# Existence for Nonautonomous Fractional Integrodifferential Equations with Nonlocal Conditions

Authors: Fang Li
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We study  the existence  of mild solution of a class of nonlinear  nonautonomous fractional integrodifferential equations
with nonlocal conditions in a separable Banach spaceX. Combining the techniques of operator semigroup, noncompactness measures, and
the fixed point theory, we obtain new existence of mild solution without the assumptions that  the nonlinearity f satisfies a Lipschitz type condition and the semigroup {exp(−tA(s))} generated by {−A(s)}s∈[0,T] is compact. An application of the abstract result is also given.


---

## Body

## 1. Introduction

In this paper, we denote thatC is a positive constant and assume that a family of closed linear operators {A(t)}t∈[0,T] satisfies the following.(A1)
The domainD(A) of {A(t)}t∈[0,T] is dense in a Banach space X and independent of t.(A2)
The operator[A(t)+λ]-1 exists in L(X) (the Banach space of all linear and bounded operators on X) for any λ with Re λ≥0 and
(1)∥[A(t)+λ]-1∥≤C|λ|+1,t∈[0,T].(A3)
There exist constantsγ∈(0,1] and C such that
(2)∥[A(t1)-A(t2)]A-1(s)∥≤C|t1-t2|γ,[A(t1)-A(t2)]lA-1(s)t1,t2,s∈[0,T].Under condition (A2), each operator-A(s), s∈[0,T], generates an analytic semigroup exp(-tA(s)), t>0, and there exists a constant C such that
(3)∥An(s)exp(-tA(s))∥≤Ctn,
where n=0,1,t>0, s∈[0,T] ([1]).Moreover,∥A(t)A-1(s)∥≤C, which follows from condition (A2), where t,s∈[0,T] and C is a positive constant independent of both t and s.This paper is concerned with existence result for nonautonomous fractional integrodifferential equations with nonlocal conditions in a separable Banach spaceX:
(4)dqv(t)dtq=-A(t)v(t)+f(t,∫0t‍K(t,s)v(s)ds,v(t)),hhhhhhhhhhhhhhhhhhhhhhhhhhht∈[0,T],v(0)=A-1(0)g(v),
where T>0, 0<q<1, {A(t)}t∈[0,T] is a family of closed linear operators in X and satisfies (A1)–(A3), K∈C(Δ,R+) with Δ={(t,s)∈R2:0≤s≤t≤T} and
(5)supt∈[0,T]∫0t‍K(t,s)ds<∞,f:[0,T]×X×X→X, g are given functions to be specified later. The fractional derivative is understood here in the Riemann-Liouville sense.Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order. The field of the application of fractional calculus is very broad. We can see it in the study of the memorial materials, earthquake analysis, robots, electric fractal network, fractional sine oscillator, electrolysis chemical, fractional capacitance theory, electrode electrolyte interface description, fractal theory, especially in the dynamic process description of porous structure, fractional controller design, vibration control of viscoelastic system and pliable structure objects, fractional biological neurons, and probability theory. For details, see the monographs of Kilbas et al. [2], Kiryakova [3], Lakshmikantham and Vatsala [4], Miller and Ross [5], Samko et al. [6] and Podlubny [7], and the references therein. Some recent contributions to the theory of fractional differential equations can be seen in [8–20] and the references therein. Among the previous researches, most of researchers focus on the case that the differential operators (possibly unbounded) in the main parts are independent of time t. However, when treating some parabolic evolution problems, it is usually assumed that the partial differential operators depend on time t (i.e., it is the case of the problems under considerations being nonautonomous), since this class of operators appears frequently in the applications (see [21] and the references therein).Moreover, since the work of Byszewski [22], the nonlocal Cauchy problems have been investigated in many papers (cf., e.g., [13–15, 20, 23–25] and the references therein). The nonlocal conditions give a better description in applications than standard ones, and the Cauchy problem with nonlocal initial condition can be applied in physics with better effect than the classical Cauchy problem with traditional initial conditions. The existence of mild solutions of nonautonomous fractional evolution equations with nonlocal conditions of the form (4) is an untreated original topic, which in fact is the main motivation of the present paper.In this paper, using a pair of evolution families{ψ(t,s)} and {φ(t,s)} associated with the semigroup {exp(-tA(s))}(t>0), we give a reasonable concept of solution to problem (4) in Section 2. Moreover, in general, the semigroup {exp(-tA(s))}(t>0) generated by -A(s)(s∈[0,T]) is not compact, so we obtain the main result based on the theory of measures of noncompactness and the condensing maps. These techniques are often used to deal with abstract integer order differential equations but rarely used in abstract fractional order differential equations(e.g., [8–20] and the references therein). We will study (4) under suitable hypotheses based on a special noncompactness measure and the properties of fixed points set of condensing operators [26, 27] and establish a new existence result for (4) without the assumptions that the nonlinearity f satisfies a Lipschitz type condition and the semigroup {exp(-tA(s))} generated by -A(s) is compact (see Theorem 14). As one can see, our result is obtained under assumptions weaker than those required previously in the similar literature. The result is new even for the case of A(t)≡A (autonomous). Moreover, an example is given to show an application of the abstract result.


