# A Nonlocal Cauchy Problem for Fractional Integrodifferential Equations

Authors: Fang Li; Jin Liang; Tzon-Tzer Lu; Huan Zhu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper is concerned with a nonlocal Cauchy problem for fractional integrodifferential equations in a separable Banach spaceX. We establish an existence theorem for mild solutions to the nonlocal Cauchy problem, by virtue of measure of noncompactness and the fixed point theorem for condensing maps. As an application, the existence of the mild solution to a nonlocal Cauchy problem for a concrete integrodifferential equation is obtained.


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## Body

## 1. Introduction

Nonlocal Cauchy problem for equations is an initial problem for the corresponding equations with nonlocal initial data. Such a Cauchy problem has better effects than the normal Cauchy problem with the classical initial data when we deal with many concrete problem coming from engineering and physics (cf., e.g., [1–10] and references therein). Therefore, the study of this type of Cauchy problem is important and significant. Actually, as we have seen from the just mentioned literature, there have been many significant developments in this field.On the other hand, fractional differential and integrodifferential equations arise from various real processes and phenomena appeared in physics, chemical technology, materials, earthquake analysis, robots, electric fractal network, statistical mechanics biotechnology, medicine, and economics. They have in recent years been an object of investigations with much increasing interest. For more information on this subject see for instance [9, 11–18] and references therein.Throughout this paper,X is a separable Banach space; L(X) is the Banach space of all linear bounded operators on X; A is the generator of an analytic and uniformly bounded semigroup {T(t)}t≥0 on X with ∥T(t)∥L(X)≤M for a constant M>0, and C([a,b],X) is the space of all X-valued continuous functions on[a,b]  with the supremum norm as follows:(1.1)‖x‖[a,b]:=max⁡{‖x(t)‖:t∈[a,b]},foranyx∈C([a,b],X).Let0<q<1, T>0, Δ={(t,s)∈[0,T]×[0,T]:t≥s}, f:[0,T]×C([0,T],X)→X, and h:Δ×C([0,T],X)→X. The nonlocal Cauchy problem for abstract fractional integrodifferential equations, with which we are concerned, is in the following form:(1.2)cDqx(t)=Ax(t)+f(t,x(t))+∫0tk(t,s)h(t,s,x(s))ds,t∈[0,T],x(0)=g(x)+x0,
where k and g are given functions to be specified later and the fractional derivative is understood in the Caputo sense, this means that, the fractional derivative is understood in the following sense:(1.3)cDqx(t):=LDq(x(t)-x(0)),t>0,0<q<1,
and where(1.4)LDqx(t):=1Γ(1-q)ddt∫0t(t-s)-qx(s)ds,t>0,0<q<1
is the Riemann-Liouville derivative of order q of x(t), where Γ(·) is the Gamma function.Our main purpose is to establish an existence theorem for the mild solutions to the nonlocal Cauchy problem based on a special measure of noncompactness under weak assumptions on the nonlinearityf and the semigroup {T(t)}t≥0 generated by A.


## 2. Existence Result and Proof

As usual, we abbreviate∥u∥Lp([0,T],R+) with ∥u∥Lp, for any u∈Lp([0,T],R+).As in [16, 17], we define the fractional integral of order q with the lower limit zero for a function f∈AC[0,∞) as(2.1)Iqf(t)=1Γ(q)∫0t(t-s)q-1f(s)ds,t>0,0<q<1,
provided the right side is point-wise defined on [0,∞).Now we recall some very basic concepts in the theory of measures of noncompactness and condensing maps (see, e.g., [19, 20]).Definition 2.1.
LetE be a Banach space, 2E the family of all nonempty subsets of E, (𝒜,≥) a partially ordered set, and α:2E→𝒜. If for every Ω∈2E:
(2.2)α(co⁡¯(Ω))=α(Ω),
then we say that α is a measure of noncompactness in E.Definition 2.2.
LetE be a Banach space, and 𝔉:Y⊆E→E is continuous. Let α be a measure of noncompactness in E such that(i)
for anyΩ0,Ω1∈2E with Ω0⊂Ω1,
(2.3)α(Ω0)≤α(Ω1);(ii)
for everya0∈E, Ω∈2E,
(2.4)α({a0}∪Ω)=α(Ω).
