# Existence, Uniqueness, and Mittag–Leffler–Ulam Stability Results for Cauchy Problem Involvingψ-Caputo Derivative in Banach and Fréchet Spaces

Authors: Choukri Derbazi; Zidane Baitiche; Mouffak Benchohra; G. N’Guérékata
Journal: International Journal of Differential Equations
 (2020)

---

## Abstract

Our aim in this paper is to investigate the existence, uniqueness, and Mittag–Leffler–Ulam stability results for a Cauchy problem involvingψ-Caputo fractional derivative with positive constant coefficient in Banach and Fréchet Spaces. The techniques used are a variety of tools for functional analysis. More specifically, we apply Weissinger’s fixed point theorem and Banach contraction principle with respect to the Chebyshev and Bielecki norms to obtain the uniqueness of solution on bounded and unbounded domains in a Banach space. However, a new fixed point theorem with respect to Meir–Keeler condensing operators combined with the technique of Hausdorff measure of noncompactness is used to investigate the existence of a solution in Banach spaces. After that, by means of new generalizations of Grönwall’s inequality, the Mittag–Leffler–Ulam stability of the proposed problem is studied on a compact interval. Meanwhile, an extension of the well-known Darbo’s fixed point theorem in Fréchet spaces associated with the concept of measures of noncompactness is applied to obtain the existence results for the problem at hand. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility of the main theorems.


---

## Body

## 1. Introduction

Fractional differential equations gained much attention due to their applications in various fields of science and engineering (see, for instance, [1–7] and the references therein). For more information about the basic theory of fractional differential equations, we can refer to the monographs [8–11] and references cited therein. Besides the classical and fractional-order differential and integral operators, there is another kind of fractional derivatives that appears in the literature called ψ-Caputo fractional derivative, which was introduced by Almeida in [12], where the kernel operator contains a special function of an arbitrary exponent. According to this idea, a wide class of well-known fractional derivatives are obtained like Caputo and Caputo–Hadamard for particular choices of ψt. Additionally, some interesting details about the ψ-fractional derivatives and integrals can be found in [13–23]. Moreover, fixed point theory is a very useful tool in the theory of the existence of solutions to functional and differential equations; the reader is advised to see references [24–29] in which many scholars turned to the existence and uniqueness of solutions for differential equations involving different kinds of fractional derivatives under various boundary conditions. On the contrary, the notion of measure of noncompactness was first introduced by Kuratowski [30] in 1930 which was further extended to general Banach spaces by Banás and Goebel [31]. Later Darbo formulated his celebrated fixed point theorem in 1955 for the case of the Kuratowski measure of noncompactness (cf. [32]) which generalizes both the classical Schauder fixed point principle and (a special variant of) Banach’s contraction mapping principle. After that, the Darbo fixed point theorem has been generalized in many different directions; we suggest some works for reference [33–36]. The reader may also consult [37–43] and references therein where several applications of the measure of noncompactness can be found.Very recently Dudek [44] proved a new fixed point theorem using the concept of measures of noncompactness in Fréchet spaces which generalize the famous Darbo’s fixed point theorem. To see more applications about the usefulness of this new fixed point theorem to prove the existence of solutions for certain classes of functional integral equations in Fréchet spaces, the reader can refer to [45–50].On the contrary, one of the important parts of the qualitative theory of linear and nonlinear differential equations is the Ulam–Hyers stability, first formulated by Hyers and Ulam in 1940 [51–53]. Furthermore, the fractional Ulam stability was introduced by Wang et al. [54]. For some recent results of stability analysis by different types of fractional derivative operator, we refer the reader to articles [55–64], as well as to the recent book by Abbas et al. [65] and the references cited therein. More recently, some authors explored another form of stability known as Mittag–Leffler–Ulam–Hyers for the solutions of fractional differential equations [66–73].Inspired by the above works, our goal is to extend the studies in [29, 37, 45, 72]. More precisely, we consider first the problem of the existence, uniqueness, and Mittag–Leffler–Ulam–Hyers stability for the following initial value problem of the fractional differential equation with constant coefficient λ>0 in Banach spaces of the form:(1)Dca+α;ψxt=λxt+ft,xt,t∈J≔a,b,xa=ϕ0,where Dca+α;ψ is the ψ-Caputo fractional derivatives such that 0<α≤1,f:J×X⟶X is a given function satisfying some assumptions that will be specified later, X is a Banach space with norm ⋅, and ϕ0∈X. Moreover, we also extend the above problem to give a uniqueness results on unbounded domains in a Banach space via Banach contraction principle coupled with Bielecki-type norm.Next, we turn our attention to the existence of solutions for the same problem (1) in the Fréchet spaces. In precise terms, we investigate the existence of solutions for the following problem:(2)Dca+α;ψxt=λxt+ft,xt,t∈J′≔a,∞,xa=ϕ0.The structure of the present work is organized as follows: in Section2, we collect some basic concepts on the fractional integrals and derivatives, auxiliary results, lemmas and notions of measures of noncompactness, and fixed point theorems that are used throughout this paper. In Section 3, based on Weissinger’s fixed point theorem combined with the Chebyshev norm, we give a uniqueness result for problem (1) on a compact interval in a Banach space. In Section 4, using the ideas of Hausdorff measure of noncompactness and Meir–Keeler condensing operator, we present the existence of solutions of IVP (1) in Banach spaces. In Section 5, we discuss the Mittag–Leffler–Ulam stability results for the problem at hand. In Section 6, we apply the Banach fixed point theorem coupled with a Bielecki-type norm to derive the uniqueness of solution on unbounded domains in a Banach space. In Section 7, we look into the existence of solutions for the IVP (2) in the Fréchet spaces via Darbo’s fixed point theorem. The last section provides a couple of examples to illustrate the applicability of the results developed.


