# Existence of Solutions in Some Interpolation Spaces for a Class of Semilinear Evolution Equations with Nonlocal Initial Conditions

Authors: Jung-Chan Chang; Hsiang Liu
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

This paper is concerned with the existence of mild and strong solutions for
a class of semilinear evolution equations with nonlocal initial conditions. The linear part
is assumed to be a (not necessarily densely defined) sectorial operator in a Banach spaceX. Considering the equations in the norm of some interpolation spaces between X and
the domain of the linear part, we generalize the recent conclusions on this topic. The
obtained results will be applied to a class of semilinear functional partial differential
equations with nonlocal conditions.


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## Body

## 1. Introduction

The purpose of this paper is to study the existence of mild and strong solutions to the following semilinear evolution problem with nonlocal initial conditions:(1)u′(t)=Au(t)+f(t,u(t)),t∈[0,T],u(0)+g(u)=u0∈X,
in a Banach space (X,∥·∥), where T>0, f and g are given X-valued functions which will be specified later, and A is a linear closed operator in X. By letting (Fu)(t)≔f(t,u(t)), we see that (1) can be considered as a special case of the following equation:
(2)u′(t)=Au(t)+(Fu)(t),t∈[0,T],u(0)+g(u)=u0∈X.
Since the nonlocal condition reflects physical phenomena more precisely than the classical initial condition u(0)=u0 does, this issue has caught an increasing interest in the past several years. For the importance of nonlocal Cauchy problems in applications, we refer to the papers [1, 2] and the references therein.The existence and uniqueness of mild and classical solutions of (1) have been investigated by many authors, see for example [1–14]. A basic approach of this problem is to define the solution operator Φ:C([0,T],X)→C([0,T],X) by
(3)(Φu)(t):=T(t)(u0-g(u))+∫0t‍T(t-s)f(s,u(s))ds,t∈[0,T],
and use various fixed-point theorems, including Schauder fixed-point theorem, Banach contraction principle, Leray-Schauder alternative, Sadovskii fixed-point theorem, and some others, to show that Φ has a fixed point, which is the mild solution of (1). To use fixed-point theorems, it is necessary that the semigroup T(t) generated by the linear part of (1) is compact; that is, T(t) is a compact operator for all t>0, so that the norm continuity of T(t) for t>0 becomes a key point in the study of the existence of mild solution. In [13], the authors studied the existence of mild solution for (1) by using the Leray-Schauder alternative. However, as pointed out by Liu [11], the proof of the main result in [13] is invalid because of the absence of compactness of the solution operator at t=0. This gap can be filled by simply assuming complete continuity on the nonlocal term g, but it is too restrictive in applications. To take away this unsatisfactory condition, Liang et al. [7] observed that in many studies of nonlocal Cauchy problems, for example [1, 6, 9], the nonlocal condition g is completely determined on [δ,T], for some δ>0; that is, such a g ignores t=0; for instance, in [9] the function g(u) is given by
(4)g(u)=∑i=0p‍ciu(ti),
where the ci are given constants, and in this case, we have measurements at t=0≤t0<t1<⋯<tp≤T rather than just at t=0. Thus, by assuming that there is a δ∈(0,T) such that
(5)g(ϕ)=g(ψ)∀ϕ,φ∈Yr≔{φ∈C([0,T],X);∥φ(t)∥≤r,∀t∈[0,T]},
with
(6)ϕ(s)=ψ(s),s∈[δ,T],
the authors use their method of utilizing fixed-point theorem twice to deduce the existence of mild solution of (1) with the nonlinear term F being Lipschitz continuous and the nonlocal term g being continuous and mapping Yr into bounded sets. Afterward, Xiao and Liang [14] initiated the study of the nonautonomous case of (1) with nondensely defined sectorial linear part. More recently, Liu and Yuan [12] gave existence results by Schauder fixed-point theorem and a limiting process, under the following hypothesis: there is aδ∈(0,T)such that(7)F(ϕ)=F(ψ),g(ϕ)=g(ψ)∀ϕ,φ∈Yr
with
(8)ϕ(s)=ψ(s),s∈[δ,T],
with the nonlinear term F being bounded by an integrable function.In the present paper, the linear partA of (1) and (2) is assumed to be a (not necessarily densely defined) sectorial operator in X, which means that there exist constants ω, M>0, and θ∈(π/2,π) such that
(9)ρ(A)⊃∑θ,ω≔{λ∈ℂ:λ≠ω,|arg(λ-ω)|<θ},∥(λ-ω)R(λ,A)∥≤M,∀λ∈∑θ,ω.
Existence results of solutions to (1) and (2) are discussed in some interpolation spaces Xα,p≔(X,D(A))α,p, where 0<α<1 and 1<p≤∞ (see [15], Chapter 2). We establish some existence results of (1) and (2) without assuming Lipschitz condition on the nonlinear term and complete continuity on the nonlocal condition. Thus, the hypotheses in this note are less restricted than relevant works and hence the obtained results are more applicable than recent conclusions on this topic.This work is organized as follows. In Section2, we collect some known notions and results on the interpolation spaces between X and D(A) and give the basic hypotheses on (2). In Section 3, we study the existence of mild solutions to (2) and (1). Section 4 is devoted to the existence of strong solutions to (2) and (1), and in Section 5, we give an example to illustrate our abstract results.