If for every bounded set Ω⊆Y which is not relatively compact,
(2.5)α(F(Ω))<α(Ω),
then we say that 𝔉 is condensing with respect to the measure of noncompactness α (or α-condensing).Definition 2.3.
Let(2.6)ϖq(σ)=1π∑n=1∞(-1)n-1σ-qn-1Γ(nq+1)n!sin⁡(nπq),σ∈(0,∞)
be a one-sided stable probability density, and
(2.7)ξq(σ)=1qσ-1-1/qϖq(σ-1/q),σ∈(0,∞).
For any z∈X, we define operators {Y(t)}t≥0 and {Z(t)}t≥0 by
(2.8)Y(t)z=∫0∞ξq(σ)T(tqσ)zdσ,Z(t)z=q∫0∞σtq-1ξq(σ)T(tqσ)zdσ.
If a continuous function x:[0,T]→X satisfies
(2.9)x(t)=Y(t)(g(x)+x0)+∫0tZ(t-s)[f(s,x(s))+a(x)(s)]ds,t∈[0,T],
then the function x is called a mild solution of (1.2).
Our main result is as follows.Theorem 2.4.
Assume that(1)
f(·,w) and h(·,·,w) are measurable for each w∈C([0,T],X); k(t,·) is measurable for each t∈[0,T];(2)
f(t,·) is continuous for a.e. t∈[0,T]; g is completely continuous; h(t,s,·) is continuous for a.e. (t,s)∈Δ; the map t→kt:=k(t,·) is continuous from [0,T] to L∞([0,T],R);(3)
there exist two positive functionsμ(·),η(·)∈Lp(0,T,R+)(p>1/q>1) and two positive functions m(·,·) and ζ(·,·) on Δ with
(2.10)sup⁡t∈[0,T]∫0tm(t,s)ds:=m*<∞,sup⁡t∈[0,T]∫0tζ(t,s)ds:=ζ*<∞,
such that
(2.11)‖f(t,w)‖≤μ(t)‖w‖(a.e.t∈[0,T]),‖h(t,s,w)‖≤m(t,s)‖w‖(a.e.(t,s)∈Δ),
for all w∈C([0,T],X), and
(2.12)χ(f(t,D))≤η(t)χ(D),(a.e.t∈[0,T]),χ(h(t,s,D))≤ζ(t,s)χ(D),(a.e.(t,s)∈Δ),
for any bounded set D⊂C([0,T],X), where χ is the Hausdorff measure of noncompactness:
(2.13)χ(Ω)=inf⁡{ɛ>0:Ωhasafiniteɛ-net}.(4)
g(·) satisfies
(2.14)‖g(x)‖≤b,∀x∈C([0,T],X),    for a positive constant b, and
(2.15)k(t):=esssup⁡⁡{|k(t,s)|,0≤s≤t}
is bounded on [0,T].
Then the mild solutions set of problem (1.2) is a nonempty compact subset of the space C([0,T],X), in the case of
(2.16)qMΓ(1+q)(p-1pq-1)(p-1)/pTq-1/p‖μ‖Lp<1.Proof.
First of all, let us prove our definition of the mild solution to problem (1.2) is well defined and reasonable. Actually, the proof is basic. We present it here for the completeness of the proof as well as the convenience of reading.
Write(2.17)a(x)(t)=∫0tk(t,s)h(t,s,x(s))ds,x̂(λ)=∫0∞e-λtx(t)dt,f̂(λ)=∫0∞e-λtf(t,x(t))dt,â(λ)=∫0∞e-λta(x)(t)dt.
Clearly, the nonlocal Cauchy problem (1.2) can be written as the following equivalent integral equation:
(2.18)x(t)=g(x)+x0+1Γ(q)∫0t(t-s)q-1[Ax(s)+f(s,x(s))+a(x)(s)]ds,t∈[0,T],
provided that the integral in (2.18) exists. Formally taking the Laplace transform to (2.18), we have
(2.19)x̂(λ)=1λ(g(x)+x0)+1λqAx̂(λ)+1λq[f̂(λ)+â(λ)].
