# Existence of Solutions for Fractional Evolution Equations with Infinite Delay and Almost Sectorial Operator

Authors: Shanshan Li; Shuqin Zhang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper discusses a class of semilinear fractional evolution equations with infinite delay and almost sectorial operator on infinite interval in Banach space. By using the properties of analytic semigroups and Schauder’s fixed-point theorem, this paper obtains the existence of mild solutions of the fractional evolution equation. Moreover, this paper also discusses the existence of mild solution when the analytic semigroup lacks compactness by Kuratowski measures of noncompactness and Darbo–Sadovskii fixed-point theorem.


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## Body

## 1. Introduction

Fractional differential models play a very important role in describing many complex phenomena such as chaotic system [1], fluid flow [2, 3], anomalous diffusion [4–7], and so on. Compared with the classical partial differential models such as [8–19], the biggest advantage of models with fractional derivatives is their global property and history memory. Delay is short for time delay, which exists widely in the objective world. In the differential equation model with delay, the function depends not only on the current state but also on the past time state, so it is more suitable to describe the process with time memory. This property of delay is very similar to that of fractional derivatives. So many researchers introduced fractional derivatives into differential equations with delay [20–24]. Evolution equation, which is a general appellation for some partial differential equations with time variable, is mainly used to describe the time-dependent state and process. Common evolution equations include the wave equation, the heat equation, Schrodinger equation, KdV equation, Navier–Stokes equation, and so on. By using the operator semigroup theory, some partial differential evolution equations can be represented to some abstract ordinary differential equations (ODEs) in some special functional spaces. At present, the research on integer-order evolution equations has been relatively perfect [25, 26], but the research on fractional-order evolution equations is still in the preliminary stage. The existence of solutions for fractional evolution equations is also the basis of the following study. The mild solution of integer-order evolution equations is defined by the constant variation method, which cannot be directly extended to fractional-order evolution equations.Li [20] studied the following fractional evolution equations with almost sectorial operator on finite interval:(1)Dtqcxt=Axt+ft,x,xt,0<q<1,t∈0,T,x0=ϕt∈B,t∈−∞,0,where Dtqc is the Caputo fractional derivative operator, the evolution operator A is an almost sectorial operator, and B is a phase space. xt is the element of B defined by xtθ=xt+θ,θ∈−∞,0. Here, xt· represents the history of state up to the present time.Baliki et al. [22] discussed a second-order evolution equation with infinite delay and obtained the existence and attractivity of mild solutions by Schauder’s fixed point as follows:(2)x″t−Atxt=ft,xt,0<q<1,t∈0,∞,x0=ϕt∈B,x′0=x˜,where At0≤t<∞ is a family of linear closed operators, xtθ=xt+θ,θ∈−∞,0, and B is a phase space. The existence of mild solutions for fractional evolution equations and evolution equations with infinite delay has been discussed in several papers (see [20, 21]). However, we find that most of the previous papers discuss the fractional evolution equations in the conventional spaces of continuous function on finite or infinite interval and in Banach space on finite interval. To our knowledge, no paper is devoted to the existence of mild solutions with infinite delay and almost sectorial operator on infinite interval on Banach space.In this paper, we consider the following fractional evolution problem with infinite time delay:(3)D0+qcxt+Axt=ft,xt,t∈0,+∞,0<q<1,xt=ϕt∈B,t∈−∞,0,where D0+qc is the Caputo fractional derivative operator, the evolution operator A is an almost sectorial operator, f is a given function which will be introduced later, and B is a phase space. For any continuous function x and any t≥0, xt is the same as in equation (1) which represents the history of state up to the present time.The rest of this paper is organized as follows. InSection 2, we recall some definitions, propositions, notations, and lemmas. In Section 3, the main results of this paper are obtained. We consider two cases: the semigroup Qt generated by operator A with compactness and without compactness. For the case that Qt is compact, we construct a special Banach space B′ and obtain the existence of global mild solution by using Schauder’s fixed-point theorem. For the case that Qt is not compact, we expand the result of Theorem 1.2.4 in Guo et al. [27] from any compact interval to infinite interval (see Lemma 10) and obtain the existence of global mild solution by applying Kuratowski measures of noncompactness theory and Darbo–Sadovskii fixed-point theorem.


