Nonlocal Problems for Fractional Differential Equations via Resolvent Operators

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.

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 𝐷^𝛼𝑢(𝑡)=𝐴𝑢(𝑡)+𝐽_𝑡^1−𝛼𝑓((𝑡𝑢(0)𝑡,𝑢)),0<𝑡≤𝑏,=𝑢₀−𝑔(𝑢), (1) where 𝐷^𝛼 is the Caputo fractional derivative of order 𝛼 with 0<𝛼<1, 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is the infinitesimal generator of a resolvent 𝑆_𝛼(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 Section 2, 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, let 𝑏>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 𝐿^𝑝([0,𝑏],𝑋) the space of 𝑋-valued Bochner integrable functions on [0,𝑏] with the norm ‖𝑓‖_𝐿𝑝=(∫^𝑏₀‖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 function 𝑓∈𝐿¹([0,𝑏],𝑋) of order 𝛼>0 is defined by 𝐽^𝛼_𝑡𝑓(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠)d𝑠, (2) where Γ is the Gamma function.

Definition 2 (see [1]). The Riemann-Liouville fractional order derivative of order 𝛼 of a function 𝑓∈𝐿¹([0,𝑏],𝑋) given on the interval [0,𝑏] is defined by 𝐷^𝛼_𝐿𝑓(𝑡)=1Γ()𝑑𝑛−𝛼𝑑𝑡^𝑛𝑛𝑡₀(𝑡−𝑠)^{𝑛−𝛼−1}𝑓(𝑠)d𝑠, (3) where 𝛼∈(𝑛−1,𝑛], 𝑛∈ℕ.

Definition 3 (see [1]). The Caputo fractional order derivative of order 𝛼 of a function 𝑓∈𝐿¹([0,𝑏],𝑋) given on the interval [0,𝑏] is defined by 𝐷^𝛼𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡₀(𝑡−𝑠)^{𝑛−𝛼−1}𝑓^(𝑛)(𝑠)d𝑠, (4) where 𝛼∈(𝑛−1,𝑛], 𝑛∈ℕ.

In the remainder of this paper, we always suppose that 0<𝛼<1 and 𝐴 is a closed and densely defined linear operator on 𝑋.

Definition 4. A family {𝑆_𝛼(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)𝑆_𝛼(𝑡) is strong continuous on ℝ₊ and 𝑆_𝛼(0)=𝐼;(S2)𝑆_𝛼(𝑡)𝐷(𝐴)⊆𝐷(𝐴) and 𝐴𝑆_𝛼(𝑡)𝑥=𝑆_𝛼(𝑡)𝐴𝑥 for all 𝑥∈𝐷(𝐴) and 𝑡≥0;(S3) the resolvent equation holds 𝑆_𝛼(𝑡)𝑥=𝑥+^𝑡₀𝑔_𝛼(𝑡−𝑠)𝐴𝑆_𝛼(𝑠)𝑥d𝑠∀𝑥∈𝐷(𝐴),𝑡≥0. (5)

For 𝜔, 𝜃∈ℝ, let (𝜔,𝜃)∶=𝜆∈ℂ∶arg(𝜆−𝜔)<𝜃. (6)

Definition 5. A resolvent 𝑆_𝛼(𝑡) is called analytic, if the function 𝑆_𝛼(⋅)∶ℝ₊→ℒ(𝑋) admits analytic extension to a sector ∑(0,𝜃₀) for some 0<𝜃₀≤𝜋/2. An analytic resolvent 𝑆_𝛼(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)‖≤𝑀₁𝑒^𝜔Re𝑧 for 𝑧∈∑(0,𝜃), where Re𝑧 denotes the real part of 𝑧.

Definition 6. A resolvent 𝑆_𝛼(𝑡) is called compact for 𝑡>0 if for every 𝑡>0, 𝑆_𝛼(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 function 𝑢∈𝐶([0,𝑏],𝑋) is called a mild solution of fractional evolution equation (1) if it satisfies 𝑢(𝑡)=𝑆_𝛼(𝑡)𝑢₀(𝑢+−𝑔)_𝑡0𝑆_𝛼(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠))d𝑠,0≤𝑡≤𝑏 (7) for every 𝑢₀∈𝑋.

