# Existence of the Mild Solutions for Delay Fractional Integrodifferential Equations with Almost Sectorial Operators

Authors: Fang Li
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper is concerned with  the existence of  mild solutions for
the fractional integrodifferential equations  with finite delay and
almost sectorial operators in a separable Banach spaceX. We obtain existence theorem for mild solutions to the above-mentioned equations, by means of measure of noncompactness and the resolvent operators associated with almost sectorial operators. As an application, the existence of mild solutions for some integrodifferential equation is
obtained.


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## Body

## 1. Introduction

Fractional differential and integrodifferential equations have received increasing attention during recent years and have been studied extensively (see, e.g., [1–15] and references therein) since they are playing an increasingly important role in engineering, physics, electrolysis chemical, fractional biological neurons, condensate physics, statistical mechanics, and so on.Moreover, the Cauchy problem for various delay equations in Banach spaces has been receiving more and more attention during the past decades (see, e.g., [3, 4, 9, 10, 14, 16–20] and references therein).We mention that much of the previous research on the fractional equations was done provided that the operator in the linear part is the infinitesimal generator of a strongly continuous operator semigroup, a compact semigroup, or an analytic semigroup, or is a Hille-Yosida operator (see, e.g., [1–4, 8–10, 12–14]). However, as presented in [15, Examples 1.1 and 1.2], for which the resolvent operators do not satisfy the required estimate to be a sectorial operator. Von Wahl in [21] first introduced examples of almost sectorial operators which are not sectorial. Recently, the study of evolution equations involving almost sectorial operators has been investigated extensively. However, much less is known about the fractional evolution equations with almost sectorial operators (see [15] and the references therein).In this paper, we are concerned with the following fractional integrodifferential equations:(1.1)cDqu(t)=Au(t)+f(t,ut)+∫0t‍g(t,s,us)ds,t∈(0,T],u(t)=ϕ(t),t∈[-r,0],
where T>0,0<q<1,and0<r<∞. The fractional derivative is understood here in the Caputo sense. X is a separable Banach space. A is an almost sectorial operator to be introduced later. Here f:[0,T]×C([-r,0],X)→X,g:Λ×C([-r,0],X)→X(Λ={(t,s)∈[0,T]×[0,T]:t≥s}),ϕ∈C([-r,0],X), where C([a,b],X) denotes the space of all continuous functions from [a,b] to X.For any continuous functionv defined on [-r,T] and any t∈[0,T], we denote by ut the element of C([-r,0],X) defined by ut(θ)=u(t+θ),θ∈[-r,0].Our paper is organized as follows. In Section2, we give out some preliminaries about fractional order operator, measure of noncompactness, and almost sectorial operators. The existence result will be established in Section 3. In Section 4, an example is given to show the application of the abstract result.


## 2. Preliminaries

Throughout this paper, we denote byX a separable Banach space with norm ∥·∥. For a linear operator A, we denote by D(A) the domain of A, by ρ(A) the resolvent set of A, and by R(z;A)=(zI-A)-1, z∈ρ(A) the resolvent of A. Moreover, we denote by L(X) the Banach space of all linear and bounded operators on X and by C([a,b],X) the space of all X-valued continuous functions on [a,b] with the supremum norm as follows:
(2.1)‖x‖[a,b]=‖x‖C([a,b],X)=sup⁡{‖x(t)‖:t∈[a,b]},foranyx∈C([a,b],X).
Moreover, we abbreviate ∥u∥Lp([0,T],R+) with ∥u∥Lp for any u∈Lp([0,T],R+).Let us recall the following known definitions. For more details, see [7, 11].Definition 2.1 (see [11]).
The fractional integral of orderq with the lower limit zero for a function f∈AC[0,∞) is defined as
(2.2)Iqf(t)=1Γ(q)∫0t‍(t-s)q-1f(s)ds,t>0,0<q<1
provided that the right side is pointwise defined on [0,∞), where Γ(·) is the gamma function.Definition 2.2 (see [11]).
The Riemann-Liouville derivative of orderq with the lower limit zero for a function f∈AC[0,∞) can be written as
(2.3)LDqf(t)=1Γ(1-q)ddt∫0t‍(t-s)-qf(s)ds,t>0,0<q<1.Definition 2.3 (see [11]).
The Caputo derivative of orderq for a function f∈AC[0,∞) can be written as
(2.4)cDqf(t)=DLq(f(t)-f(0)),t>0,0<q<1.Remark 2.4.
(1) Iff(t)∈C1[0,∞), then
(2.5)cDqf(t)=1Γ(1-q)∫0t‍(t-s)-qf′(s)ds=I1-qf′(t),t>0,0<q<1;
(2) the Caputo derivative of a constant is equal to zero.We will need the following facts from the theory of measures of noncompactness and condensing maps (see, e.g., [22, 23]).Definition 2.5.
LetE be a Banach space, 2E the family of all nonempty subsets of E, (𝒜,≥) a partially ordered set and ν:2E→𝒜. If for every Ω∈2E:
(2.6)ν(co¯(Ω))=ν(Ω),
then we say that ν is a measure of noncompactness (MNC) in E.
