# On Nonlinear Neutral Fractional Integrodifferential  Inclusions with Infinite Delay

Authors: Fang Li; Ti-Jun Xiao; Hong-Kun Xu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Of concern is a class of nonlinear neutral fractional integrodifferential inclusions with infinite delay in Banach spaces. A theorem about the existence of mild solutions to the fractional integrodifferential inclusions is obtained based on  Martelli’s fixed point theorem. An example is given to illustrate the existence result.


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## Body

## 1. Introduction

As have been seen, the field of the application of fractional calculus is very broad. For instance, 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. The mathematical modeling and simulation of systems and processes, based on the description of their properties in terms of fractional derivatives, naturally leads to differential equations of fractional-order. The main feature of fractional order differential equation is containing the noninteger order derivative. It can effectively describe the memory and transmissibility of many natural phenomena. These differential equations have been studied by many researchers (cf., e.g., [1–11] and references therein).As an generalization of differential equations, differential inclusions have also been investigated (cf., e.g., [1, 7, 12, 13] and references therein). Moreover, equations with delay are often more useful to describe concrete systems than those without delay. So the study of these equations has been attracted so much attention (cf., e.g., [1, 4, 8, 12, 14–21] and references therein).In this paper, we pay our attention to the investigation of the existence of mild solutions to the following fractional integrodifferential inclusions of neutral type with infinite delay in a Banach spaceX:
(1.1)Dq(x(t)-g(t,xt))∈A(x(t)-g(t,xt))+∫0tK(t,s)F(s,x(s),xs)ds,t∈[0,T],x0=ϕ∈P,
where 0<q<1, the fractional derivative is understood in the Caputo sense ([2], see Definition 2.3 in Section 2),𝒫  is an admissible phase space, xt:(-∞,0]→X defined by(1.2)xt(θ)=x(t+θ),forθ∈(-∞,0],T>0, g:[0,T]×𝒫→X, A generates a compact and uniformly bounded semigroup S(·) on X which implies that there exists M≥1 such that(1.3)‖S(t)‖≤M,∀t≥0,K:[0,T]×[0,T]→R, ϕ belongs to 𝒫 with(1.4)ϕ(0)=0,
and F is a multivalued map to be specified later.


## 2. Preliminaries

Throughout this paper,X is a Banach space with norm ∥·∥, L(X) is the Banach space of all linear continuous operators on X, J:=[0,T], and C(J,X) (C([0,∞),X)) is the space of all X-valued continuous functions on J([0,∞)).Moreover, we abbreviate∥u∥L1(J,R+)as ∥u∥L1, for any u∈L1(J,R+).We use the notation𝔅(X) to denote the family of all nonempty subsets of X. Let 𝔅bd(X),𝔅cl⁡(X),𝔅cp(X),𝔅cv(X),and𝔅cp,cv(X) denote, respectively, the family of all nonempty bounded, closed, compact, convex, and compact-convex subsets of X.See the following definition about admissible phase space according to [8, 14–21].Definition 2.1.
A linear space𝒫 consisting of functions from R- into X with norm ∥·∥𝒫 is called an admissible phase space if 𝒫 has the following properties.(H1)
For anyt0∈R and a>0, if x:(-∞,t0+ah]→X is continuous on [t0,t0+a] and xt0∈𝒫, then xt∈𝒫, xt is continuous in t∈[t0,t0+a], and(2.1)‖x(t)‖≤C¯‖xt‖P,
for a positive constant C¯. (H2)
There exists a continuous functionC1(t)>0 and a locally bounded function C2(t)≥0 in t≥0 such that(2.2)‖xt‖P≤C1(t-t0)max⁡s∈[t0,t]‖x(s)‖+C2(t-t0)‖xt0‖P
for t∈[t0,t0+a] and x as in (H1).(H3)
The space(𝒫,∥·∥𝒫) is complete.Remark 2.2.
(H1) is equivalent to that for anyt0∈R and a>0, if x:(-∞,t0+a]→X is continuous on [t0,t0+a] and xt0∈𝒫, then xt∈𝒫, xt is continuous in t∈[t0,t0+a], and
(2.3)‖ϕ(0)‖≤C¯‖ϕ‖P,∀ϕ∈P
for a positive constant C¯.Now we recall some very basic concepts in the fractional calculus theory. For more details see, for example, [2, 9, 11].We set forβ≥0,
(2.4)g{β}(t)={1Γ(β)tβ-1,t>0,0,t≤0,
and g0(t)=0, where Γ(·) is the Gamma function.Definition 2.3.