## 2. Preliminaries

As usual, for a Banach spaceX, L(X) is the space of linear operators on X with operator norm denoted by ∥·∥L(X). If A satisfies (9), then it is known that the family of Dunford integrals
(10)T(t)=12πi∫ω+γr,ηeλt(λ-A)-1dλ,
where γr,η is the curve {λ∈ℂ:|argλ|=η,|λ|≥r}∪{λ∈ℂ:|argλ|≤η,|λ|=r}, is an analytic semigroup satisfying the following properties (see [15], Proposition 2.1.1, p. 35).(A1)
T(t)x∈D(Ak) for each t>0, x∈X and k∈ℕ. Moreover, AkT(t)x=T(t)Akx for all t≥0 and x∈D(Ak).(A2)
T(t)T(s)=T(t+s) for all t,s≥0.(A3)
There are positive constantsM0,M1,M2,… such that
(11)∥T(t)∥L(X)≤M0eωt,t>0,∥tk(A-ωI)kT(t)∥L(X)≤Mkeωt,t>0.(A4)
The functiont↦T(t) belongs to C∞((0,∞),L(X)) and
(12)dkdtkT(t)=AkT(t),t>0.
Moreover, it has an analytic extension in the sector
(13){λ∈ℂ:λ≠0,|argλ|<θ-π2}.
We see by (A3), that the family (T(t))t≥0 is uniformly bounded on the interval [0,T]; that is,
(14)M≔supt∈[0,T]∥T(t)∥<∞.
For α∈(0,1), p∈(1,∞], and t∈[s,T], we denote by Xα,p the interpolation space
(15)(X,D)α,p≔{x∈X:themappingt⟼t1-α-1/pAT(t)xbelongstoLp(0,1)t1-α-1/p}
with norm
(16)|x|α,p=∥x∥+∥θ1-α-1/pAT(·)x∥p,
where θλ:[0,T]→ℝ is given by θλ(t)=tλ.LetE be the Banach space C([0,T],Xα,p) endowed with the supnorm given by
(17)∥u∥E≔sup0≤t≤T|u(t)|α,pforx∈E,
and for any δ∈(0,T), set Eδ:=C([δ,T],Xα,p).By (2.2.1) of [15] (p.44), we have
(18)Cn≔sup0<t≤T∥tnAnT(t)∥L(X)<∞,∀n∈ℕ.
And hence the analytic semigroup (T(t))t≥0 also satisfies the following property.(A5)
For everyδ>0(19)Cδ≔supt∈[δ,T]∥AT(t)∥L(X)≤δ-1C1<∞.
Moreover, by (2.2.20) of [15] (see the last line of p.51), there is a constant C>0 such that(A6)
∥AnT(t)∥L(X,Xα,p)≤Ct-(n+α),0<t≤T,n∈ℕ∪{0}.The following conditions are basic assumptions of this paper.(H1)
The operator family(T(t))t≥0 is compact; that is, T(t) maps bounded subsets of X into relatively compact subsets of X for all t>0.(H2)
The functionF:E→L1([0,T],X) is continuous and for each k∈ℕ, there exists a positive function hk:[0,T]→ℝ+ such that
(1)
s↦hk(s)/(t-s)α is integrable on [0,t] for each t∈[0,T],(2)
sup∥u∥E<k∥(Fu)(t)∥≤hk(t) for almost all t∈[0,T],(3)
liminfk→∞(1/k)(supt∈[0,T]∫0t‍(hk(s)/(t-s)α)ds)=γ,(4)
limε→0(supt∈[0,T]∫t-εt‍(hk(s)/(t-s)α)ds)=0.(H3)
The functiong:L1([0,T],Xα,p)→Xα,p is continuous and there exist constants L>0 and β∈[0,1] such that
(20)|g(u)|α,p≤L(∥u∥E+1)β
for u∈E. Moreover, let ξ be the number defined by
(21)ξ:=lim∥u∥E→∞L(∥u∥E+1)β∥u∥E;
that is, ξ=L if β=1 and ξ=0 if β<1.We will argue that the hypothesis (H2) is reasonable. In fact, consider a functionf:[0,T]×Xα,p→X satisfying that there exist ℒ>0 and ϑ∈(0,1) such that
(22)∥f(t,x)-f(s,y)∥≤ℒ(|t-s|ϑ+|x-y|α,p),∀t,s∈[0,T],x,y∈Xα,p,
and define (Fu)(t):=f(t,u(t)). It is easy to see that F satisfies (H2) by letting
(23)hk(t)≔ksup∥u∥E<k∥f(·,u(·))∥L∞([0,t],X).
Moreover, one can choose the functions hk to be in L∞(0,T) and find easily that there is a lot more functions satisfying (H2).


## 3. Mild Solutions

Definition 1.
Letu:[0,T]→X be a continuous function.(i)
u is called a mild solution of (1) on [0,T] if u satisfies
(24)u(t)=T(t)(u0-g(u))+∫0t‍T(t-s)f(s,u(s))ds
for t∈[0,T].(ii)
u is called a mild solution of (2) on [0,T] if u satisfies
(25)u(t)=T(t)(u0-g(u))+∫0t‍T(t-s)(Fu)(s)ds
for t∈[0,T].The partAα,p of A in Xα,p is defined by
(26)D(Aα,p)={x∈D(A):Ax∈Xα,p},Aα,px=Ax,∀x∈D(Aα,p).
By Proposition 2.2.7 of [15], we see that Aα,p is sectorial in Xα,p. We denote by (Tα,p(t))t≥0 the analytic semigroup generated by Aα,p on Xα,p. It is clear that Tα,p(t)x=T(t)x for all x∈Xα,p and T(t)x∈D(A)⊂Xα,p for all x∈X and t>0.Remark 2.
It is known thatXα,p is contained in D(A)¯ ([15], Corollary 2.2.3 (ii), p.47) and that t↦T(t)x is continuous at 0 if x∈D(A)¯ ([15], Proposition 2.1.4 (i), p.38).Before stating and proving the main results, we need some lemmas.Lemma 3.