## 2. Preliminaries and Background Materials

In this section, we present some basic notations, definitions, and preliminary results, which will be used throughout this paper.LetJ≔a,b0<a<b<∞ be a finite interval and ψ:J⟶ℝ be an increasing function with ψ′t≠0, for all t∈J, and let CJ,X be the Banach space of all continuous functions x from J into X with the supremum (uniform) norm:(3)x∞=supt∈Jxt.A measurable functionx:J⟶X is Bochner integrable if and only if x is Lebesgue integrable.ByL1J,X, we denote the space of the Bochner integrable functions x:J⟶X, with the norm(4)x1=∫abxtdt.Now, we define the Hausdorff measure of noncompactness and give some of its important properties.Definition 1.
(see [31]). Let X be a Banach space and B a bounded subset of X. Then the Hausdorff measure of noncompactness of B is defined by(5)χB=infε>0:Bcan be covered by finitely many balls with radius<ε.
To discuss the problem in this paper, we need the following lemmas.Lemma 1.
LetA,B⊂X be bounded. Then the Hausdorff measure of noncompactness has the following properties. For more details and the proof of these properties see [31]:(1)
χA=0⟺A is relatively compact(2)
A⊂B⟹χA≤χB(3)
χA∪B=maxχA,χB(4)
χA=χA¯=χconvA, where A¯ and convA represent the closure and the convex hull of A, respectively(5)
χA+B≤χA+χB, where A+B=x+y:x∈A,y∈B(6)
χβA≤βχA, for any β∈ℝ
Now, we recall some fixed point theorems that will be used laterTheorem 1 (Weissinger’s fixed point theorem [74]).
AssumeE,d to be a nonempty complete metric space and let βj≥0 for every j∈ℕ such that ∑j=0∞βj converges. Furthermore, let the mapping T:E⟶E satisfy the following inequality:(6)dTju,Tjv≤βjdu,v,for every j∈ℕ and every u,v∈E. Then, T has a unique fixed point u∗. Moreover, for any v0∈E, the sequence Tjv0j=1∞ converges to this fixed point u∗.
On the contrary, in 1969, the concepts of the Meir–Keeler contraction mapping were introduced by Meir and Keeler.Definition 2 (see [75]).
LetE,d be a metric space. Then a mapping T on E is said to be a Meir–Keeler contraction (MKC, for short); if for any ε>0, there exists δ>0 such that(7)ε≤du,v<ε+δ⟹dTu,Tv<ε,∀u,v∈E.
In [34], the authors defined the notion of the Meir–Keeler condensing operator on a Banach space and gave some fixed point results.Definition 3 (see [34]).
LetC be a nonempty subset of a Banach space X and μ arbitrary measure of noncompactness on X. We say that an operator T:C⟶C is a Meir–Keeler condensing operator if for any ε>0, there exists δ>0 such that(8)ε≤μΩ<ε+δ⟹μTΩ<ε,for any bounded subset Ω of C.
The following fixed point theorem with respect to the Meir–Keeler condensing operator which is introduced by Aghajani et al. [34] plays a key role in the proof of our main results.Theorem 2 (see [34]).
LetΩ be a nonempty, bounded, closed, and convex subset of a Banach space X. Also, let μ be an arbitrary measure of noncompactness on X. If T:Ω⟶Ω is a continuous and Meir–Keeler condensing operator, then T has at least one fixed point and the set of all fixed points of T in Ω is compact.
The following lemmas are needed in our argument.Lemma 2 (see [76]).
LetX be a Banach space. If B⊂CJ,X is bounded, then χBt≤χCB for any t∈J, where Bt=xt:x∈B,t∈J, and χC is the Hausdorff measure of noncompactness defined on the bounded sets of CJ,X. Furthermore if B is equicontinuous, then t⟶χBt is continuous on J, and(9)χCB=maxt∈JχBt.Lemma 3 (see [77]).
LetX be a Banach space and let B⊂X be bounded. Then for each ε, there is a sequence xnn=1∞⊂B, such that(10)χB≤2χxnn=1∞+ε.
We callB⊂L1J,X uniformly integrable if there exists η∈L1J,ℝ+ such that(11)xt≤ηt,for allx∈Band a.e.t∈J.Lemma 4 (see [78]).
Ifxnn=1∞⊂L1J,X is uniformly integrable, then t⟼χxntn=1∞ is measurable, and(12)χ∫atxnsdsn=1∞≤2∫atχxnsn=1∞ds.
Before introducing the basic facts on fractional operators, we recall three types of functions that are important in fractional calculus: the gamma, beta, and Mittag–Leffler functionsDefinition 4 (see [79]).
The gamma function, or the second-order Euler integral, denotedΓ⋅ is defined as(13)Γα=∫0+∞e−ttα−1dt,α>0.Definition 5 (see [79]).
The beta function, or the first-order Euler function, can be defined as(14)Bα,β=∫011−tα−1tβ−1dt,α,β>0.
We use the following formula which expresses the beta function in terms of the gamma function:(15)Bα,β=ΓαΓβΓα+β,α,β>0.
The next function is a direct generalization of the exponential series.Definition 6 (see [79]).
The one-parameter Mittag–Leffler functionEα⋅ is defined as(16)Eαz=∑k=0∞zkΓαk+1,z∈ℝ,α>0.