Therefore, if the related integrals exist, then we obtain
(2.20)x̂(λ)=λq-1(λq-A)-1(g(x)+x0)+(λq-A)-1[f̂(λ)+â(λ)]=λq-1∫0∞e-λqsT(s)(g(x)+x0)ds+∫0∞e-λqsT(s)[f̂(λ)+â(λ)]ds=q∫0∞(λt)q-1e-(λt)qT(tq)(g(x)+x0)dt+∫0∞e-(λτ)qqτq-1T(τq)(∫0∞e-λt(f(t,x(t))+a(x)(t))dt)dτ=-1λ∫0∞(ddte-(λt)q)T(tq)(g(x)+x0)dt+∫0∞∫0∞e-λτσqτq-1ϖq(σ)T(τq)(∫0∞e-λt(f(t,x(t))+a(x)(t))dt)dσdτ=∫0∞∫0∞e-λtσσϖq(σ)T(tq)(g(x)+x0)dσdt+q∫0∞∫0∞e-λθθq-1σqϖq(σ)T(θqσq)(∫0∞e-λt(f(t,x(t))+a(x)(t))dt)dσdθ=∫0∞e-λt[∫0∞ϖq(σ)T(tqσq)(g(x)+x0)dσ]dt+q∫0∞(∫0∞∫t∞e-λτ(τ-t)q-1σqϖq(σ)T((τ-t)qσq)(f(t,x(t))+a(x)(t))dτdt)dσ=∫0∞e-λt[∫0∞ϖq(σ)T(tqσq)(g(x)+x0)dσ]dt+q∫0∞(∫0∞∫0τe-λτ(τ-t)q-1σqϖq(σ)T((τ-t)qσq)(f(t,x(t))+a(x)(t))dtdτ)dσ=∫0∞e-λt[∫0∞ϖq(σ)T(tqσq)(g(x)+x0)dσ]dt+∫0∞e-λt[q∫0t∫0∞(t-s)q-1σqϖq(σ)T((t-s)qσq)(f(s,x(s))+a(x)(s))dσds]dt.
Now using the uniqueness of the Laplace transform (cf., e.g., [21, Theorem  1.1.6]), we deduce that
(2.21)x(t)=∫0∞ϖq(σ)T(tqσq)(g(x)+x0)dσ+q∫0t∫0∞(t-s)q-1σqϖq(σ)T((t-s)qσq)f(s,x(s))dσds+q∫0t∫0∞(t-s)q-1σqϖq(σ)T((t-s)qσq)a(x)(s)dσds=∫0∞ξq(σ)T(tqσ)(g(x)+x0)dσ+q∫0t∫0∞σ(t-s)q-1ξq(σ)T((t-s)qσ)f(s,x(s))dσds+q∫0t∫0∞σ(t-s)q-1ξq(σ)T((t-s)qσ)a(x)(s)dσds.
Consequently, we see that the mild solution to problem (1.2) given by Definition 2.3 is well defined.
Next, we define the operatorℱ:C([0,T],X)→C([0,T],X) as follows:
(2.22)(Fx)(t)=Y(t)(g(x)+x0)+∫0tZ(t-s)[f(s,x(s))+a(x)(s)]ds,t∈[0,T].
It is clear that the operator ℱ is well defined.
The operatorℱ can be written in the form ℱ=ℱ1+ℱ2, where the operators ℱi,i=1,2 are defined as follows:
(2.23)(F1x)(t)=Y(t)(g(x)+x0),t∈[0,T],(F2x)(t)=∫0tZ(t-s)[f(s,x(s))+a(x)(s)]ds,t∈[0,T].
The following facts will be used in the proof. (1)
(2.24)∫0∞ξq(σ)dσ=1,
which implies that
(2.25)‖Y(t)‖≤Const;(2)
(2.26)∫0∞σνξq(σ)dσ=∫0∞1σqνϖq(σ)dσ=Γ(1+ν)Γ(1+qν),ν∈(0,1],
which implies that(2.27)‖Z(t)‖≤qMΓ(1+q)tq-1,t>0.
Let {xn}n∈N⊂C([0,T],X) such that
(2.28)lim⁡n→∞⁡‖xn-x‖[0,T]=0,
for an x∈C([0,T],X). Then by the assumptions, we know that for almost every t∈[0,T] and (t,s)∈Δ:
(2.29)lim⁡n→∞f(t,xn(t))=f(t,x(t)),lim⁡n→∞h(t,s,xn(s))=h(t,s,x(s)).