If (H1) is satisfied, then the following conclusions hold.(i)
(Tα,p(t))t≥0 is norm-continuous for t>0.(ii)
If1-α-1/p≥0, then (Tα,p(t))t≥0 is compact.(iii)
If{xn}n=1∞ is a bounded sequence in X and 1-α-1/p≥0, then for any t>0, {Tα,p(t)xn}n=1∞ has a convergent subsequence in Xα,p.Proof.
(i) Since the operator family(T(t))t≥0 is, by (H1), compact for t>0, then (T(t))t>0 is norm-continuous for t>0. Note that
(27)|Tα,p(t)x-Tα,p(s)x|α,p=∥[T(t)-T(s)]x∥+∥θ1-α-1/pAT(·)[T(t)-T(s)]x∥p≤∥T(t)-T(s)∥L(X)∥x∥+∥T(t)-T(s)∥L(X)∥θ1-α-1/pAT(·)x∥p=∥T(t)-T(s)∥L(X)|x|α,p,∀t,s∈(0,∞),x∈Xα,p,
which shows the norm continuity of (Tα,p(t))t≥0 for t>0. A similar argument also shows that
(28)∥Tα,p(t)∥L(Xα,p)≤∥T(t)∥L(X)
for all t≥0.
(ii) For the compactness of(Tα,p(t))t≥0 for t>0, it is sufficient to show that the sequence {Tα,p(t)xn} has a convergent subsequence in Xα,p for each sequence {xn} in the unit ball of Xα,p and t>0. Since
(29)∥xn∥≤|xn|α,p=∥xn∥+∥θ1-α-1/pAT(·)x∥p≤1,
then there is, by the compactness of T(t/2), a subsequence {xnk} of {xn} and an x0∈X so that
(30)∥T(t2)xnk-x0∥⟶0asnk⟶∞.
Thus, the boundedness of AT(t/2) implies that
(31)∥AT(t)xnk-AT(t2)x0∥=∥AT(t2)[T(t2)xnk-x0]∥⟶0asnk⟶∞.
If p∈(1,∞) and 1-α-1/p≥0, then (30), (31), and the Lebesgue dominated convergence theorem insure that
(32)|Tα,p(t)xnk-T(t2)x0|α,p=∥[T(t)xnk-T(t2)x0]∥+(∫01‍∥s1-αAT(s)[T(t)xnk-T(t2)x0]∥pdss)1/p=∥T(t2)[T(t2)xnk-x0]∥+(∫01‍∥s1-α-1/pT(s)[AT(t)xnk-AT(t2)x0]∥pds)1/p⟶0
as nk→∞. On the other hand, using (14) and (31), we have
(33)|Tα,p(t)xnk-T(t2)x0|α,∞=∥[T(t)xnk-T(t2)x0]∥+∥θ1-αAT(·)[T(t)xnk-T(t2)x0]∥∞=∥T(t2)[T(t2)xnk-x0]∥+sups∈[0,T]∥θ1-αAT(s)[T(t)xnk-T(t2)x0]∥≤∥T(t2)[T(t2)xnk-x0]∥+MT1-α∥AT(t2)∥∥T(t2)xnk-x0∥⟶0
as nk→∞. Since T(t/2)x∈D(A)⊂Xα,p, the assertion follows.
(iii) This follows immediately from the proof of (ii).In the sequel, we assume thatα∈(0,1) and p∈(1,∞] satisfy 1-α-1/p≥0. To see the existence of mild solution of nonlocal problem (2), we define the mapping Φ on E by
(34)(Φu)(t)≔T(t)(u0-g(u))+∫0t‍T(t-s)(Fu)(s)ds,t∈[0,T],
and for each k∈ℕ, let
(35)Bk={u∈E:|u(t)|α,p≤k,t∈[0,T]}.
We have the following results.Lemma 4.
Assume that the conditions (H1)–(H3) are satisfied. Ifu0∈Xα,p and, in addition,(H4)
(36)Mξ+Cγ<1 also holds, then ΦBk⊂Bk for some k∈ℕ.Proof.
If this were not the case, then for eachk∈ℕ, there would exist uk∈Bk and tk∈[0,T] such that |(Φuk)(tk)|α,p>k and hence we see by (14), (28), (H2), (H3), and (A6) that
(37)k<|(Φuk)(tk)|α,p≤|T(t)(u0-g(uk))|α,p+∫0tk‍|T(tk-s)(Fuk)(s)|α,pds≤∥T(t)∥L(Xα,p)|u0-g(uk)|α,p+∫0tk‍∥T(tk-s)∥L(X,Xα,p)∥(Fuk)(s)∥ds≤M(|u0|α,p+L(∥uk∥E+1)β)+C∫0tk‍hk(s)(tk-s)αds.
Dividing on both sides by k and taking the lower limit as ∥uk∥E→∞ we obtain
(38)Mξ+Cγ≥1
which contradicts (H4). Hence ΦBk⊂Bk for some k∈ℕ.The following lemma is useful in the proof of our main result in this section.Lemma 5.
Assume that the conditions (H1)–(H4) are satisfied. Ifu0∈Xα,p and, in addition, (H5)
there exists aδ∈(0,T), such that F(u)=F(v), g(u)=g(v) for any u,v∈Bk with u(s)=v(s), s∈[δ,T],
then the nonlocal problem (2) has at least one mild solution in Bk.Proof.
