# Nonlocal Problems for Fractional Differential Equations via Resolvent Operators

Authors: Zhenbin Fan; Gisèle Mophou
Journal: International Journal of Differential Equations
 (2013)

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## Abstract

We discuss the continuity of analytic resolvent in the uniform operator topology and then obtain the compactness of Cauchy operator by means of the analytic resolvent method. Based on this result, we derive the existence of mild solutions for nonlocal fractional differential equations when the nonlocal item is assumed to be Lipschitz continuous and neither Lipschitz nor compact, respectively. An example is also given to illustrate our theory.


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## Body

## 1. Introduction

In this paper, we are concerned with the existence of mild solutions for a fractional differential equation with nonlocal conditions of the form(1)Dαu(t)=Au(t)+Jt1-αf(t,u(t)),0<t≤b,u(0)=u0-g(u),
where Dα is the Caputo fractional derivative of order α with 0<α<1, A:D(A)⊂X→X is the infinitesimal generator of a resolvent Sα(t), t≥0, X is a real Banach space endowed with the norm ∥·∥, and f and g are appropriate continuous functions to be specified later.The theory of fractional differential equations has received much attention over the past twenty years, since they are important in describing the natural models such as diffusion processes, stochastic processes, finance, and hydrology. Many notions associated with resolvent are developed such as integral resolvent, solution operators,α-resolvent operator functions, (a,k)-regularized resolvent, and α-order fractional semigroups. All of these notions play a central role in the study of Volterra equations, especially the fractional differential equations. Concerning the literature, we refer the reader to the books [1, 2], the recent papers [3–20], and the references therein.On the other hand, abstract differential equations with nonlocal conditions have also been studied extensively in the literature, since it is demonstrated that the nonlocal problems have better effects in applications than the classical ones. It was Byszewski and Lakshmikantham [21] who first studied the existence and uniqueness of mild solutions for nonlocal differential equations. And the main difficulty in dealing with the nonlocal problem is how to get the compactness of solution operator at zero, especially when the nonlocal item is only assumed to be Lipschitz continuous or continuous. Many authors developed different techniques and methods to solve this problem. For more details on this topic, we refer to [10, 11, 22–33] and references therein.In this paper, we combine the above two directions and study the nonlocal fractional differential equation (1) governed by operator A generating an analytic resolvent. A standard approach in deriving the mild solution of (1) is to define the solution operator Q. Then, conditions are given such that some fixed point theorems such as Browder’s and Schauder’s fixed point theorems can be applied to get a fixed point for solution operator Q, which gives rise to a mild solution of (1). The key step of using this approach is to prove the compactness of Cauchy operator G associated with solution operator Q. When the operator A generates a compact semigroup, it is well known that the Cauchy operator G is also compact. However, to the best of our knowledge, it is unknown when A generates a compact resolvent. The main difficulty of this problem lies in the fact that there is no property of semigroups for resolvent.To this end, we will first discuss the continuity of resolvent in the uniform operator topology in this paper. In fact, we prove that the compact analytic resolvent is continuous in the uniform operator topology. Based on this result, we can prove the compactness of Cauchy operator. As a consequence, we obtain the existence of mild solutions for (1) when the nonlocal item is Lipschitz continuous. At the same time, we also derive the existence of mild solutions for (1) without the Lipschitz or compact assumption on the nonlocal item g by using the techniques developed in [24, 30]. Actually, we only assume that g is continuous on C([0,b],X) and g is completely determined on [δ,b] for some small δ>0 or g is continuous on C([0,b],X) with L1([0,b],X) topology (see Corollaries 17–19).This paper has four sections. In Section2, we recall some definitions on Caputo fractional derivatives, analytic resolvent, and mild solutions to (1). In Section 3, we prove the compactness of Cauchy operator. Finally, in Section 4 we establish the existence of mild solutions of (1) when the nonlocal item satisfies different conditions. An example is also given in this section.