Therefore, for sufficiently large n, we have
(2.30)‖f(t,xn(t))-f(t,x(t))‖≤μ(t)(1+2‖x‖[0,T]),‖h(t,s,xn(s))-h(t,s,x(s))‖≤m(t,s)(1+2‖x‖[0,T]),‖∫0tk(t,s)h(t,s,xn(s))ds-∫0tk(t,s)h(t,s,x(s))ds‖≤m*k*(1+2‖x‖[0,T]),
where
(2.31)k*:=sup⁡t∈[0,T]k(t).
Hence,
(2.32)lim⁡n→∞‖∫0tk(t,s)h(t,s,xn(s))ds-∫0tk(t,s)h(t,s,x(s))ds‖=0.
Thus,
(2.33)‖F2xn-F2x‖[0,T]⟶0,asn⟶∞,
since (2.27) implies that
(2.34)‖∫0tZ(t-s){f(s,xn(s))+∫0sk(s,τ)h(s,τ,xn(τ))dτ-[f(s,x(s))+∫0sk(s,τ)h(s,τ,x(τ))dτ]}ds∫0t‖≤qMΓ(1+q)∫0t(t-s)q-1[∫0s∫0s‖f(s,xn(s))-f(s,x(s))‖+‖∫0sk(s,τ)(h(s,τ,xn(τ))-h(s,τ,x(τ)))dτ‖]ds.
By (2.33) and our assumptions, we see that ℱ is continuous.
Sinceχ is the Hausdorff measure of noncompactness in X, we know that χ 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,
(2.35)χ(Ω0)≤χ(Ω1);(ii)
for everya0∈E, Ω∈2E,
(2.36)χ({a0}∪Ω)=χ(Ω);(iii)
for every relatively compact setD⊂E, Ω∈2E,
(2.37)χ({D}∪Ω)=χ(Ω);(iv)
for eachΩ0, Ω1∈2E,
(2.38)χ(Ω0+Ω1)≤χ(Ω0)+χ(Ω1);(v)
χ(Ω)=0 is equivalent to the relative compactness of Ω.Noting that for any ψ∈L1([0,T],X), we have
(2.39)lim⁡L→+∞sup⁡t∈[0,T]∫0te-L(t-s)ψ(s)ds=0.
So, there exists a positive constant L such that
(2.40)qMΓ(1+q)sup⁡t∈[0,T]∫0t(t-s)q-1η(s)e-L(t-s)ds=L1<13,qMk*ζ*Γ(1+q)sup⁡t∈[0,T]∫0t(t-s)q-1e-L(t-s)ds=L2<13,qMΓ(1+q)sup⁡t∈[0,T]∫0t(t-s)q-1(μ(s)+m*k*)e-L(t-s)ds=L3<13.
For every bounded subset Ω⊂C([0,T],X), we define
(2.41)mod⁡c(Ω)≔lim⁡δ→0sup⁡v∈Ωmax⁡|t1-t2|≤δ‖v(t1)-v(t2)‖,Ψ(Ω)≔sup⁡⁡t∈[0,T](e-Ltχ(Ω(t))),α(Ω)≔(Ψ(Ω),mod⁡c(Ω)).
Then modc(Ω) is the module of equicontinuity of Ω, and α is a measure of noncompactness in the space C([0,T],X) with values in the cone R+2.
LetΩ⊂C([0,T],X) be a nonempty, bounded set such that
(2.42)α(F(Ω))≥α(Ω).
By the assumptions and the continuity of T(t) in the uniform operator topology for t>0, we get
(2.43)mod⁡c(F1Ω)=0.
Clearly,
(2.44)‖f(s,x(s))‖+‖a(x)(s)‖≤(μ(s)+m*k*)‖x‖[0,T].
Let δ>0, t1,t2∈(0,T] such that 0<t1-t2≤δ and x∈Ω. Then
(2.45)‖∫0t1Z(t1-s)[f(s,x(s))+a(x)(s)]ds-∫0t2Z(t2-s)[f(s,x(s))+a(x)(s)]ds‖≤‖x‖[0,T](∫0t2‖Z(t1-s)-Z(t2-s)‖(μ(s)+m*k*)ds+∫t2t1‖Z(t1-s)‖(μ(s)+m*k*)ds)≤q‖x‖[0,T][∫0t2∫0∞|(t1-s)q-1-(t2-s)q-1|σξq(σ)‖T((t1-s)qσ)‖(μ(s)+m*k*)dσds+∫0t2∫0∞(t2-s)q-1σξq(σ)‖T((t1-s)qσ)-T((t2-s)qσ)‖×(μ(s)+m*k*)dσds∫0t2∫0∞]+qM‖x‖[0,T]Γ(1+q)∫t2t1(t1-s)q-1(μ(s)+m*k*)ds≤qM‖x‖[0,T]Γ(1+q)[∫0t2|(t1-s)q-1-(t2-s)q-1|(μ(s)+m*k*)dσds+∫t2t1(t1-s)q-1(μ(s)+m*k*)ds]+∫0t2∫0∞(t2-s)q-1σξq(σ)‖T((t1-s)qσ)-T((t2-s)qσ)‖(μ(s)+m*k*)dσds.