## 2. Preliminaries

Throughout this paper, we setJ=[0,T], a compact interval in R. We denote by X a separable Banach space with norm ∥·∥, 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:
(6)∥x∥[a,b]=∥x∥C([a,b],X)=sup{∥x(t)∥:t∈[a,b]},lhforanyx∈C([a,b],X).
Moreover, we abbreviate ∥u∥Lp([0,T],R+) with ∥u∥Lp, for any u∈Lp([0,T],R+).We set(7)Gv(t):=∫0t‍K(t,s)v(s)ds,G*:=supt∈J∫0t‍K(t,s)ds<∞.Next, we recall the definition of the Riemann-Liouville integral.Definition 1 (see [6]).
The fractional (arbitrary) order integral of the functionF∈L1(R+,R) of order α>0 is defined by
(8)IαF(t)=1Γ(α)∫0t‍(t-s)α-1F(s)ds,
where Γ is the Gamma function. Moreover, Iα1Iα2=Iα1+α2, for all α1,α2>0.Remark 2.
We have (1)Iα:L1[0,T]→L1[0,T] [6].(2) Obviously, forF∈L1(J,R), it follows from Definition 1 that
(9)∫0t‍∫0η‍(t-η)q-1(η-s)γ-1F(s)dsdη∫0t‍∫0η‍=B(q,γ)∫0t‍(t-s)q+γ-1F(s)ds,
where B(q,γ) is a beta function.Definition 3 (see [7]).
The Riemann-Liouville derivative of orderq with the lower limit zero for a function F∈AC[0,∞) can be written as
(10)LDtqF(t)=1Γ(1-q)ddt∫0t‍(t-s)-qF(s)ds,=hhhl1Γ(1-q)t>0,0<q<1.
Based on the work in [12], we give the following definition of the operator family {ψ(t,s)}.Definition 4.
Letξq be a probability density function defined on (0,∞) such that its Laplace transform is given by
(11)∫0∞‍e-σxξq(σ)dσ=∑j=0∞‍(-x)jΓ(1+qj),0<q≤1,x>0.
We define operator families {ψ(t,s)} by the semigroup exp(-tA(s)) associated with A(s) as follows:
(12)ψ(t,s)=q∫0∞‍θtq-1ξq(θ)exp(-tqθA(s))dθ.By using the family{ψ(t,s)}, we denote
(13)φ1(t,s)=[A(t)-A(s)]ψ(t-s,s),φk+1(t,s)=∫st‍φk(t,θ)φ1(θ,s)dθ,k=1,2,…,
and construct the family {φ(t,s)} by
(14)φ(t,s)=∑k=1∞‍φk(t,s).Lemma 5 (see [12]).
The operator-valued functionsψ(t-s,s) and A(t)ψ(t-s,s) are continuous in uniform topology in the variables t, s, where 0≤s≤t-ε, 0≤t≤T, for any ε>0. Clearly,
(15)∥ψ(t-s,s)∥≤C(t-s)q-1.Moreover, we have(16)∥φ(t,s)∥≤C(t-s)γ-1.A mild solution of (4) can be defined as follows.Definition 6.
A functionv∈C(J,X) satisfying the equation
(17)v(t)=A-1(0)g(v)+∫0t‍ψ(t-s,s)f(s,Gv(s),v(s))ds+∫0t‍ψ(t-s,s)U(s)g(v)ds+∫0t‍∫0η‍ψ(t-η,η)φ(η,s)f(s,Gv(s),v(s))dsdη
is called a mild solution of (4), where
(18)U(t)=-A(t)A-1(0)-∫0t‍φ(t,s)A(s)A-1(0)ds,(19)∥U(t)∥≤C+Ctγ.We will need the following facts from the theory of measures of noncompactness and condensing maps (see, e.g., [26, 27]) which are used later in this paper.Definition 7.
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,
(20)ν(co¯(Ω))=ν(Ω),
then we say that ν is a measure of noncompactness (MNC) in E.As an example of the MNC, we may consider the Hausdorff MNC:(21)χ(Ω)=inf{ε>0:Ωhasafiniteε-net}.We know thatχ is monotone, nonsingular, invariant with respect to union with compact sets, algebraically semiadditive, and regular. This means that (i)
for anyΩ0,Ω1∈2E with Ω0⊂Ω1, χ(Ω0)≤χ(Ω1),(ii)
for anya0∈E, Ω∈2E, χ({a0}∪Ω)=χ(Ω),(iii)
for every relatively compact setD⊂E, Ω∈2E, χ({D}∪Ω)=χ(Ω),(iv)
for eachΩ0,Ω1∈2E, χ(Ω0+Ω1)≤χ(Ω0)+χ(Ω1),(v)
ν(Ω)=0 is equivalent to the relative compactness of Ω.In Section3, we will establish an existence result to the problem (4) using the following assertion about χ-estimates for a multivalued integral (Theorem 4.2.3 of [27]).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
(22)∥𝒢(t)∥:=sup{∥𝔤∥:𝔤∈𝒢(t)}≤ϱ(t)a.e.t∈[0,h~].Proposition 8.
For an integrable, integrably bounded multifunction𝒢:[0,h~]→2X where X is a separable Banach space, let
(23)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 and ν a monotone nonsingular MNC in E.Definition 9.
A continuous map𝔅: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
(24)ν(𝔅(Ω))≱ν(Ω).The application of the topological degree theory for condensing maps (see, e.g., [26, 27]) yields the following fixed point principle which will be used later.Theorem 10.
Let𝔐 be a bounded convex closed subset of E and 𝔅:𝔐→𝔐 a ν-condensing map. Then fix 𝔅={x:x=𝔅(x)} is nonempty.