## 2. Preliminaries

Throughout this paper, letb>0 be fixed, and let ℕ, ℝ, and ℝ+ be the set of positive integers, real numbers, and nonnegative real numbers, respectively. We denote by X the Banach space with the norm ∥·∥, C([0,b],X) the space of all X-valued continuous functions on [0,b] with the norm ∥u∥=sup{∥u(t)∥,t∈[0,b]}, and Lp([0,b],X) the space of X-valued Bochner integrable functions on [0,b] with the norm ∥f∥Lp=(∫0b∥f(t)∥p‍dt)1/p, where 1≤p<∞. Also, we denote by ℒ(X) the space of bounded linear operators from X into X endowed with the norm of operators.Now, let us recall some basic definitions and results on fractional derivative and fractional differential equations.Definition 1 (see [1]).
The fractional order integral of the functionf∈L1([0,b],X) of order α>0 is defined by
(2)Jtαf(t)=1Γ(α)∫0t‍(t-s)α-1f(s)ds,
where Γ is the Gamma function.Definition 2 (see [1]).
The Riemann-Liouville fractional order derivative of orderα of a function f∈L1([0,b],X) given on the interval [0,b] is defined by
(3)DLαf(t)=1Γ(n-α)dndtn∫0t‍(t-s)n-α-1f(s)ds,
where α∈(n-1,n], n∈ℕ.Definition 3 (see [1]).
The Caputo fractional order derivative of orderα of a function f∈L1([0,b],X) given on the interval [0,b] is defined by
(4)Dαf(t)=1Γ(n-α)∫0t‍(t-s)n-α-1f(n)(s)ds,
where α∈(n-1,n], n∈ℕ.In the remainder of this paper, we always suppose that0<α<1 and A is a closed and densely defined linear operator on X.Definition 4.
A family{Sα(t)}t≥0⊆ℒ(X) of bounded linear operators in X is called a resolvent (or solution operator) generating by A if the following conditions are satisfied:(S1)
Sα(t) is strong continuous on ℝ+ and Sα(0)=I;(S2)
Sα(t)D(A)⊆D(A) and ASα(t)x=Sα(t)Ax for all x∈D(A) and t≥0;(S3)
the resolvent equation holds(5)Sα(t)x=x+∫0t‍gα(t-s)ASα(s)xds∀x∈D(A),t≥0.Forω, θ∈ℝ, let
(6)∑‍(ω,θ):={λ∈ℂ:|arg(λ-ω)|<θ}.Definition 5.
A resolventSα(t) is called analytic, if the function Sα(·):ℝ+→ℒ(X) admits analytic extension to a sector ∑‍(0,θ0) for some 0<θ0≤π/2. An analytic resolvent Sα(t) is said to be of analyticity type (ω0,θ0) if for each θ<θ0 and ω>ω0 there is M1=M1(ω,θ) such that ∥S(z)∥≤M1eωRez for z∈∑‍(0,θ), where Rez denotes the real part of z.Definition 6.
A resolventSα(t) is called compact for t>0 if for every t>0, Sα(t) is a compact operator.According to Proposition  1.2 in [2], we can give the following definition of mild solutions for (1).Definition 7.
A functionu∈C([0,b],X) is called a mild solution of fractional evolution equation (1) if it satisfies
(7)u(t)=Sα(t)[u0-g(u)]+∫0t‍Sα(t-s)f(s,u(s))ds,0≤t≤b
for every u0∈X.Next, we introduce the Hausdorff measure of noncompactnessβ(·) defined on each bounded subset Ω of Banach space Y by
(8)β(Ω)=inf{ε>0;Ωhasafiniteε-netinY}.Some basic properties ofβ(·) are given in the following lemma.Lemma 8 (see [34]).
LetY be a real Banach space and let B,C⊆Y be bounded; then the following properties are satisfied:(1)
B is precompact if and only if β(B)=0;(2)
β(B)=β(B¯)=β(convB), where B¯ and convB mean the closure and convex hull of B, respectively;(3)
β(B)≤β(C) when B⊆C;(4)
β(B+C)≤β(B)+β(C), where B+C={x+y;x∈B,y∈C};(5)
β(B∪C)≤max{β(B),β(C)};(6)
β(λB)=|λ|β(B) for any λ∈ℝ;(7)
if the mapQ:D(Q)⊆Y→Z is Lipschitz continuous with constant k, then β(QB)≤kβ(B) for any bounded subset B⊆D(Q), where Z is a Banach space.The mapQ:W⊆Y→Y is said to be a β-contraction if there exists a positive constant k<1 such that β(QC)≤kβ(C) for any bounded closed subset C⊆W, where Y is a Banach space.Lemma 9 ([34], Darbo-Sadovskii).
IfW⊆Y is bounded closed and convex, the continuous map Q:W→W is a β-contraction, then the map Q has at least one fixed point in W.