Letf∈L1(0,∞;X) and α≥0. Then the expression
(2.5)Iαf(t):=(g{α}*f)(t)=1Γ(α)∫0t(t-s)α-1f(s)ds,t>0,α>0
with I0f(t)=f(t) is called Riemann-Liouville integral of order α of f.Definition 2.4.
Letf(t)∈Cm-1([0,∞);X), g{m-α}*f∈Wm,1(I,X) (m∈ℕ,0≤m-1<α<m). The Caputo fractional derivative of order α of f is defined by
(2.6)Dαf(t)=DmIm-α(f(t)-∑i=0m-1f(i)(0)gi+1(t)),
where Dm:=dm/dtm.The following concepts are also very basic, which will be used later.A multivalued mapG:X→𝔅(X) is convex (closed) valued if G(x) is convex (closed) for all x∈X. G is bounded on bounded sets if(2.7)G(B)=⋃x∈BG(x)
is bounded in X for all B∈𝔅bd(X), that is,(2.8)sup⁡x∈B{sup⁡{‖y‖:y∈G(x)}}<∞.A multivalued mapG:J→𝔅cl⁡(X)is said to be measurable if for each x∈X the function Y:J→R defined by(2.9)Y(t)=d(x,G(t))=inf⁡{‖x-z‖:z∈G(t)}
is measurable.If for eachx∈X, the set G(x) is a nonempty, closed subset of X, and for each open set B of X containing G(x), there exists an open neighborhood V of x such that G(V)⊆B, then G is called upper semicontinuous (u.s.c.) on X.If for everyB∈𝔅bd(X), G(B) is relatively compact, then G is said to be completely continuous.If the multivalued mapG is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, that is,(2.10)xn⟶x*,yn⟶y*,yn∈G(xn)implyy*∈G(x*).
We say that G has a fixed point if there is some x∈X such that x∈G(x).For more details on multivalued maps we refer to the book by Deimling [22].The following is the multivalued version of the fixed-point theorem due to Martelli [23].Lemma 2.5.
LetX be a Banach space and let N:X→𝔅cp,cv(X) be an upper semicontinuous is bounded; then N has a fixed point and completely continuous multivalued map. If the set
(2.11)Ω:={y∈X:λy∈Nyforsomeλ>1}
is bounded, then N has a fixed point.Following Liang and Xiao [14, 15], let 𝒫[0,T] be the set defined by(2.12)P[0,T]={x:(-∞,T]⟶X:x|J∈C(J,X),x0∈P}.Let∥·∥T be the norm of 𝒫[0,T] defined by(2.13)‖y‖T=‖y0‖P+max⁡{‖y(s)‖:0≤s≤T},y∈P[0,T].Based on the work in [7, 11], we set(2.14)Q(t)=∫0∞ξq(σ)S(tqσ)dσ,R(t)=q∫0∞σtq-1ξq(σ)S(tqσ)dσ,
and ξq is a probability density function defined on (0,∞)(see [7]) such that(2.15)ξq(σ)=1qσ-1-1/qϖq(σ-1/q)≥0,
where(2.16)ϖq(σ)=1π∑n=1∞(-1)n-1σ-qn-1Γ(nq+1)n!sin⁡(nπq),σ∈(0,∞).Remark 2.6.
It is not difficult to verify that forv∈[0,1],
(2.17)∫0∞σvξq(σ)dσ=∫0∞σ-qvϖq(σ)dσ=Γ(1+v)Γ(1+qv).
Then, we can see
(2.18)‖R(t)‖≤qMΓ(1+q)tq-1,t>0.We define the mild solution to problem (1.1) as follows.Definition 2.7.
A functionx∈𝒫[0,T] satisfying the equation
(2.19)x(t)={ϕ(t),t∈(-∞,0],-Q(t)g(0,ϕ)+g(t,xt)+∫0t∫0sR(t-s)K(s,τ)f(τ)dτds,t∈J,
is called a mild solution of problem (1.1), where
(2.20)f∈SF,x={f∈L1(J,X):f(t)∈F(t,x(t),xt)fora.e.t∈J}.Remark 2.8.
(1) Since we only consider the following case:(2.21)ϕ(0)=0,
we define the mild solution to problem (1.1) in the way as mentioned before.