## 2. Preliminaries

In this section, we introduce some notations, definitions, lemmas, and preliminary facts that will be used in the rest of this paper. LetE,· be a Banach space. Denote BE as the space of all bounded linear operators from E to itself with norm ‖·‖BE.Definition 1.
(see [28, 29]). Let −1<γ<0 and 0<ω<π/2. Denote by ΘωγE all the linear closed operators A:DA⊂E⟶E which satisfy(1)
σA⊂Sω=z∈C∖0,argz≤ω∪0.(2)
For everyω<μ<π, there exists a constant Cμ such that(4)Rz,A≤Cμzγfor allz∈C\Sμ.
A linear operatorA will be called an almost sectorial operator on E if A∈ΘωγE.
Define the power ofA as(5)Aβ=12πi∫ΓθzβRz,Adz,β>1+γ,where Γθ=R+eiθ∪R+e−iθ is an appropriate path oriented counterclockwise and ω<θ<μ. Then, the linear power space Xβ≔DAβ can be defined and Xβ is a Banach space with the graph norm xβ=Aβx,x∈DAβ.
Next, let us introduce the semigroup associated withA. If A is an almost sectorial operator, then A generates an analytic semigroup Qt of growth order 1+γ as follows:(6)Qt=12πi∫Γθe−tzRz,Adz,t∈Sπ/2−ω0,where Γθ=R+eiθ∪R+e−iθ is oriented counterclockwise and ω<θ<μ<π/2−argt. Sπ/2−ω0 is the open sector z∈C∖0,argz<π/2−ω. Furthermore, Qt satisfies the following properties.Proposition 1.
(see [28, 29]). Let A∈ΘωγE with −1<γ<0 and 0<ω<π/2. Then, the following properties remain true:(1)
Qt is analytic in Sπ/2−ω0 and dn/dtnQt=−AnQt,t∈Sπ/2−ω0.(2)
The functional equation holds:Qs+t=QsQt for all s,t∈Sπ/2−ω0.(3)
There is a constantC0=C0γ>0 such that Qt≤C0t−γ−1, t>0.(4)
The rangeRQt of Qtt∈Sπ/2−ω0 is contained in DA∞. Particularly, RQt⊂DAβ for all β∈C with Reβ>0:(7)AβQtx=12πi∫Γθzβe−tzRz,Axdz,t∈Sπ/2−ω0,x∈E,and there exists a constantC′=C′γ,β>0 such that for all t>0,(8)AβQt≤C′t−γ+Reβ−1.(5)
Ifβ>1+γ, then DAβ⊂∑Q=x∈E,limt⟶0Qtx=x.
By Theorem 3.13 in Periago [28], if A is an almost sectorial operator, then for every λ∈C with Reλ>0,(9)Rλ,−A=∫0+∞e−λtQtdt.
LetX be the following set:(10)X≔x:R⟶Xβ,x0,+∞∈C0,+∞,Xβ,limt⟶+∞e−ktxt=0,x0∈B,where x0,+∞ is the restriction of x on 0,+∞ and k is a constant.
In this paper, we use an axiomatic definition of the phase spaceB. B,·B is a seminormed linear space of functions mapping −∞,0 into E and satisfies the following axioms which are introduced by Hale and Kato in [30].(A)
Ifx:−∞,b⟶E,b>0 is continuous on 0,b and x0∈B, then for any t∈0,b, the following conditions hold:(i)
xt∈B.(ii)
There exists a positive constantH such that x≤HxtB.(iii)
There exist positive continuous functionsK·,M· independent of x· such that(11)xtB≤Ktsup0≤s≤txsβ+Mtx0B.(B)
For the functions in (A), xt is a B-value continuous function on 0,b.(C)