## 3. Main Result

We need the hypotheses as follows.(H1)
Functionf:J×X×X→X satisfies that f(·,u,w):J→X is measurable for all (u,w)∈X×X and f(t,·,·):X×X→X is continuous for a.e. t∈J, and there exists a function μ(·)∈Lp¯(J,R+)(p¯>1/q>1) such that
(25)∥f(t,u,w)∥≤μ(t)(∥u∥+∥w∥),(t,v,w)∈J×X×X,for almost allt∈J.(H2)
There exists a functionδ∈Lp(J,R+) such that, for any bounded set D⊂X,
(26)χ(f(t,GD,D))≤δ(t)χ(D),a.e.t∈[0,T],wherep>max{1/q,1/γ}>1.(H3)
The functiong is completely continuous and there exists a positive constant b such that
(27)∥g(v)∥≤b,v∈C(J,X).Define the operatorℱ:C(J,X)→C(J,X) as follows:
(28)(ℱv)(t)=A-1(0)g(v)+∫0t‍ψ(t-s,s)U(s)g(v)ds+∫0t‍ψ(t-s,s)f(s,Gv(s),v(s))ds+∫0t‍∫0σ‍ψ(t-σ,σ)φ(σ,s)+∫0t‍∫0σ‍×f(s,Gv(s),v(s))dsdσ.
It is clear that the operator ℱ is well defined. For some MNC ν, we will show that the operator ℱ is ν-condensing on every bounded subset of C(J,X). To this end, we divided the proof into three propositions.Proposition 11.
The operatorℱ is continuous.Proof.
Let{vn}n∈N be a sequence such that vn→v  in C(J,X) as n→∞. Since f satisfies (H1), for almost every t∈J and (t,s)∈Δ, we have
(29)f(t,Gvn(t),vn(t))⟶f(t,Gv(t),v(t)),asn⟶∞.
For t∈J, we can prove that ℱ is continuous. In fact, noting (H1) and vn→v in C(J,X), we know that there exists ε>0 such that ∥vn-v∥J≤ε for n sufficiently large. Therefore, we get
(30)∥f(t,Gvn(t),vn(t))-f(t,Gv(t),v(t))∥≤μ(t)(G*+1)[∥vn-v∥J+2∥v∥J]≤μ(t)(G*+1)[ε+2∥v∥J].
In view of (15) and (16), we obtain
(31)∫0t∥ψ(t-s,s)[f(s,Gvn(s),vn(s))hhhhhhhhhhh-f(s,Gv(s),v(s))]∥dshhh+∫0t∫0σ∥ψ(t-σ,σ)φ(σ,s)hhhhhhhhlhh×[f(s,Gvn(s),vn(s))hhhhhhhhhhhhhh-f(s,Gv(s),v(s))]∥ds≤C∫0t‍(t-s)q-1hhhhhhhh×∥f(s,Gvn(s),vn(s))hhhhhhhhhhhh-f(s,Gv(s),v(s))∥dshhh+C2∫0t‍∫0σ‍(t-σ)q-1(σ-s)γ-1hhhhhhhhhhhh×∥f(s,Gvn(s),vn(s))hhhhhhhhhhhhhhhh-f(s,Gv(s),v(s))∥ds⟶0,asn⟶∞.