## 3. Compactness of Cauchy Operators

Let Cauchy operatorG:C([0,b],X)→C([0,b],X) be defined by
(9)(Gf)(t)=∫0t‍Sα(t-s)f(s)ds,t∈[0,b].
If Sα(t) is a compact C0-semigroup, it is well known that G is compact. However, it is unknown in case of compact resolvent. The main difficulty is that the resolvent does not have the property of semigroups. Thus, it seems to be more complicated to prove the compactness of Cauchy operator. Here, we will first discuss the continuity of resolvent in the uniform operator topology. Then, we can give the positive answer to the above problem.In the remainder of this paper, we always assume thatM=supt∈[0,b]∥Sα(t)∥<+∞.Lemma 10.
SupposeSα(t) is a compact analytic resolvent of analyticity type (ω0,θ0). Then the following are hold:(i)
limh→0∥Sα(t+h)-Sα(t)∥=0 for t>0;(ii)
limh→0+∥Sα(t+h)-Sα(h)Sα(t)∥=0 for t>0;(iii)
limh→0+∥Sα(t)-Sα(h)Sα(t-h)∥=0 for t>0.Proof.
(i) LetSα(t) be an analytic resolvent of analyticity type (ω0,θ0), and let t>0 be given. Then, by means of Cauchy integral formula, we have
(10)Sα′(t)=12πi∫|z-t|=tsinθ‍Sα(z)(z-t)2dz.
Thus, for any ω>ω0, 0<θ<θ0, there exists a constant M1=M1(ω,θ) such that
(11)∥Sα′(t)∥≤M1(2π)-1∫-ππ‍eω(t+tsinθcosγ)(tsinθ)-1dγ≤M1e2ωt(tsinθ)-1.
Now, let ∥x∥≤1 and |h|<t. It follows from (11) that there exists a constant M′ such that
(12)∥Sα(t+h)x-Sα(t)x∥=∥∫tt+h‍Sα′(s)xds∥≤M′∥x∥|∫tt+h‍s-1ds|≤M′|ln(t+h)-lnt|⟶0ash⟶0,
which implies that Sα(t) is continuous in the uniform operator topology for t>0, that is,
(13)limh→0∥Sα(t+h)-Sα(t)∥=0fort>0.
(ii) Let∥x∥≤1, t>0, and ε>0 be given. Since Sα(t) is compact, the set Wt:={Sα(t)x:∥x∥≤1} is also compact. Thus, there exists a finite family {Sα(t)x1,Sα(t)x2,…,Sα(t)xm}⊂Wt such that for any x with ∥x∥≤1, there exists xi (1≤i≤m) such that
(14)∥Sα(t)x-Sα(t)xi∥≤ε3(M+1).
From the strong continuity of Sα(t), t≥0, there exists 0<h1<min{t,b} such that
(15)∥Sα(t)xi-Sα(h)Sα(t)xi∥≤ε3,∀0≤h≤h1,1≤i≤m.
On the other hand, from (i), there exists 0<h2<min{t,b} such that
(16)∥Sα(t+h)x-Sα(t)x∥≤ε3,∀0≤h≤h2,∥x∥≤1.
Thus, for 0≤h≤min{h1,h2} and ∥x∥≤1, it follows from (14)–(16) that
(17)∥Sα(t+h)x-Sα(h)Sα(t)x∥≤∥Sα(t+h)x-Sα(t)x∥+∥Sα(t)x-Sα(t)xi∥+∥Sα(t)xi-Sα(h)Sα(t)xi∥+∥Sα(h)Sα(t)xi-Sα(h)Sα(t)x∥≤∥Sα(t+h)x-Sα(t)x∥+(M+1)∥Sα(t)x-Sα(t)xi∥+∥Sα(t)xi-Sα(h)Sα(t)xi∥≤ε3+ε3+ε3≤ε,
which implies that limh→0+∥Sα(t+h)-Sα(h)Sα(t)∥=0 for all t>0.
(iii) Lett>0 and 0<h<min{t,b}. Then, there exists M>0 such that