The spaceB is complete.Definition 2.
(see [31, 32]). Let f∈L10,+∞,E and q>0; then,(12)I0+qft=1Γq∫0tt−sq−1fsdsis called the Riemann–Liouville fractional integral of order q.Definition 3.
(see [31, 32]). The Caputo fractional derivative of order q>0 of the function f:0,+∞⟶E is given by(13)D0+qcft=1Γn−q∫0tt−sn−q−1fnsds,where n is the smallest integer greater than or equal to q, provided that the right side is well defined on 0,+∞.Lemma 1.
(see [31, 32]). For all f,g∈Lq0,+∞,E,1≤q<∞,(14)I0+qf∗g=I0+qf∗g.
Next, we will introduce the mild solution of equation (3). Shu et al. [33] define the mild solution of equation (3) as(15)xt=Sqtϕ0+∫0tt−sq−1Pqt−sfs,xsds,where Sqt and Pqt have the following expressions and Γ is an appropriate path in ρ−A.(16)Sqt=12πi∫Γeλtλq−1Rλq,−Adλ,Pqt=t1−q2πi∫ΓeλtRλq,−Adλ.
Using the properties of the Mittag-Leffler function (for more details, we refer the readers to [32]),(17)Eα,βz=∑k=0+∞zkΓαk+β=12πi∫Γλα−βeαλα−zdλ,where Γ is the same path as in (4) (see [32]), the above operators Sqt and Pqt can be represented as the generalized Mittag-Leffler-type functions:(18)Sqt=Eq,1−tqA=Eq−tqA,Pqt=Eq,q−tqA.
Moreover, Wang et al. [29] and Zhou et al. [34–36] introduced the function of Wright-type Mqz:(19)Mqz=∑n=1+∞−zn−1n−1!Γ1−nq,0<q<1,z∈C,and obtained another expression of Sqt, Pqt:(20)Sqt=∫0+∞MqsQtqsds,Pqt=∫0+∞qsMqsQtqsds.
In fact, these three expressions ((16)–(20)) are equivalent in the case that t>0 and A∈ΘωγE. Therefore, in this paper, we use the same expression of Sqt, Pqt as Wang et al. in [29] and Zhou et al. in [34–36]. Then, the global mild solution of problem (3) is given in the following definition.Definition 4.
A functionx:R⟶X is called a global mild solution to the problem (3), if xt∈CR,X and(21)xt=Sqtϕ0+∫0tt−sq−1Pqt−sfs,xsds,t∈0,+∞,ϕt,t∈−∞,0.Lemma 2.
(see [29]). For any fixed t>0, Sqt and Pqt are linear and bounded operators and there exist constants Cs and Cp such that for all x∈E,(22)Sqtx≤Cst−q1+γx,Pqtx≤Cpt−q1+γx.Lemma 3.
(see [29]). For t>0, operators Sqt and Pqt are continuous in the uniform operator topology. Moreover, for every r>0, the continuity is uniform on r,+∞.Lemma 4.
(see [29]). Let 0<β<1−γ; then,(1)
Fort>0, the range RPqt of Pqt is contained in DAβ.(2)
For allx∈DA and t>0, ASqtx≤Ct−q1+γAx, where C is a constant depending on γ, q.Remark 1.
Moreover, for allx∈DAβ0<β<1−γ and t>0,(23)AβSqtx≤Cst−q1+γAβx,AβPqtx≤Cpt−q1+γAβx,that is,(24)Sqtxβ≤Cst−q1+γxβ,Pqtxβ≤Cpt−q1+γxβ.Lemma 5.
(see [29]). Let β>1+γ; then, limt⟶0+Sqtx=x for all x∈DAβ.