Next, we introduce the Hausdorff measure of noncompactness 𝛽(⋅) defined on each bounded subset Ω of Banach space 𝑌 by 𝛽(Ω)=inf{𝜀>0;Ωhasafinite𝜀-netin𝑌}. (8)

Some basic properties of 𝛽(⋅) are given in the following lemma.

Lemma 8 (see [34]). Let 𝑌 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 map 𝑄∶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 map 𝑄∶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). If 𝑊⊆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 operator 𝐺∶𝐶([0,𝑏],𝑋)→𝐶([0,𝑏],𝑋) be defined by 𝐺𝑓(𝑡)=^𝑡₀𝑆_𝛼(𝑡−𝑠)𝑓(𝑠)d𝑠,𝑡∈[0,𝑏]. (9) If 𝑆_𝛼(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 that 𝑀=sup_{𝑡∈[0,𝑏]}‖𝑆_𝛼(𝑡)‖<+∞.

Lemma 10. Suppose 𝑆_𝛼(𝑡) is a compact analytic resolvent of analyticity type (𝜔₀,𝜃₀). Then the following are hold:(i)lim_ℎ→0‖𝑆_𝛼(𝑡+ℎ)−𝑆_𝛼(𝑡)‖=0 for 𝑡>0;(ii)lim_ℎ→0⁺‖𝑆_𝛼(𝑡+ℎ)−𝑆_𝛼(ℎ)𝑆_𝛼(𝑡)‖=0 for 𝑡>0;(iii)lim_ℎ→0⁺‖𝑆_𝛼(𝑡)−𝑆_𝛼(ℎ)𝑆_𝛼(𝑡−ℎ)‖=0 for 𝑡>0.