(2) For generalϕ(0), one can define the mild solution to problem (1.1) similarly and obtain the same conclusion by the similar arguments given in this paper. So we only pay attention to the essential case:
(2.22)ϕ(0)=0.


## 3. Results and Proofs

We will require the following assumptions.(A1)
F:J×X×𝒫→𝔅cp,cv(X);  (t,v,w)→F(t,v,w) is measurable with respect to t for each (v,w)∈X×𝒫; for every t∈J, the map F(t,·,·):X×𝒫→𝔅cp,cv(X) is u.s.c., and the set(3.1)SF,v={f∈L1(J,X):f(t)∈F(t,v(t),vt)fora.e.t∈J}is nonempty.(A2)
There exist two functionsμi∈L1(J,R+)(i=1,2) such that(3.2)‖F(t,v,w)‖∶=sup⁡{‖f‖:f∈F(t,v,w)}≤μ1(t)‖v‖+μ2(t)‖w‖P,(t,v,w)∈J×X×P.(A3)
There exist positive constantsa andb such that(3.3)‖g(t,φ̃)‖≤a‖φ̃‖P+b,fort∈J,φ̃∈P.(A4)
For eacht∈J, K(t,·) is measurable on [0,t] and(3.4)K(t)=esssup⁡{|K(t,s)|,0≤s≤t}is bounded onJ. The map t→K(t,*) is continuous from J to L∞(J,R).The following lemma will be used in the proof of our main result.Lemma 3.1 (see [24]).
LetI be a compact real interval and let E be a Banach space. Let F be a multivalued map satisfying hypothesis (A1) and let Υ be a linear continuous mapping from L1(I,E)→C(I,E). Then,
(3.5)Υ∘SF:C(I,E)⟶Bcp,cv(C(I,E)),x⟼(Υ∘SF)(x)=Υ(SF,x)
is a closed graph operator in C(I,E)×C(I,E).To prove the main result, we consider the multivalued map𝒩:𝒫[0,T]→𝔅(𝒫[0,T]) defined by
(3.6)N(x)(t)={ρ∈P[0,T]:ρ(t)={ϕ(t),t∈(-∞,0],-Q(t)g(0,ϕ)+g(t,xt)+∫0t∫0sR(t-s)K(s,τ)f(τ)dτds,t∈J,},
where f∈SF,x.It is clear that the fixed points of𝒩 are mild solutions to problem (1.1).Forϕ∈𝒫, we define the function
(3.7)y(t)={ϕ(t),t∈(-∞,0],0,t∈J,
then y∈𝒫[0,T].Setx(t)=u(t)+y(t), t∈(-∞,T].It is obvious thatx satisfies (2.19) if and only if u satisfies u0=0 and for t∈J,
(3.8)u(t)=-Q(t)g(0,ϕ)+g(t,ut+yt)+∫0t∫0sR(t-s)K(s,τ)f(τ)dτds,
where
(3.9)f∈SF,u={f∈L1(J,X):f(t)∈F(t,u(t)+y(t),ut+yt)fora.e.t∈J}.
Let
(3.10)P0[0,T]={u∈P[0,T]:u0=0}.
For any u∈𝒫0[0,T],
(3.11)‖u‖T=‖u0‖P+max⁡{‖u(s)‖:0≤s≤T}=max⁡{‖u(s)‖:0≤s≤T}.
Thus (𝒫0[0,T],∥·∥T) is a Banach space.Set(3.12)Br={u∈P0[0,T]:‖u‖T≤r},forr≥0.
For u∈Br, from Definition 2.1, we have
(3.13)‖ut+yt‖P≤‖ut‖P+‖yt‖P≤C1(t)max⁡0≤τ≤t‖u(τ)‖+C2(t)‖u0‖P+C1(t)max⁡0≤τ≤t‖y(τ)‖+C2(t)‖y0‖P≤C1*r+C2*‖ϕ‖P:=r′,
where
(3.14)Ci*=sup⁡t∈J⁡Ci(t)(i=1,2).