(18)∥Sα(t)-Sα(h)Sα(t-h)∥≤∥Sα(t)-Sα(t+h)∥+∥Sα(t+h)-Sα(h)Sα(t)∥+∥Sα(h)Sα(t)-Sα(h)Sα(t-h)∥≤∥Sα(t)-Sα(t+h)∥+∥Sα(t+h)-Sα(h)Sα(t)∥+M∥Sα(t)-Sα(t-h)∥.
It follows from (i) and (ii) that
(19)limh→0+∥Sα(t)-Sα(h)Sα(t-h)∥=0fort>0.
This completes the proof.Lemma 11.
SupposeSα(t) is a compact analytic resolvent of analyticity type (ω0,θ0). Then Cauchy operator G defined by (9) is a compact operator.Proof.
We will showG:C([0,b],X)→C([0,b],X) is compact by using the Arzela-Ascoli theorem. Let Dl={f∈C([0,b],X):∥f∥≤l} be any bounded subset of C([0,b],X).
First, we claim that the setGDl is equicontinuous on C([0,b],X). In fact, let 0≤t1≤t2≤b and f∈Dl; then we have
(20)I:=∥(Gf)(t2)-(Gf)(t1)∥=∥∫0t2‍Sα(t2-s)f(s)ds-∫0t1‍Sα(t1-s)f(s)ds∥≤∫0t1‍∥Sα(t2-s)-Sα(t1-s)∥∥f(s)∥ds+∫t1t2‍∥Sα(t2-s)f(s)∥ds≤l∫0t1‍∥Sα(t2-s)-Sα(t1-s)∥ds+Ml(t2-t1).
If t1=0, it is easy to see that
(21)limt2→0I=0,uniformlyforf∈Dl.
If 0<t1<b, for 0<δ<t1, we have
(22)I≤l∫0t1-δ‍∥Sα(t2-s)-Sα(t1-s)∥ds+l∫t1-δt1‍∥Sα(t2-s)-Sα(t1-s)∥ds+Ml(t2-t1)≤blsups∈[0,t1-δ]∥Sα(t2-s)-Sα(t1-s)∥+2Mlδ+Ml(t2-t1).
Note that from Lemma 10(i), we know Sα(t) is operator norm continuous uniformly for t∈[δ,b]. Combining this and the arbitrariness of δ with the above estimation on I, we can conclude that
(23)lim|t1-t2|→0I=0,uniformlyforf∈Dl.
Thus, GDl is equicontinuous on C([0,b],X).
Next, we will show that the set{(Gf)(t):f∈Dl} is precompact in X for every t∈[0,b]. It is easy to see that the set {(Gf)(0):f∈Dl} is precompact in X. Now, let 0<t≤b be given and 0<ε<t. Then
(24){Sα(ε)∫0t-ε‍Sα(t-s-ε)f(s)ds:f∈Dl}
is precompact since Sα(ε) is compact. Moreover, for arbitrary ε<δ<t, we have
(25)∥Sα(ε)∫0t-ε‍Sα(t-s-ε)f(s)ds-∫0t-ε‍Sα(t-s)f(s)ds∥≤l∫0t-ε‍∥Sα(ε)Sα(t-s-ε)-Sα(t-s)∥ds≤l∫0t-δ‍∥Sα(ε)Sα(t-s-ε)-Sα(t-s)∥ds+l∫t-δt-ε‍∥Sα(ε)Sα(t-s-ε)-Sα(t-s)∥ds≤l∫0t-δ‍∥Sα(ε)Sα(t-s-ε)-Sα(t-s)∥ds+lδ(M2+M).
From Lemma 10(iii), we know
(26)Sα(ε)Sα(t-s-ε)-Sα(t-s)⟶0,asε⟶0fors∈[0,t-δ].
Then, it follows from the Lebesgue dominated convergence theorem and the arbitrariness of δ that
(27)limε→0∥Sα(ε)∫0t-ε‍Sα(t-s-ε)f(s)ds-∫0t-ε‍Sα(t-s)f(s)ds∥=0.
On the other hand,
(28)∥Sα(ε)∫0t-ε‍Sα(t-s-ε)f(s)ds-∫0t‍Sα(t-s)f(s)ds∥≤∥Sα(ε)∫0t-ε‍Sα(t-s-ε)f(s)ds-∫0t-ε‍Sα(t-s)f(s)ds∥+∥∫0t-ε‍Sα(t-s)f(s)ds-∫0t‍Sα(t-s)f(s)ds∥≤∥Sα(ε)∫0t-ε‍Sα(t-s-ε)f(s)ds-∫0t-ε‍Sα(t-s)f(s)ds∥+lεM.
Thus,
(29)limε→0∥Sα(ε)∫0t-ε‍Sα(t-s-ε)f(s)ds-∫0t‍Sα(t-s)f(s)ds∥=0,
which implies that {(Gf)(t):f∈Dl} is precompact in X by using the total boundedness, and therefore G is compact in view of Arzela-Ascoli theorem.


