# Existence of Mild Solutions for a Class of Fractional Evolution Equations with Compact Analytic Semigroup

Authors: He Yang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper deals with the existence of mild solutions for a class of fractional evolution equations with compact analytic semigroup. We prove the existence of mild solutions, assuming that the nonlinear part satisfies some local growth conditions in fractional power spaces. An example is also given to illustrate the applicability of abstract results.


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## Body

## 1. Introduction

The differential equations involving fractional derivatives in time have recently been proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics, economics, and science. Numerous applications can be found in electrochemistry, control, porous media, electromagnetic, see for example, [1–5] and references therein. Hence the study of such equations has become an object of extensive study during recent years, see [6–23] and references therein.In this paper, we consider the existence of the following fractional evolution equation:(1.1)Dqu(t)+Au(t)=f(t,u(t),Gu(t)),t∈J=[0,T],u(0)=x0,
where Dq is the Caputo fractional derivative of order q∈(0,1), -A is the infinitesimal generator of a compact analytic semigroup S(·) of uniformly bounded linear operators,  f is the nonlinear term and will be specified later, and (1.2)Gu(t)=∫0tK(t,s)u(s)ds
is a Volterra integral operator with integral kernel K∈C(Δ,R+), Δ={(t,s):0≤s≤t≤T},R+=[0,+∞). Throughout this paper, we denote by K*:=max(t,s)∈ΔK(t,s)In some existing articles, the fractional differential equations were treated under the hypothesis that nonlinear term satisfies Lipschitz conditions or linear growth conditions. It is obvious that these conditions are not easy to be verified sometimes. To make the things more applicable, in this work, we will prove the existence of mild solutions for (1.1) under some new conditions. More precisely, the nonlinear term only satisfies some local growth conditions (see conditions (H1) and (H2)). These conditions are much weaker than Lipschitz conditions and linear growth conditions. The main techniques used here are fractional calculus, theory of analytic semigroup, and Schauder fixed point theorem.The rest of this paper is organized as follows. In Section2, some preliminaries are given on the fractional power of the generator of a compact analytic semigroup and the definition of mild solutions of (1.1). In Section 3, we study the existence of mild solutions for (1.1). In Section 4, an example is given to illustrate the applicability of abstract results obtained in Section 3.


## 2. Preliminaries

In this section, we introduce some basic facts about the fractional power of the generator of a compact analytic semigroup and the fractional calculus that are used throughout this paper.LetX be a Banach space with norm ∥·∥. Throughout this paper, we assume that -A:D(A)⊂X→X is the infinitesimal generator of a compact analytic semigroup S(t)(t≥0) of uniformly bounded linear operator in X, that is, there exists M≥1 such that ∥S(t)∥≤M for all t≥0. Without loss of generality, let 0∈ρ(-A), where ρ(-A) is the resolvent set of -A. Then for any α>0, we can define A-α by
(2.1)A-α:=1Γ(α)∫0∞‍t(α-1)S(t)dt.It follows that eachA-α is an injective continuous endomorphism of X. Hence we can define Aα by Aα:=(A-α)-1, which is a closed bijective linear operator in X. It can be shown that each Aα has dense domain and that D(Aβ)⊂D(Aα) for 0≤α≤β. Moreover, Aα+βx=AαAβx=AβAαx for every α,β∈ℝ and x∈D(Aμ) with μ:=max{α,β,α+β}, where A0=I,I is the identity in X. (For proofs of these facts, we refer to the literature [24–26]).We denote byXα the Banach space of D(Aα) equipped with norm ∥x∥α=∥Aαx∥ for x∈D(Aα), which is equivalent to the graph norm of Aα. Then we have Xβ↪Xα for 0≤α≤β≤1 (with X0=X), and the embedding is continuous. Moreover, Aα has the following basic properties.Lemma 2.1 (see [24]).