Forα=1, this function coincides with the series expansion of ez, i.e.,(17)E1z=∑k=0∞zkΓk+1=∑k=0∞zkk!=ez.Definition 7 (see [79]).
The two-parameter Mittag–Leffler functionEα,β⋅ is defined as(18)Eα,βz=∑k=0∞zkΓαk+β,α,β>0andz∈ℝ.
Now, we give some results and properties from the theory of fractional calculus. We begin by definingψ-Riemann–Liouville fractional integrals and derivatives, in what follows.Definition 8 (see [2, 12]).
Forα>0, the left-sided ψ-Riemann–Liouville fractional integral of order α for an integrable function x:J⟶ℝ with respect to another function ψ:J⟶ℝ that is an increasing differentiable function such that ψ′t≠0, for all t∈J, is defined as follows:(19)ℐa+α;ψxt=1Γα∫atψ′sψt−ψsα−1xsds,where Γ is the gamma function.
Note that equation (19) is reduced to the Riemann–Liouville and Hadamard fractional integrals when ψt=t and ψt=lnt, respectively.
The integer order of the differential operatorxψ1 with respect to another function ψ:J⟶ℝ that is an increasing differentiable function such that ψ′t≠0, for all t∈J is defined by(20)xψ1t=1ψ′tddtxt.
Furthermore, forn∈ℕ, we use the symbol xψn to indicate the n-th composition of xψ1 with itself; that is, we put(21)xψnt≔xψ1xψn−1t=1ψ′tddtnxt,n≥2.Definition 9 (see [12]).
Letn∈ℕ and let ψ,x∈CnJ,ℝ, be two functions such that ψ is increasing and ψ′t≠0, for all t∈J. The left-sided ψ–Riemann–Liouville fractional derivative of a function x of order α is defined by(22)Da+α;ψxt=xψnℐa+n−α;ψxt=1Γn−α1ψ′tddtn∫atψ′sψt−ψsn−α−1xsds,where n=α+1.Definition 10 (see [12]).
Letn∈ℕ and let ψ,x∈CnJ,ℝ, be two functions such that ψ is increasing and ψ′t≠0, for all t∈J. The left-sided ψ-Caputo fractional derivative of x of order α is defined by(23)Dca+α;ψxt=ℐa+n−α;ψxψnt,where n=α+1 for α∉ℕ and n=α for α∈ℕ.
From the definition, it is clear that(24)Dca+α;ψxt=∫atψ′sψt−ψsn−α−1Γn−αxψnsds,ifα∉ℕ,xψnt,ifα∈ℕ.
This generalization (24) yields the Caputo fractional derivative operator when ψt=t. Moreover, for ψt=lnt, it gives the Caputo–Hadamard fractional derivative.
Some basic properties are listed in the following lemma.Lemma 5 (see [2, 12]).
Letα,β>0, and x∈CJ,ℝ. Then for each t∈J, we have(1)
Dca+α;ψℐa+α;ψxt=xt(2)
ℐa+α;ψDca+α;ψxt=xt−xa,0<α≤1(3)
ℐa+α;ψ(ψt−ψa)β−1=Γβ/Γβ+α(ψt−ψa)β+α−1(4)
Dca+α;ψ(ψt−ψa)β−1=Γβ/Γβ−α(ψt−ψa)β−α−1(5)
Dca+α;ψ(ψt−ψa)k=0,for allk∈0,…,n−1,n∈NRemark 1.
Note that for an abstract functionx:J⟶X, the integrals which appear in the previous definitions are taken in Bochner’s sense (see, for instance, [80]).
In the sequel, we will make use of the following generalizations of Grönwall’s lemmasTheorem 3 (see [23]).
Letu,v be two integrable functions and w continuous, with domain J. Let ψ∈C1J,ℝ+ be an increasing function such that ψ′t≠0,∀t∈J. Assume that(1)
u and v are nonnegative(2)
w is nonnegative and nondecreasing.
If(25)ut≤vt+wt∫atψ′sψt−ψsα−1usds,t∈J,then(26)ut≤vt+∫at∑n=0∞wtΓαnΓnαψ′sψt−ψsnα−1vsds,t∈J.Corollary 1 (see [23]).
Under the hypotheses of Theorem3, let v be a nondecreasing function on J. Then, we have(27)ut≤vtEαΓαwtψt−ψaα,t∈J.where Eα⋅ is a Mittag–Leffler function with one parameter.Lemma 6 (see [14]).
Letα,β>0. Then for all t∈J, we have(28)ℐa+α;ψEαβψt−ψaα=1βEαβψt−ψaα−1.Remark 2.
Observe that from Lemma6 if β=1, we can get the following inequality:(29)ℐa+α;ψEαψt−ψaα≤Eαψt−ψaα.
For the existence of solutions for the problem (1), we need the following lemma.Lemma 7.
Letf∈CJ×ℝ,ℝ. Then x∈CJ,ℝ is the solution of(30)Dca+α;ψxt=λxt+ft,xt,t∈J,xa=ϕ0∈ℝ,if and only if it is the solution of the integral equation:(31)xt=ϕ0+ℐa+α;ψλxt+ft,xt=ϕ0+λ∫atψ′sψt−ψsα−1Γαxsds+∫atψ′sψt−ψsα−1Γαfs,xsds.Proof.
Letxt be a solution of the problem (30). Define ht=λxt+ft,xt. Then,(32)Dca+α;ψxt=ht,0<α≤1.
Since(33)Dca+α;ψxt=ℐa+1−α;ψ1ψ′tddtxt=ht,0<α≤1,taking the ψ-Riemann–Liouville fractional integral of order α to the above equation, we get(34)ℐa+1;ψ1ψ′tddtxt=ℐa+α;ψht,0<α≤1.
Since(35)ℐa+1;ψ1ψ′tddtxt=ℐa+1ddtxt=xt−xa,we get(36)xt=ϕ0+ℐa+α;ψht.
Using the definition ofht, we obtain equation (31). Conversely, suppose that xt is the solution of the equation (31). Then, it can be written as(37)xt=ϕ0+ℐa+α;ψht,where ht=λxt+ft,xt. Since ht is continuous and ϕ0 is constant, operating the ψ-Caputo fractional differential operator Dca+α;ψ on both sides of equation (37), we obtain(38)Dca+α;ψxt=Dca+α;ψϕ0+Dca+α;ψℐa+α;ψht.
Using Lemma5, the following is obtained:(39)Dca+α;ψxt=λxt+ft,xt.
From equation (37), we get xa=ϕ0. This proves that xt is the solution of the Cauchy problem (30) which completes the proof.
Now, we are ready to present our main results.