## 3. Main Results

In this section, our main purpose is to establish sufficient conditions for the existence of global mild solutions to problem (3) in X. Assume that:Hf:0,+∞×B⟶Xβ,1+γ<β<1−γ is continuous and satisfies(25)ft,xβ≤pte−ktxB,where pt is a nonnegative and continuous function on 0,+∞ and here exists a big enough k>0 such that(i)
For anyt≥0,(26)Cpe−kt∫0tt−s−qγ−1psKsds≤12,(ii)
limt⟶+∞e−kt∫0tt−s−qγ−1psKsds=0,limt⟶+∞e−kt∫0tt−s−qγ−1psMsds=0.In order to obtain the existence of global mild solution of problem (3), we transform it into a fixed-point problem. For any ϕ0∈Xβ, define the operator T^:X⟶X as(27)T^xt=Sqtϕ0+∫0tt−sq−1Pqt−sfs,xsds,t∈0,+∞,ϕt,t∈−∞,0.Letzt:R⟶X be the function(28)zt=Sqtϕ0,t∈0,+∞,ϕt,t∈−∞,0,and xt=yt+zt,t∈R. It is easy to know that xt satisfies (21) if and only if(29)yt=∫0tt−sq−1Pqt−sfs,ys+zsds,t∈0,+∞,y0=0,t∈−∞,0.Define the setB′≔y∈X:y0=0∈B endowed with seminorm ·b:(30)yb=y0B+supt≥0e−ktytβ=supt≥0e−ktytβ.Thus,B′,·b is a Banach space. Define the operator T:B′⟶B′ as(31)Tyt=∫0tt−sq−1Pqt−sfs,ys+zsds,t∈0,+∞,y0=0,t∈−∞,0.Consequently, the operatorT^:X⟶X having a fixed point in X is equivalent to the operator T:B′⟶B′ having a fixed point in B′.Lemma 6.
Assume that conditionH is valid; then, there exists a constant r>0 such that(32)Cpe−kt∫0tt−s−qγ−1e−kspsKsM+MsϕBds≤r2,where M satisfies supt>0Sqtϕ0β≤M. Consider Br≔y∈B′,yb≤r; then, for any ϕ0∈Xβ, the operator T:Br⟶Br is continuous.Proof.
Byϕ0∈Xβ and Lemma 5 (1), there exists 0<δ1<T, and for any t∈0,δ1, such that Sqtϕ0−ϕ0β<ε. for any t≥δ1, Sqtϕ0β≤Csϕ0βδ1−q1+γ. Therefore, there exists a constant M>0 such that supt>0Sqtϕ0β≤M.
For anyyt∈Br, 0<s<t, note that(33)ys+zsB≤ysB+zsB,≤Kseksyb+Kssup0<τ≤sSqτϕ0β+MsϕB,≤Kseksyb+KsM+MsϕB,≔ηs.
Then, by conditionH and Remark 1, we have(34)e−ktTytβ≤e−kt∫0tt−sq−1Pqt−sfs,ys+zsβds,≤Cpe−kt∫0tt−s−qγ−1e−kspsηsds,≤Cpe−kt∫0tt−s−qγ−1e−kspsKsdsyb,+Cpe−kt∫0tt−s−qγ−1e−kspsKsM+Ms‖ϕ‖Bds,≤r2+r2=r,which implies that Tyb≤r and T:Br⟶Br.
Next, we will prove the continuity ofT. Let yntn=1∞∈Br and yn−yb⟶0 as n⟶∞ for any t≥0. Then, for any t>0, by the continuity of f,(35)e−ktTynt−Tytβ≤Cpe−kt∫0tt−s−qγ−1fs,ysn−fs,ysβds⟶0n⟶∞,which implies that Tynt−Tytb⟶0 as n⟶∞. Therefore, the continuity of T is proved.Lemma 7.
Assume that conditionH is satisfied; then, for any ϕ0∈Xβ,(1)
e−ktTyt,y∈B′ is equicontinuous on any compact interval of 0,+∞.(2)
For any givenε>0, there exists a constant T>0 such that e−ktTytβ<ε for any t≥T and y∈B′.Proof.
(1) Without loss of generality, we take0,T⊂0,+∞ as the compact interval and 0≤t1<t2≤T. Firstly, for t1=0,t1<t2≤T and any y∈B′, according to the continuity of ps and ηs, we have(36)e−kt1Tyt1−e−kt2Tyt2β≤Cpe−kt2∫0t2t2−s−qγ−1e−kspsηsds⟶0t2⟶0.
Next, for0<t1<t2≤T, by Lemma 2 and Remark 1, we have(37)e−kt1Tyt1−e−kt2Tyt2β,≤e−kt2∫t1t2t2−sq−1Pqt2−sfs,ys+zsβds,+e−kt1−e−kt2∫0t1t2−sq−1Pqt2−sfs,ys+zsβds,+e−kt1∫0t1t1−sq−1−t2−sq−1Pqt2−sfs,ys+zsβds,+e−kt1∫0t1t1−sq−1Pqt2−s−Pqt1−sfs,ys+zsβds,≤Cpe−kt2∫t1t2t2−s−qγ−1e−kspsηsds,+Cpe−kt1−e−kt2∫0t1t2−s−qγ−1e−kspsηsds,+Cpe−kt1∫0t1t1−sq−1−t2−s−qγ−1e−kspsηsds,+sups∈0,t1−δPqt2−s−Pqt1−sBEe−kt1∫0t1−δt1−s−qγ−1e−kspsηsds,+e−kt1∫t1−δt1t1−sq−1Pqt2−s−Pqt1−sfs,ys+zsβds,≔I11t+I12t+I13t+I14t+I15t.
ForI11t, I12t, and I14t by the continuity of ps, ηs, e−ks, and Pqs, we have I11t, I12tI14t⟶0 as t2⟶t1, δ⟶0. For I13t and I15t, note that(38)I1it≤2Cpe−kt1∫0t1t1−s−qγ−1e−kspsηsds,i=3,5.
Then, by using Lebesgue’s dominated convergence theorem, we haveI13t,I15t⟶0 as t2⟶t1, δ⟶0. Therefore, for any 0≤t1<t2≤T and y∈B′, Tyt1−Tyt2b⟶0 as t2⟶t1, δ⟶0.
(2) By conditionH, for big enough T>0,(39)e−kt∫0tt−s−qγ−1e−kspsηsds<1Cpε.
Then, for anyt≥T, y∈B′, we have(40)e−ktTytβ≤Cpe−kt∫0tt−s−qγ−1e−kspsηsds<ε.