A MNCν is called:  (i)
monotone ifΩ0,Ω1∈2E, Ω0⊂Ω1 implies ν(Ω0)≤ν(Ω1);(ii)
nonsingular ifν({a0}∪Ω)=ν(Ω) for every a0∈E,Ω∈2E;(iii)
invariant with respect to union with compact sets ifν({D}∪Ω)=ν(Ω) for every relatively compact set D⊂E, Ω∈2E.If𝒜 is a cone in a normed space, we say that the MNC ν is;(iv)
algebraically semiadditive ifν(Ω0+Ω1)≤ν(Ω0)+ν(Ω1) for each Ω0,Ω1∈2E;(v)
regular ifν(Ω)=0 is equivalent to the relative compactness of Ω;(vi)
real if𝒜 is [0,+∞) with the natural order.As an example of the MNC possessing all these properties, we may consider the Hausdorff MNC:(2.7)χ(Ω)=inf⁡{ε>0:Ωhasafiniteε-net}.Now, letG:[0,h]→2E be a multifunction. It is called:  (i)
integrable, if it admits a Bochner integrable selection𝔤:[0,h]→E,𝔤(t)∈G(t) for a.e. t∈[0,h];(ii)
integrably bounded, if there exists a functionϱ∈L1([0,h],E) such that
(2.8)‖G(t)‖:=sup⁡{‖g‖:g∈G(t)}≤ϱ(t)a.e.t∈[0,h].We present the following assertion aboutχ-estimates for a multivalued integral [23, Theorem 4.2.3].Proposition 2.6.
For an integrable, integrably bounded multifunctionG:[0,h]→2X, where X is a separable Banach space, let
(2.9)χ(G(t))≤q(t),fora.e.t∈[0,h],
where 𝔮∈L+1([0,h]). Then, χ(∫0t‍G(s)ds)≤∫0t‍𝔮(s)ds for all t∈[0,h].LetE be a Banach space, ν a monotone nonsingular MNC in E.Definition 2.7.
A continuous map𝔉:Y⊆E→E is called condensing with respect to a MNC ν (or ν-condensing) if for every bounded set Ω⊆Y which is not relatively compact, we have
(2.10)ν(F(Ω))≱ν(Ω).
The following fixed point principle (see, e.g., [22, 23]) will be used later.Theorem 2.8.
Let𝔐 be a bounded convex closed subset of E and 𝔉:𝔐→𝔐 a ν-condensing map. Then, Fix𝔉={x:x=𝔉(x)} is nonempty.Theorem 2.9.
LetV⊂E be a bounded open neighborhood of zero and 𝔉:V¯→E a ν-condensing map satisfying the boundary condition:
(2.11)x≠λ¯F(x),
for all x∈∂V and 0<λ¯≤1. Then, Fix𝔉 is a nonempty compact set.To prove the main result, we will need the following generalization of Gronwall's lemma for singular kernels [24, Lemma 7.1.1].Lemma 2.10.
Letx,y:[0,T]→[0,+∞) be continuous functions. If y(·) is nondecreasing and there are constants 𝔞>0 and 0<ϑ<1 such that
(2.12)x(t)≤y(t)+a∫0t‍(t-s)-ϑx(s)ds,
then there exists a constant κ=κ(ϑ) such that
(2.13)x(t)≤y(t)+κa∫0t‍(t-s)-ϑy(s)ds,foreacht∈[0,T].Next, we recall the knowledge of almost sectorial operator, for more details, we refer to [25, 26].Let-1<γ<0 and Sμ0 with 0<μ<π be the open sector:
(2.14){z∈C∖{0}:|argz|<μ},
and Sμ its closure, that is,
(2.15)Sμ={z∈C∖{0}:|argz|≤μ}∪{0}.Let us recall the following definition.Definition 2.11.
Let-1<γ<0 and 0<ω<π/2. By  Θωγ(X), we denote the family of all linear closed operators A:D(A)⊂X→X which satisfy:(1)
σ(A)⊂Sω={z∈C∖{0}:|argz|≤ω}∪{0};(2)
for everyω<μ<π, there exists a constant Cμ such that
(2.16)‖R(z;A)‖L(X)≤Cμ|z|γ,forallz∈C∖Sμ.A linear operatorA will be called an almost sectorial operator on X if A∈Θωγ(X).Remark 2.12.
LetA∈Θωγ(X), then the definition implies that 0∈ρ(A).Remark 2.13 (see [15]).
From [26], note in particular that if A∈Θωγ(X), then A generates a semigroup T(t) with a singular behavior at t=0 in a sense, called semigroup of growth 1+γ. Moreover, the semigroup T(t) is analytic in an open sector of the complex plane C, but the strong continuity fails at t=0 for data which are not sufficiently smooth.We denote the semigroup associated withA by T(t). For t∈Sπ/2-ω0,
(2.17)T(t)=e-tz(A)=12πi∫Γθ‍e-tzR(z;A)dz
forms an analytic semigroup of growth order 1+γ, where ω<θ<μ<π/2-|argt|, the integral contour Γθ:={R+eiθ}∪{R+e-iθ} is oriented counter-clockwise ([15, 26]).We have the following proposition onT(t) [26, Theorem 3.9].Proposition 2.14.