It is not hard to see that the right-hand side of (2.45) tend to 0 as t2→t1. Thus, the set {(ℱ2x)(·):x∈Ω} is equicontinuous, then modc(ℱ2Ω)=0. Combining with (2.43), we have modc(ℱΩ)=0, which implies modc(Ω)=0 from (2.42). Next, we show that Ψ(Ω)=0.
It is easy to see that(2.46)Ψ(F1Ω)=0.
For any t∈[0,T], we define
(2.47)F̂2(Ω)(t):={∫0tZ(t-s)f(s,x(s))ds:x∈Ω}.
We consider the multifunction s∈[0,t]⊸G(s):
(2.48)G(s)={Z(t-s)f(s,x(s)):x∈Ω}.
Obviously, G is integrable, that is, G admits a Bochner integrable selection 𝔤:[0,h]→E, and
(2.49)g(t)∈G(t),fora.e.t∈[0,h].
From (2.27) and our assumptions, it follows that G is integrably bounded, that is, there exists a function ϱ∈L1([0,h],E) such that
(2.50)‖G(t)‖:=sup⁡⁡{‖g‖:g∈G(t)}≤ϱ(t),a.e.t∈[0,h].
Moreover, we have the following estimate for a.e. s∈[0,t]:
(2.51)χ(G(s))≤qMΓ(1+q)(t-s)q-1χ(f(s,Ω(s)))≤qMΓ(1+q)(t-s)q-1η(s)χ(Ω(s))=qMΓ(1+q)(t-s)q-1η(s)eLse-Lsχ(Ω(s))≤qMΓ(1+q)(t-s)q-1η(s)eLsΨ(Ω).
Therefore, since X is a separable Banach space, we know by [20, Theorem  4.2.3] that
(2.52)χ(F̂2(Ω)(t))=χ(∫0tG(s)ds)≤qMΓ(1+q)∫0t(t-s)q-1η(s)eLsds⋅Ψ(Ω).
So
(2.53)sup⁡t∈[0,T](e-Ltχ(F̂2(Ω)(t)))≤qMΓ(1+q)sup⁡t∈[0,T]∫0t(t-s)q-1η(s)e-L(t-s)ds⋅Ψ(Ω)=L1Ψ(Ω).
Similarly, if we set
(2.54)F̃2(Ω)(t)={∫0tZ(t-s)a(x)(s)ds:x∈Ω},
then we see that the multifunction s∈[0,t]⊸G̃(s),
(2.55)G̃(s)={Z(t-s)a(x)(s):x∈Ω}
is integrable and integrably bounded. Thus, we obtain the following estimate for a.e. s∈[0,t]:
(2.56)χ(G̃(s))≤qMk*ζ*Γ(1+q)(t-s)q-1eLsΨ(Ω),sup⁡t∈[0,T](e-Ltχ(F̃2(Ω)(t)))≤qMk*ζ*Γ(1+q)sup⁡t∈[0,T]∫0t(t-s)q-1e-L(t-s)ds⋅Ψ(Ω)=L2Ψ(Ω).
Now, from (2.53) and (2.56), it follows that
(2.57)Ψ(F(Ω))≤Ψ(F1(Ω))+Ψ(F2(Ω))≤(L1+L2)Ψ(Ω)=L̃Ψ(Ω),
where 0<L̃<1. Then by (2.42), we get Ψ(Ω)=0. Hence α(Ω)=(0,0). Thus, Ω is relatively compact due to the regularity property of α. This means that ℱ is α-condensing.
Let us introduce in the spaceC([0,T],X) the equivalent norm defined as
(2.58)‖x‖*=sup⁡t∈[0,T](e-Lt‖x(t)‖).
Consider the set
(2.59)Br={x∈C([0,T],X):‖x‖*≤r}.