A
α has the following properties.(i)
S(t):X→Xα for each t>0 and α≥0.(ii)
AαS(t)x=S(t)Aαx for each x∈D(Aα) and t≥0.(iii)
For everyt>0,AαS(t) is bounded in X and there exists Mα>0 such that
(2.2)∥AαS(t)∥≤Mαt-α.(iv)
A-α is a bounded linear operator for 0≤α≤1 in X.In the following, we denote byC(J,Xα) the Banach space of all continuous functions from J into Xα with supnorm given by ∥u∥C=supt∈J∥u(t)∥α for u∈C(J,Xα). From Lemma 2.1(iv), since A-α is a bounded linear operator for 0≤α≤1, there exists a constant Cα such that ∥A-α∥≤Cα for 0≤α≤1.For anyt≥0, denote by Sα(t) the restriction of S(t) to Xα. From Lemma 2.1(i) and (ii), for any x∈Xα, we have
(2.3)∥S(t)x∥α=∥Aα·S(t)x∥=∥S(t)·Aαx∥≤∥S(t)∥·∥Aαx∥=∥S(t)∥·∥x∥α,∥S(t)x-xα∥=∥Aα·S(t)x-Aαx∥=∥S(t)·Aαx-Aαx∥→0
as t→0. Therefore, S(t)(t≥0) is a strongly continuous semigroup in Xα, and ∥Sα(t)∥α≤∥S(t)∥ for all t≥0. To prove our main results, the following lemma is also needed.Lemma 2.2 (see [27]).
S
α
(
t
)
(
t
≥
0
) is an immediately compact semigroup in Xα, and hence it is immediately norm-continuous.Let us recall the following known definitions in fractional calculus. For more details, see [16–20, 23].Definition 2.3.
The fractional integral of orderσ>0 with the lower limits zero for a function f is defined by
(2.4)Iσf(t)=1Γ(σ)∫0t‍(t-s)σ-1f(s)ds,t>0,
where Γ is the gamma function.
The Riemann-Liouville fractional derivative of ordern-1<σ<n with the lower limits zero for a function f can be written as
(2.5)LDσf(t)=1Γ(n-σ)dndtn∫0t‍(t-s)n-σ-1f(s)ds,t>0,n∈ℕ.
Also the Caputo fractional derivative of order n-1<σ<n with the lower limits zero for a function f∈Cn[0,∞) can be written as
(2.6)Dσf(t)=1Γ(n-σ)∫0t‍(t-s)n-σ-1f(n)(s)ds,t>0,n∈ℕ.Remark 2.4.
(
1
) The Caputo derivative of a constant is equal to zero.
(
2
) If f is an abstract function with values in X, then integrals which appear in Definition 2.3 are taken in Bochner's sense.Lemma 2.5 (see [12]).
A measurable functionh:J→X is Bochner integrable if ∥h∥ is Lebesgue integrable.Forx∈X, we define two families {U(t)}t≥0 and {V(t)}t≥0 of operators by
(2.7)U(t)x=∫0∞‍ηq(θ)S(tqθ)xdθ,V(t)x=q∫0∞‍θηq(θ)S(tqθ)xdθ,0<q<1,
Where
(2.8)ηq(θ)=1qθ-1-(1/q)ρq(θ-(1/q)),ρq(θ)=1π∑n=1∞(-1)n-1θ-qn-1Γ(nq+1)n!sin(nπq),θ∈(0,∞),ηq is a probability density function defined on (0,∞), which has properties ηq(θ)≥0 for all θ∈(0,∞) and
(2.9)∫0∞‍ηq(θ)dθ=1,∫0∞‍θηq(θ)dθ=1Γ(q+1).
The following lemma follows from the results in [7, 11–13].Lemma 2.6.
The operatorsU and V have the following properties.(i)
For fixedt≥0 and any x∈Xα, we have
(2.10)∥U(t)x∥α≤M∥x∥α,∥V(t)x∥α≤qMΓ(1+q)∥x∥α=MΓ(q)∥x∥α.(ii)
The operatorsU(t) and V(t) are strongly continuous for all t≥0.(iii)
U(t) and V(t) are norm-continuous in X for t>0.(iv)
U(t) and V(t) are compact operators in X for t>0.(v)
For everyt>0, the restriction of U(t) to Xα and the restriction of V(t) to Xα are norm-continuous.(vi)
For everyt>0, the restriction of U(t) to Xα and the restriction of V(t) to Xα are compact operators in Xα.Based on an overall observation of the previous related literature, in this paper, we adopt the following definition of mild solution of (1.1).Definition 2.7.
By a mild solution of (1.1), we mean a function u∈C(J,Xα) satisfying
(2.11)u(t)=U(t)x0+∫0t‍(t-s)q-1V(t-s)f(s,u(s),Gu(s))ds
for all t∈J.