Proof. (i) Let 𝑆_𝛼(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 𝑆^′_𝛼(𝑡)=12𝜋𝑖_{|𝑧−𝑡|=𝑡sin𝜃}𝑆_𝛼(𝑧)(𝑧−𝑡)²d𝑧. (10) Thus, for any 𝜔>𝜔₀, 0<𝜃<𝜃₀, there exists a constant 𝑀₁=𝑀₁(𝜔,𝜃) such that 𝑆^′_𝛼(𝑡)≤𝑀₁(2𝜋)^−1𝜋_−𝜋𝑒^{𝜔(𝑡+𝑡sin𝜃cos𝛾)}(𝑡sin𝜃)⁻¹d𝛾≤𝑀₁𝑒^2𝜔𝑡(𝑡sin𝜃)⁻¹. (11) Now, let ‖𝑥‖≤1 and |ℎ|<𝑡. It follows from (11) that there exists a constant 𝑀^′ such that 𝑆_𝛼(𝑡+ℎ)𝑥−𝑆_𝛼(𝑡)𝑥=_𝑡^𝑡+ℎ𝑆^′_𝛼(𝑠)𝑥d𝑠≤𝑀^′‖𝑥‖_𝑡^𝑡+ℎ𝑠⁻¹d𝑠≤𝑀^′|ln(𝑡+ℎ)−ln𝑡|⟶0asℎ⟶0, (12) which implies that 𝑆_𝛼(𝑡) is continuous in the uniform operator topology for 𝑡>0, that is, lim_ℎ→0𝑆_𝛼(𝑡+ℎ)−𝑆_𝛼(𝑡)=0for𝑡>0. (13)
(ii) Let ‖𝑥‖≤1, 𝑡>0, and 𝜀>0 be given. Since 𝑆_𝛼(𝑡) is compact, the set 𝑊_𝑡∶={𝑆_𝛼(𝑡)𝑥∶‖𝑥‖≤1} is also compact. Thus, there exists a finite family {𝑆_𝛼(𝑡)𝑥₁,𝑆_𝛼(𝑡)𝑥₂,…,𝑆_𝛼(𝑡)𝑥_𝑚}⊂𝑊_𝑡 such that for any 𝑥 with ‖𝑥‖≤1, there exists 𝑥_𝑖 (1≤𝑖≤𝑚) such that 𝑆_𝛼(𝑡)𝑥−𝑆_𝛼(𝑡)𝑥_𝑖≤𝜀3().𝑀+1 (14) From the strong continuity of 𝑆_𝛼(𝑡), 𝑡≥0, there exists 0<ℎ₁<min{𝑡,𝑏} such that 𝑆_𝛼(𝑡)𝑥_𝑖−𝑆_𝛼(ℎ)𝑆_𝛼(𝑡)𝑥_𝑖≤𝜀3,∀0≤ℎ≤ℎ₁,1≤𝑖≤𝑚. (15) On the other hand, from (i), there exists 0<ℎ₂<min{𝑡,𝑏} such that 𝑆_𝛼(𝑡+ℎ)𝑥−𝑆_𝛼(𝑡)𝑥≤𝜀3,∀0≤ℎ≤ℎ₂,‖𝑥‖≤1. (16) Thus, for 0≤ℎ≤min{ℎ₁,ℎ₂} and ‖𝑥‖≤1, it follows from (14)–(16) that 𝑆_𝛼(𝑡+ℎ)𝑥−𝑆_𝛼(ℎ)𝑆_𝛼(𝑡)𝑥≤𝑆_𝛼(𝑡+ℎ)𝑥−𝑆_𝛼(𝑡)𝑥+𝑆_𝛼(𝑡)𝑥−𝑆_𝛼(𝑡)𝑥_𝑖+𝑆_𝛼(𝑡)𝑥_𝑖−𝑆_𝛼(ℎ)𝑆_𝛼(𝑡)𝑥_𝑖+𝑆_𝛼(ℎ)𝑆_𝛼(𝑡)𝑥_𝑖−𝑆_𝛼(ℎ)𝑆_𝛼(𝑡)𝑥≤𝑆_𝛼(𝑡+ℎ)𝑥−𝑆_𝛼(𝑡)𝑥+(𝑀+1)𝑆_𝛼(𝑡)𝑥−𝑆_𝛼(𝑡)𝑥_𝑖+𝑆_𝛼(𝑡)𝑥_𝑖−𝑆_𝛼(ℎ)𝑆_𝛼(𝑡)𝑥_𝑖≤𝜀3+𝜀3+𝜀3≤𝜀, (17) which implies that lim_ℎ→0⁺‖𝑆_𝛼(𝑡+ℎ)−𝑆_𝛼(ℎ)𝑆_𝛼(𝑡)‖=0 for all 𝑡>0.
(iii) Let 𝑡>0 and 0<ℎ<min{𝑡,𝑏}. Then, there exists 𝑀>0 such that 𝑆_𝛼(𝑡)−𝑆_𝛼(ℎ)𝑆_𝛼(≤𝑆𝑡−ℎ)_𝛼(𝑡)−𝑆_𝛼(𝑡+ℎ)+𝑆_𝛼(𝑡+ℎ)−𝑆_𝛼(ℎ)𝑆_𝛼(𝑡+𝑆_𝛼)(ℎ)𝑆_𝛼(𝑡)−𝑆_𝛼(ℎ)𝑆_𝛼(≤𝑆𝑡−ℎ)_𝛼(𝑡)−𝑆_𝛼(𝑡+ℎ)+𝑆_𝛼(𝑡+ℎ)−𝑆_𝛼(ℎ)𝑆_𝛼(𝑡𝑆)+𝑀_𝛼(𝑡)−𝑆_𝛼(𝑡−ℎ). (18) It follows from (i) and (ii) that lim_ℎ→0+𝑆_𝛼(𝑡)−𝑆_𝛼(ℎ)𝑆_𝛼(𝑡−ℎ)=0for𝑡>0. (19) This completes the proof.