Next, we show that there exists some r>0 such that ℱBr⊂Br. Suppose on the contrary that for each r>0 there exist xr(·)∈Br, and some t∈[0,T] such that ∥(ℱxr)(t)∥*>r.
From the assumptions, we have(2.60)‖(F1xr)(t)‖*≤M(b+‖x0‖).
Moreover,
(2.61)‖(F2xr)(t)‖≤∫0t‖Z(t-s)[f(s,xr(s))+a(xr)(s)]‖ds≤qMΓ(1+q)∫0t(t-s)q-1(μ(s)‖xr(s)‖+m*k*eLs‖xr‖*)ds=qMΓ(1+q)[∫0t(t-s)q-1μ(s)eLse-Ls‖xr(s)‖ds+m*k*∫0t(t-s)q-1eLsds⋅‖xr‖*]≤qMrΓ(1+q)∫0t(t-s)q-1(μ(s)+m*k*)eLsds.
Therefore,
(2.62)r<sup⁡t∈[0,T](e-Lt‖(Fxr)(t)‖)≤M(b+‖x0‖)+qMrΓ(1+q)sup⁡t∈[0,T]∫0t(t-s)q-1(μ(s)+m*k*)e-L(t-s)ds.
Dividing both sides of (2.62) by r, and taking r→∞, we have
(2.63)qMΓ(1+q)sup⁡t∈[0,T]∫0t(t-s)q-1(μ(s)+m*k*)e-L(t-s)ds≥1.
This is a contradiction. Hence for some positive number r, ℱBr⊂Br. According to the following known fact.
Let𝔐be a bounded convex closed subset ofEand𝔉:𝔐→𝔐aα-condensing map. ThenFix𝔉={x:x=𝔉(x)}is nonempty.
we see that problem (1.2) has at least one mild solution.
Next, forc∈(0,1], we consider the following one-parameter family of maps:
(2.64)H:[0,1]×C([0,T],X)⟶C([0,T],X)(c,x)⟶H(c,x)=cF(x).
We will demonstrate that the fixed point set of the family ℋ,
(2.65)Fix⁡H={x∈H(c,x)forsomec∈(0,1]}
is a priori bounded. Indeed, let x∈Fixℋ, for t∈[0,T], we have
(2.66)‖x(t)‖≤M‖g(x)+x0‖+∫0t‖Z(t-s)‖‖f(s,x(s))+a(x)(s)‖ds≤M(b+‖x0‖)+qMΓ(1+q)[∫0t(t-s)q-1μ(s)‖x(s)‖ds+m*k*∫0t(t-s)q-1sup⁡τ∈[0,s]‖x(τ)‖ds].
Noting that the Hölder inequality, we have(2.67)∫0t(t-s)q-1μ(s)ds≤(p-1pq-1)(p-1)/p⋅t(pq-1)/p⋅‖μ‖Lp≤(p-1pq-1)(p-1)/p⋅Tq-1/p⋅‖μ‖Lp.
Therefore, from (2.66), we obtain
(2.68)‖x(t)‖≤M(b+‖x0‖)+qMΓ(1+q)(p-1pq-1)(p-1)/pTq-1/p‖μ‖Lpsup⁡s∈[0,t]‖x(s)‖+qMm*k*Γ(1+q)∫0t(t-s)q-1sup⁡τ∈[0,s]‖x(τ)‖ds.
We denote
(2.69)y(t):=sup⁡s∈[0,t]∥x(s)∥.
Let t̃∈[0,t] such that y(t)=∥x(t̃)∥. Then, by (2.68), we can see
(2.70)y(t)≤M(b+‖x0‖)+qMΓ(1+q)(p-1pq-1)(p-1)/pTq-1/p‖μ‖Lpy(t)+qMm*k*Γ(1+q)∫0t(t-s)q-1y(s)ds.
By a generalization of Gronwall’s lemma for singular kernels ([22, Lemma  7.1.1]), we deduce that there exists a constant κ=κ(q) such that
(2.71)y(t)≤M(b+‖x0‖)1-(qM/Γ(1+q))((p-1)/(pq-1))(p-1)/pTq-1/p‖μ‖Lp+κM(b+‖x0‖)(qMm*k*/Γ(1+q))(1-(qM/Γ(1+q))((p-1)/(pq-1))(p-1)/pTq-1/p‖μ‖Lp)2∫0t(t-s)q-1ds≤M(b+‖x0‖)1-(qM/Γ(1+q))((p-1)/(pq-1))(p-1)/pTq-1/p‖μ‖Lp+κM(b+‖x0‖)(qMm*k*/Γ(1+q))Tqq(1-(qM/Γ(1+q))((p-1)/(pq-1))(p-1)/pTq-1/p‖μ‖Lp)2:=w.