Therefore, the fact∥A-1(0)∥≤C (by (A2)), (H3), and the Lebesgue dominated convergence theorem ensure that
(32)∥(ℱvn)(t)-(ℱv)(t)∥⟶0,asn⟶∞.
Therefore, we deduce that
(33)limn→∞∥ℱvn-ℱv∥J=0.Proposition 12.
The operatorℱ transforms bounded sets into equicontinuous ones.Proof.
For anyr>0, we set Br={v∈X:∥v∥J≤r}. Let 0<t2<t1<T and v∈Br. Then
(34)∥(ℱv)(t1)-(ℱv)(t2)∥≤∑i=16‍Ii,
where
(35)I1=∫0t2‍∥[ψ(t1-s,s)-ψ(t2-s,s)]U(s)g(v)∥ds,I2=∫t2t1‍∥ψ(t1-s,s)U(s)g(v)∥ds,I3=∫0t2‍∥[ψ(t1-s,s)-ψ(t2-s,s)]×f(s,Gv(s),v(s))∥ds,I4=∫t2t1‍∥ψ(t1-s,s)f(s,Gv(s),v(s))∥ds,I5=∫0t2‍∫0σ‍∥[ψ(t1-σ,σ)-ψ(t2-σ,σ)]h×φ(σ,s)f(s,Gv(s),v(s))∥dsdσ,I6=∫t2t1‍∫0σ‍∥ψ(t1-σ,σ)φ(σ,s)f(s,Gv(s),v(s))∥dsdσ.
For I1,
(36)I1=(∫0t2-ε‍+∫t2-εt2‍)×∥[ψ(t1-s,s)-ψ(t2-s,s)]U(s)g(v)∥ds=I1′+I1′′.
It follows from Lemma 5, (19), and (H3) that I1′→0 as t2→t1. For I1′′, from (15) and (19),
(37)I1′′≤C2b{∫t2-εt2‍[(t1-s)q-1+(t2-s)q-1](1+sγ)ds}⟶0,ast2⟶t1,ε⟶0.
Similarly, I3,I5→0, as t2→t1 and ε→0. For I2, from (15) and (19), we have
(38)I2≤b∫t2t1‍C2(t1-s)q-1(1+sγ)ds⟶0,ast2⟶t1.
Similarly, I4,I6→0, as t2→t1.
So, the set{(ℱv)(·):v∈Br} is equicontinuous.Proposition 13.
The operatorℱ is ν-condensing.Proof.
Noting that, for anyϕ∈L1(J,X), we have
(39)limL→+∞supt∈J∫0t‍e-L(t-s)ϕ(s)ds=0,
so, we can take the appropriate L such that
(40)C2(p-1pγ-1)(p-1)/p×∥δ∥Lpsupt∈J∫0t‍(t-σ)q-1σγ-1/pe-L(t-σ)dσ=L1<12,(41)Csupt∈J∫0t‍(t-s)q-1δ(s)e-L(t-s)ds=L2<12,(42)C(G*+1)supt∈J∫0t‍(t-s)q-1μ(s)hhhhhhhhhhhhhh×(1+CB(q,γ)(t-s)γ)e-L(t-s)ds=L3<1.
For every bounded subsetΩ⊂C(J,X), we consider the measure of noncompactness ν in the space C(J,X) with values in the cone R+2 of the following way:
(43)ν(Ω)=(Ψ(Ω),modc(Ω)),
where modc(Ω) is the module of equicontinuity of Ω given by
