# Nonlocal Conformable-Fractional Differential Equations with a Measure of Noncompactness in Banach Spaces

Authors: Mohamed Bouaouid; Mohamed Hannabou; Khalid Hilal
Journal: Journal of Mathematics
 (2020)

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## Abstract

This paper deals with the existence of mild solutions for the following Cauchy problem:dαxt/dtα=Axt+ft,xt,x0=x0+gx,t∈0,τ, where dα./dtα is the so-called conformable fractional derivative. The linear part A is the infinitesimal generator of a uniformly continuous semigroup Ttt≥0 on a Banach space X, f and g are given functions. The main result is proved by using the Darbo–Sadovskii fixed point theorem without assuming the compactness of the family Ttt>0 and the Lipshitz condition on the nonlocal part g.


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## Body

## 1. Introduction

Many dynamical processes in physics, biology, economics, and other areas of applications can be governed by abstract ordinary differential evolution equations of the following form:(1)x˙t=Axt+ft,xt.Unfortunately, the classical derivativex˙t appearing in equation (1) is local and cannot model the dynamical processes with memory. Hence, in order to avoid this shortcoming of classical derivative, many authors try to replace the classical derivative by a fractional derivative [1–4] because fractional derivatives have been proved that they are a very good way to model many phenomena with memory in various fields of science and engineering [5–9]. In consequence, many researchers pay attention to form the best definition of fractional derivative. Recently, a novel definition named conformable fractional derivative is introduced in [10]. This new fractional derivative quickly becomes the subject of many contributions in several areas of science [11–22]. Motivated by the better effect of the fractional derivative and simple properties of the conformable fractional derivative, we consider model (1) in the framework of conformable fractional calculus. Precisely, we study the following Cauchy problem:(2)dαxtdtα=Axt+ft,xt,x0=x0+gx,t∈0,τ,where dα./dtα is the conformable fractional derivative of the order α∈0,1. The linear part A is the infinitesimal generator of a uniformly continuous semigroup Ttt≥0 on a Banach space X,.. For more details about semigroup theory, we refer to [23]. The nonlinear part f:0,τ×X⟶X is a given function. The initial condition x0=x0+gx means the nonlocal condition [24]. For physical interpretations of this condition, we can see [25, 26]. The nonlocal condition attracts the attention of many authors in several works [27, 28]. The vector x0 is an element of X and g:C⟶X is a given function, with C is the space of continuous functions x. defined from 0,τ into X. Throughout this paper, we endow the space C with the norm xc=supt∈0,τxt. It is well known that the space C,.c is a Banach space. We also denote by . the norm in the space ℒX of bounded operators defined form X into itself.Our goal in this paper is to prove the existence of mild solutions for the Cauchy problem (2) by means of the Darbo–Sadovskii fixed point theorem without assuming the compactness of the family Ttt>0 and the Lipshitz condition on the nonlocal part g.The content of this paper is organized as follows. In section2, we recall some preliminary facts on the conformable fractional calculus and measure of noncompactness. Section 3 is devoted to prove the main result.