LetBk(δ)≔{u∈C([δ,T],Xα,p):|u(t)|α,p≤k,forallt∈[δ,T]} where δ is the number defined in (H5). For u∈Bk(δ), find v∈Bk such that u(t)=v(t) for all t∈[δ,T] and define 𝔉:Bk(δ)→L([0,T],X) and 𝔤:Bk(δ)→Xα,p by (𝔉u)(t)=(Fv)(t) for t∈[0,T] and 𝔤(u)=g(v), respectively. By (H2), (H3), and (H5), we see that 𝔉 and 𝔤 are well-defined, continuous and satisfy
(39)∥(𝔉u)(t)∥≤hk(t),a.e.t∈[0,T],∀u∈Bk(δ),supu∈Bk(δ)∥𝔤(u)∥=supv∈Bk|g(v)|α,p≤L(k+1)β<∞.
Now let us define a mapping Φδ on Bk(δ) as follows:
(40)(Φδu)(t)≔T(t)(u0-𝔤(u))+∫0t‍T(t-s)(𝔉u)(s)ds,t∈[δ,T].
We will show that Φδ has a fixed point in Bk(δ). To see this, note first that Φδ is continuous by the continuity of 𝔉 and 𝔤. Then, in view of Schauder’s fixed point theorem, it suffices to show that the set {Φδu:u∈Bk(δ)} is relatively compact in C([δ,T],Xα,p). Thus, we will prove that for each t∈[δ,T], both sets
(41){Tα,p(t)(u0-𝔤(u));u∈Bk(δ)},V(t)≔{∫0t‍T(t-s)(𝔉u)(s)ds;u∈Bk(δ)}
are relatively compact in Xα,p, and that both
(42){Tα,p(·)(u0-𝔤(u)):u∈Bk(δ)},{∫0·‍Tα,p(·-s)(𝔉u)(s)ds:u∈Bk(δ)}
are equicontinuous families of functions on [δ,T].
In fact, from the compactness ofTα,p(t) for t∈[δ,T] it follows that {Tα,p(t)(u0-𝔤(u));u∈Bk(δ)} is relatively compact in Xα,p for t∈[δ,T]. Moreover, for each t∈[δ,T], ε∈(0,t), and u∈Bk(δ), consider the mapping defined by
(43)(Pεu)(t)=Tα,p(ε)∫0t-ε‍Tα,p(t-ε-s)(𝔉u)(s)ds.
Since Vε(t)={(Pεu)(t);u∈Bk(δ)} is relatively compact in Xα,p by the compactness of Tα,p(ε) and since
(44)|∫0t‍T(t-s)(𝔉u)(s)ds-(Pεu)(t)|α,p≤∫t-εt‍∥T(t-s)∥L(X,Xα,p)∥(𝔉u)(s)∥ds≤C∫t-εt‍hk(s)(t-s)αds,
then we see by (H2) that there are relative compact sets arbitrarily close to V(t) and hence V(t) is also relatively compact in Xα,p. Therefore, the set {(Φδu)(t):u∈Bk(δ)} is relatively compact in Xα,p for each t∈[δ,T].
Now, by the norm continuity ofTα,p(t) for t>0, we see that
(45)|[Tα,p(t+h)-Tα,p(t)](u0-𝔤(u))|α,p⟶0,ash⟶0
independently of u∈Bk, and hence {Tα,p(·)(u0-𝔤(u)):u∈Bk(δ)} is an equicontinuous family of functions on [δ,T]. Finally, let ϵ>0 be arbitrarily small and we see by (A6) with n=0 that
(46)|∫0t+h‍T(t+h-s)(𝔉u)(s)ds-∫0t‍T(t-s)(𝔉u)(s)ds|α,p≤∫0t-ϵ‍|[T(t+h-s)-T(t-s)](𝔉u)(s)|α,pds+∫t-ϵt‍|[T(t+h-s)-T(t-s)](𝔉u)(s)|α,pds+∫tt+h‍|T(t+h-s)(𝔉u)(s)|α,pds≤∫0t-ϵ‍∥T(t-s2+h)-T(t-s2)∥L(Xα,p)×∥T(t-s2)∥L(X,Xα,p)∥(𝔉u)(s)∥ds+∫t-ϵt‍(∥T(t+h-s)-T(t-s)∥L(X,Xα,p))∥(𝔉u)(s)∥ds+∫tt+h‍∥T(t+h-s)∥L(X,Xα,p)∥(𝔉u)(s)∥ds≤2αC∫0t-ϵ‍∥T(t-s2+h)-T(t-s2)∥L(Xα,p)hk(s)(t-s)αds+C∫t-ϵt‍hk(s)(t+h-s)αds+C∫t-ϵt‍hk(s)(t-s)αds+C∫tt+h‍hk(s)(t+h-s)αds≤2αC∫0t-ϵ‍∥T(t-s2+h)-T(t-s2)∥L(Xα,p)hk(s)(t-s)αds+C{∫t-ϵt+h‍hk(s)(t+h-s)αds-∫t-ϵt‍hk(s)(t-s)αds+2∫t-ϵt‍hk(s)(t-s)αds}≤2αC∫0t-ϵ‍∥T(t-s2+h)-T(t-s2)∥L(Xα,p)hk(s)(t-s)αds+C{∫0t-ε‍[θ-α*hk(t+h)-θ-α*hk(t)]+∫0t-ε‍[hk(s)(t-s)α-hk(s)(t+h-s)α]ds+2∫t-ϵt‍hk(s)(t-s)αds}
which is, by the norm continuity of Tα,p(t) for t>0 and dominated convergence theorem, arbitrarily small independently of u∈Bk as h→0. Therefore,
(47){∫0·‍Tα,p(·-s)(𝔉u)(s)ds:u∈Bk(δ)}
is an equicontinuous family of functions on [δ,T].
Consequently, by Arzela-Ascoli’s theorem, it follows that{Φδu:u∈Bk(δ)} is relatively compact in Eδ. Thus there is a function φ∈Bk(δ) such that Φδφ=φ; that is,
(48)φ(·)=T(·)(u0-𝔤(φ))+∫0·‍T(·-s)(𝔉φ)(s)ds∈Bk(δ),
or
(49)φ(t)=T(t)(u0-𝔤(φ))+∫0t‍T(t-s)(𝔉φ)(s)ds,∀t∈[δ,T].