Define the operator
(3.15)Ñ:P0[0,T]⟶B(P0[0,T])
by
(3.16)Ñ(u)(t)={∫0th∈P0[0,T]:h(t)=-Q(t)g(0,ϕ)+g(t,ut+yt)+∫0t∫0sR(t-s)K(s,τ)f(τ)dτds,t∈J},
where f∈SF,u.We can see that if𝒩̃ has a fixed point in 𝒫0[0,T], then 𝒩 has a fixed point in 𝒫[0,T] which is a mild solution of problem (1.1).Assume the following.(A5)
The functiong:J×𝒫→X is completely continuous, and for every bounded set B∈𝒫0[0,T], the set {t→g(t,ut):u∈B} is equicontinuous in X.Then we can deduce that𝒩̃ has a fixed point under the assumptions (A1)–(A5). For this purpose, we will show that the multivalued operator 𝒩̃ is completely continuous, u.s.c. with convex values. The proof of this conclusion will be given by proving the following six propositions.Proposition 3.2.
𝒩̃u is convex for each u∈𝒫0[0,T].Proof.
Forh1(t),h2(t)∈𝒩̃u, there exist f1,f2∈SF,u such that for each t∈J we have
(3.17)hi(t)=-Q(t)g(0,ϕ)+g(t,ut+yt)+∫0t∫0sR(t-s)K(s,τ)fi(τ)dτds,i=1,2.
Let β∈[0,1]. Then for each t∈J, we get
(3.18)βh1(t)+(1-β)h2(t)=-Q(t)g(0,ϕ)+g(t,ut+yt)+∫0t∫0sR(t-s)K(s,τ)(βf1(τ)+(1-β)f2(τ))dτds.
Since F has convex values, SF,u is convex, we see that
(3.19)βh1(t)+(1-β)h2(t)∈Ñu.Proposition 3.3.
𝒩̃ maps bounded sets into bounded sets in 𝒫0[0,T].Proof.
Letu∈Br. If h¯∈𝒩̃u, then there exists f∈SF,u such that
(3.20)h¯(t)=-Q(t)g(0,ϕ)+g(t,ut+yt)+∫0t∫0sR(t-s)K(s,τ)f(τ)dτds,fort∈J.
In view of (A3) and (3.13),
(3.21)‖g(t,ut+yt)‖≤ar′+b.
Hence from (A2), (A3), and (3.13), it follows that
(3.22)‖h¯(t)‖≤‖-Q(t)g(0,ϕ)‖+‖g(t,ut+yt)‖+qMK*Γ(1+q)∫0t(t-s)q-1∫0s[μ1(τ)‖u(τ)+y(τ)‖+μ2(τ)‖uτ+yτ‖P]dτds≤M(a‖ϕ‖P+b)+ar′+b+qMK*Γ(1+q)(r‖μ1‖L1+r′‖μ2‖L1)∫0t(t-s)q-1ds≤M(a‖ϕ‖P+b)+ar′+b+MTqK*Γ(1+q)(r‖μ1‖L1+r′‖μ2‖L1)=:ω,
where
(3.23)K*=sup⁡t∈JK(t).
Therefore, for each h¯∈𝒩̃(Br), we have
(3.24)‖h¯‖T≤ω.Proposition 3.4.
𝒩̃ maps bounded sets into equicontinuous sets in 𝒫0[0,T].Proof.
Leth∈𝒩̃u for u∈Br, and let 0<t2<t1≤T. Then we have
(3.25)‖h(t1)-h(t2)‖≤‖Q(t1)-Q(t2)‖⋅‖g(0,ϕ)‖+‖g(t1,ut1+yt1)-g(t2,ut2+yt2)‖+‖∫0t1∫0sR(t1-s)K(s,τ)f(τ)dτds-∫0t2∫0sR(t2-s)K(s,τ)f(τ)dτds‖=I1+I2+I3.
It follows from the continuity of S(t) in the uniform operator topology for t>0 that
(3.26)I1tendsto0,ast2⟶t1.
The equicontinuity of g ensures that
(3.27)I2tendsto0,ast2⟶t1.
For I3, we obtain
(3.28)I3≤K*∫0t2∫0s‖R(t1-s)-R(t2-s)‖‖f(τ)‖dτds+K*∫t2t1∫0s‖R(t1-s)‖‖f(τ)‖dτds≤K*r*(∫0t2‖R(t1-s)-R(t2-s)‖ds+qMΓ(1+q)∫t2t1(t1-s)q-1ds)≤qr*K*∫0t2∫0∞σ‖[(t1-s)q-1-(t2-s)q-1]ξq(σ)S((t1-s)qσ)‖dσds+qr*K*∫0t2∫0∞σ(t2-s)q-1ξq(σ)‖S((t1-s)qσ)-S((t2-s)qσ)‖dσds+qMr*K*Γ(1+q)∫t2t1(t1-s)q-1ds≤qMr*K*Γ(1+q)∫0t2|(t1-s)q-1-(t2-s)q-1|ds+qr*K*∫0t2∫0∞σ(t2-s)q-1ξq(σ)‖S((t1-s)qσ)-S((t2-s)qσ)‖dσds+Mr*K*Γ(1+q)(t1-t2)q,
where
(3.29)r*:=r‖μ1‖L1+r′‖μ2‖L1.