## 3. Existence of Mild Solutions

In this section, we give the existence theorems of mild solutions of (1.1). The discussions are based on fractional calculus and Schauder fixed point theorem. Our main results are as follows.Theorem 3.1.
Assume that the following condition onf is satisfied.(H1)
There exists a constantβ∈[α,1] such that f:J×Xα×Xα→Xβ satisfies:(i)
for each(x,y)∈Xα×Xα, the function f(·,x,y):J→Xβ is measurable;(ii)
for eacht∈J, the function f(t,·,·):Xα×Xα→Xβ is continuous;(iii)
for anyr>0, there exists a function gr∈L∞(J,ℝ+) such that
(3.1)sup∥x∥α≤r,∥y∥α≤K*Tr∥f(t,x,y)∥β≤gr(t),t∈J,and there is a constant γ>0 such that
(3.2)liminfr→+∞1r∫0t‍gr(s)(t-s)1-qds≤γ<+∞.
If x0∈Xα and MCβ-αγ<Γ(q), then (1.1) has at least one mild solution.Proof.
Define an operatorQ by
(3.3)(Qu)(t)=U(t)x0+∫0t‍(t-s)q-1V(t-s)f(s,u(s),Gu(s))ds,t∈J.
It is not difficult to verify that Q:C(J,Xα)→C(J,Xα). We will use Schauder fixed point theorem to prove that Q has fixed points in C(J,Xα).
For anyr>0, let Br:={u∈C(J,Xα):∥u(t)∥α≤r,t∈J}. We first show that there is a positive number r such that Q(Br)⊂Br. If this were not the case, then for each r>0, there would exist ur∈Br and tr∈J such that ∥(Qur)(tr)∥α>r. Thus, from Lemma 2.6(i) and (H1)(iii), we see that
(3.4)r<∥(Qur)(tr)∥α≤∥U(tr)x0∥α+∫0tr(tr-s)q-1‍∥V(tr-s)f(s,ur(s),Gur(s))∥αds≤M∥x0∥α+∫0tr‍(tr-s)q-1∥Aα-βV(tr-s)·Aβf(s,ur(s),Gur(s))∥ds≤M∥x0∥α+MCβ-αΓ(q)∫0tr‍(tr-s)q-1gr(s)ds.
Dividing on both sides by r and taking the lower limit as r→+∞, we have
(3.5)MCβ-αγ≥Γ(q),
which is a contradiction. Hence Q(Br)⊂Br for some r>0.
To complete the proof, we separate the rest of proof into the following three steps.
Step 1. Q:Br→Br is continuous.
Let{un}⊂Br with un→u∈Br as n→∞. From the assumption (H1)(ii), for each s∈J, we have
(3.6)f(s,un(s),Gun(s))→f(s,u(s),Gu(s))
as n→∞. Since ∥f(s,un(s),Gun(s))-f(s,u(s),Gu(s))∥β≤2gr(s), by the Lebesgue dominated convergence theorem, for each t∈J, we have
(3.7)∥(Qun)(t)-(Qu)(t)∥α≤∫0t‍(t-s)q-1∥V(t-s)[f(s,un(s),Gun(s))-f(s,u(s),Gu(s))]∥αds≤∫0t‍(t-s)q-1∥Aα-βV(t-s)·Aβ[f(s,un(s),Gun(s))-f(s,u(s),Gu(s))]∥ds≤MCβ-αΓ(q)∫0t‍(t-s)q-1∥f(s,un(s),Gun(s))-f(s,u(s),Gu(s))∥βds→0
as n→∞, which implies that Q:Br→Br is continuous.
Step 2. (QBr)(t):={(Qu)(t):u∈Br} is relatively compact in Xα for all t∈J.