LetA∈Θωγ(X) with -1<γ<0 and 0<ω<π/2. Then the following properties remain true:(i)
T(t) is analytic in Sπ/2-ω0 and
(2.18)dndtnT(t)=(-A)nT(t),forallt∈Sπ/2-ω0;(ii)
the functional equationT(s+t)=T(s)T(t) for all s,t∈Sπ/2-ω0 holds;(iii)
there exists a constantC0=C0(γ)>0 such that
(2.19)‖T(t)‖L(X)≤C0t-γ-1,forallt>0;(iv)
ifβ>1+γ, then D(Aβ)⊂ΣT={x∈X:limt→0;t>0T(t)x=x}.Clearly, we note that the condition (ii) of the Proposition2.14 does not satisfy for t=0 or s=0.The relation between the resolvent operators ofA and the semigroup T(t) is characterized by.Proposition 2.15 (see [26, Theorem 3.13]).
LetA∈Θωγ(X) with -1<γ<0 and 0<ω<π/2. Then for every λ∈C with Reλ>0, one has
(2.20)R(λ;A)=∫0∞‍e-λtT(t)dt.Based on the work in [15], we define operator families {𝒮q(t)}|t∈Sπ/2-ω0 and {𝒫q(t)}|t∈Sπ/2-ω0 by
(2.21)Sq(t)x=∫0∞‍Ψq(σ)T(σtq)xdσ,t∈Sπ/2-ω0,x∈X,Pq(t)x=∫0∞‍qσΨq(σ)T(σtq)xdσ,t∈Sπ/2-ω0,x∈X,
where Ψq(z) with 0<q<1 is a function of Wright-type (cf. e.g., [15]) as follows:
(2.22)Ψq(z):=∑n=0∞‍(-z)nn!Γ(-qn+1-q)=1π∑n=1∞‍(-z)n(n-1)!Γ(nq)sin⁡(nπq),z∈C.We collect some basic properties onΨq(z). For more details, we refer to ([7, 11, 15, 25]).Proposition 2.16.
For-1<r~<∞, λ>0, the following results hold: (1)
Ψq(t)≥0,  t>0;(2)
∫0∞‍q/tq+1Ψq(1/tq)e-λtdt=e-λq;(3)
∫0∞‍Ψq(t)tr~dt=Γ(1+r~)/Γ(1+qr~).Theorem 2.17 (see [15, Theorem 3.1]).
For each fixedt∈Sπ/2-ω0, 𝒮q(t) and 𝒫q(t) are linear and bounded operators on X. Moreover, for all t>0, -1<γ<0, 0<q<1,
(2.23)‖Sq(t)x‖≤C0Γ(-γ)Γ(1-q(1+γ))t-q(1+γ)‖x‖,x∈X,‖Pq(t)x‖≤qC0Γ(1-γ)Γ(1-qγ)t-q(1+γ)‖x‖,x∈X.Theorem 2.18 (see [15, Theorem 3.2]).
Fort>0, 𝒮q(t) and 𝒫q(t) are continuous in the uniform operator topology. Moreover, for every r~>0, the continuity is uniform on [r~,∞).Remark 2.19 (see [15, Theorem 3.4]).
Letβ>1+γ. Then for all x∈D(Aβ),
(2.24)lim⁡t→0;t>0Sq(t)x=x.Next, we will present the definition of mild solution of problem (1.1).According to Definitions2.1–2.3, we can rewrite problem (1.1) in the equivalent integral equation:
(2.25)u(t)=ϕ(0)+1Γ(q)∫0t‍(t-s)q-1[Au(s)+f(s,us)+a(u)(s)]ds,t∈(0,T],u(t)=ϕ(t),t∈[-r,0]
provided that the integral in (2.25) exists, where
(2.26)a(u)(t)=∫0t‍g(t,s,us)ds.Set(2.27)u^(λ)=∫0∞‍e-λtu(t)dt,f^(λ)=∫0∞‍e-λtf(t,ut)dt,g^(λ)=∫0∞‍e-λta(u)(t)dt,
formally applying the Laplace transform to (2.25), we get
(2.28)u^(λ)=1λϕ(0)+1λqAu^(λ)+1λq[f^(λ)+g^(λ)],
then
(2.29)(λq-A)u^(λ)=λq-1ϕ(0)+[f^(λ)+g^(λ)],
thus
(2.30)u^(λ)=λq-1(λq-A)-1ϕ(0)+(λq-A)-1[f^(λ)+g^(λ)]=λq-1∫0∞‍e-λqsT(s)ϕ(0)ds+∫0∞‍e-λqsT(s)[f^(λ)+g^(λ)]ds
provided that the integral in (2.30) exists.Then, using Proposition2.16(2), we have
(2.31)λq-1∫0∞‍e-λqsT(s)ϕ(0)ds=q∫0∞‍(λt)q-1e-(λt)qT(tq)ϕ(0)dt=-1λ∫0∞‍(ddte-(λt)q)T(tq)ϕ(0)dt=∫0∞∫0∞qσqΨq(1σq)e-λtσT(tq)ϕ(0)dσdt=∫0∞‍e-λt[∫0∞‍qσq+1Ψq(1σq)T(tqσq)ϕ(0)dσ]dt=∫0∞‍e-λt[∫0∞‍Ψq(τ)T(tqτ)ϕ(0)dτ]dt,∫0∞‍e-λqsT(s)f^(λ)ds=∫0∞‍e-(λτ)qqτq-1T(τq)(∫0∞‍e-λtf(t,ut)dt)dτ=∫0∞∫0∞e-λτσqσq+1Ψq(1σq)qτq-1T(τq)(∫0∞‍e-λtf(t,ut)dt)dσdτ=q2∫0∞∫0∞e-λθθq-1σ2q+1Ψq(1σq)T(θqσq)(∫0∞‍e-λtf(t,ut)dt)dθdσ=q2∫0∞‍(∫0∞‍∫t∞‍e-λτ(τ-t)q-1σ2q+1Ψq(1σq)T((τ-t)qσq)f(t,ut)dτdt)dσ=q2∫0∞‍(∫0∞‍∫0τ‍e-λτ(τ-t)q-1σ2q+1Ψq(1σq)T((τ-t)qσq)f(t,ut)dtdτ)dσ=∫0∞‍e-λt[q∫0t‍∫0∞‍(t-s)q-1σΨq(σ)T((t-s)qσ)f(s,us)dσds]dt.