## 4. Nonlocal Problems

In this section, we always assume thatu0∈X and that the operator A generates a compact analytic resolvent Sα(t) of analyticity type (ω0,θ0), and we will prove the existence of mild solutions of (1) when the nonlocal item g is assumed to be Lipschitz continuous and neither Lipschitz nor compact, respectively.Letr be a fixed positive real number and
(30)Wr:={u∈C([0,b],X):∥u∥≤r}.
Clearly, Wr is a bounded closed and convex set. We make the following assumptions.(H1)
f:[0,b]×X→X is continuous.(H2)
g:C([0,b],X)→X is Lipschitz continuous with Lipschitz constant k such that Mk<1.Under these assumptions, we can prove the first main result in this paper.Theorem 12.
Assume that conditions (H1) and (H2) are satisfied. Then the nonlocal problem (1) has at least one mild solution provided that
(31)M[∥u0∥+supu∈Wr∥g(u)∥+bsups∈[0,b],u∈Wr∥f(s,u(s))∥]≤r.Proof.
We consider the solution operatorQ:C([0,b],X)→C([0,b],X) defined by
(32)(Qu)(t)=Sα(t)[u0-g(u)]+∫0t‍Sα(t-s)f(s,u(s))ds:=(Q1u)(t)+(Q2u)(t),t∈[0,b].
It is easy to see that the fixed point of Q is the mild solution of nonlocal Cauchy problem (1). Subsequently, we will prove that Q has a fixed point by using Lemma 9 (Darbo-Sadovskii’s fixed point theorem).
Firstly, we prove that the mappingQ is continuous on C([0,b],X). For this purpose, let {un}n≥1 be a sequence in C([0,b],X) with limn→∞un=u in C([0,b],X). Then
(33)∥Qun-Qu∥≤M[∥g(un)-g(u)∥+bsups∈[0,b]∥f(s,un(s))-f(s,u(s))∥].
By the continuity of f and g, we deduce that Q is continuous on C([0,b],X).
Secondly, we claim thatQWr⊆Wr, where Wr is defined by (30). In fact, for any u∈Wr, by (31), we have
(34)∥Qu∥≤M(∥u0∥+∥g(u)∥+∫0b‍∥f(s,u(s))∥ds)≤M(∥u0∥+supu∈Wr∥g(u)∥+bsups∈[0,b],u∈Wr∥f(s,u(s))∥)≤r,
which implies that Q maps Wr into itself.
Now, according to Lemma9, it remains to prove that Q is a β-contraction in Wr. From condition (H2), we get that Q1:Wr→C([0,b],X) is Lipschitz continuous with constant Mk. In fact, for u,v∈Wr, by (H2), we have
(35)∥Q1u-Q1v∥≤M∥g(u)-g(v)∥≤Mk∥u-v∥.
Thus, it follows from Lemma 8-(30) that β(Q1Wr)≤Mkβ(Wr).
For operatorQ2:Wr→C([0,b],X), we have Q2u=Gfu, where fu(·)=f(·,u(·)) is continuous on [0,b] and G is the Cauchy operator defined by (9). Thus, in view of Lemma 11, we know that Q2 is compact on C([0,b],X), and hence β(Q2Wr)=0. Consequently,
(36)β(QWr)≤β(Q1Wr)+β(Q2Wr)≤Mkβ(Wr).
Since Mk<1, the mapping Q is a β-contraction on Wr. By Darbo-Sadovskii’s fixed point theorem, the operator Q has a fixed point in Wr, which is just the mild solution of nonlocal Cauchy problem (1).Now, we give the following technical condition on functiong.(H)
g:C([0,b],X)→X is continuous, and the set g(conv¯QWr) is precompact, where conv¯B denotes the convex closed hull of set B⊆C([0,T],X), and Q is given by (32).Remark 13.
It is easy to see that condition (H) is weaker than the compactness and convexity ofg. The same hypothesis can be seen from [24, 30], where the authors considered the existence of mild solutions for semilinear nonlocal problems of integer order when A is a linear, densely defined operator on X which generates a C0-semigroup. After the proof of our main results, we will give some special types of nonlocal item g which is neither Lipschitz nor compact, but satisfies the condition (H) in the next Corollaries.Theorem 14.