### 3.1. The Case ThatQt Is Compact

In this section, we assume thatQt is compact for t>0, i.e., Qt is a compact operator for every t>0.Lemma 8.
LetZ⊆B′ be a bounded set; then, Z is relatively compact in B′ if the following conditions hold:(1)
The setyt,y∈Z is equicontinuous on any compact interval of 0,+∞ and for any t≥0, yt,y∈Z is relatively compact in X.(2)
For any givenε>0, there exists a constant T=Tε>0 such that e−ktytβ<ε for any t≥T and yt∈Z.Proof.
It is sufficient to prove thatZ is totally bounded. We consider the compact interval 0,T of 0,+∞. Define(41)Z0,T≔yt:yt∈Z,t∈0,T,with norm yb1≔sup0≤t≤Te−ktytβ; then, condition (1) combined with Arzelà–Ascoli theorem in Banach space indicates that Z0,T is relatively compact. Therefore, for any ε>0, there exist finitely many balls Bεyi such that Z0,T⊂∪i=1nBεyi, where yi∈B′.(42)Bεyi=yt∈Z0,T,y−yib1=sup0≤t≤Te−ktyt−yitβ≤ε.
Hence, for anyyt∈Z, there exists an i∈1,2,…,n such that y0,T∈Bεyi, i.e., for t∈0,T,(43)e−ktyt−yitβ≤ε.
Moreover, fort∈T,+∞, with conditions (3) and (43),(44)e−ktyt−yitβ,≤e−ktyt−e−kTyTβ+e−kTyT−e−kTyiTβ+e−kTyiT−e−ktyitβ,≤5ε.
Therefore, by (43) and (44), we have yt−yitb≤5ε for any t≥0. Then, Z can be covered by balls B5εyi=yt∈Z,y−yib≤5ε. Consequently, Z is totally bounded and the process is complete.Theorem 1.
Assume that conditionH holds; then, for ϕ0∈Xβ, problem (3) has at least one global mild solution in Br.Proof.
We aim to prove this theorem by using Schauder’s fixed-point theorem. In view of Lemma6, T:Br⟶Br and T is continuous, so we just need to prove that for any bounded subset V⊂Br, TV is relatively compact in X. Then, it is easy to prove that TV satisfies all conditions in Lemma 8.
Consider Lemma6; we have proved that Tyb=supt≥0e−ktTytβ≤r for any y∈Br which implies Ty,y∈Br is uniformly bounded. By Lemma 7, Ty,y∈B′ is equicontinuous on any compact interval 0,T of 0,+∞ and e−ktTytβ<ε for any t≥T and y∈B′. Then, it remains to show that Vt=Tyt,yt∈V is relatively compact in X for any t∈0,T.
It is easy to know thatV0=0 is compact in X. Let t∈0,T be fixed and for any ε∈0,t, δ>0, we define an operator Tεδ on V by the formula(45)Tεδyt=∫0t−ε∫δ+∞qθt−sq−1MqθQt−sqθfs,ys+zsdθds,=Qεqδ∫0t−ε∫δ+∞qθt−sq−1MqθQt−sqθ−εqδfs,ys+zsdθds,where y∈V. Under the compactness of Qεqδεqδ>0 and the boundedness of(46)∫0t−ε∫δ+∞qθt−sq−1MqθQt−sqθ−εqδfs,ys+zsdθds,we obtain that the set Vεδt=Tεδyt,y∈V is relatively compact in X for any ε∈0,t and δ>0. Moreover, for any y∈V, t>0, we have(47)e−ktTyt−Tεδytβ,≤qe−kt∫0t∫0δθt−sq−1Mqθt−sqθfs,ys+zsdθdsβ,+e−kt∫t−εt∫δ+∞qθt−sq−1MqθQt−sqθfs,ys+zsdθdsβ,≤qCpe−kt∫0tt−s−qγ−1e−kspsηsds∫0δθ−γMqθdθ,+Cpe−kt∫t−εtt−s−qγ−1e−kspsηsds.
According to∫0+∞θrMqθdθ=Γ1+r/Γ1+qr and condition H, we have(48)∫0δθ−γMqθdθ⟶0,∫t−εtt−s−qγ−1psηsds⟶0,asε⟶0,δ⟶0,which implies Tyt−Tεδytb⟶0 as ε⟶0,δ⟶0.
Therefore, the relatively compact setVεδt is arbitrarily close to the set Vt. Hence, for any t∈0,T, the set Vt, t∈0,T is also relatively compact in X.
Hence,T:Br⟶Br is a completely continuous operator. So, by Schauder’s fixed-point theorem, T has at least one fixed point in Br which implies that problem (3) has at least one global mild solution in Br.