Hence, supt∈[0,T]∥x(t)∥≤w.
Now we consider a closed ball:(2.72)BR={x∈C([0,T],X):‖x‖[0,T]≤R}⊂C([0,T],X).
We take the radius R>0 large enough to contain the set Fixℋ inside itself. Moreover, from the proof above, ℱ:BR→C([0,T],X) is α-condensing. Consequently, the following known fact implies our conclusion: Let V⊂Ebe a bounded open neighborhood of zero and𝔉:V¯→Eaα-condensing map satisfying the boundary condition:(2.73)x≠λ¯F(x),for allx∈∂V and 0<λ¯≤1. Then,Fix𝔉is nonempty compact.


## 3. Example

In this section, letX=L2([0,π]), we consider the following nonlocal Cauchy problem for an integrodifferential problem:(3.1)∂tqu(t,ξ)=∂2∂ξ2u(t,ξ)+1ktk⋅u(t,ξ)1+u(t,ξ)+∫0t(t-s)sin⁡⁡(s⋅u(s,ξ)t)ds,t∈(0,1]u(t,0)=u(t,π)=0,t∈[0,1]u(0,ξ)=∑i=0j∫0πci(ξ,y)u(ti,y)1+u(ti,y)dy+u0(ξ),
where ∂tq is the Caputo fractional partial derivative of order 0<q<1; ξ∈[0,π]; k>0 is a constant to be specified later;
(3.2)u0(ξ)∈X;j∈N+;0<t0<t1<⋯<tj<1;ci(·,·)(i=0,1,…,j) are continuous functions and there exists a positive constant b such that(3.3)∑i=0j∫0π‖ci(ξ,y)‖dy≤b.
For t∈(0,1], ξ∈[0,π], we set(3.4)x(t)(ξ)=u(t,ξ),g(x)(ξ)=∑i=0j∫0πci(ξ,y)x(ti)(y)1+x(ti)(y)dy,k(t,s)=t-s,h(t,s,x(s))(ξ)=sin⁡⁡(s⋅x(s)(ξ)t),f(t,x(t))(ξ)=1ktk⋅x(t)(ξ)1+x(t)(ξ).
On the other hand, it is known that the operator A (Au=u′′ with D(A)=H2([0,π])∩H01([0,π])) generates an analytic semigroup and uniformly bounded semigroup {T(t)}t≥0 on X with ∥T(t)∥L(X)≤1. Therefore, (3.1) is a special case of (1.2).Moreover, we have(1)
for allt∈(0,1],
(3.5)‖f(t,x)‖≤1ktk‖x‖:=μ(t)‖x‖;(2)
for anyw,w̃∈X,
(3.6)‖f(t,w)-f(t,w̃)‖≤1ktk‖w-w̃‖,
that is, for any bounded set D⊂X,
(3.7)χ(f(t,D))≤1ktkχ(D),
for a.e. t∈[0,1];(3)
for almost all(t,s)∈Δ,
(3.8)‖h(t,s,x)‖=‖sin⁡⁡(s⋅x(s)(ξ)t)‖≤m(t,s)‖x‖,
where m(t,s):=s/t, and
(3.9)m*=sup⁡t∈[0,1]∫0tm(t,s)ds=sup⁡t∈[0,1]∫0tstds=23;(4)
(3.10)‖h(t,s,w)-h(t,s,w̃)‖≤st‖w-w̃‖,
that is, for any bounded set D⊂X,
(3.11)χ(h(t,s,D))≤ζ(t,s)χ(D),
where ζ(t,s):=s/t, and
(3.12)sup⁡t∈[0,1]∫0tζ(t,s)ds=23.
Therefore, Theorem 2.4 implies that the problem (3.1) has at least a mild solution when(3.13)q⋅((p-1)/(pq-1))(p-1)/pΓ(1+q)‖μ‖Lp<1.



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*Source: 901942-2012-05-13.xml*