## 3. Uniqueness Result with respect to the Chebyshev Norm and Weissinger’s Fixed Point Theorem

First of all, we define what we mean by a solution of equation (1).Definition 11.
A functionx∈CJ,X is said to be a solution of equation (1) if x satisfies the equation Dca+α;ψxt=λxt+ft,xt on J and the condition xa=ϕ0.Theorem 4.
Let the following assumptions hold:(H1) The functionf:a,b×X⟶X is continuous.(H2) There exists a constantL>0 such that(40)ft,x−ft,y≤Lx−y,for any x,y∈X and t∈J.
Then there exists a unique solution of equation (1) on J.Proof.
In view of Lemma7, we introduce an operator T:CJ,X⟶CJ,X associated with equation (1) as follows:(41)Txt=ϕ0+λ∫atψ′sψt−ψsα−1Γαxsds+∫atψ′sψt−ψsα−1Γαfs,xsds.
Clearly, the fixed points of the operatorT are solutions of equation (1). Weissinger’s fixed point theorem will be used to prove that T has a fixed point. For this reason, we shall show that T is a contraction. Let x,y∈CJ,X. Then, for every n∈ℕ and t∈J, using (H2), we have(42)Tnxt−Tnyt=‖TTn−1xt−TTn−1yt≤∫atψ′sψt−ψsα−1Γαfs,Tn−1xs−fs,Tn−1ysds+λ∫atψ′sψt−ψsα−1ΓαTn−1xs−Tn−1ysds≤L+λ∫atψ′sψt−ψsα−1ΓαTn−1xs−Tn−1ysds≤L+λ2∫atψ′sψt−ψs2α−1Γ2αTn−2xs−Tn−2ysds⋮≤L+λn∫atψ′sψt−ψsnα−1Γnαxs−ysds.
Therefore, we conclude(43)Tnx−Tny∞≤L+λψb−ψaαnΓnα+1x−y∞,for each n∈ℕ and all x,y∈CJ,X.
Now let(44)βn=L+λψb−ψaαnΓnα+1.
By Definition (6), we have(45)∑n=0∞βn=∑n=0∞L+λψb−ψaαnΓnα+1=EαL+λψb−ψaα.
Therefore, the existence of the unique fixed point ofT follows from Weissinger’s fixed point theorem. That is, (1)fd1 has a unique solution. This completes the proof.