Lemma 11. Suppose 𝑆_𝛼(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 show 𝐺∶𝐶([0,𝑏],𝑋)→𝐶([0,𝑏],𝑋) is compact by using the Arzela-Ascoli theorem. Let 𝐷_𝑙={𝑓∈𝐶([0,𝑏],𝑋)∶‖𝑓‖≤𝑙} be any bounded subset of 𝐶([0,𝑏],𝑋).
First, we claim that the set 𝐺𝐷_𝑙 is equicontinuous on 𝐶([0,𝑏],𝑋). In fact, let 0≤𝑡₁≤𝑡₂≤𝑏 and 𝑓∈𝐷_𝑙; then we have 𝐼∶=𝐺𝑓𝑡₂−𝐺𝑓𝑡₁=^𝑡2₀𝑆_𝛼𝑡₂−𝑠𝑓(𝑠)d𝑠−^𝑡1₀𝑆_𝛼𝑡₁𝑓(𝑠)≤−𝑠d𝑠_𝑡10𝑆_𝛼𝑡₂−𝑠−𝑆_𝛼𝑡₁−𝑠𝑓(𝑠)d𝑠+^𝑡2_𝑡1𝑆_𝛼𝑡₂−𝑠𝑓(𝑠)d𝑠≤𝑙^𝑡1₀𝑆_𝛼𝑡₂−𝑠−𝑆_𝛼𝑡₁−𝑠d𝑠+𝑀𝑙𝑡₂−𝑡₁. (20) If 𝑡₁=0, it is easy to see that lim_𝑡2→0𝐼=0,uniformlyfor𝑓∈𝐷_𝑙. (21) If 0<𝑡₁<𝑏, for 0<𝛿<𝑡₁, we have 𝐼≤𝑙^𝑡1₀^−𝛿𝑆_𝛼𝑡₂−𝑠−𝑆_𝛼𝑡₁−𝑠d𝑠+𝑙^𝑡1_𝑡1−𝛿𝑆_𝛼𝑡₂−𝑠−𝑆_𝛼𝑡₁−𝑠d𝑠+𝑀𝑙𝑡₂−𝑡₁≤𝑏𝑙sup_{𝑠∈[0,𝑡1−𝛿]}𝑆_𝛼𝑡₂−𝑠−𝑆_𝛼𝑡₁𝑡−𝑠+2𝑀𝑙𝛿+𝑀𝑙₂−𝑡₁. (22) Note that from Lemma 10(i), we know 𝑆_𝛼(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 lim_{|𝑡1−𝑡2|→0}𝐼=0,uniformlyfor𝑓∈𝐷_𝑙. (23) Thus, 𝐺𝐷_𝑙 is equicontinuous on 𝐶([0,𝑏],𝑋).
Next, we will show that the set {(𝐺𝑓)(𝑡)∶𝑓∈𝐷_𝑙} is precompact in 𝑋 for every 𝑡∈[0,𝑏]. It is easy to see that the set {(𝐺𝑓)(0)∶𝑓∈𝐷_𝑙} is precompact in 𝑋. Now, let 0<𝑡≤𝑏 be given and 0<𝜀<𝑡. Then 𝑆_𝛼(𝜀)₀^𝑡−𝜀𝑆_𝛼(𝑡−𝑠−𝜀)𝑓(𝑠)d𝑠∶𝑓∈𝐷_𝑙 (24) is precompact since 𝑆_𝛼(𝜀) is compact. Moreover, for arbitrary 𝜀<𝛿<𝑡, we have 𝑆_𝛼(𝜀)₀^𝑡−𝜀𝑆_𝛼(𝑡−𝑠−𝜀)𝑓(𝑠)d𝑠−₀^𝑡−𝜀𝑆_𝛼(𝑡−𝑠)𝑓(𝑠)d𝑠≤𝑙₀^𝑡−𝜀𝑆_𝛼(𝜀)𝑆_𝛼(𝑡−𝑠−𝜀)−𝑆_𝛼(𝑡−𝑠)d𝑠≤𝑙₀^𝑡−𝛿𝑆_𝛼(𝜀)𝑆_𝛼(𝑡−𝑠−𝜀)−𝑆_𝛼(𝑡−𝑠)d𝑠+𝑙^𝑡−𝜀_𝑡−𝛿𝑆_𝛼(𝜀)𝑆_𝛼(𝑡−𝑠−𝜀)−𝑆_𝛼(𝑡−𝑠)d𝑠≤𝑙₀^𝑡−𝛿𝑆_𝛼(𝜀)𝑆_𝛼(𝑡−𝑠−𝜀)−𝑆_𝛼(𝑡−𝑠)d𝑠+𝑙𝛿𝑀²+𝑀. (25) From Lemma 10(iii), we know 𝑆_𝛼(𝜀)𝑆_𝛼(𝑡−𝑠−𝜀)−𝑆_𝛼()[].𝑡−𝑠⟶0,as𝜀⟶0for𝑠∈0,𝑡−𝛿 (26) Then, it follows from the Lebesgue dominated convergence theorem and the arbitrariness of 𝛿 that lim_𝜀→0𝑆_𝛼(𝜀)₀^𝑡−𝜀𝑆_𝛼()𝑓(𝑠)−𝑡−𝑠−𝜀d𝑠_0𝑡−𝜀𝑆_𝛼(𝑡−𝑠)𝑓(𝑠)d𝑠=0. (27) On the other hand, 𝑆_𝛼(𝜀)₀^𝑡−𝜀𝑆_𝛼(𝑡−𝑠−𝜀)𝑓(𝑠)d𝑠−^𝑡₀𝑆_𝛼()𝑓(𝑠)≤𝑆𝑡−𝑠d𝑠_𝛼(𝜀)₀^𝑡−𝜀𝑆_𝛼()𝑓(𝑠)−𝑡−𝑠−𝜀d𝑠_0𝑡−𝜀𝑆_𝛼()𝑓(𝑠)+𝑡−𝑠d𝑠_0𝑡−𝜀𝑆_𝛼(𝑡−𝑠)𝑓(𝑠)d𝑠−^𝑡₀𝑆_𝛼()𝑓(𝑠)≤𝑆𝑡−𝑠d𝑠_𝛼(𝜀)₀^𝑡−𝜀𝑆_𝛼()𝑓(𝑠)−𝑡−𝑠−𝜀d𝑠_0𝑡−𝜀𝑆_𝛼(𝑡−𝑠)𝑓(𝑠)d𝑠+𝑙𝜀𝑀. (28) Thus, lim_𝜀→0𝑆_𝛼(𝜀)₀^𝑡−𝜀𝑆_𝛼()𝑓(𝑠)−𝑡−𝑠−𝜀d𝑠_𝑡0𝑆_𝛼(𝑡−𝑠)𝑓(𝑠)d𝑠=0, (29) which implies that {(𝐺𝑓)(𝑡)∶𝑓∈𝐷_𝑙} 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 that 𝑢₀∈𝑋 and that the operator 𝐴 generates a compact analytic resolvent 𝑆_𝛼(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.