## 2. Preliminaries

Recalling some preliminary facts on the conformable fractional calculus.Definition 1 (see [10]).
Letα∈0,1. The conformable fractional derivative of order α of a function x. for t>0 is defined as follows:(3)dαxtdtα=limε⟶0xt+εt1−α−xtε.
Fort=0, we adopt the following definition:(4)dαx0dtα=limt⟶0+dαxtdtα.
The fractional integralIα. associated with the conformable fractional derivative is defined by(5)Iαxt=∫0tsα−1xsds.Theorem 1 (see [10]).
Ifx. is a continuous function in the domain of Iα., then we have(6)dαIαxtdtα=xt.Definition 2 (see [8]).
The Laplace transform of a functionx. is defined by(7)ℒxtλ≔∫0+∞e−λtxtdt,λ>0.
It is remarkable that the above transform is not compatible with the conformable fractional derivative. For this, the adapted transform is given by the following definition.Definition 3 (see [11]).
The fractional Laplace transform of orderα∈0,1 of a function x. is defined by(8)ℒαxtλ≔∫0+∞tα−1e−λtα/αxtdt,λ>0.
The following proposition gives us the actions of the fractional integral and the fractional Laplace transform on the conformable fractional derivative, respectively.Proposition 1 (see [11]).
Ifx. is a differentiable function, then we have the following results:(9)Iαdαx.dtαt=xt−x0,ℒαdαxtdtαλ=λℒαxtλ−x0.
According to [15], we have the following remark.Remark 1.
For two functionsx. and y., we have(10)ℒαxtααλ=ℒxtλ,ℒα∫0tsα−1xtα−sααysdsλ=ℒxtλℒαytλ.
Now, we recall some concepts on the Hausdorff measure of noncompactness.Definition 4 (see [29, 30]).
For a bounded set B in a Banach spaceX, the Hausdorff measure of noncompactness σ is defined as(11)σB=infɛ>0:Bcan be covered by a  finite number of balls with radiiɛ.
The following lemma presents some basic properties of the Hausdorff measure of noncompactness.Lemma 1 (see [29, 30]).
LetX be a Banach space and B,C⊆X be bounded. Then, the following properties hold.(1)
B is precompact if and only ifσB=0;(2)
σB=σB¯=σconvB, where B¯ and convB mean the closure and convex hull of B, respectively;(3)
σB≤σC, where B⊆C;(4)
σB+C≤σB+σC, where B+C=x+y:x∈B,y∈C;(5)
σB∪C≤maxσB,σC;(6)
σλB=λσB for any λ∈ℝ, when X be a real Banach space;(7)
If the operatorQ:DQ⊆X⟶Y is Lipschitz continuous with constant k ≥ 0 then we have ρQB≤kσB for any bounded subset B⊆DQ, where Y is another Banach space and ρ represents the Hausdorff measure of noncompactness in Y.Definition 5 (see [30]).
The operatorQ:DQ⊆X⟶X is said to be a σ-contraction if there exists a positive constant k<1 such that σQB≤kσB for any bounded closed subset B⊆DQ.Lemma 2 (see [29, 30] (Darbo–Sadovskii theorem)).
LetB⊂X be a bounded, closed, and convex set. If Q:B⟶B is a continuous and σ-contraction operator. Then, Q has at least one fixed point in B.Lemma 3 (see [31, 32]).
LetD⊂X be a bounded set, then there exists a countable set D0⊂D such that σD≤2σD0.
We denote byσc the Hausdorff measure of noncompactness in the space C of continuous functions x. defined from 0,τ into X.Lemma 4 (see [33]).
LetD0≔xn⊂C be a countable set, then(1)
σD0t≔σxnt is Lebesgue integral on 0,τ,(2)
σ∫0τD0sds≤2∫0τσD0sds, whereσ∫0τD0sds≔σ∫0τxnsdsLemma 5 (see [29]).
LetD⊂C be bounded and equicontinuous, then(1)
σDt is continuous on 0,τ,(2)
σcD=maxt∈0,τσDt


