−1tρ−sρα−1ψs−ξsds≤εKφ+Kφ21−KφLφt.
Therefore, we have(102)ψt−ξt≤εKφ+Kφ21−KφLφt+ρ1−αΓα∫atsρ−1tρ−sρα−1ψs−ξsds.
Setpt=ψt−ξt,qt=εKφ+Kφ2/1−KφLφt,rt=1/Γα. Then, equation (102) becomes(103)pt≤qt+rtρ1−α∫atsρ−1tρ−sρα−1psds.
By using the Gronwall inequality in1, (103) yields that(104)pt≤qt+∫at∑i=1∞rtΓαiΓiαsρ−1tρ−sρiα−1qsds.
Furthermore, since the functionqt is nondecreasing on a,b, we obtain(105)pt≤qtEα,1rtΓαtρ−aρρα.
Consequently, we obtain(106)ψt−ξt≤εKφ+Kφ21−KφLφtEα,1tρ−aρρα,∀t∈a,b.
Then, by Definition 1-(iii) we can deduce problem (16) is Ulam–Hyers–Rassias–Mittag-Leffler stable, where Cf,φ≔Kφ+Kφ2/1−KφLφt and βf≔1.


## 5. Conclusion

In this work, the global existence and uniqueness of the solution to the Caputo-generalized fractional differential equation are investigated. We discussed the stability in various Ulam–Hyers–Mittag-Leffler’s types via the successive approximation method.



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