## 4. Existence Result via Meir–Keeler Condensing Operators

In this section, we can weaken the condition (H2) to a linear growth condition. But now Theorem2 that we apply will only guarantee the existence not also the uniqueness of the solution.Theorem 5.
Assume that the hypothesis (H1) holds. Furthermore, we impose the following:(H3) There exist continuous functionsμ,ν:J⟶ℝ+ such that(46)ft,x≤μt+νtx,for anyx∈X and t∈J.(H4) For each bounded setB⊂X, and each t∈J, the following inequality holds:(47)χft,B≤νtχB.
Then the problem (1) has at least one solution defined on J provided that(48)4ν∗+λℓα,ψ<1,where(49)ν∗=supt∈Jνt,ℓα,ψ=ψb−ψaαΓα+1.Proof.
Consider the operatorT defined by equation (41) and define a bounded closed convex set(50)Ωr=x∈CJ,X:x∞≤r.with(51)r≥ϕ0+ℓα,ψμ∗1−ν∗+λℓα,ψ,μ∗=supt∈Jμt.
We shall show that the operatorT satisfies all the assumptions of Theorem 2. We split the proof into four steps:Step 1. The operatorT maps the set Ωr into itself. By the assumption (H3), we have(52)Txt≤ϕ0+∫atψ′sψt−ψsα−1Γαfs,xsds+λ∫atψ′sψt−ψsα−1Γαxsds≤ϕ0+∫atψ′sψt−ψsα−1Γαμs+νs+λxsds≤ϕ0+μ∗+ν∗+λx∞∫atψ′sψt−ψsα−1Γαds≤ϕ0+ψb−ψaαΓα+1μ∗+ν∗+λr=ϕ0+ℓα,ψμ∗+ν∗+λr≤r.Thus(53)Tx∞≤r.This proves thatT transforms the ball Ωr into itself.Step 2. The operatorT is continuous. Suppose that xn is a sequence such that xn⟶x in Ωr as n⟶∞. It is easy to see that fs,xns⟶fs,xs,asn⟶+∞ due to the continuity of f. On the contrary, taking (H3) into consideration, we get the following inequality:(54)ψ′sψt−ψsα−1fs,xns−fs,xs≤2μs+νsrψ′sψt−ψsα−1.We notice that since the functions⟼2μs+νsrψ′sψt−ψsα−1 is the Lebesgue integrable over a,t. This fact together with the Lebesgue dominated convergence theorem implies that(55)∫atψ′sψt−ψsα−1Γαfs,xns−fs,xsds⟶0asn⟶+∞.It follows thatTxn−Tx∞⟶0asn⟶+∞, which implies the continuity of the operator T.Step 3.TΩr is equicontinuous.For anya<t1<t2<b and x∈Ωr, we get(56)Txt2−Txt1≤∫at1ψ′sψt1−ψsα−1−ψt2−ψsα−1Γαλxs+fs,xsds+∫t1t2ψ′sψt2−ψsα−1Γαλxs+fs,xsds≤μ∗+ν∗+λrΓα+1ψt1−ψaα+2ψt2−ψt1α−ψt2−ψaα≤2μ∗+ν∗+λrΓα+1ψt2−ψt1α,where we have used the fact thatψt1−ψaα−ψt2−ψaα≤0. Therefore,(57)Txt2−Txt1≤2μ∗+ν∗+λrΓα+1ψt2−ψt1α.Ast2⟶t1, the right-hand side of the above inequality tends to zero independently of x∈Ωr. Hence, we conclude that TΩr⊆CJ,X is bounded and equicontinuous.Step 4. Now, we prove thatT:Ωr⟶Ωr is a Meir–Keeler condensing operator. To do this, suppose ε>0 is given. We will prove that there exists δ>0 such that(58)ε≤χCB<ε+δ⟹χCTB<ε,for anyB⊂Ωr.For every bounded and equicontinuous subsetB⊂Ωr and ε′>0 using Lemma 3 and the properties of χ, there exist sequences xnn=1∞⊂B such that(59)χTBt≤2χ∫atψ′sψt−ψsα−1Γαλxnsn=1∞+fs,xnsn=1∞ds+ε′.Next, by Lemma4 and (H4), we have(60)χTBt≤4∫atψ′sψt−ψsα−1Γαλχxnsn=1∞+χfs,xnsn=1∞ds+ε′≤4ν∗+λΓα∫atψ′sψt−ψsα−1χxnsn=1∞ds+ε′≤4ν∗+λψb−ψaαΓα+1χCB+ε′=4ν∗+λℓα,ψχCB+ε′.As the last inequality is true, for everyε′>0, we infer(61)χTBt≤4ν∗+λℓα,ψχCB.SinceTB⊂Ωr is bounded and equicontinuous, we know from Lemma 3 that(62)χCTB=maxt∈JχTBt.Therefore, we have(63)χCTB≤4ν∗+λℓα,ψχCB.Observe that from the last estimates(64)χCTB≤4ν∗+λℓα,ψχCB<ε⟹χCB<14ν∗+λℓα,ψε.Let us now take(65)δ=1−4ν∗+λℓα,ψ4ν∗+λℓα,ψε,so we get(66)ε≤χB<ε+δ.which means thatT:Ωr⟶Ωr is a Meir–Keeler condensing operator. It follows from Theorem 2 that the operator T defined by (41) has at least one fixed point x∈Ωr, which is just the solution of the initial value problem (1). This completes the proof of Theorem 5.