It follows from (2.9) and (3.3) that (QBr)(0)={(Qu)(0):u∈Br}={x0} is compact in Xα. Hence it is only necessary to consider the case of t>0. For each t∈(0,T],ϵ∈(0,t), and any δ>0, we define a set (Qϵ,δBr)(t) by
(3.8)(Qϵ,δBr)(t):={(Qϵ,δu)(t):u∈Br},
where
(3.9)(Qϵ,δu)(t)=∫δ∞‍ηq(θ)S(tqθ)dθx0+q∫0t-ϵ‍(t-s)q-1∫δ∞‍θηq(θ)S((t-s)qθ)dθ·f(s,u(s),Gu(s))ds=S(ϵqδ)[∫δ∞‍∫δ∞ηq(θ)S(tqθ-ϵqδ)dθx0+q∫0t-ϵ‍(t-s)q-1∫δ∞‍θηq(θ)S((t-s)qθ-ϵqδ)dθ·f(s,u(s),Gu(s))ds].
Then the set(Qϵ,δBr)(t) is relatively compact in Xα since by Lemma 2.2, the operator Sα(ϵqδ) is compact in Xα. For any u∈Br and t∈(0,T], from the following inequality:(3.10)∥(Qu)(t)-(Qϵ,δu)(t)∥α≤∥∫0δ‍ηq(θ)S(tqθ)dθx0∥α+∥q∫0t‍(t-s)q-1∫0δ‍θηq(θ)S((t-s)qθ)dθ·f(s,u(s),Gu(s))ds∫0t‍∥α+∥q∫0t‍(t-s)q-1∫δ∞‍θηq(θ)S((t-s)qθ)dθ·f(s,u(s),Gu(s))ds-q∫0t-ϵ‍(t-s)q-1∫δ∞‍θηq(θ)S((t-s)qθ)dθ·f(s,u(s),Gu(s))ds∫0δ‍∥α≤M∥x0∥α∫0δ‍ηq(θ)dθ+qMCβ-α∥gr∥L∞∫0t‍(t-s)q-1ds∫0δ‍θηq(θ)dθ+qMCβ-α∥gr∥L∞∫t-ϵt‍(t-s)q-1ds∫0∞‍θηq(θ)dθ≤M∥x0∥α∫0δ‍ηq(θ)dθ+MCβ-αTq∥gr∥L∞×∫0δ‍θηq(θ)dθ+MCβ-α∥gr∥L∞Γ(q+1)ϵq.
One can obtain that the set (QBr)(t) is relatively compact in Xα for all t∈(0,T). And since it is compact at t=0, we have the relatively compactness of (QBr)(t) in Xα for all t∈J.
Step 3. QBr:={Qu∈C(J,Xα):u∈Br} is equicontinuous.
Forτ∈[0,T), by (3.3), we have
(3.11)∥(Qu)(τ)-(Qu)(0)∥α≤∥U(τ)x0-x0∥α+∥∫0τ‍(τ-s)q-1V(τ-s)f(s,u(s),Gu(s))ds∥α≤∥U(τ)-I∥·∥x0∥α+MCβ-α∥gr∥L∞Γ(q+1)τq.
Hence it is only necessary to consider the case of t>0. For 0<t1<t2≤T, by Lemma 2.1 and Lemma 2.6(i), we have
(3.12)∥(Qu)(t2)-(Qu)(t1)∥α≤∥U(t2)x0-U(t1)x0∥α+∥∫0t2‍(t2-s)q-1V(t2-s)f(s,u(s),Gu(s))ds-∫0t1‍(t1-s)q-1V(t1-s)f(s,u(s),Gu(s))ds∥α≤∥U(t2)-U(t1)∥·∥x0∥α+∥∫t1t2‍(t2-s)q-1V(t2-s)f(s,u(s),Gu(s))ds∥α+∥∫0t1‍[(t2-s)q-1-(t1-s)q-1]V(t2-s)f(s,u(s),Gu(s))ds∥α+∥∫0t1‍(t1-s)q-1f(s,u(s),Gu(s))[V(t2-s)-V(t1-s)]ds∥α≤∥U(t2)-U(t1)∥·∥x0∥α+MCβ-α∥gr∥L∞Γ(q+1)(t2-t1)q+MCβ-α∥gr∥L∞Γ(q+1)[t2q-t1q-(t2-t1)q]+∥∫0t1‍(t1-s)q-1[V(t2-s)-V(t1-s)]·f(s,u(s),Gu(s))ds∥α≜I1+I2+I3+I4.