## 3. Main Result

We first give the definition of mild solutions for the Cauchy problem (2). To do so, applying the fractional Laplace transform in equation (2), we obtain(12)λℒαxtλ=x0+gx+Aℒαxtλ+ℒαft,xtλ.Then, one has(13)ℒαxtλ=λ−A−1x0+gx+λ−A−1ℒαft,xtλ.Using the inverse fractional Laplace transform combined with Remark1, we obtain(14)xt=Ttααx0+gx+∫0tsα−1Ttα−sααfs,xsds.Motivate by the above calculus, we can introduce the following definition.Definition 6.
A functionx∈C is called a mild solution of the Cauchy problem (2) if(15)xt=Ttααx0+gx+∫0tsα−1Ttα−sααfs,xsds.
To obtain the existence of mild solutions, we will need the following assumptions:H1 The function ft,.:X⟶X is continuous, and for all r>0 there exists a function μr∈L∞0,τ,ℝ+ such that supx≤rft,x≤μrt, for all t∈0,τ.H2 The function f.,x:0,τ⟶X is continuous, for all x∈X.H3 The function g:C⟶X is continuous and compact.H4 There exist positive constants a and b such that gx≤axc+b, for all x∈C.H5 There exists a positive constant L such that σft,D0≤LσD0, for any countable set D0⊂X and t∈0,τ.Theorem 2.
Assume thatH1−H5 hold, then the Cauchy problem (2) has at least one mild solution provided that(16)supt∈0,τTtααmaxa,4Lταα<1.Proof.
In order to use the Darbo–Sadovskii fixed point theorem, we putBr≔x∈C,xc≤r for r>0 and define the operator Γ:C⟶C by(17)Γxt=Ttααx0+gx+∫0tsα−1Ttα−sααfs,xsds.
The proof will be given in four steps.Step 1.
Prove that there exists a radiusδ>0 such that Γ:Bδ⟶Bδ.
Letx∈C, we have(18)Γxt≤Ttααx0+gx+∫0tsα−1Ttα−sααfs,xsds.
Taking the supremum, we obtain(19)Γxc≤supt∈0,τTtααx0+gx+∫0τsα−1fs,xsds.
Using assumptionH4, we deduce that(20)Γxc≤supt∈0,τTtααx0+axc+b+∫0τsα−1fs,xsds.
Hence, it suffices to considerδ as a solution of the following inequality:(21)supt∈0,τTtααx0+ar+b+τααμrL∞0,τ,ℝ+≤r.
Precisely, we can chooseδ such that(22)δ≥supt∈0,τTtα/α1−asupt∈0,τTtα/αx0+b+τααμδL∞0,τ,ℝ+.Step 2.
Prove thatΓ:Bδ⟶Bδ is continuous.
Letxn⊂Bδ such that xn⟶x in Bδ. We have(23)Γxnt−Γxt=Ttααgxn−gx+∫0tsα−1Ttα−sααfs,xns−fs,xsds.
Then, by using a direct computation, we obtain(24)Γxn−Γxc≤supt∈0,τTtααgxn−gx+∫0τsα−1fs,xns−fs,xsds.
Using assumptionH1, we get sα−1fs,xns−fs,xs≤2μδssα−1 and fs,xns⟶fs,xs as n⟶+∞.
The Lebesgue dominated convergence theorem proves that∫0τsα−1fs,xns−fs,xsds⟶0 as n⟶+∞. According to continuity of the function g, we deduce that limn⟶+∞gxn−gx=0. Hence, Γ is continuous.Step 3.
Prove thatΓBδ is equicontinuous.
Forx∈Bδ and t1,t2∈0,τ such that t1<t2. We have(25)Γxt2−Γxt1=Tt2α−t1αα−ITt1ααx0+gx+∫0t1sα−1Tt1α−sααfs,xsds+∫t1t2sα−1Tt2α−sααfs,xsds.
By using assumptionsH1 and H4, we obtain(26)Γxt2−Γxt1≤x0+aδ+b+τααμδL∞0,τ,ℝ+supt∈0,τTtααTt2α−t1αα−I+supt∈0,τTtααμδL∞0,τ,ℝ+t2α−t1αα.
The above inequality combined with the uniform continuity of the familyTtt≥0 proves that ΓBδ is equicontinuous on 0,τ.Step 4.
Prove thatΓ:Bδ⟶Bδ is a σc-contraction operator.
LetD⊂Bδ, then by Lemma 3 there exists a countable set D0 such that D0=xn⊂D. Hence, Γ(D0) becomes a countable subset of ΓD. Thus, Lemma 3 proves that σcΓD≤2σcΓD0. Since ΓD0 is bounded and equicontinuous, then by using Lemma 5, we obtain(27)σcΓD0=maxt∈0,τσΓD0t.
Then, one has(28)σcΓD≤2σcΓD0=2maxt∈0,τσΓD0t=2maxt∈0,τσTtααx0+gD0+∫0tsα−1Ttα−sααfs,D0sds.
By using point4 of Lemma 1, we deduce that(29)σcΓD≤2maxt∈0,τσTtααx0+gD0+σ∫0tsα−1Ttα−sααfs,D0sds.
Sinceg is compact, then Ttα/αx0+gD0 is relatively compact. Hence, using point 1 of Lemma 1 in the above inequality, we obtain(30)σcΓD≤2maxt∈0,τσ∫0tsα−1Ttα−sααfs,D0sds.
In view of Lemma4, we get(31)σcΓD≤4maxt∈0,τ∫0tsα−1σTtα−sααfs,D0sds.
Next, point7 of Lemma 1 shows that(32)σcΓD≤4supt∈0,τTtααmaxt∈0,τ∫0tsα−1σfs,D0sds.
By using assumptionH5, we obtain(33)σcΓD≤4Lsupt∈0,τTtααmaxt∈0,τ∫0tsα−1σD0sds.
Hence, by using a direct computation combined with point2 of Lemma 5, we obtain(34)σcΓD≤4Lsupt∈0,τTtαασcD∫0τsα−1ds=4Lτααsupt∈0,τTtαασcD.
In consequence, we have(35)σcΓD≤4Lτααsupt∈0,τTtαασcD.
Since4Lτα/αsupt∈0,τTtα/α<1, then Γ is a σc-contraction operator.
In conclusion, Lemma2 shows that Γ has at least one fixed point, which is a mild solution of the Cauchy problem (2).Remark 2.
We note that Theorem2 improves Theorem 3 in [18] because in Theorem 2 we have not imposed the compactness of the family Ttt>0 and the Lipshitz condition on the nonlocal part g.


## 4. Conclusion

Without imposing the compactness condition on the semigroup family and the Lipschitz condition on the nonlocal condition, we have proved the existence of mild solutions for a class of conformable-fractional differential equations with nonlocal conditions in a Banach space. The main result is obtained by means of semigroup theory combined with the Darbo–Sadovskii fixed point theorem.



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