## 5. Mittag–Leffler–Ulam–Hyers Stability Analysis

In this section, we discuss the Mittag–Leffler–Ulam–Hyers stability analysis of the solutions to the proposed problem (1).Now, we consider the Mittag–Leffler–Ulam–Hyers stability for problem (1).Letε,λ>0 and Φ:J⟶ℝ+ be a continuous function. We consider the following inequalities:(67)Dca+α;ψxt−λxt−ft,xt≤εEαψt−ψaα,t∈J,(68)Dca+α;ψxt−λxt−ft,xt≤εΦtEαψt−ψaα,t∈J.Definition 12 (see [71, 73]).
Equation (1) is Mittag–Leffler–Ulam–Hyers stable, with respect to Eαψt−ψaα if there exists a real number c>0 such that for each ε>0 and for each solution y∈CJ,X of the inequality (67), there exists a solution x∈CJ,X of equation (1) with(69)yt−xt≤cεEαψt−ψaα,t∈J.Definition 13 (see [71, 73]).
Equation (1) is generalized Mittag–Leffler–Ulam–Hyers stable, with respect to Eαψt−ψaα if there exists ω:Cℝ+,ℝ+ with ω0=0 such that for each ε>0 and for each solution y∈CJ,X of the inequality (67), there exists a solution x∈CJ,X of equation (1) with(70)yt−xt≤ωεEαψt−ψaα,t∈J.Remark 3 (see [71, 73]).
It is clear that Definition12⟹ Definition 13,Remark 4 (see [71, 73]).
A functiony∈CJ,X is a solution of the inequality (67) if and only if there exists a function z∈CJ,X (which depends on solution y) such that(i)
zt≤εEαψt−ψaα,t∈J(ii)
Dca+α;ψyt=λyt+ft,yt+zt,t∈J
Now we are ready to state our Mittag–Leffler–Ulam–Hyers stability of solution to the problem (1). The arguments are based on the Grönwall inequality equation (27).Theorem 6 Assume that (H1) and (H2) hold.
Then problem (1) is Mittag–Leffler–Ulam–Hyers stable on J and consequently generalized Mittag–Leffler–Ulam–Hyers stable.Proof.
Letε,λ>0 and let y∈CJ,X be a function which satisfies the inequality (67) and let x∈CJ,X be the unique solution of the following problem:(71)Dca+α;ψxt=λxt+ft,xt,t∈J≔a,b,xa=ϕ0.
By Lemma7, we have(72)xt=ϕ0+∫atψ′sψt−ψsα−1Γαλxs+fs,xsds.
Since we have assumed thaty is a solution of the inequality (67), we have the following by Remark 4:(73)Dca+α;ψyt=λyt+ft,yt+zt,t∈J≔a,b,ya=ϕ0.
Again by Lemma7, we have(74)yt=ϕ0+∫atψ′sψt−ψsα−1Γαλys+fs,ys+zsds.
On the contrary, we have, for eacht∈J,(75)yt−xt≤∫atψ′sψt−ψsα−1Γαzsds+∫atψ′sψt−ψsα−1Γαfs,ys−fs,xsds+λ∫atψ′sψt−ψsα−1Γαys−xsds.
Hence, using Remark2 and part (i) of Remark 4 and (H2), we can get(76)yt−xt≤εEαψt−ψaα+L+λ∫atψ′sψt−ψsα−1Γαys−xsds.
Applying Corollary1 (the Grönwall inequality equation (27)) to the above inequality with ut=yt−xt,vt=εEαψt−ψaα, and wt=L+λ/Γα. Since vt is a nondecreasing function on J, we conclude that(77)yt−xt≤vtEαL+λψt−ψaα,t∈J,which yields(78)yt−xt≤Eαψt−ψaαEαL+λψb−ψaαε,t∈J.
Taking for simplicity(79)c=EαL+λψb−ψaα,then (78) becomes(80)yt−xt≤cεEαψt−ψaα,t∈J.
In consequence, it follows that(81)y−x∞≤cεEαψt−ψaα.
Thus, the problem (1) is Mittag–Leffler–Ulam–Hyers stable. Furthermore, if we set ωε=cε;ω0=0, then the problem (1) is generalized Mittag–Leffler–Ulam–Hyers stable. This completes the proof.


## 6. Uniqueness Result with respect to the Bielecki Norms and Banach’s Fixed Point Theorem on an Unbounded Domain

In this section, we present an uniqueness result concerning the problem (1) on unbounded domain, i.e., in the case J′=[a,+∞).Theorem 7.
Iff:J′×X⟶X be a continuous function that satisfies the Lipschitz condition with respect to the second variable, i.e., there exists a positive constant M such that(82)ft,x−ft,y≤Mx−y,for all t∈J′ and each x,y∈X.
Then, the problem (1) possesses a unique solution defined on J′=[a,+∞).Proof.
Consider the Banach spaceCJ′,X equipped with a Bielecki norm type ⋅B defined as below:(83)xB≔supt∈J′xtEαβψt−ψaα,where β>0 will be chosen later and Eα⋅ is the Mittag–Leffler function which is given in Definition 6. (for more properties on the Bielecki-type norm, see [25, 28]).
Consider the operatorA:CJ′,X⟶CJ′,X defined by(84)Axt=ϕ0+∫atGψαt,sfs,xsds.t∈J′,where(85)Gψαt,s=ψ′sψt−ψsα−1Γα.
Now, we prove that the operatorA is a contraction mapping on CJ′,X with respect to the Bielecki norm. To this end, we apply the Banach fixed point theorem to prove that A has a fixed point. Given x,y∈CJ′,X and t∈J′, using the inequality (82), and Lemma 6, we can get(86)Axt−Ayt≤∫atGψαt,sMxs−ysEαβψs−ψaαEαβψs−ψaαds+λ∫atGψαt,sxs−ysEαβψs−ψaαEαβψs−ψaαds≤M+λβEαβψt−ψaα−1x−yB.
Hence, we have(87)Ax−AyB≤M+λβ1−1Eαβψt−ψaαx−yB.
Note thatEα⋅ is a monotone increasing function on J′, then we get(88)Ax−AyB≤M+λβx−yB.
Since we can chooseβ>0 sufficiently large such that(89)M+λβ<1,it follows that the mapping A is a contraction with respect to the Bielecki norm. Hence, by the Banach fixed point theorem, A has a unique fixed point which is a unique solution of the initial value problem (1) in the space CJ′,X. This completes the proof.