Similarly, we have
(2.32)∫0∞‍e-λqsT(s)g^(λ)ds=∫0∞‍e-λt[q∫0t‍∫0∞‍(-s)q-1σΨq(σ)T((-s)qσ)a(u)(s)dσds]dt.
Thus, from (2.30)–(2.32), we obtain
(2.33)u^(λ)=∫0∞‍e-λt[∫0∞‍Ψq(τ)T(tqτ)ϕ(0)dτ+q∫0t‍∫0∞‍(t-s)q-1σΨq(σ)T((t-s)qσ)f(s,us)dσds+q∫0t‍∫0∞‍(t-s)q-1σΨq(σ)T((t-s)qσ)a(u)(s)dσds]dt.We invert the last Laplace transform to obtain(2.34)u(t)=∫0∞‍Ψq(τ)T(tqτ)ϕ(0)dτ+q∫0t‍∫0∞‍(t-s)q-1σΨq(σ)T((t-s)qσ)f(s,us)dσds+q∫0t‍∫0∞‍(t-s)q-1σΨq(σ)T((t-s)qσ)a(u)(s)dσds=Sq(t)ϕ(0)+∫0t‍(t-s)q-1Pq(t-s)f(s,us)ds+∫0t‍(t-s)q-1Pq(t-s)a(u)(s)ds.Then from the above induction, whenϕ(0)∈D(Aβ) with β>1+γ, we can give the following definition of the mild solution of (1.1).Definition 2.20.
A continuous functionu:[-r,T]→X satisfying the equation:
(2.35)u(t)={ϕ(t),t∈[-r,0],Sq(t)ϕ(0)+∫0t‍(t-s)q-1Pq(t-s)f(s,us)ds+∫0t‍(t-s)q-1Pq(t-s)a(u)(s)ds,t∈[0,T]
is called a mild solution of (1.1).Remark 2.21 (see [15], Remark 4.1).
(1)  In general, since the operator𝒮q(t) is singular at t=0, solutions to problem (1.1) are assumed to have the same kind of singularity at t=0 as the operator 𝒮q(t).
(2)  Whenϕ(0)∈D(Aβ) with β>1+γ, it follows from Remark 2.19 that the mild solution is continuous at t=0.


## 3.  Main Result

Throughout this section, letA be an operator in the class Θωγ(X) and -1<γ<0,0<ω<π/2. Moreover, we require the following assumptions:  (Hf)
(1)f:[0,T]×C([-r,0],X)→X satisfies f(·,y):[0,T]→X is measurable for all y∈C([-r,0],X) and f(t,·):C([-r,0],X)→X is continuous for a.e. t∈[0,T], and there exists a positive function μ(·)∈Lp([0,T],R+)(p>-1/(qγ)>1/q>1) such that
(3.1)‖f(t,y)‖≤μ(t)‖y‖[-r,0],for almost allt∈[0,T];(2) there exists a nondecreasing functionη∈Lp([0,T],R+) such that for any bounded set D⊂C([-r,0],X):
(3.2)χ(f(t,D))≤η(t)sup⁡θ∈[-r,0]χ(D(θ)),a.e.t∈[0,T].(Hg)
(1)g:Λ×C([-r,0],X)→X and g(t,s,·):C([-r,0],X)→X is continuous for a.e. (t,s)∈Λ, and for each y∈C([-r,0],X), the function g(·,·,y):Λ→X is measurable. Moreover, there exists a function m:Λ→R+ with supt∈[0,T]∫0t‍m(t,s)ds:=m*<∞ such that
(3.3)‖g(t,s,y)‖≤m(t,s)‖y‖[-r,0],for almost all(t,s)∈Λ;(2) for any bounded setD⊂C([-r,0],X), there exists a function ξ:Λ→R+ such that
(3.4)χ(g(t,s,D))≤ξ(t,s)sup⁡θ∈[-r,0]χ(D(θ)),wheresupt∈[0,T]∫0t‍ξ(t,s)ds:=ξ*<∞.Theorem 3.1.
LetA∈Θωγ(X) with -1<γ<0, 0<ω<π/2. Assume that (Hf) and (Hg) are satisfied. Then for every ϕ(0)∈D(Aβ) with β>1+γ, the mild solution set of problem (1.1) is a nonempty compact subset of the space C([-r,T],X), provided that
(3.5)M(lp,qT-(1+pqγ)/p‖μ‖Lp+-m*T-qγqγ)<1,
where M:=(qC0Γ(1-γ))/Γ(1-qγ) and lp,q:=((p-1)/(-pqγ-1))(p-1)/p.Proof.
We define the operatorℱ:C([-r,T],X)→C([-r,T],X) in the following way:
(3.6)(Fu)(t)={ϕ(t),t∈[-r,0],Sq(t)ϕ(0)+∫0t‍(t-s)q-1Pq(t-s)f(s,us)ds+∫0t‍(t-s)q-1Pq(t-s)a(u)(s)ds,t∈[0,T].