From Lemma 2.6(v), we see that I1→0 as t2→t1 independently of u∈Br. From the expressions of I2 and I3, it is clear that I2→0 and I3→0as t2→t1 independently of u∈Br. For any ϵ∈(0,t1), we have
(3.13)I4≤∥∫0t1-ϵ‍(t1-s)q-1[V(t2-s)-V(t1-s)]·f(s,u(s),Gu(s))ds∥α+∥∫t1-ϵt1‍(t1-s)q-1[V(t2-s)-V(t1-s)]·f(s,u(s),Gu(s))ds∥α≤1qCβ-α∥gr∥L∞(Tq+ϵq)sup0≤s≤t1-ϵ∥V(t2-s)-V(t1-s)∥+2MCβ-α∥gr∥L∞Γ(q+1)ϵq.
It follows from Lemma2.6(v) that I4→0 as t2→t1 and ϵ→0 independently of u∈Br. Therefore, we prove that QBr is equicontinuous.
Thus, the Arzela-Ascoli theorem guarantees thatQ is a compact operator. By the Schauder fixed point theorem, the operator Q has at least one fixed point u* in Br, which is a mild solution of (1.1). This completes the proof.Remark 3.2.
In assumption(H1)(iii), if the function gr(t) is independent of t, then we can easily obtain a constant γ>0 satisfying (3.2). For example, if there is a constant af>0 such that
(3.14)∥f(t,x,y)∥β≤af(1+∥x∥α+∥y∥α)
for all x,y∈Xα and t∈J, then for any r>0, x,y∈Xα with ∥x∥α≤r,∥y∥α≤K*Tr, we have ∥f(t,x,y)∥β≤af+af(1+K*T)r≜gr(t), where gr(t) is independent of t. Thus, γ:=(1/q)afTq(1+K*T)>0 is the constant in (3.2).
More generally, iff satisfies the following condition:(H2)
there is a constantβ∈[α,1] such that f:J×Xα×Xα→Xβ satisfies:(i)
for each(x,y)∈Xα×Xα, the function f(·,x,y):J→Xβ is measurable,(ii)
for anyr>0, there exists a function ℓ∈L∞(J,ℝ+) such that
(3.15)∥f(t,x1,y1)-f(t,x2,y2)∥β≤ℓ(t)(∥x1-x2∥α+∥y1-y2∥α)for any xi,yi∈Xα with ∥xi∥α≤r,∥yi∥α≤K*Tr(i=1,2) and t∈J, then we have the following existence and uniqueness theorem.Theorem 3.3.
Assume that the condition(H2) is satisfied. If x0∈Xα and MCβ-αTq(1+K*T)∥ℓ∥L∞<Γ(q+1), then (1.1) has a unique mild solution.Proof.
For anyr>0, if x,y∈Xα with ∥x∥α≤r,∥y∥α≤K*Tr, then from (H2)(ii), we have
(3.16)∥f(t,x,y)∥β≤ℓ(t)(1+K*T)r+b(t)≜gr(t),
where  b(t)=∥f(t,0,0)∥β. Therefore, the condition (H1)(iii) is satisfied with γ=((1+K*T)Tq∥ℓ∥L∞)/q. By Theorem 3.1, (1.1) has at least one mild solution u*∈Br.
Letu1,u2∈Br be the solutions of (1.1). We show that u1≡u2. Since u1(t)=(Qu1)(t) and u2(t)=(Qu2)(t) for all t∈J, we have
(3.17)∥u1(t)-u2(t)∥α=∥(Qu1)(t)-(Qu2)(t)∥α≤∫0t‍(t-s)q-1∥V(t-s)[f(s,u1(s),Gu1(s))-f(s,u2(s),Gu2(s))]∥αds=∫0t‍(t-s)q-1∥Aα-βV(t-s)·Aβ[f(s,u1(s),Gu1(s))-f(s,u2(s),Gu2(s))]∥ds≤MCβ-αΓ(q)∫0t‍(t-s)q-1·∥f(s,u1(s),Gu1(s))-f(s,u2(s),Gu2(s))∥βds≤MCβ-αΓ(q)∫0t‍(t-s)q-1·ℓ(s)(∥u1(s)-u2(s)∥α+∥Gu1(s)-Gu2(s)∥α)ds≤MCβ-α∥ℓ∥L∞Γ(q)(1+K*T)∫0t‍(t-s)q-1·∥u1(s)-u2(s)∥αds.