Now, define a functionψ on [0,T] by
(50)ψ(t)=T(t)(u0-𝔤(φ))+∫0t‍T(t-s)(𝔉φ)(s)ds,ddddddddddddddddddddddddddddd∀t∈[0,T].
Then ψ=φ on [δ,T] and we have ψ∈Bk. By the definition of 𝔉, 𝔤, and (H5), we see that 𝔉φ=Fφ=Fψ, 𝔤(φ)=g(φ)=g(ψ) and consequently,
(51)ψ(t)=T(t)(u0-g(ψ))+∫0t‍T(t-s)(Fψ)(s)ds,ffffffffffffffffffffffffffffffffffffffffffffffdddfff∀t∈[0,T];
that is, ψ is a mild solution of (2), and the proof is completed.For our main result in this section, we introduce in the following a family of nonlocal Cauchy problems. Firstly, we define, for eachδ∈(0,T), an operator ℬδ on E by
(52)(ℬδu)(t)≔{u(δ),u(t),0≤t≤δ,δ≤t≤T,∀u∈E.
It is clear that ℬδ is bounded on E and ∥ℬδ∥L(E)≤1, and hence ℬδBk⊂Bk. Now, for each δ∈[0,T], we define gδ:E→Xα,p by
(53)gδ(u)=g(ℬδu),u∈E,
and Fδ:E→L([0,T],X) by
(54)Fδu=F(ℬδu),u∈E,
and consider the nonlocal Cauchy problems:
(NCPδ)u′(t)=Au(t)+(Fδu)(t),t∈[0,T],u(0)+gδ(u)=u0∈Xα,p.
In view of (52), (53), and (54), the following result is an immediate corollary of Lemma 5.Lemma 6.
Suppose that (H1)–(H4) are satisfied. Then for anyδ∈(0,T], the problem (NCPδ) has at least one mild solution in Bk.Theorem 7.
Suppose that the hypotheses (H1)–(H4) are satisfied. Ifu0∈Xα,p, then the problem (2) has at least one mild solution in Bk for some k∈ℕ.Proof.
Choose a decreasing sequence{δn}n∈ℕ⊂(0,T) so that limn→∞δn=0 and then by Lemma 6, we see that for each n∈ℕ,
(55)un(t)=T(t)(u0-gδn(un))+∫0t‍T(t-s)(Fδnun)(s)ds,t∈[0,T],
or
(56)un=T(·)(u0-g(ℬδnun))+∫0·‍T(·-s)(F(ℬδnun))(s)ds∈Bk
is a mild solution for the nonlocal Cauchy problem (NCPδn).
Since the sequence{ℬδnun}n∈ℕ lies in Bk, then a similar argument as in the proof of Lemma 5 (see (44)–(46) shows that
(57)foranyt∈[0,T],theset{un(t):n∈ℕ,t∈[0,T]}⊆Xα,p,
that
(58)foreacht∈[0,T],theset{∫0t‍T(t-s)F(ℬδnun)(s)ds}n∈ℕisrelativelycompactinXα,p,
and that
(59){∫0·‍T(·-s)F(ℬδnun)(s)ds}n∈ℕisanequicontinuoussequenceoffunctionson[0,T].
Hence it follows by Ascoli-Arzela theorem that
(60){∫0·‍T(·-s)F(ℬδnun)(s)ds}n∈ℕisrelativelycompactinE.
Now, let {ϵn}n∈ℕ⊂(0,T) be a decreasing sequence such that limn→∞ϵn=0 and let {un0}n0∈ℕ be a subsequence of {un}n∈ℕ. Then, a similar argument as in the proof of Lemma 5 insures that {Tα,p(·)(u0-g(ℬδn0un0))}n0∈ℕ is an equicontinuous sequence of functions on [ϵ1,T]. Thus, Ascoli-Arzela theorem guarantees that the sequence
(61){Tα,p(·)(u0-g(ℬδn0un0))}n0∈ℕisrelativelycompactinC([ϵ1,T],Xα,p).
Thus by (60) and (61), we see that {un0}n0∈ℕ is relatively compact in C([ϵ1,T],Xα,p), and hence we can select a subsequence of {un0}n0∈ℕ, denoted by {un1}n1∈ℕ which is a Cauchy sequence in C([ϵ1,T],Xα,p). By a similar process, we can select a subsequence of {un1}n1∈ℕ, denoted by {un2}n2∈ℕ which is a Cauchy sequence in C([ϵ2,T],Xα,p). Repeat the above argument and use a diagonal argument to obtain a subsequence of {un0}n0∈ℕ, denoted by {vn}n∈ℕ. Then for every t∈(0,T], {vn(t)}n∈ℕ is a Cauchy sequence in Xα,p and so, we can define the function v∞ by
(62)v∞(t)={0,t=0;limn→∞vn(t),0<t≤T.
It is clear that v∞ is strongly measurable, v∞∈Bk and
(63)∫0T‍|vn(t)-v∞(t)|α,pdt≤2Tk.
It therefore follows by Lebesgue’s dominated convergence theorem that there is a subsequence {τn}n∈ℕ of {δn}n∈ℕ such that
(64)∫0T‍|(ℬτnvn)(t)-v∞(t)|α,pdt≤∫0T‍|(ℬτnvn)(t)-vn(t)|α,pdt+∫0T‍|vn(t)-v∞(t)|α,pdt⟶0
as n→∞. This shows that the sequence
(65){T(·)(u0-g(ℬδnun))}n∈ℕisrelativelycompactinE.