## 7. An Existence Result in Fréchet Spaces via Darbo’s Fixed Point Theorem

By using Darbo’s fixed point theorem, we give in this section our last existence theorem concerning the IVP (2) in the Fréchet spaces.Firstly, we need to fix the notation. LetJ′=[a,+∞) and let Jn≔a,n,n∈ℕ∗.In this section, we letE≔CJ′,X to be the Fréchet space of all continuous functions x from J′ into X, equipped with the family of seminorms:(90)xn=supt∈Jnxt,n∈ℕ∗.Definition 14.
A nonempty subsetB⊂E is said to be bounded if(91)supx∈Bxn<∞,n∈ℕ∗.
Next we present some facts concerning the notion of a sequence of measures of noncompactness in the Fréchet spaces [44, 50].
In what follows,F will be a real Fréchet space. If B is a nonempty subset of F, then B¯ and ConvB denote the closure and the closed convex closure of B, respectively. Also, we denote by mF the family of all nonempty and bounded subsets of F and nF the family of all relatively compact subsets of F.Definition 15.
A family of functionsμnn∈ℕ where μn:mF⟶ℝ+=[0,∞) is said to be a family of measures of noncompactness in the real Fréchet space F if it satisfies the following conditions:(1)
The familyKerμn=B∈mF;μnB=0,forn∈ℕ is nonempty and Kerμn∈nF(2)
B1⊂B2⟹μnB1≤μnB2, for all B1,B2∈mF and n∈ℕ∗(3)
μnConvB=μnB, for all B∈mF and n∈ℕ(4)
IfBi is a sequence of closed subsets of mF such that Bi+1⊂Bi for i=1,2,3,…, and limi⟶∞μnBi=0 for each n∈ℕ, then B∞≔∩i=1∞Bi≠∅
The following lemmas are needed in our argument.Lemma 8 (see [77]).
IfB is a bounded subset of a Fréchet space F, then for each ε>0, there is a sequence xkk=1∞⊂B, such that(92)μnB≤2μnxkk=1∞+ε,for eachn∈ℕ∗.Lemma 9 (see [78]).
Ifxkk=1∞⊂L1Jn,X is uniformly integrable, then μnxkk=1∞ is measurable, for n∈ℕ∗ and(93)μn∫atxksdsk=1∞≤2∫atμnxksk=1∞ds,for each t∈Jn.Definition 16.
LetΩ be a nonempty subset of a Fréchet space F, and let A:Ω⟶F be a continuous operator which transforms bounded subsets of Ω onto bounded ones. One says that A satisfies the Darbo condition with constants knn∈ℕ∗ with respect to a family of measures of noncompactness μnn∈ℕ∗, if(94)μnAB≤knμnB,for each bounded set B⊂Ω and n∈ℕ∗.
Ifkn<1,n∈ℕ∗, then A is called a contraction with respect to μnn∈ℕ∗.
The following generalization of the classical Darbo fixed point theorem for the Fréchet spaces plays a key role in the proof of our main results.Theorem 8.
(see [44]). Let Ω be a nonempty, bounded, closed, and convex subset of a Fréchet space F and let A:Ω⟶Ω be a continuous mapping. Suppose that A is a contraction with respect to a family of measures of noncompactness μnn∈ℕ∗. Then A has at least one fixed point in the set Ω.Definition 17.
A functionx∈CJ′,X is said to be a solution of equation (2) if x satisfies the equation Dca+α;ψxt=ft,xt on J′, and the condition xa=ϕ0.
Now, we shall prove the following theorem concerning the existence of solutions of problem (2).Theorem 9.
Letf:J′×X⟶X be a continuous function such that the following assumptions hold:(H5) There exists a continuous functionsp:J′⟶ℝ+ such that(95)ft,x≤pt1+x,for allt∈J′,and eachx∈X.(H6) For each bounded setD⊂X and for each t∈J′, we have(96)μft,D≤ptμD,where μ is a measure of noncompactness on the Banach space X.
Forn∈ℕ∗, let(97)pn∗=supt∈Jnpt,ℓn,ψ=ψn−ψaαΓα+1.
Define onE the family of measures of noncompactness by(98)μnD=supt∈JnμDt,where D is a bounded and equicontinuous set of E, and Dt=xt∈X:x∈D,t∈Jn.
If(99)4pn∗+λℓn,ψ<1,for each n∈ℕ∗, then the problem (2) has at least one solution.Proof.
Consider the operatorA defined by (84), but in the Fréchet space E≔CJ′,X.
Note that, the fixed points of the operatorA are solutions of the problems (2). For any n∈ℕ∗, let Rn be a positive real number with(100)Rn≥ϕ0+ℓn,ψpn∗1−pn∗+λℓn,ψ,and we consider the ball(101)BRn=x∈E:xn≤Rn.
Notice thatBRn is closed, convex, and bounded subset of the Fréchet space E. We shall show that the operator A satisfies all the assumptions of Theorem 8. We split the proof into three steps.Step 1. The operatorA maps the set BRn into itself. For any n∈ℕ∗, and each x∈BRn and t∈Jn, by (H5), we have(102)Axt≤ϕ0+∫atGψαt,sps1+xs+λxsds≤ϕ0+ψn−ψaαΓα+1pn∗+pn∗+λxn≤ϕ0+ℓn,ψpn∗+pn∗+λRn≤Rn.Thus(103)Axn≤Rn.This proves thatA transforms the ball BRn into itself.Step 2. The operatorA is continuous. Suppose that xkk∈ℕ is a sequence such that xk⟶x in BRn as k⟶∞. Then for each t∈Jn, we have(104)Axnt−At≤∫atGψαt,sfs,xks−fs,xsds+λ∫atGψαt,sxks−xsds.Sincexk⟶x as k⟶+∞, the Lebesgue dominated convergence theorem implies that(105)Axk−Axn⟶0ask⟶+∞,which implies the continuity of the operatorA.Step 3. Our aim in this step is to show thatA is μn contraction on BRn. For every bounded equicontinuous subset D⊂BRn and ε>0 using Lemma 8, there exist sequences xkk=1∞⊂D such that for all t∈Jn, we have(106)μADt≤2μ∫atGψαt,sλxnsn=1∞+fs,xnsn=1∞ds+ε.Next, by Lemma9 and (H6), we have(107)μADt≤4∫atGψαt,sμλxnsn=1∞+fs,xnsn=1∞ds+ε≤4∫atGψαt,sps+λμxksk=1∞ds+ε≤4pn∗ψn−ψaαΓα+1μnD+ε=4pn∗+λℓn,ψμnD+ε.As the last inequality is true, for everyε>0, we infer(108)μADt≤4pn∗+λℓn,ψμnD.Thus(109)μnAD≤4pn∗+λℓn,ψμnD.Using the condition (99), we claim that A is a kn contraction on BRn. It follows from Theorem 8 that the operator A defined by (84) has at least one fixed point x∈BRn, which is just the solution of initial value problem (2). This completes the proof of Theorem 9.