It is clear that the operator ℱ is well defined.
We define(3.7)ϕ^(t)={ϕ(t),t∈[-r,0],Sq(t)ϕ(0),t∈[0,T].
Let u(t)=v(t)+ϕ^(t). It is easy to see that v satisfies v0=0 and
(3.8)v(t)=∫0t‍(t-s)q-1Pq(t-s)f(s,vs+ϕ^s)ds+∫0t‍(t-s)q-1Pq(t-s)a(v+ϕ^)(s)ds,t∈[0,T]
if and only if u satisfies
(3.9)u(t)=Sq(t)ϕ(0)+∫0t‍(t-s)q-1Pq(t-s)f(s,us)ds+∫0t‍(t-s)q-1Pq(t-s)a(u)(s)ds,t∈[0,T]
and u(t)=ϕ(t),t∈[-r,0].
Letℱ^:C([-r,T],X)→C([-r,T],X) be an operator defined by (ℱ^v)(t)=0 for t∈[-r,0] and
(3.10)(F^v)(t)=∫0t‍(t-s)q-1Pq(t-s)f(s,vs+ϕ^s)ds+∫0t‍(t-s)q-1Pq(t-s)a(v+ϕ^)(s)ds,t∈[0,T].
Clearly the operatorℱ^ has a fixed point is equivalent to ℱ having one.
We define𝒞:={v∈C([-r,T],X):v0=0}⊂C([-r,T],X). Next we will prove that ℱ^ has a fixed point on 𝒞.
Let{vn}n∈N be a sequence such that vn→v in 𝒞 as n→∞. Since f satisfies (Hf)(1) and g satisfies (Hg)(1), for almost every t∈[0,T] and (t,s)∈Λ, we get
(3.11)f(t,vtn+ϕ^t)⟶f(t,vt+ϕ^t),asn⟶∞,g(t,s,vsn+ϕ^s)⟶g(t,s,vs+ϕ^s),asn⟶∞.
Noting that(3.12)‖vt+ϕ^t‖[-r,0]≤sup⁡t∈[-r,0]‖v(t)‖+sup⁡t∈[0,T]‖v(t)‖+sup⁡t∈[-r,0]‖ϕ^(t)‖+sup⁡t∈[0,T]‖ϕ^(t)‖=sup⁡t∈[0,T]‖v(t)‖+sup⁡t∈[-r,0]‖ϕ(t)‖+sup⁡t∈[0,T]‖Sq(t)ϕ(0)‖:=α.
Therefore,
(3.13)‖f(t,vt+ϕ^t)‖≤μ(t)‖vt+ϕ^t‖[-r,0]≤αμ(t),
and
(3.14)‖a(v+ϕ^)(t)‖≤∫0t‍‖g(t,s,vs+ϕ^s)‖ds≤∫0t‍m(t,s)‖vs+ϕ^s‖[-r,0]ds≤m*α.
Sincevn→v in 𝒞, it follows that there exists ε>0 such that ∥vn-v∥[0,T]≤ε for n sufficiently large. Moreover, noting that ∥vtn-vt∥[-r,0]≤∥vn-v∥[0,T], we have
(3.15)‖f(t,vtn+ϕ^t)-f(t,vt+ϕ^t)‖≤μ(t)‖vtn+ϕ^t‖[-r,0]+αμ(t)≤μ(t)‖vtn-vt‖[-r,0]+2αμ(t)≤(ε+2α)μ(t).
Similarly,(3.16)‖g(t,s,vsn+ϕ^s)-g(t,s,vs+ϕ^s)‖≤(ε+2α)m(t,s),‖∫0t‍g(t,s,vsn+ϕ^s)ds-∫0t‍g(t,s,vs+ϕ^s)ds‖≤m*(ε+2α).
It follows from the Lebesgue's Dominated Convergence Theorem that
(3.17)‖∫0t‍g(t,s,vsn+ϕ^s)ds-∫0t‍g(t,s,vs+ϕ^s)ds‖⟶0,asn⟶∞,
and from (2.23), we have
(3.18)‖∫0t‍(t-s)q-1Pq(t-s)[f(s,vsn+ϕ^s)-f(s,vs+ϕ^s)]ds‖+‖∫0t‍(t-s)q-1Pq(t-s)[∫0s‍g(s,τ,vτn+ϕ^τ)dτ-∫0s‍g(s,τ,vτ+ϕ^τ)dτ]ds‖≤M∫0t‍(t-s)-qγ-1[‖∫0s‍g(s,τ,vτn+ϕ^τ)dτ-∫0s‍g(s,τ,vτ+ϕ^τ)dτ‖‖f(s,vsn+ϕ^s)-f(s,vs+ϕ^s)‖+‖∫0s‍g(s,τ,vτn+ϕ^τ)dτ-∫0s‍g(s,τ,vτ+ϕ^τ)dτ‖]ds⟶0,asn⟶∞.
Therefore, we obtain that
(3.19)lim⁡n→∞‖F^vn-F^v‖[0,T]=0,
then we see that ℱ^ is continuous.