By (60) and (65), we see the relative compactness of {un}n∈ℕ in E. Thus, there is a subsequence of {un}n∈ℕ, denoted by {uν}ν∈ℕ and a function u∞∈E such that
(66)limν→∞∥uν-u∞∥E=0.
It is clear that u∞∈Bk. Since
(67)∥ℬδνuν-u∞∥E=sup0≤t≤T|ℬδνuν(t)-u∞(t)|α,p≤sup0≤t≤δν|uν(δν)-u∞(t)|α,p+supδν≤t≤T|uν(t)-u∞(t)|α,p≤|uν(δν)-u∞(δν)|α,p+sup0≤t≤δν|u∞(δν)-u∞(t)|α,p+∥uν-u∞∥E≤2∥uν-u∞∥E+sup0≤t≤δν|u∞(δν)-u∞(t)|α,p,
then (66) and the uniform continuity of u∞ imply that limν→∞∥ℬδνuν-u∞∥E=0. By taking limits in (56), we see that u∞ is a mild solution of (2) and this completes the proof.We will consider a case more generally; that is, the nonlocal conditiong is defined on E rather than L1([0,T],X).Theorem 8.
Suppose that the hypotheses (H1) and (H2) are satisfied and, in addition, that there hold the following conditions.(H6)
The functiong:E→Xα,p is continuous and there exist constants L>0 and β∈[0,1] such that
(68)|g(u)|α,p≤L(∥u∥E+1)β
for u∈E. Moreover, the real number ξ is defined by
(69)ξ:=lim∥u∥E→∞L(∥u∥E+1)β∥u∥E.(H7)
limϵ→0supϕ∈Bk∥g(ϕ)-g(ϕϵ)∥E=0, where ϕϵ(t)={ϕ(ϵ),0≤t≤ϵ,ϕ(t),ϵ≤t≤T.
If the inequality (H4) holds andu0∈Xα,p, then the problem (2) has at least one mild solution in Bk for some k∈ℕ.Proof.
Let{un}n∈ℕ be the sequence defined as in the proof of Theorem 7. By the same arguments as in the proofs of Lemma 4 and Theorem 7, we see that ΦBk⊂Bk for some k∈ℕ, (60) and (61) also hold and for every subsequence {un0}n0∈ℕ of {un}n∈ℕ, and there exist a subsequence {vn}n=1∞ and a function v∞:(0,T]→Xα,p such that v∞ is continuous on (0,T] and for every ϵk(70)limn→∞maxt∈[ϵk,T]|vn(t)-v∞(t)|α,p=0,orlimn→∞∥ℬϵk[vn-v∞]∥E=0.
Let ε>0 be given. It follows by (H7) and (70) that there is a δ>0 such that
(71)|g(ϕ)-g(ψ)|α,p<ε4wheneverϕ=ψon[δ,T]
and that for every k∈ℕ there is an Nk such that n>Nk implies that
(72)|g(ℬϵk[vn-v∞])|α,p<ε4.
Choose K large enough so that ϵK<δ and define φ:[0,T]→Xα,p by
(73)φ(t)={v∞(ϵK),0≤t≤ϵK,v∞(t),ϵK≤t≤T.
Thus, (H7), (71), and (72) insures that
(74)|g(vn)-g(vm)|α,p≤|g(vn)-g(ℬϵKvn)|α,p+|g(ℬϵKvn)-g(φ)|α,p+|g(φ)-g(BϵKvm)|α,p+|g(ℬϵKvm)-g(vm)|α,p<ε.
And hence by the continuity of g and compactness of Tα,p(t) for t>0, (65) is also valid in this case. Therefore, the same argument as in the last paragraph of the proof of Theorem 7 shows the existence of mild solution for (2).Now, we consider the nonlocal Cauchy problem (1). For this, we introduce the following hypothesis.(H8)
The functionf:[0,T]×Xα,p→X satisfies the Carathéodory condition; that is, f(·,x) is measurable for each x∈X and f(t,·) is continuous for any t∈[0,T]. Moreover, for each k∈ℕ, there exists a function hk:[0,T]→ℝ+ such that
(1)
s↦hk(s)/(t-s)α is integrable on [0,t] for each t∈[0,T],(2)
sup∥u∥E<k∥F(t,u(t))∥≤hk(t) for almost all t∈[0,T],(3)
liminfk→∞(1/k)(supt∈[0,T]∫0t‍(hk(s)/(t-s)α)ds)=γ,(4)
limε→0(supt∈[0,T]∫t-εt‍(hk(s)/(t-s)α)ds)=0. The following existence results are immediate corollaries of Theorems 7 and 8.Theorem 9.
(i) Suppose that (H1), (H3), and (H8) are satisfied. If the inequality (H4) also holds andu0∈Xα,p, then (1) has at least one mild solution in Bk for some k∈ℕ.
(ii) Suppose that (H1) and (H6)–(H8) are satisfied. If the inequality (H4) also holds andu0∈Xα,p, then (1) has at least one mild solution in Bk for some k∈ℕ.Proof.
By letting(75)(Fu)(t)=f(t,u(t)),t∈[0,T]
one sees that (H8) implies (H2), and hence we obtain the existence results of (1) by Theorems 7 and 8.


## 4. The Existence of Strong Solutions

Definition 10.
A mild solutionu of (1) is called a strong solution if u is continuously differentiable on (0,T] and satisfies (1). The strong solution of (2) is defined in a similar way.In the following, we establish existence of strong solutions for (2).Theorem 11.