(44)modc(Ω)=limϵ→0supv∈Ωmax|t1-t2|≤ϵ∥v(t1)-v(t2)∥,Ψ(Ω)=supt∈J(e-Ltχ(Ω(t))).
LetΩ⊂C(J,X) be a nonempty, bounded set such that
(45)ν(ℱ(Ω))≥ν(Ω).
For anyt∈[0,η], we set
(46)ℱ^1(Ω)(t)={∫0t‍φ(t,s)f(s,Gv(s),v(s))ds:v∈Ω}.
We consider the multifunctions∈[0,t]⊸H(s),
(47)H(s)={φ(t,s)f(s,Gv(s),v(s)):v∈Ω}.
Obviously, H is integrable, and from (16) and (H1) it follows that H is integrably bounded. Moreover, noting (H2), we have the following estimate for a.e. s∈[0,t]:
(48)χ(H(s))≤C(t-s)γ-1χ(f(s,GΩ(s),Ω(s)))≤C(t-s)γ-1δ(s)χ(Ω(s))=C(t-s)γ-1δ(s)eLse-Lsχ(Ω(s))≤C(t-s)γ-1δ(s)eLsΨ(Ω).
Applying Proposition 8, we have
(49)χ(ℱ^1(Ω)(t))=χ(∫0t‍H(s)ds)≤C∫0t‍(t-s)γ-1δ(s)eLsds·Ψ(Ω).
Set
(50)ℱ^2(Ω)(σ)={ψ(t-σ,σ)∫0σ‍φ(σ,s)=lll×∫0σ‍φ(σ,s)f(s,Gv(s),v(s))ds:v∈Ω∫0σ‍}.
Then from (49),
(51)χ(ℱ^2(Ω)(σ))≤C2(t-σ)q-1∫0σ‍(σ-s)γ-1δ(s)eLsds·Ψ(Ω),hhhhhhhhhlhhhhhhhhhhhhhhhhhhha.e.σ∈[0,t].
Furthermore
(52)χ(∫0t‍ℱ^2(Ω)(σ)dσ)≤C2∫0t‍(t-σ)q-1∫0σ‍(σ-s)γ-1δ(s)eLsdsdσ·Ψ(Ω)≤C2∫0t‍(t-σ)q-1eLσ(∫0σ‍(σ-s)γ-1δ(s)ds)dσ·Ψ(Ω)≤C2(p-1pγ-1)(p-1)/p×∥δ∥Lp∫0t‍(t-σ)q-1σγ-1/peLσdσ·Ψ(Ω).
Therefore, combining with (40), we have
(53)supt∈J(e-Ltχ(∫0t‍ℱ^2(Ω)(σ)dσ))≤C2(p-1pγ-1)(p-1)/p∥δ∥Lp×supt∈J∫0t‍(t-σ)q-1σγ-1/pe-L(t-σ)dσ·Ψ(Ω)=L1Ψ(Ω).
For anyt∈J, we set
(54)ℱ^3(Ω)(t)={∫0t‍ψ(t-s,s)f(s,Gv(s),v(s))ds:v∈Ω}
and consider the multifunction s∈[0,t]⊸H~(s):
(55)H~(s)={ψ(t-s,s)f(s,Gv(s),v(s)):v∈Ω}.
Obviously, H~ is integrable, and from (15) and (H1) it follows that H~ is integrably bounded. Moreover, for a.e. s∈[0,t], we can obtain
(56)χ(H~(s))≤C(t-s)q-1δ(s)eLsΨ(Ω),χ(ℱ^3(Ω)(t))=χ(∫0t‍H~(s)ds)≤C∫0t‍(t-s)q-1δ(s)eLsds·Ψ(Ω)
by the similar technique used in (49)–(53); then
(57)supt∈J(e-Ltχ(ℱ^3(Ω)(t)))≤Csupt∈J∫0t‍(t-s)q-1δ(s)e-L(t-s)ds·Ψ(Ω)=L2Ψ(Ω).