## 8. Examples

In this section, we give a couple of examples to illustrate the usefulness of our main result.Let(110)X=c0=x=x1,x2,…,xk,…:xk⟶0k⟶∞.be the Banach space of real sequences converging to zero, endowed its usual norm(111)x∞=supk≥1xk.Example 1.
Consider the following initial value problem of a fractional differential posed inc0:(112)DCH1+0.5xt=0.5xt+ft,xt,t∈1,e,u1=0.5,0.25,…,0.5n,….
Note that, this problem is a particular case of IVP (1), where(113)α=0.5,a=1,b=e,λ=0.5,ψt=lnt,and f:1,e×c0⟶c0 given by(114)ft,x=1t+11k2+arctanxkk≥1,fort∈1,e,x=xkk≥1∈c0.
It is clear that condition (H1) holds. Moreover, for anyx,y∈c0 and t∈1,e, we have(115)ft,x−ft,y≤12x−y,where we have used the trigonometric identity arctanu−arctanv=arctanu−v/1+uv and arctanu≤u,∀u,v∈ℝ+. Hence condition (H2) holds with L=1/2. Therefore, by Theorem 4, the IVP (112) has a unique solution x∈C1,e,c0. Moreover, the inequality appearing in (67) has the following expression:(116)DCH1+0.5xt−0.5xt−ft,xt≤εE0.5lnt,ε>0,t∈1,e.
By Theorem6, the IVP (112) is Mittag–Leffler–Ulam–Hyers stable with(117)y−x∞≤cεEαψt−ψaα,where(118)c=E0.51≃5.0090.
Let now(119)X=ℓ1=x=x1,x2,…,xk,…,∑k=1∞xk<∞,be the Banach space with the norm(120)x=∑k=1∞xk.Example 2.
Consider the following initial value problem of a fractional differential posed inℓ1:(121)Dc0+0.9xt=0.1xt+ft,xt,t∈0,1,x0=1,0,…,0,….
In this case, we take(122)α=0.9,a=0,b=1,λ=0.1,ψt=t,and f:0,1×ℓ1⟶ℓ1 given by(123)ft,x=1et+912k+xkx+1k≥1,fort∈0,1,x=xkk≥1∈ℓ1.
It is clear that condition (H1) holds and as(124)ft,x≤1et+91+x,x,y∈ℓ1.
Hence condition (H3) holds withμt=νt=1/et+9; we get easily μ∗=ν∗=0.1. On the contrary, for any bounded set B⊂ℓ1, we have(125)χft,B≤1et+9χB,for anyt∈0,1.
Hence, (H4) is satisfied. Now, we shall check that condition (48) is satisfied. Indeed(126)4ν∗+λℓα,ψ≃0.8318<1,r≥ϕ0+ℓα,ψμ∗1−ν∗+λℓα,ψ≃1.3938.
Thenr can be chosen as r=1.4. Consequently, all the hypothesis of Theorem 5 are satisfied, and we conclude that the IVP (121) has at least one solution x∈C0,1,ℓ1.Example 3.
Consider the following scalar initial value problem:(127)Dc0+0.5,ψxt=2xt+1t2+11+sinxt,t∈0,+∞,u0=2,where(128)α=0.5,a=0,λ=2,ψt=t2,X=R,ft,x=1t2+11+sinx,fort∈0,+∞,x∈ℝ.
Clearly,f:0,+∞×ℝ⟶ℝ is continuous. On the contrary, for any t∈0,+∞,x,y∈ℝ, we have(129)ft,x−ft,y≤x−y.
Hence the inequality (82) holds with M=1. Moreover, if we choose, β>3, it follows that the mapping A is a contraction. Hence by Theorem 7, the IVP (127) has a unique solution on 0,+∞.Example 4.
Let us consider problem (2) with specific data:(130)α=12,a=0,λ=18,X=c0,ϕ0=1,0,…,0,…,and E≔C0,+∞,X be the Fréchet space of all continuous functions x from 0,+∞ into X, equipped with the family of seminorms:(131)xn=supt∈0,nxt,n∈ℕ∗.
In order to illustrate Theorem9, we take ψt=σt where σt is the sigmoid function [19] which can be expressed as in the following form:(132)σt=11+e−t,and a convenience of the sigmoid function is its derivative:(133)σ′t=σt1−σt.
Taking alsof:0,+∞×c0⟶c0 given by(134)ft,x=σt81k+1+ln1+xktk≥1,fort∈0,+∞,x=xkk≥1∈c0.
The hypothesis (H5) is satisfied withpt=σt/8. So, for any n∈ℕ∗, we have(135)pn∗=supt∈0,npt=σn8,4pn∗+λℓn,ψ=σn+1σn−1/2π≤2π<1.
It follows from Theorem9 that the problem (2) with the data (130), (132), and (134) has at least one solution defined on ℝ+.



---

*Source: 6383916-2020-10-14.xml*