(i) Suppose that the hypotheses (H1) and (H3) are satisfied, and in addition, that there holds the following conditions.(H9)
The functionF:E→L1([0,T],X) is continuous and for each k∈ℕ, there exists a hk:[0,T]→ℝ+ such that
(1)
hk is locally bounded function on (0,T],(2)
θ-α*hk∈L1([0,T],ℝ+) is locally Hölder continuous on (0,T],(3)
sup∥u∥E<k∥(Fu)(t)∥≤hk(t) for almost all t∈[0,T],(4)
liminfk→∞(1/k)(supt∈[0,T]∫0t‍(hk(s)/(t-s)α)ds)=γ,(5)
limε→0(supt∈[0,T]∫t-εt‍(hk(s)/(t-s)α)ds)=0.(H10)
Ifu∈E is locally Hölder continuous on (0,T], then Fu is locally Hölder continuous on (0,T].
If the inequality (H4) also holds andu0∈Xα,p, then (2) has a strong solution on [0,T].
(ii) Suppose that the hypotheses (H1), (H6), (H7), (H9), and (H10) are satisfied. If the inequality (H4) also holds andu0∈Xα,p, then (2) has a strong solution on [0,T].Proof.
It is clear that (H9) implies (H2). Thus, the conditions (H1)–(H4) hold by assumption and hence Theorem7 shows that if u0∈Xα,p, then (2) has a mild solution on [0,T]; that is, there is a u∈E such that
(76)u(t)≔T(t)(u0-g(u))+∫0t‍T(t-s)(Fu)(s)ds
for t∈[0,T]. We will show that this mild solution is a strong solution. In view of [16] (p.113, Corollary 4.3.3), it suffices to prove that Fu is locally Hölder continuous on (0,T].
From (76), for 0<t1<t2≤T, one derives the following:
(77)|u(t2)-u(t1)|α,p≤|[T(t2)-T(t1)](u0-g(u))|α,p+|∫0t1‍[T(t2-s)-T(t1-s)](Fu)(s)ds|α,p+|∫t1t2‍T(t2-s)(Fu)(s)ds|α,p.
By (18), (A5), and (A6) we have
(78)|[T(t2)-T(t1)](u0-g(u))|α,p=|∫t1t2‍AT(s)(u0-g(u))ds|α,p≤∫0t2-t1‍|AT(t1+s)(u0-g(u))|α,pds≤∫0t2-t1‍∥T(s)∥L(X,Xα,p)∥AT(t1)(u0-g(u))∥ds≤∫0t2-t1‍(Cs-α)(t1-1C1)∥u0-g(u)∥ds=CC1t1(1-α)∥u0-g(u)∥(t2-t1)1-α,
that
(79)|∫0t1‍[T(t2-s)-T(t1-s)](Fu)(s)ds|α,p≤∫0t1‍∫0t2-t1‍|AT(τ+t1-s)(Fu)(s)|α,pdτds≤∫0t1‍∫0t2-t1‍∥AT(τ+t1-s)∥L(X,Xα,p)∥(Fu)(s)∥dτds≤C∫0t1‍∫0t2-t1‍hk(s)(τ+t1-s)1+αdτds≤-Cα∫0t1‍[1(τ+t1-s)α|0t2-t1]hk(s)ds≤Cα∫0t1‍[1(t1-s)α-1(t2-s)α]hk(s)ds,
where we use (A6) with n=1 and that
(80)|∫t1t2‍T(t2-s)(Fu)(s)ds|α,p≤∫t1t2‍|T(t2-s)(Fu)(s)|α,pds≤∫t1t2‍∥T(t2-s)∥L(X,Xα,p)∥(Fu)(s)∥ds≤C∫t1t2‍hk(s)(t2-s)αds.
Since hk is locally bounded and θ-α*hk is locally Hölder continuous on (0,T] by (H9), then
(81)Cα∫0t1‍[1(t1-s)α-1(t2-s)α]hk(s)ds+C∫t1t2‍hk(s)(t2-s)αds=Cα[(θ-α*hk)(t1)-(θ-α*hk)(t2)]+C(1α+1)∫t1t2‍hk(s)(t2-s)αds≤Cα|(θ-α*hk)(t2)-(θ-α*hk)(t1)|+C(1α+1)∫t1t2‍Hk(t2-s)αds
for some Hk>0. This shows that u is locally Hölder continuous on (0,T] and so, by (H10), Fu is locally Hölder continuous on (0,T]. This completes the proof of (i) and the assertion (ii) follows in a similar argument.Now, we establish existence results for strong solutions of the nonlocal Cauchy problem (1) and for this purpose, we introduce the following hypotheses.(H11)
The functionf:[0,T]×Xα,p→X satisfies the Carathéodory condition; that is, f(·,x) is measurable for each x∈X and f(t,·) is continuous for any t∈[0,T]. Moreover, for each k∈ℕ, there exists a function hk:[0,T]→ℝ+ such that
(1)
hk is locally bounded function on (0,T],(2)
θ-α*hk∈L1([0,T],ℝ+) is locally Hölder continuous on (0,T],(3)
sup∥u∥E<k∥f(t,u(t))∥≤hk(t) for almost all t∈[0,T],(4)
liminfk→∞(1/k)(supt∈[0,T]∫0t(hk(s)/(t-s)α)‍ds)=γ, and(5)
limε→0(supt∈[0,T]∫t-εt‍(hk(s)/(t-s)α)ds)=0.(H12)
Ifu∈E is locally Hölder continuous on (0,T], then f(t,u(t)) is locally Hölder continuous on (0,T].Theorem 12.
(i) Suppose that (H1), (H3), (H11), and (H12) are satisfied. If the inequality (H4) also holds andu0∈Xα,p, then (1) has at least one strong solution in Bk for some k∈ℕ.
(ii) Suppose that (H1), (H6), (H7), (H11), and (H12) are satisfied. If the inequality (H4) also holds andu0∈Xα,p, then (1) has at least one strong solution in Bk for some k∈ℕ.Proof.