Now, from (53) and (57), L>0 can be chosen so that
(58)Ψ(ℱ(Ω))≤(L1+L2)Ψ(Ω)=L~Ψ(Ω),
where 0<L~<1. Then from (45), we have Ψ(Ω)=0.
Further, from Proposition12 we know that modc(ℱ(Ω))=0 and integrating with (45) one yields modc(Ω)=0. Hence ν(Ω)=(0,0). The regularity property of ν implies the relative compactness of Ω. Now, it follows from Definition 9 that ℱ is ν-condensing.Theorem 14.
Assume that (H1), (H2), and (H3) are satisfied; then problem (4) has at least one mild solution on the interval [0,T].Proof.
Let us introduce in the spaceC(J,X) the equivalent norm defined as
(59)∥v∥*=supt∈J(e-Lt∥v(t)∥).
Consider the set
(60)Br={v∈C(J,X):∥v∥*≤r}.
Next, we show that there exists somer>0 such that ℱBr⊂Br. Suppose on the contrary that for each r>0 there exist vr(·)∈Br and some t∈J such that ∥(ℱvr)(t)∥*>r.
Combining with (H1)–(H3), Remark2(2), and (15), (16) and (19), we have
(61)∥(ℱvr)(t)∥≤∥A-1(0)g(vr)∥+∫0t‍∥ψ(t-s,s)U(s)g(vr)∥ds+∫0t‍∥ψ(t-s,s)f(s,Gvr(s),vr(s))∥ds+∫0t‍∫0σ‍∥ψ(t-σ,σ)φ(σ,s)×f(s,Gvr(s),vr(s))∥dsdσ≤Cb+C2b∫0t‍(t-s)q-1(1+sγ)ds+CG*∫0t‍(t-s)q-1μ(s)eLsds·∥vr∥*+C∫0t‍(t-s)q-1μ(s)∥vr(s)∥ds+C2G*B(q,γ)∫0t‍(t-s)q+γ-1μ(s)eLsds·∥vr∥*+C2B(q,γ)∫0t‍(t-s)q+γ-1μ(s)∥vr(s)∥ds≤Cb+C2b∫0t‍(t-s)q-1(1+sγ)ds+C(G*+1)·r∫0t‍(t-s)q-1μ(s)eLsds+C2(G*+1)·rB(q,γ)∫0t‍(t-s)q+γ-1μ(s)eLsds.
Therefore
(62)r<supt∈J(e-Lt∥(ℱvr)(t)∥)≤supt∈JM(t)+C(G*+1)·rsupt∈J∫0t‍(t-s)q-1μ(s)e-L(t-s)ds+C2(G*+1)·rB(q,γ)×supt∈J∫0t‍(t-s)q+γ-1μ(s)e-L(t-s)ds,
where M(t)=Cb(1+C∫0t‍(t-s)q-1(1+sγ)ds).
Dividing both sides of (62) by r and taking r→∞, we have
(63)C(G*+1)supt∈J∫0t‍(t-s)q-1μ(s)gg×(1+CB(q,γ)(t-s)γ)e-L(t-s)ds≥1.
This contradicts (42). Hence for some positive number r, ℱBr⊂Br. From Proposition 13 it follows that ℱ is ν-condensing and we apply Theorem 10 to complete the proof.