### 3.2. The Case ThatQt Is Not Compact

In this section, we assume thatQt is not compact. In the following, α and αB′ denote the Kuratowski measures of noncompactness of bounded sets in Xβ and in B′. For more details about Kuratowski measures of noncompactness, we refer the readers to [27]. Assume that:H∗ There exists mt∈L0,+∞,0,+∞ such that I0+qm exists and for any bounded set V⊂B,(49)αft,V≤mte−ktsup−∞<τ≤0αVτ,and for any t≥0,(50)Cpe−kt∫0tt−s−qγ−1msds<1.Lemma 9.
(see [27]). If V⊂CJ,E is bounded and equicontinuous, then αVt is continuous and(51)α∫Jytdt,y∈V≤∫JαVtdt,where J is any compact interval of 0,+∞.Lemma 10.
LetV be a bounded set in B′. Suppose that Vt is equicontinuous on any compact interval 0,T of 0,+∞ and for any t≥T, ε>0, and y∈V,(52)e−ktytβ<ε.
Then, for eachVt=yt,y∈V,(53)αB′V=supt≥0e−ktαVt.Proof.
First, we prove thatαB′V≥supt≥0e−ktαVt. For the above given ε>0, t≥0, there exists a partition V=∪j=1nVj such that(54)diamVj<αB′V+ε,for anyj=1,2,…,n.
Then,Vt=∪j=1nVjt. For any u,v∈Vj, t≥0,(55)e−ktut−vtβ≤diamVj<αB′V+ε.
Therefore,diamVjt≤ektαB′V+ε which implies(56)supt≥0e−ktαVt≤αB′V,by the arbitrariness of ε.
Next, we show thatαB′V≤supt≥0e−ktαVt. By the equicontinuity of Vt on 0,T, there exists a partition 0=t0<t1<⋯<tm=T such that(57)e−kti′yti′−e−kti″yti″β<ε,for any ti′,ti″∈ti,ti+1, y∈V,i=0,1,…,m−1. Let Ii=ti,ti+1, i=0,1,…,m−1 and Im=tm,+∞; then, by (51) and (57),(58)e−kti′yti′−e−kti″yti″β<2ε,for anyy∈V,ti′,ti″∈Ii,i=0,1,…,m.
For eachi∈0,1,…,m, there exists a division V=∪j=1nVji such that Vti′=∪j=1nVjiti′ and(59)diamVjiti′<αVti′+2ε,j=1,2,…,n.
LetY be the finite set of all maps i⟶γi of 0,1,…,m into 1,2,…,n. For γ∈Y,(60)Zγ≔y∈V,yti′∈Vγiiti′,i=0,1,…,m,so V=yt,y∈Zγ,γ∈Y. For any u,v∈Zγ and t≥0, there exists i∈0,1,…,m such that t∈Ii; then,(61)e−ktut−vtβ,≤e−ktut−e−kti′uti′β+e−kti′uti′−e−kti′vti′β+e−ktvt−e−kti′vti′β,<αVti′+6ε.
Therefore,diamZγ≤αVti′+6ε. Since ε>0 is arbitrary, we have(62)αB′V≤supt≥0e−ktαVt.Lemma 11.
(see [27]). Let D be a bounded, closed, and convex subset of Banach space E. If the operator T:D⟶D is a strict set contraction, then T has a fixed point in D.Remark 2.
A bounded and continuous operatorT:D⟶E is called a strict set contraction if there is a constant 0≤λ<1 such that αTV≤λαV for any bounded set V⊂D.Theorem 2.
Assume that conditionsH,H∗ are satisfied; then, for ϕ0∈Xβ, problem (3) has a global mild solution in Br.Proof.
LetV be an arbitrary bounded set in Br. According to Lemmas 6 and 7, we know that T:Br⟶Br is bounded and continuous and Ty·,y∈V is equicontinuous on 0,T and e−ktTytβ<ε for any t≥T, y∈V, ε>0. Then, by Lemma 3.6, it follows that(63)αB′TV=supt≥0e−ktαTVt.
Consider Lemma9 and condition H∗; let any t≥0 be fixed, and for the above ε>0, we have(64)e−ktαTVt=e−ktα∫0tt−sq−1Pqt−sfs,ys+zsds,y∈V,≤e−kt∫0tαt−sq−1Pqt−sfs,ys+zs,y∈Vds,≤Cpe−kt∫0tt−s−qγ−1αfs,ys+zs,y∈Vds,≤Cpe−kt∫0tt−s−qγ−1mse−kssup0≤τ≤sαVτds,≤Cpe−kt∫0tt−s−qγ−1msdsαB′V,which implies that αB′TV≤λαB′V where λ≔Cpe−kt∫0tt−s−qγ−1msds<1. Then, T is a strict set contraction.
Consequently, by Lemma11, T has a fixed point in Br which implies that problem (3) has a global mild solution in Br. The proof process is completed.