Assume that conditions (H1) and (H) are satisfied. Then the nonlocal problem (1) has at least one mild solution provided that (31) holds.Proof.
We will prove thatQ has a fixed point by using Schauder’s fixed point theorem. According to the proof of Theorem 12, we have proven that Q:Wr→Wr is continuous. Next, we will prove that there exists a set W⊆Wr such that Q:W→W is compact.
For this purpose, let0<t≤b and 0<δ<t. It is easy to see that the set
(37){Sα(δ)Sα(t-δ)[u0-g(u)]:u∈Wr}
is precompact since Sα(δ) is compact. On the other hand, by Lemma 10(iii), we obtain
(38)limδ→0∥Sα(t)[u0-g(u)]-Sα(δ)Sα(t-δ)[u0-g(u)]∥=0,∀u∈Wr,
which implies that Q1Wr(t) is precompact in X by using the total boundedness. Next, we claim Q1Wr is equicontinuous on [η,b] for any small positive number η. In fact, for u∈Wr and η≤t1≤t2≤b, we have
(39)∥(Q1u)(t2)-(Q1u)(t1)∥≤∥Sα(t2)-Sα(t1)∥(∥u0∥+∥g(u)∥).
By Lemma 10(i), Sα(t) is operator norm continuous for t>0. Thus Sα(t) is operator norm continuous uniformly for t∈[η,b], and hence
(40)lim|t2-t1|→0∥(Q1u)(t2)-(Q1u)(t1)∥=0,uniformlyforu∈Wr,
that is, Q1Wr is equicontinuous on [η,b]. Note that Q2 is compact by Lemma 11. Therefore, we have proven that QWr(t) is precompact for every t∈(0,b] and QWr is equicontinuous on [η,b] for any small positive number η.
Now, letW=conv¯QWr, we get that W is a bounded closed and convex subset of C([0,b],X) and QW⊆W. It is easy to see that QW(t) is precompact in X for every t∈(0,b] and QW is equicontinuous on [η,b] for any small positive number η. Moreover, we have that g(W)=g(conv¯QWr) is precompact due to condition (H).
Thus, we can now claim thatQ:W→W is compact. In fact, it is easy to see that Q1W(0)=u0-g(W) is precompact since g(W)=g(conv¯QWr) is precompact. It remains to prove that Q1W is equicontinuous on [0,b]. To this end, let u∈W and 0≤t1<t2≤b; then we have
(41)∥(Q1u)(t2)-(Q1u)(t1)∥≤∥Sα(t2)-Sα(t1)∥∥u0-g(u)∥.
In view of the compactness of g(W) and the strong continuity of Sα(t) on [0,b], we obtain the equicontinuous of Q1W on [0,b]. Thus, Q1:W→C([0,b],X) is compact by Arzela-Ascoli theorem, and hence Q:W→W is also compact. Now, Schauder’s fixed point theorem implies that Q has a fixed point on W, which gives rise to a mild solution of nonlocal problem (1).The following theorem is a direct consequence of Theorem14.Theorem 15.
Assume that conditions (H1) and (H) are satisfied for eachr>0. If
(42)∥g(u)∥∥u∥⟶0,∥u∥⟶∞,∥f(t,x)∥∥x∥⟶0,∥x∥⟶∞
uniformly for t∈[0,b], then the nonlocal problem (1) has at least one mild solution.Remark 16.
It is easy to see that if there exist constantsL1, L2>0 and γ1, γ2∈[0,1) such that
(43)∥g(u)∥≤L1(1+∥u∥)γ1,∥f(t,x)∥≤L2(1+∥x∥)γ2
for t∈[0,b], then conditions (42) are satisfied.Next, we will give special types of nonlocal itemg which is neither Lipschitz nor compact, but satisfies condition (H).We give the following assumptions.(H3)
g:C([0,b],X)→X is a continuous mapping which maps Wr into a bounded set, and there is a δ=δ(r)∈(0,b) such that g(u)=g(v) for any u,v∈Wr with u(s)=v(s), s∈[δ,b].(H4)
g:(C([0,b],X),∥·∥L1)→X is continuous.Corollary 17.