Let us consider the MNCν in the space 𝒞 with values in the cone R+3 of the following way: for every bounded subset Ω⊂𝒞,
(3.20)ν(Ω)=(sup⁡t∈[-r,0]χ(Ω(t)),Φ(Ω),mod⁡c(Ω)),
where modc(Ω) is the module of equicontinuity of Ω given by
(3.21)mod⁡c(Ω)=lim⁡δ→0sup⁡v∈Ωmax⁡|t1-t2|≤δ‖v(t1)-v(t2)‖,Φ(Ω)=sup⁡t∈[0,T](e-Ltsup⁡s∈[0,t]χ(Ω(s))),
where L>0 is a constant chosen so that
(3.22)Msup⁡t∈[0,T]∫0t‍(t-s)-qγ-1η(s)e-L(t-s)ds=L1<1,(3.23)Mξ*sup⁡t∈[0,T]∫0t‍(t-s)-qγ-1e-L(t-s)ds=L2<1.
Noting that for any ψ∈L1([0,T],X), we have
(3.24)lim⁡L→+∞sup⁡t∈[0,T]∫0t‍e-L(t-s)ψ(s)ds=0,
so, we can take the appropriate L to satisfy (3.22) and (3.23).
Next, we show that the operatorℱ^ is ν-condensing on every bounded subset of 𝒞.
LetΩ⊂𝒞 be a nonempty, bounded set for which
(3.25)ν(F^(Ω))≥ν(Ω).
Noting that
(3.26)sup⁡t∈[-r,0]χ(F^(Ω)(t))=0
and (3.25), we can see that supt∈[-r,0]χ(Ω(t))=0.
Next, we estimateΦ(Ω). For any t∈[0,T], we set
(3.27)F^1(Ω)(t)={∫0t‍(t-s)q-1Pq(t-s)f(s,vs+ϕ^s)ds:v∈Ω}.
We consider the multifunction s∈[0,t]⊸H(s):
(3.28)H(s)={(t-s)q-1Pq(t-s)f(s,vs+ϕ^s):v∈Ω}.
Obviously, H is integrable, and from (2.23), (Hf)(1), and (3.13), it follows that H is integrably bounded. Moreover, noting that (Hf)(2), we have the following estimate for a.e. s∈[0,t]:
(3.29)χ(H(s))=χ({(t-s)q-1Pq(t-s)f(s,vs+ϕ^s):v∈Ω})=χ((t-s)q-1Pq(t-s)f(s,Ωs+ϕ^s))≤M(t-s)-qγ-1η(s)sup⁡τ∈[0,s]χ(Ω(τ))=M(t-s)-qγ-1η(s)eLse-Lssup⁡τ∈[0,s]χ(Ω(τ))≤M(t-s)-qγ-1η(s)eLsΦ(Ω).
Applying Proposition 2.6, we have
(3.30)χ(F^1(Ω)(t))=χ(∫0t‍H(s)ds)≤M∫0t‍(t-s)-qγ-1η(s)eLsds⋅Φ(Ω).
Therefore, from (3.22), we have
(3.31)sup⁡t∈[0,T](e-Ltsup⁡s∈[0,t]χ(F^1(Ω)(s)))≤Msup⁡t∈[0,T]∫0t‍(t-s)-qγ-1η(s)e-L(t-s)ds⋅Φ(Ω)=L1Φ(Ω).
Similarly, if we set(3.32)F^2(Ω)(t)={∫0t‍(t-s)q-1Pq(t-s)a(v+ϕ^)(s)ds:v∈Ω},
then we can see that the multifunction s∈[0,t]⊸H~(s),
(3.33)H~(s)={(t-s)q-1Pq(t-s)a(v+ϕ^)(s):v∈Ω}
is integrable, and from (2.23), (Hg)(1), and (3.14), it follows that H~ is integrably bounded. Moreover, noting that (Hg)(2), Proposition 2.6, and (3.23), we have the following estimate for a.e. s∈[0,t]:
(3.34)χ(H~(s))≤Mξ*(t-s)-qγ-1eLsΦ(Ω),sup⁡t∈[0,T](e-Ltsup⁡s∈[0,t]χ(F^2(Ω)(s)))≤Mξ*sup⁡t∈[0,T]∫0t‍(t-s)-qγ-1e-L(t-s)ds⋅Φ(Ω)=L2Φ(Ω).
Now, from (3.31) and (3.34), L>0 can be chosen so that
(3.35)Φ(F^(Ω))≤(L1+L2)Φ(Ω)=L~Φ(Ω),
where 0<L~<1.
From (3.25), we have Φ(Ω)=0. Next, we will prove that  modc(Ω)=0.
Letδ>0, t1,t2∈(0,T] such that 0<t2-t1≤δ and v∈Ω, noting that (Hf)(1) and (Hg)(1), we obtain
(3.36)‖∫0t1‍(t1-s)q-1Pq(t1-s)[f(s,vs+ϕ^s)+a(v+ϕ^)(s)]ds-∫0t2‍(t2-s)q-1Pq(t2-s)[f(s,vs+ϕ^s)+a(v+ϕ^)(s)]ds‖≤‖∫0t1‍[(t2-s)q-1-(t1-s)q-1]Pq(t2-s)[f(s,vs+ϕ^s)+a(v+ϕ^)(s)]ds‖+‖∫t1t2‍(t2-s)q-1Pq(t2-s)[f(s,vs+ϕ^s)+a(v+ϕ^)(s)]ds‖+‖∫0t1‍(t1-s)q-1[Pq(t2-s)-Pq(t1-s)][f(s,vs+ϕ^s)+a(v+ϕ^)(s)]ds‖=I1+I2+I3.