By using the Gronwall-Bellman inequality (see [14, Theorem 1]), we can deduce that ∥u1(t)-u2(t)∥α=0 for all t∈J, which implies that u1≡u2. Hence (1.1) has a unique mild solution u*∈Br. This completes the proof.Remark 3.4.
In Theorem3.3, we only assume that f satisfies a local Lioschitz condition (see condition (H2)), and an existence and uniqueness result is obtained. If f(t,u,v)≡f(t,u):J×Xα→X, then the assumption (H2) deletes the linear growth condition (3) of assumption (Hf) in [12]. Therefore, the Theorem 3.3 extends and improves the main result in [12].


## 4. An Example

Assume thatX=L2[0,π] equipped with its natural norm and inner product defined, respectively, for all u,v∈L2[0,π], by
(4.1)∥u∥X=(∫0π‍|u(x)|2dx)1/2,〈u,v〉=∫0π‍u(x)v(x)¯dx.
Consider the following fractional partial differential equation:
(4.2)∂1/2∂t1/2u(x,t)-∂2∂x2u(x,t)=g(x,t,u(x,t),∫0t‍K(t,s)u(x,s)ds),t∈[0,T],x∈[0,π]u(0,t)=u(π,t)=0,t∈[0,T],u(x,0)=u0(x),x∈[0,π],
where T>0 is a constant.Let the operatorA:D(A)⊂X→X be defined by
(4.3)D(A):={v∈X:v′′∈X,v(0)=v(π)=0},Au=-∂2u∂x2.
It is well known that A has a discrete spectrum with eigenvalues of the form n2,n∈ℕ, and corresponding normalized eigenfunctions given by zn=(2/π)sin(nx). Moreover, -A generates a compact analytic semigroup S(t)(t≥0) in X, and
(4.4)S(t)u=∑n=1∞‍e-n2t〈u,zn〉zn.
It is not difficult to verify that ∥S(t)∥ℒ(X)≤e-t for all t≥0. Hence, we take M=1.The following results are also well known.(I)
The operatorA can be written as
(4.5)Au=∑n=1∞‍n2〈u,zn〉znfor every u∈D(A).(II)
The operatorA1/2 is given by
(4.6)A1/2u=∑n=1∞‍n〈u,zn〉znfor each u∈D(A1/2):={v∈X:∑n=1∞n<v,zn>zn∈X} and ∥A-(1/2)∥ℒ(X)=1.Lemma 4.1 (see [28]).
Ifm∈D(A1/2), then m is absolutely continuous, m′∈X and ∥m′∥X=∥A1/2m∥X.LetX1/2=(D(A1/2),∥·∥1/2), where ∥x∥1/2:=∥A1/2x∥X for all x∈D(A1/2). Assume that g:[0,π]×[0,T]×ℝ×ℝ→ℝ satisfies the following conditions.(i)
For each(x,t)∈[0,π]×[0,T], the function g(x,t,·,·) is continuous.(ii)
For each(ξ,η)∈ℝ2, the function g(·,·,ξ,η) is measurable.(iii)
For eacht∈[0,T] and ξ,η∈ℝ, g(·,t,ξ,η) is differentiable, and (∂/∂x)g(x,t,ξ,η)∈X.(iv)
g(0,·,·,·)=g(π,·,·,·)=0.(v)