Assume that conditions (H1) and (H3) are satisfied. Then the nonlocal problem (1) has at least one mild solution on [0,b] provided that (31) holds.Proof.
Let(44)(QWr)δ={u∈C([0,b],X);u(t)=v(t)fort∈[δ,b],u(t)=u(δ)fort∈[0,δ),wherev∈QWr}.
From the proof of Theorem 14, we know that (QWr)δ is precompact in C([0,b],X). Moreover, by condition (H3), g(conv¯QWr)=g(conv¯(QWr)δ) is also precompact in C([0,b],X). Thus, all the hypotheses in Theorem 14 are satisfied. Therefore, there is at least one mild solution of nonlocal problem (1).Corollary 18.
Let condition (H1) be satisfied. Suppose thatg(u)=∑j=1p‍cju(tj), where cj are given positive constants, and 0<t1<t2<⋯<tp≤b. Then the nonlocal problem (1) has at least one mild solution on [0,b] provided that
(45)M[∥u0∥+∑j=1p‍cjr+bsups∈[0,b],u∈Wr∥f(s,u(s))∥]≤r.Proof.
It is easy to see that the mappingg with g(u)=∑j=1p‍cju(tj) satisfies condition (H3). And all the conditions in Corollary 17 are satisfied. So the conclusion holds.Corollary 19.
Assume that conditions (H1) and (H4) are satisfied. Then the nonlocal problem (1) has at least one mild solution on [0,b] provided that (31) holds.Proof.
According to Theorem14, it is sufficient to prove that the hypothesis (H) is satisfied. For arbitrary ϵ>0, there exists 0<δ<b such that ∫0δ‍∥u(s)∥ds<ϵ for all u∈QWr. Let
(46)(QWr)δ={u∈C([0,b],X);u(t)=v(t)fort∈[δ,b],u(t)=u(δ)fort∈[0,δ),wherev∈QWr}.
From the proof of Theorem 14, we know that (QWr)δ is precompact in C([0,b],X), which implies that (QWr)δ is precompact in L1([0,b],X). Thus, QWr is precompact in L1([0,b],X) as it has an ϵ-net (QWr)δ. By condition g:(C([0,b],X),∥·∥L1)→X is continuous and conv¯QWr⊆(L)conv¯QWr, it follows that condition (H) is satisfied, where (L)conv¯B denotes the convex and closed hull of B in L1([0,b],X). Therefore, the nonlocal problem (1) has at least one mild solution on [0,b].Finally, we give a simple example to illustrate our theory.Example 20.
Consider the following fractional partial differential heat equation inℝ:
(47)Dtαw(t,x)=∂2w(t,x)∂x2+Jt1-αF(t,w(t,x)),0<x<1,0<t<1,w(t,0)=w(t,1),wx′(t,0)=wx′(t,1),0≤t≤1,w(0,x)=w0(x)-∑j=1p‍cjw(tj,x),0<x<1,0<tj≤1,
where 0<α<1 and Dtα denotes the Caputo fractional derivative.
We introduce the abstract frame as follows. LetX={u∈C([0,1],ℝ):u(0)=u(1)}. Let A be the linear operator in X defined by D(A)={u∈X:u′,u′′∈X} and for u∈D(A), Au=u′′. Then, it is well known that A generates a compact C0 semigroup T(t) for t>0 on X. From the subordination principle [6, Theorems  3.1 and 3.3], A also generates a compact resolvent Sα(t) for t>0.
Assume thatf:[0,1]×X→X is a continuous function defined by f(t,z)(x)=F(t,z(x)), 0≤x≤1 and g:C([0,1],X)→X is also a continuous function defined by g(u)(x)=w0(x)-∑j=1p‍cju(tj)(x), 0≤t, x≤1, where u(t)(x):=w(t,x).
Under these assumptions, the fractional partial differential heat equation (47) can be reformulated as the abstract problem (1). If the inequality
(48)M[∥w0∥+∑j=1p‍cjr+sups∈[0,1],u∈Wr∥f(s,u(s))∥]≤r
holds for some constant r>0, there exists at least one mild solution for fractional equation (47) in view of Corollary 18.



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