By letting(82)(Fu)(t)=f(t,u(t)),t∈[0,T]
one sees that (H11) implies (H2), and hence we obtain the existence results of (1) by using a similar argument as the proof of Theorem 11.


## 5. An Example

In this section, we consider the following system:(83)∂u∂tu(t,x)=∂2u(t,x)∂x2+f(t,u(t,x),∂u(t,x)∂x),vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvt∈[0,T],u(t,0)=u(t,1)=0,t∈[0,T],u(0,x)+∫0T‍k1(t,x)∫01‍k2(u(t,r),∂u(t,r)∂r)drdt=ϕ0(x),x∈[0,1],
where u:[0,T]×[0,1]→ℝ, f:[0,T]×ℝ×ℝ→ℝ, k1:[0,T]×[0,1]→ℝ, and k2:ℝ×ℝ→ℝ. This system will be solved by transforming it into (2). To do this, let X=C([0,1],ℝ) and let D={ϕ:ϕ∈C2([0,1],ℝ);ϕ(0)=ϕ(1)=0}. Define A:D→X by Au=u′′. By [15], it is well known that A is a sectorial operator on X and A generates a semigroup (T(t))t≥0 on X. The nonautonomous cases have been studied by Xiao and Liang [14] and they have shown there that T(t) is a compact operator for each t>0. Choose α∈(1/2,1) and again by [15] (P.107, (3.1.76)) one sees that Xα,∞=C02α([0,1],ℝ). Suppose that f, k1, and k2 satisfy the following conditions.(1)f:[0,T]×ℝ×ℝ→ℝ is continuous and there is a function h∈L∞([0,T],ℝ) such that |f(t,x,y)|≤h(t) for t∈[0,T] and x, y∈ℝ.(2)k1:[0,T]×[0,1]→ℝ is continuous and for each t∈[0,T], the mapping s↦k1(t,s) belongs to Xα,∞=C02α([0,1],ℝ).(3)k2:ℝ×ℝ→ℝ is continuous and uniformly bounded. Moreover, there is a constant L>0 such that |k2(x1,y1)-k2(x2,y2)|≤L(|x1-x2|+|y1-y2|) for (xi,yi)∈ℝ2 and i=1,2.LetE denote the space C([0,T],Xα,∞) equipped the supnorm ∥·∥E and for each u∈E or L1([0,T],Xα,∞), let u(·,·):[0,T]×[0,1]→ℝ be given by u(t,s):=(u(t))(s), for all t∈[0,T] and s∈[0,1]. Consider the following two operators.(i)
F:E→L1([0,T],X) is defined by
(84)[(Fu)(t)](s)=f(t,u(t,s),D2u(t,s)),t∈[0,T],s∈[0,1],
where D2 denotes the partial derivative with respect to the second variable.(ii)
g:L1([0,T],Xα,∞)→Xα,∞ is defined by
(85)g(u)(s)=∫0T‍∫01‍k1(t,s)k2(u(t,r),D2u(t,r))drdt,s∈[0,1].From assumptions (1)–(3), it is easy to see that these operators are well defined.Now, (83) can be transformed into
(86)du(t)dt=Au(t)+(F(u))(t),t∈[0,T],u(0)+g(u)=ϕ0.
We will show that F and g satisfy the hypotheses (H2) and (H3), respectively. In fact, it follows from assumption (1) that
(87)sup∥u∥E<k∥(Fu)(t)∥≤h(t),liminfk→01k∫0t‍h(s)(t-s)αds=γ=0
for each t∈[0,T] and k∈ℕ. From assumptions (2) and (3), it is easy to see that g is continuous and satisfies (H3) with ξ=0. Hence (H4) also holds. So, Theorem 7 ensures the following.Theorem 13.
If the conditions (1)–(3) hold andϕ0∈C02α([0,1],ℝ), then (86) has a mild solution.For the existence of strong solutions to (86), the following assumption is also needed.(4) There are constantsL′>0 and η∈(0,1] such that
(88)|f(t1,x1,y1)-f(t2,x2,y2)|≤L′(|t1-t2|η+|x1-x2|η+|y1-y2|η)
for (t1,x1,y1), (t2,x2,y2)∈[0,T]×ℝ×ℝ.Next, fort2>t1>0, we have
(89)|θ-α*h(t2)-θ-α*h(t1)|=|∫0t2‍h(s)(t2-s)αds-∫0t1‍h(s)(t1-s)αds|≤|∫0t1‍h(s)(t2-s)αds-∫0t1‍h(s)(t1-s)αds|+|∫t1t2‍h(s)(t2-s)αds|≤∫0t1‍|1(t2-s)α-1(t1-s)α|h(s)ds+∫t1t2‍h(s)(t2-s)αds≤∥h∥∞∫0t1‍[1(t1-s)α-1(t2-s)α]ds+∥h∥∞∫t1t2‍1(t2-s)αds=∥h∥∞1-α[-(t1-s)1-α|0t1+(t2-s)1-α|0t1-(t2-s)1-α|t1t2]=∥h∥∞1-α[t11-α+(t2-t1)1-α-t21-α+(t2-t1)1-α].
By Mean Value Theorem, it is easy to see that
(90)t21-α-t11-α=1-αδα(t2-t1)
for some δ∈(t1,t2). So, it easy to see that θ-α*h is locally Hölder continuous on (0,T]. Hence, (H9) holds. Finally, from (4), it is easy to see that (H10) holds. So from Theorem 11, we have the following.Theorem 14.
If the conditions (1)–(4) hold andϕ0∈C02α([0,1],ℝ), then (86) has a strong solution.



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*Source: 451252-2013-08-12.xml*