## 4. Example

In this section, setX=L2([0,1],R), and we consider the following integrodifferential problem:
(64)∂q∂tqu(t,ξ)=a(t,ξ)∂2u∂ξ2(t,ξ)+sin(∫0t‍(t-s)u(s,ξ)ds)+1ktk·u(t,ξ)1+u(t,ξ),u(t,0)=u(t,1)=0,u(0,ξ)=(a(0,ξ))-1×∑i=0j‍∫01‍ci(ξ,y)u(ti,y)1+u(ti,y)dy,
where ∂q/∂tq is the Riemann-Liouville fractional partial derivative of order 0<q<1, t∈[0,1], ξ∈[0,1], k>0 is a constant to be specified later. Also 0<t0<t1<⋯<tj<1(j∈N+), ci(·,·)(i=0,1,…,j) are continuous functions and there exists a positive constant b such that
(65)∑i=0j‍∫01‍∥ci(ξ,y)∥dy≤b.a(t,ξ) is a continuous function and is uniformly Hölder continuous in t; that is, there exist C>0 and γ¯∈(0,1) such that
(66)∥a(t1,ξ)-a(t2,ξ)∥≤C|t1-t2|γ¯,hhhhhlξ∈[0,1],0≤t1≤t2≤1.
Define A(t) by
(67)D(A(t))=H2(0,1)∩H01(0,1),A(t)u=-a(t,ξ)u′′.
Then -A(s) generates an analytic semigroup exp(-tA(s)) satisfying assumptions (A1)–(A3) ([28]).Fort∈(0,1], ξ∈[0,1], we set
(68)v(t)(ξ)=u(t,ξ),g(v)(ξ)=∑i=0j‍∫01‍ci(ξ,y)v(ti)(y)1+v(ti)(y)dy,K(t,s)=t-s,f(t,Gv(s),v(t))(ξ)=sin(∫0t‍(t-s)v(s)(ξ)ds)=l+1ktk·v(t)(ξ)1+v(t)(ξ),
where 1/tk∈Lp([0,1],R+)(p>max{1/q,1/γ¯}) and
(69)(Gv(t))(ξ)=∫0t‍(t-s)v(s)(ξ)ds.
Now G*=supt∈[0,1]∫0t‍(t-s)ds=1/2<∞.Then (64) can be reformulated as the abstract (4).Moreover,(70)∥f(t,Gv(t),v(t))(ξ)∥≤∥Gv(t)∥+1ktk∥v(t)∥,≤μ(t)(∥Gv(t)∥+∥v(t)∥),hhhhlhhhhhla.e.t∈[0,1],
where μ(t)=max{1,1/ktk}.Forv1,v2∈X, we have
(71)∥f(t,Gv1,v1)-f(t,Gv2,v2)∥≤∫0t‍(t-s)∥v1(s)-v2(s)∥ds+1ktk∥v1-v2∥.
Therefore, for any bounded set D⊂X, we have
(72)χ(f(t,GD,D))≤δ(t)χ(D),fora.e.t∈[0,1],
where δ(t)=max{1/2,1/ktk}. Now, Theorem 14 implies that the problem (64) has at least a mild solution.


## 5. Conclusion

This paper deals with the existence of mild solution of a class of nonlinear nonautonomous fractional integrodifferential equations with nonlocal conditions in an abstract space. Sufficient conditions for the existence of mild solution are derived with the help of the fixed point theorem for condensing maps. An example is provided to illustrate the obtained result.



---

*Source: 723453-2013-11-17.xml*