Clearly, the first term and third term on the right-hand side of (3.28) tend to 0 as t2→t1. The second term on the right-hand side of (3.28) tends to 0 as t2→t1 as a consequence of the continuity of S(t) in the uniform operator topology for t>0.
Thus the set{𝒩̃u:u∈Br} is equicontinuous.Proposition 3.5.
(𝒩̃Br)(t) is relatively compact for each t∈J, where
(3.30)(ÑBr)(t)={h(t):h∈Ñ(Br)}.Proof.
Fixt∈(0,T]. For arbitrary 0<ɛ<t and arbitrary δ>0, write
(3.31)hɛ,δ(t)=-Q(t)g(0,ϕ)+g(t,ut+yt)+q∫0t-ɛ(t-s)q-1∫δ∞σξq(σ)S((t-s)qσ)∫0sK(s,τ)f(τ)dτdσds=-Q(t)g(0,ϕ)+g(t,ut+yt)+qS(ɛqδ)∫0t-ɛ(t-s)q-1∫δ∞σξq(σ)S((t-s)qσ-ɛqδ)∫0sK(s,τ)f(τ)dτdσds,
where f∈SF,u. Since S(t) is compact for each t∈(0,T] and (A5), the set
(3.32)Uɛ,δ={hɛ,δ(t):h∈Ñ(Br)}
is relatively compact. Moreover,
(3.33)‖h(t)-hɛ,δ(t)‖≤q∫0t-ɛ(t-s)q-1∫0δσξq(σ)S((t-s)qσ)∫0sK(s,τ)f(τ)dτdσds+q∫t-ɛt(t-s)q-1∫0∞σξq(σ)S((t-s)qσ)∫0sK(s,τ)f(τ)dτdσds≤MK*r*Tq∫0δσξq(σ)dσ+Mr*K*ɛqΓ(1+q),
which implies that (𝒩̃Br)(t) is relatively compact.Now, it follows from Propositions3.3–3.5 and the Ascoli-Arzela theorem that(3.34)Ñ:P0[0,T]⟶B(P0[0,T])is completely continuous.Proposition 3.6.
𝒩̃ has a closed graph.Proof.
Suppose that(3.35)un⟶u*,hn∈Ñunwithhn⟶h*.
We claim that
(3.36)h*∈Ñu*.
In fact, the assumption hn∈𝒩̃un implies that there exists fn∈SF,un such that
(3.37)hn(t)=-Q(t)g(0,ϕ)+g(t,unt+yt)+∫0t∫0sR(t-s)K(s,τ)fn(τ)dτds,t∈J.
We will show that there exists f*∈SF,u* such that
(3.38)h*(t)=-Q(t)g(0,ϕ)+g(t,u*t+yt)+∫0t∫0sR(t-s)K(s,τ)f*(τ)dτds,t∈J.
Obviously, as n→∞, we have
(3.39)‖(hn(t)+Q(t)g(0,ϕ)-g(t,unt+yt))-(h*(t)+Q(t)g(0,ϕ)-g(t,u*t+yt))‖⟶0.
Consider the following linear continuous operator:
(3.40)Υ:L1(J,X)⟶C(J,X),f⟼Υ(f)(t)=∫0t∫0sR(t-s)K(s,τ)f(τ)dτds.
By virtue of Lemma 3.1, we know that Υ∘SF is a closed graph operator. Moreover, we get
(3.41)hn(t)+Q(t)g(0,ϕ)-g(t,unt+yt)∈Υ(SF,un).
Since un→u* and hn→h*, it follows from Lemma 3.1 that
(3.42)h*(t)+Q(t)g(0,ϕ)-g(t,u*t+yt)=∫0t∫0sR(t-s)K(s,τ)f*(τ)dτds,
for some f*∈SF,u*.Now, we can conclude that𝒩̃ is a completely continuous multivalued map, u.s.c. with convex values. Next, we give the existence result of problem (1.1).Theorem 3.7.