## 3.1. The Case ThatQt Is Compact

In this section, we assume thatQt is compact for t>0, i.e., Qt is a compact operator for every t>0.Lemma 8.
LetZ⊆B′ be a bounded set; then, Z is relatively compact in B′ if the following conditions hold:(1)
The setyt,y∈Z is equicontinuous on any compact interval of 0,+∞ and for any t≥0, yt,y∈Z is relatively compact in X.(2)
For any givenε>0, there exists a constant T=Tε>0 such that e−ktytβ<ε for any t≥T and yt∈Z.Proof.
It is sufficient to prove thatZ is totally bounded. We consider the compact interval 0,T of 0,+∞. Define(41)Z0,T≔yt:yt∈Z,t∈0,T,with norm yb1≔sup0≤t≤Te−ktytβ; then, condition (1) combined with Arzelà–Ascoli theorem in Banach space indicates that Z0,T is relatively compact. Therefore, for any ε>0, there exist finitely many balls Bεyi such that Z0,T⊂∪i=1nBεyi, where yi∈B′.(42)Bεyi=yt∈Z0,T,y−yib1=sup0≤t≤Te−ktyt−yitβ≤ε.
Hence, for anyyt∈Z, there exists an i∈1,2,…,n such that y0,T∈Bεyi, i.e., for t∈0,T,(43)e−ktyt−yitβ≤ε.
Moreover, fort∈T,+∞, with conditions (3) and (43),(44)e−ktyt−yitβ,≤e−ktyt−e−kTyTβ+e−kTyT−e−kTyiTβ+e−kTyiT−e−ktyitβ,≤5ε.
Therefore, by (43) and (44), we have yt−yitb≤5ε for any t≥0. Then, Z can be covered by balls B5εyi=yt∈Z,y−yib≤5ε. Consequently, Z is totally bounded and the process is complete.Theorem 1.
Assume that conditionH holds; then, for ϕ0∈Xβ, problem (3) has at least one global mild solution in Br.Proof.
We aim to prove this theorem by using Schauder’s fixed-point theorem. In view of Lemma6, T:Br⟶Br and T is continuous, so we just need to prove that for any bounded subset V⊂Br, TV is relatively compact in X. Then, it is easy to prove that TV satisfies all conditions in Lemma 8.
Consider Lemma6; we have proved that Tyb=supt≥0e−ktTytβ≤r for any y∈Br which implies Ty,y∈Br is uniformly bounded. By Lemma 7, Ty,y∈B′ is equicontinuous on any compact interval 0,T of 0,+∞ and e−ktTytβ<ε for any t≥T and y∈B′. Then, it remains to show that Vt=Tyt,yt∈V is relatively compact in X for any t∈0,T.
It is easy to know thatV0=0 is compact in X. Let t∈0,T be fixed and for any ε∈0,t, δ>0, we define an operator Tεδ on V by the formula(45)Tεδyt=∫0t−ε∫δ+∞qθt−sq−1MqθQt−sqθfs,ys+zsdθds,=Qεqδ∫0t−ε∫δ+∞qθt−sq−1MqθQt−sqθ−εqδfs,ys+zsdθds,where y∈V. Under the compactness of Qεqδεqδ>0 and the boundedness of(46)∫0t−ε∫δ+∞qθt−sq−1MqθQt−sqθ−εqδfs,ys+zsdθds,we obtain that the set Vεδt=Tεδyt,y∈V is relatively compact in X for any ε∈0,t and δ>0. Moreover, for any y∈V, t>0, we have(47)e−ktTyt−Tεδytβ,≤qe−kt∫0t∫0δθt−sq−1Mqθt−sqθfs,ys+zsdθdsβ,+e−kt∫t−εt∫δ+∞qθt−sq−1MqθQt−sqθfs,ys+zsdθdsβ,≤qCpe−kt∫0tt−s−qγ−1e−kspsηsds∫0δθ−γMqθdθ,+Cpe−kt∫t−εtt−s−qγ−1e−kspsηsds.
According to∫0+∞θrMqθdθ=Γ1+r/Γ1+qr and condition H, we have(48)∫0δθ−γMqθdθ⟶0,∫t−εtt−s−qγ−1psηsds⟶0,asε⟶0,δ⟶0,which implies Tyt−Tεδytb⟶0 as ε⟶0,δ⟶0.
Therefore, the relatively compact setVεδt is arbitrarily close to the set Vt. Hence, for any t∈0,T, the set Vt, t∈0,T is also relatively compact in X.
Hence,T:Br⟶Br is a completely continuous operator. So, by Schauder’s fixed-point theorem, T has at least one fixed point in Br which implies that problem (3) has at least one global mild solution in Br.