There exist the functionsℓ1,ℓ0∈L∞([0,T],ℝ+) such that
(4.7)|∂∂xg(x,t,ξ,η)|≤ℓ1(t)(|ξ|+|η|)+ℓ0(t)for all (x,t,ξ,η)∈[0,π]×[0,T]×ℝ×ℝ.Definef(t,u(t),Gu(t))(x)=g(x,t,u(x,t),∫0t‍K(t,s)u(x,s)ds). Then, for each ϕ∈X1/2, from assumptions (iii) and (iv), we have
(4.8)〈f(t,ϕ,Gϕ),zn〉=∫0π‍g(x,t,ϕ(x,t),∫0t‍K(t,s)ϕ(x,s)ds)·2πsin(nx)dx=1n∫0π‍(∂∂xg(x,t,ϕ(x,t),∫0t‍K(t,s)ϕ(x,s)ds))·2πcos(nx)dx.This implies from(II) that f:[0,T]×X1/2×X1/2→X1/2. Moreover, for any r>0, by Minkowski inequality, assumption (v) and Lemma 4.1, we have
(4.9)sup∥ϕ∥1/2≤r∥f(t,ϕ,Gϕ)∥1/2=sup∥ϕ∥1/2≤r∥∂∂xg(x,t,ϕ(x,t),∫0t‍K(t,s)ϕ(x,s)ds)∥X=sup∥ϕ∥1/2≤r(∫0π‍|∂∂xg(x,t,ϕ(x,t),∫0t‍K(t,s)ϕ(x,s)ds)|2dx)1/2≤sup∥ϕ∥1/2≤r(∫0π[ℓ1(t)(|ϕ(x,t)|+|∫0t‍K(t,s)ϕ(x,s)ds|)‍+ℓ0(t)∫0π‍]2dx∫0t‍)1/2≤sup∥ϕ∥1/2≤r[ℓ1(t)(∥ϕ∥X+K*T∥ϕ∥X)+ℓ0(t)]=sup∥ϕ∥1/2≤r[(1+K*T)ℓ1(t)∥A-(1/2)·A1/2ϕ∥X+ℓ0(t)]≤(1+K*T)rℓ1(t)+ℓ0(t)≜gr(t).Therefore,f satisfies the condition (H1) with γ=2T1/2(1+K*T)∥ℓ1∥(L∞). Thus, (4.2) has at least one mild solution provided that γ<π due to Theorem 3.1.Assume furthermore that the functiong satisfies the following:(vi)
for anyr>0, there exists a function ℓ2∈L∞([0,T],ℝ+) such that
(4.10)|∂∂xg(x,t,ξ1,η1)-∂∂xg(x,t,ξ2,η2)|≤ℓ2(t)(|ξ1-ξ2|+|η1-η2|)for (x,t,ξ1,η1),(x,t,ξ2,η2)∈[0,π]×[0,T]×ℝ×ℝ with |ξi|≤r and |ηi|≤K*Tr,i=1,2.Then for eachϕ1,ϕ2∈X1/2, by Lemma 4.1, we have
(4.11)∥f(t,ϕ1,Gϕ1)-f(t,ϕ2,Gϕ2)∥1/2=∥A1/2[f(t,ϕ1,Gϕ1)-f(t,ϕ2,Gϕ2)]∥X=∥∂∂xg(x,t,ϕ1(x,t),∫0t‍K(t,s)ϕ1(x,s)ds)-∂∂xg(x,t,ϕ2(x,t),∫0t‍K(t,s)ϕ2(x,s)ds)∥X=(∫0π‍|∂∂xg(x,t,ϕ1(x,t),∫0t‍K(t,s)ϕ1(x,s)ds)-∂∂xg(x,t,ϕ2(x,t),∫0t‍K(t,s)ϕ2(x,s)ds)|2dx)1/2≤(∫0t‍∫0π‍|ℓ2(t)(∫0π‍|ϕ1(x,t)-ϕ2(x,t)|+|∫0t‍K(t,s)ϕ1(x,s)ds-∫0t‍K(t,s)ϕ2(x,s)ds|)|2dx)1/2≤ℓ2(t)[-∫0t‍K(t,s)ϕ2(x,s)ds|2dx)1/2(∫0π‍|ϕ1(x,t)-ϕ2(x,t)|2dx)1/2+(∫0π‍|∫0t‍K(t,s)ϕ1(x,s)ds-∫0t‍K(t,s)ϕ2(x,s)ds|2dx)1/2]=ℓ2(t)(∥ϕ1-ϕ2∥X+∥Gϕ1-Gϕ2∥X)=ℓ2(t)(∥A-(1/2)·A1/2(Gϕ1-Gϕ2)∥X∥A-(1/2)·A1/2(ϕ1-ϕ2)∥X+∥A-(1/2)·A1/2(Gϕ1-Gϕ2)∥X)≤ℓ2(t)(∥A1/2(ϕ1-ϕ2)∥X+∥A1/2(Gϕ1-Gϕ2)∥X)=ℓ2(t)(∥ϕ1-ϕ2∥1/2+∥Gϕ1-Gϕ2∥1/2).
This shows that f satisfies the condition (H2). Hence by Theorem 3.3, the mild solution of (4.2) is unique.



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*Source: 903518-2012-11-06.xml*