Assume that (A1)–(A5) are satisfied; then there exists a mild solution of (1.1) on (-∞,T] provided that aC1*<1.Proof.
Define(3.43)Ω:={u∈P0[0,T]:λu∈Ñu,forsomeλ>1}.
Then, according to the previous propositions and discussions, we see that we only need to prove that the set Ω is bounded.
Takeu∈Ω. Then there exists f∈SF,u such that(3.44)u(t)=λ-1(-Q(t)g(0,ϕ)+g(t,ut+yt)+∫0t∫0sR(t-s)K(s,τ)f(τ)dτds).
It follows from Definition 2.1 and (A2) that
(3.45)‖u(t)‖<M(a‖ϕ‖P+b)+a(C1*max⁡0≤τ≤t‖u(τ)‖+C2*‖ϕ‖P)+b+qMΓ(1+q)∫0t(t-s)q-1∫0s|K(s,τ)|‖f(τ)‖dτds≤M1+aC1*max⁡0≤τ≤t‖u(τ)‖+qMK*Γ(1+q)∫0t(t-s)q-1∫0s[μ1(τ)‖u(τ)+y(τ)‖+μ2(τ)‖uτ+yτ‖P]dτds≤M1+aC1*max⁡0≤τ≤t‖u(τ)‖+qMK*Γ(1+q)(∫0t(t-s)q-1∫0sμ1(τ)‖u(τ)‖dτds+∫0t(t-s)q-1∫0sμ2(τ)‖uτ‖Pdτds+C2*∫0t(t-s)q-1∫0sμ2(τ)‖ϕ‖Pdτds)≤θ1+aC1*max⁡0≤τ≤t‖u(τ)‖+qMK*‖μ1‖L1Γ(1+q)∫0t(t-s)q-1max⁡0≤τ≤s‖u(τ)‖ds+qMK*‖μ2‖L1C1*Γ(1+q)∫0t(t-s)q-1max⁡0≤τ≤s‖u(τ)‖ds=θ1+aC1*max⁡0≤τ≤t‖u(τ)‖+θ2∫0t(t-s)q-1max⁡0≤τ≤s‖u(τ)‖ds,
where
(3.46)M1∶=M(a‖ϕ‖P+b)+aC2*‖ϕ‖P+b,θ1∶=M1+MK*Γ(1+q)‖μ2‖L1C2*Tq‖ϕ‖P,θ2∶=qMK*(‖μ1‖L1+C1*‖μ2‖L1)Γ(1+q).
Denote
(3.47)κ(t):=max⁡0≤s≤t‖u(s)‖,
and let t̃∈[0,t] such that κ(t)=∥u(t̃)∥. Then, by (3.45), we get
(3.48)κ(t)≤θ1+aC1*κ(t)+θ2∫0t(t-s)q-1κ(s)ds.
Furthermore,
(3.49)κ(t)≤θ11-aC1*+θ21-aC1*∫0t(t-s)q-1κ(s)ds.
It is known from [25, Lemma  7.1.1] that, for any continuous functions v,w:J→[0,+∞), if w(·) is nondecreasing and there are constants a¯>0 and 0<α¯<1 such that
(3.50)v(t)≤w(t)+a¯∫0t(t-s)-α¯v(s)ds,
then there exists a constant k=k(α¯) such that
(3.51)v(t)≤w(t)+ka¯∫0t(t-s)-α¯w(s)ds,foreacht∈J.
By virtue of this general fact and (3.49), we see that there exists a constant k̃=k̃(q) such that
(3.52)κ(t)≤θ11-aC1*+k̃θ21-aC1*∫0t(t-s)q-1θ11-aC1*ds≤θ11-aC1*[1+k̃θ2Tqq(1-aC1*)]=:ζ.
Therefore ∥u∥T≤ζ. This means that the set Ω is bounded.
Thus, it follows from Lemma2.5 that 𝒩̃ has a fixed point in 𝒫0[0,T]. Then 𝒩 has a fixed point which gives rise to a mild solution to problem (1.1).Example 3.8.
SetX=L2([0,π],R) and define A by
(3.53)D(A)=H2(0,π)∩H01(0,π),Au=u′′.
Then A generates a compact, analytic semigroup S(·) of uniformly bounded linear operators, and ∥S(t)∥≤1 (see [26] for more related information).