## 3.2. The Case ThatQt Is Not Compact

In this section, we assume thatQt is not compact. In the following, α and αB′ denote the Kuratowski measures of noncompactness of bounded sets in Xβ and in B′. For more details about Kuratowski measures of noncompactness, we refer the readers to [27]. Assume that:H∗ There exists mt∈L0,+∞,0,+∞ such that I0+qm exists and for any bounded set V⊂B,(49)αft,V≤mte−ktsup−∞<τ≤0αVτ,and for any t≥0,(50)Cpe−kt∫0tt−s−qγ−1msds<1.Lemma 9.
(see [27]). If V⊂CJ,E is bounded and equicontinuous, then αVt is continuous and(51)α∫Jytdt,y∈V≤∫JαVtdt,where J is any compact interval of 0,+∞.Lemma 10.
LetV be a bounded set in B′. Suppose that Vt is equicontinuous on any compact interval 0,T of 0,+∞ and for any t≥T, ε>0, and y∈V,(52)e−ktytβ<ε.
Then, for eachVt=yt,y∈V,(53)αB′V=supt≥0e−ktαVt.Proof.
First, we prove thatαB′V≥supt≥0e−ktαVt. For the above given ε>0, t≥0, there exists a partition V=∪j=1nVj such that(54)diamVj<αB′V+ε,for anyj=1,2,…,n.
Then,Vt=∪j=1nVjt. For any u,v∈Vj, t≥0,(55)e−ktut−vtβ≤diamVj<αB′V+ε.
Therefore,diamVjt≤ektαB′V+ε which implies(56)supt≥0e−ktαVt≤αB′V,by the arbitrariness of ε.
Next, we show thatαB′V≤supt≥0e−ktαVt. By the equicontinuity of Vt on 0,T, there exists a partition 0=t0<t1<⋯<tm=T such that(57)e−kti′yti′−e−kti″yti″β<ε,for any ti′,ti″∈ti,ti+1, y∈V,i=0,1,…,m−1. Let Ii=ti,ti+1, i=0,1,…,m−1 and Im=tm,+∞; then, by (51) and (57),(58)e−kti′yti′−e−kti″yti″β<2ε,for anyy∈V,ti′,ti″∈Ii,i=0,1,…,m.
For eachi∈0,1,…,m, there exists a division V=∪j=1nVji such that Vti′=∪j=1nVjiti′ and(59)diamVjiti′<αVti′+2ε,j=1,2,…,n.
LetY be the finite set of all maps i⟶γi of 0,1,…,m into 1,2,…,n. For γ∈Y,(60)Zγ≔y∈V,yti′∈Vγiiti′,i=0,1,…,m,so V=yt,y∈Zγ,γ∈Y. For any u,v∈Zγ and t≥0, there exists i∈0,1,…,m such that t∈Ii; then,(61)e−ktut−vtβ,≤e−ktut−e−kti′uti′β+e−kti′uti′−e−kti′vti′β+e−ktvt−e−kti′vti′β,<αVti′+6ε.
Therefore,diamZγ≤αVti′+6ε. Since ε>0 is arbitrary, we have(62)αB′V≤supt≥0e−ktαVt.Lemma 11.
(see [27]). Let D be a bounded, closed, and convex subset of Banach space E. If the operator T:D⟶D is a strict set contraction, then T has a fixed point in D.Remark 2.
A bounded and continuous operatorT:D⟶E is called a strict set contraction if there is a constant 0≤λ<1 such that αTV≤λαV for any bounded set V⊂D.Theorem 2.
Assume that conditionsH,H∗ are satisfied; then, for ϕ0∈Xβ, problem (3) has a global mild solution in Br.Proof.
LetV be an arbitrary bounded set in Br. According to Lemmas 6 and 7, we know that T:Br⟶Br is bounded and continuous and Ty·,y∈V is equicontinuous on 0,T and e−ktTytβ<ε for any t≥T, y∈V, ε>0. Then, by Lemma 3.6, it follows that(63)αB′TV=supt≥0e−ktαTVt.
Consider Lemma9 and condition H∗; let any t≥0 be fixed, and for the above ε>0, we have(64)e−ktαTVt=e−ktα∫0tt−sq−1Pqt−sfs,ys+zsds,y∈V,≤e−kt∫0tαt−sq−1Pqt−sfs,ys+zs,y∈Vds,≤Cpe−kt∫0tt−s−qγ−1αfs,ys+zs,y∈Vds,≤Cpe−kt∫0tt−s−qγ−1mse−kssup0≤τ≤sαVτds,≤Cpe−kt∫0tt−s−qγ−1msdsαB′V,which implies that αB′TV≤λαB′V where λ≔Cpe−kt∫0tt−s−qγ−1msds<1. Then, T is a strict set contraction.
Consequently, by Lemma11, T has a fixed point in Br which implies that problem (3) has a global mild solution in Br. The proof process is completed.


## 4. Conclusions

In this paper, we investigated a class of fractional evolution equations with infinite delay and almost sectorial operator on unbounded domains in Banach space. We considered the case of compact semigroups and noncompact semigroups and obtained sufficient conditions of the existence of global mild solutions.



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