Using (2.23), (3.13), and (3.14), we have
(3.37)I1≤αM∫0t1‍|(t2-s)q-1-(t1-s)q-1|(t2-s)-q(1+γ)[μ(s)+m*]ds.
Clearly, I1 tends to zero as t2→t1. Similarly, for I2, we have
(3.38)I2≤αM∫t1t2‍(t2-s)-qγ-1[μ(s)+m*]ds⟶0,ast2⟶t1.
ForI3,  for ε>0 small enough, noting that (2.23), (3.13), and (3.14), we have
(3.39)I3≤α∫0t1-ε‍(t1-s)q-1‖Pq(t2-s)-Pq(t1-s)‖L(X)[μ(s)+m*]ds+α∫t1-εt1‍(t1-s)q-1‖Pq(t2-s)-Pq(t1-s)‖L(X)[μ(s)+m*]ds≤αsup⁡s∈[0,t1-ε]‖Pq(t2-s)-Pq(t1-s)‖L(X)⋅∫0t1-ε‍(t1-s)q-1[μ(s)+m*]ds+αM∫t1-εt1‍((t1-s)q-1(t1-s)q(γ+1)+(t1-s)q-1(t2-s)q(γ+1))[μ(s)+m*]ds,
it follows from Theorem 2.18 that I3 tends to zero as t2→t1 and ε→0.
For the case when0=t1<t2≤T, we can see
(3.40)‖∫0t2‍(t2-s)q-1Pq(t2-s)[f(s,vs+ϕ^s)+a(v+ϕ^)(s)]ds‖≤αM∫0t2‍(t2-s)-qγ-1[μ(s)+m*]ds⟶0,ast2⟶0.
Thus, the set{(ℱ^v)(·):v∈Ω} is equicontinuous, then modc(ℱ^Ω)=0. From (3.25), we get that  modc(Ω)=0. Hence ν(Ω)=(0,0,0).
The regularity property ofν implies the relative compactness of Ω. Now, it follows from Definition 2.7 that ℱ^ is ν-condensing.
Consider the set(3.41)Bρ={v∈C:‖v‖[0,T]≤ρ}.
Next, we show that there exists some ρ>0 such that ℱ^Bρ⊂Bρ. Suppose on the contrary that for each ρ>0 there exist vρ(·)∈Bρ and some t∈[0,T] such that ∥(ℱ^vρ)(t)∥[0,T]>ρ.
Noting the Hölder inequality, we have(3.42)∫0t‍(t-s)-1-qγμ(s)ds≤t(-(1+pqγ))/plp,q‖μ‖Lp≤T(-(1+pqγ))/plp,q‖μ‖Lp.
By (2.23), (Hf)(1), (Hg)(1), and (3.42), we have
(3.43)‖(F^vρ)(t)‖=‖∫0t‍(t-s)q-1Pq(t-s)[f(s,vsρ+ϕ^s)+a(vρ+ϕ^)(s)]ds‖≤M[‖vρ‖[0,T]+M~]∫0t‍(t-s)-qγ-1[μ(s)+m*]ds≤M(ρ+M~)⋅[lp,qT(-(1+pqγ))/p‖μ‖Lp+-m*T-qγqγ],
where M~=∥ϕ∥[-r,0]+supt∈[0,T]∥𝒮q(t)ϕ(0)∥.
Then,(3.44)ρ<‖(F^vρ)(t)‖[0,T]≤M[ρ+M~]⋅[lp,qT-(1+pqγ)/p‖μ‖Lp+-m*T-qγqγ].
Dividing both sides of (3.44) by ρ, and taking ρ→∞, we have
(3.45)M(lp,qT-(1+pqγ)/p‖μ‖Lp+-m*T-qγqγ)≥1.
This contradicts (3.5). Hence for some positive number ρ, ℱ^Bρ⊂Bρ. According to Theorem 2.8, problem (1.1) has at least one mild solution.
Next, forc∈(0,1], we consider the following one-parameter family of maps:
(3.46)H:[0,1]×C⟶C(c,v)⟶H(c,v)=cF^(v).
We will demonstrate that the fixed point set of the family ℋ:
(3.47)Fix⁡H={v∈H(c,v)forsomec∈(0,1]}
is a priori bounded. In fact, let v∈Fixℋ, for t∈[0,T], we have
(3.48)‖v(t)‖≤∫0t‍‖(t-s)q-1Pq(t-s)[f(s,vs+ϕ^s)+a(v+ϕ^)(s)]‖ds≤M∫0t‍(t-s)-qγ-1[μ(s)+m*]⋅[sup⁡τ∈[0,s]‖v(τ)‖+M~]ds≤M[sup⁡s∈[0,t]‖v(s)‖∫0t‍(t-s)-qγ-1μ(s)ds+m*∫0t‍(t-s)-qγ-1sup⁡τ∈[0,s]‖v(τ)‖ds]+a1≤a1+a2sup⁡s∈[0,t]‖v(s)‖+a3∫0t‍(t-s)-qγ-1sup⁡τ∈[0,s]‖v(τ)‖ds,
where
(3.49)a1=MM~⋅(lp,qT-(1+pqγ)/p‖μ‖Lp+-m*T-qγqγ),a2=Mlp,qT-(1+pqγ)/p‖μ‖Lp,a3=Mm*.