Consider the following Cauchy problem for a fractional integrodifferential conclusion:(3.54)∂q∂tq(v(t,ξ)-∫-∞tγ(s-t)v(s,ξ)ds)∈∂2∂ξ2(v(t,ξ)-∫-∞tγ(s-t)v(s,ξ)ds)+∫0t(t-s)∫-∞sη(s,τ-s,ξ)H(s,v(τ,ξ))dτds,t∈[0,1],v(t,0)-∫-∞tγ(s-t)v(s,0)ds=0,v(t,π)-∫-∞tγ(s-t)v(s,π)ds=0,v(θ,ξ)=v0(θ,ξ),-∞<θ≤0,
where 0<q<1, ξ∈[0,π], v0:(-∞,0]×[0,π]→R is a continuous function and H:[0,1]×R→𝔅(R) is a u.s.c. multivalued map with compact convex values.
Letϖ<0, define the space
(3.55)P={φ∈C((-∞,0],X):lim⁡θ→-∞eϖθφ(θ)existsin⁡X}
endowed with the norm
(3.56)‖φ‖P=sup⁡-∞<θ≤0{eϖθ‖φ(θ)‖L2}.
Clearly, we can see that 𝒫 is an admissible phase space which satisfies (H1)–(H3) with
(3.57)C1(t)=max⁡{1,e-ϖt},C2(t)=e-ϖt.
For t∈[0,1], ξ∈[0,π], and φ∈𝒫, let
(3.58)x(t)(ξ)=v(t,ξ),ϕ(θ)(ξ)=v0(θ,ξ),θ∈(-∞,0],g(t,φ)(ξ)=∫-∞0γ(θ)φ(θ)(ξ)dθ,K(t,s)=t-s,F(t,x(t),φ)(ξ)=∫-∞0η(t,θ,ξ)H(t,φ(θ)(ξ))dθ.
Then problem (3.54) can be written in the abstract form (1.1).Furthermore, we assume the following.(1)
The functionγ:(-∞,0]→R is continuous and(3.59)M2∶=(-12ϖ∫-∞0γ2(θ)dθ)1/2<∞.(2)
There exists a continuous functionv1(t) such that(3.60)|H(t,φ)|≤v1(t)‖φ(θ)‖L2.(3)
The functionη(t,θ,ξ)≥0 is continuous in [0,1]×(-∞,0]×[0,π] and(3.61)∫-∞0η(t,θ,ξ)e-ϖθdθ=p(t,ξ)<∞.
Then, we can obtain(3.62)‖F(t,x(t),φ)‖L2=(∫0π|∫-∞0η(t,θ,ξ)H(t,φ(θ)(ξ))dθ|2dξ)1/2≤(∫0π(∫-∞0η(t,θ,ξ)v2(t)‖φ(θ)‖L2dθ)2dξ)1/2=(∫0π(∫-∞0η(t,θ,ξ)v2(t)e-ϖθeϖθ‖φ(θ)‖L2dθ)2dξ)1/2≤(∫0π(∫-∞0η(t,θ,ξ)e-ϖθdθ)2dξ)1/2⋅v2(t)⋅‖φ‖P≤(∫0πp2(t,ξ)dξ)1/2⋅v2(t)⋅‖φ‖P=p(t)v2(t)‖φ‖P,,
where p(t)=∥p(t,·)∥L2.Moreover,(3.63)‖g(t,φ)‖L2=(∫0π(∫-∞0γ(θ)φ(θ)(ξ)dθ)2dξ)1/2≤(∫0π(∫-∞0γ2(θ)dθ)⋅(∫-∞0φ2(θ)(ξ)dθ)dξ)1/2=(∫-∞0γ2(θ)dθ)1/2(∫0π∫-∞0φ2(θ)(ξ)dθdξ)1/2=(∫-∞0γ2(θ)dθ)1/2(∫-∞0‖φ(θ)‖L22dθ)1/2=(∫-∞0γ2(θ)dθ)1/2(∫-∞0e-2ϖθe2ϖθ‖φ(θ)‖L22dθ)1/2≤(∫-∞0γ2(θ)dθ)1/2(∫-∞0e-2ϖθ[sup⁡-∞<θ≤0eϖθ‖φ(θ)‖L2]2dθ)1/2=(∫-∞0γ2(θ)dθ)1/2(∫-∞0e-2ϖθdθ)1/2‖φ‖P=M2‖φ‖P.
Therefore, by virtue of Theorem 3.7, problem (3.54) has a mild solution when e-ϖM2<1.



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*Source: 916543-2012-05-09.xml*