We denote thatx(t):=sups∈[0,t]∥v(s)∥. Let t~∈[0,t] such that x(t)=∥v(t~)∥. Then, by (3.48), we can see
(3.50)x(t)≤a1+a2x(t)+a3∫0t‍(t-s)-qγ-1x(s)ds.
By Lemma 2.10, there exists a constant κ such that
(3.51)x(t)≤a11-a2+κa1a3(1-a2)2∫0t‍(t-s)-qγ-1ds≤a11-a2-κa1a3T-qγqγ(1-a2)2:=ϖ.
Hence, supt∈[0,T]∥v(t)∥≤ϖ.
Now, we consider a closed ball as follows:(3.52)BR={v∈C:‖v‖[0,T]≤R}⊂C.
We take the radius R>0 large enough to contain the set Fixℋ inside itself. Moreover, from the proof above-mentioned, ℱ^:BR→𝒞 is ν-condensing, and it remains to apply Theorem 2.9.


## 4. Application

Example 4.1.
Let(4.1)A^=(-iΔ+σ)1/2,D(A^)=W1,3(R2)(aSobolevspace)
be as in [15, Example 6.3], in which, the authors demonstrate that A^∈Θωγ(L3(R2)) for some 0<ω<π/2 and γ=-1/6. We denote the semigroup associated with A^ by T(t) and ∥T(t)∥≤C0t-5/6(C0 is a constant).
LetX=L3(R2), we consider the following integrodifferential problem:
(4.2)∂tqw(t,x)=A^w(t,x)+1t2k∫0t‍(t-s)(1/(2k))-1∫-r0‍ζ1(θ)⋅sin⁡(|wt(θ,x)|sk)dθds+∫0t‍(t-s)k∫-r0‍ζ2(θ)ws(θ,x)dθds,w(θ,x)=w0(θ,x),-r≤θ≤0,
where ∂tq is the Caputo fractional partial derivative of order 0<q<1, t∈(0,1], r>0, x∈R2 and wt(θ,x)=w(t+θ,x). ζi:[-r,0]→R(i=1,2), and ∫-r0‍|ζi(θ)|dθ<∞(i=1,2), k>1 is a constant.
Forx∈R2 and φ∈C([-r,0],X), we set
(4.3)u(t)(x)=w(t,x),ϕ(θ)(x)=w0(θ,x),θ∈[-r,0],f(t,φ)(x)=1t2k∫0t‍(t-s)(1/(2k))-1∫-r0‍ζ1(θ)⋅sin⁡(|φ(θ)(x)|sk)dθds,g(t,s,φ)(x)=(t-s)k∫-r0‍ζ2(θ)φ(θ)(x)dθ.
Then the above equation (4.2) can be reformulated as the abstract (1.1).
Fort∈(0,1], we can see that(4.4)‖f(t,φ)‖≤1t2k(∫0t‍(t-s)(1/(2k))-1skds⋅∫-r0‍|ζ1(θ)|dθ)⋅‖φ‖[-r,0]:=μ(t)⋅‖φ‖[-r,0],
clearly, μ(t)=tkB(1/(2k),k+1)∫-r0‍|ζ1(θ)|dθ∈Lp([0,1],R+)(p>6/q), where B(·,·) is a beta function.
For anyφ,φ~∈C([-r,0],X),
(4.5)‖f(t,φ)(x)-f(t,φ~)(x)‖≤1t2k∫0t‍(t-s)(1/(2k))-1skds⋅∫-r0‍|ζ1(θ)|‖φ(θ)(x)-φ~(θ)(x)‖dθ.
Therefore, for any bounded set D⊂C([-r,0],X), we have
(4.6)χ(f(t,D))≤μ(t)sup⁡-r≤θ≤0χ(D(θ)),a.e.t∈[0,1].
Moreover, for almost all(t,s)∈Λ:
(4.7)‖g(t,s,φ)(x)‖=‖(t-s)k∫-r0‍ζ2(θ)φ(θ)(x)dθ‖≤m(t,s)‖φ‖[-r,0],
where m(t,s):=(t-s)k∫-r0‍|ζ2(θ)|dθ. Then, we obtain
(4.8)m*=sup⁡t∈[0,1]∫0t‍m(t,s)ds=∫-r0‍|ζ2(θ)|dθ⋅sup⁡t∈[0,1]∫0t‍(t-s)kds=1k+1∫-r0‍|ζ2(θ)|dθ,‖g(t,s,φ)(x)-g(t,s,φ~)(x)‖≤(t-s)k∫-r0‍|ζ2(θ)|‖φ(θ)(x)-φ~(θ)(x)‖dθ.
Hence, for any bounded set D⊂C([-r,0],X), we have
(4.9)χ(g(t,s,D))≤ξ(t,s)sup⁡-r≤θ≤0χ(D(θ)),
where ξ(t,s):=(t-s)k∫-r0‍|ζ2(θ)|dθ, and supt∈[0,1]∫0t‍ξ(t,s)ds=1/(k+1)∫-r0‍|ζ2(θ)|dθ.
Suppose further that there exists constantM*∈(0,1) such that
(4.10)qC0Γ(1-γ)Γ(1-qγ)(lp,q‖μ‖Lp+-m*qγ)<M*<1,
then for ϕ(0)∈D(A^β)(β>5/6), problem (4.2) has at least a mild solution by Theorem 3.1.



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*Source: 729615-2012-07-30.xml*