Let 𝑟 be a fixed positive real number and 𝑊_𝑟∶={𝑢∈𝐶([0,𝑏],𝑋)∶‖𝑢‖≤𝑟}. (30) Clearly, 𝑊_𝑟 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 𝑀𝑢₀+sup_{𝑢∈𝑊𝑟}𝑔(𝑢)+𝑏sup_{[]𝑠∈0,𝑏,𝑢∈𝑊𝑟}𝑓(𝑠,𝑢(𝑠))≤𝑟. (31)

Proof. We consider the solution operator 𝑄∶𝐶([0,𝑏],𝑋)→𝐶([0,𝑏],𝑋) defined by (𝑄𝑢)(𝑡)=𝑆_𝛼(𝑡)𝑢₀−𝑔(𝑢)+^𝑡₀𝑆_𝛼()𝑓((𝑠𝑄𝑡−𝑠𝑠,𝑢))d𝑠∶=₁𝑢(𝑡)+𝑄₂𝑢(𝑡),𝑡∈[0,𝑏]. (32) 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 mapping 𝑄 is continuous on 𝐶([0,𝑏],𝑋). For this purpose, let {𝑢_𝑛}_𝑛≥1 be a sequence in 𝐶([0,𝑏],𝑋) with lim_𝑛→∞𝑢_𝑛=𝑢 in 𝐶([0,𝑏],𝑋). Then 𝑄𝑢_𝑛𝑔𝑢−𝑄𝑢≤𝑀_𝑛−𝑔(𝑢)+𝑏sup_{[]𝑠∈0,𝑏}𝑓𝑠,𝑢_𝑛(𝑠)−𝑓(𝑠,𝑢(𝑠)). (33) By the continuity of 𝑓 and 𝑔, we deduce that 𝑄 is continuous on 𝐶([0,𝑏],𝑋).
Secondly, we claim that 𝑄𝑊_𝑟⊆𝑊_𝑟, where 𝑊_𝑟 is defined by (30). In fact, for any 𝑢∈𝑊_𝑟, by (31), we have ‖𝑄𝑢‖≤𝑀𝑢₀+𝑔(𝑢)+^𝑏₀𝑓((𝑠𝑢𝑠,𝑢))d𝑠≤𝑀₀+sup_{𝑢∈𝑊𝑟}𝑔(𝑢)+𝑏sup_{[]𝑠∈0,𝑏,𝑢∈𝑊𝑟}𝑓(𝑠,𝑢(𝑠))≤𝑟, (34) which implies that 𝑄 maps 𝑊_𝑟 into itself.
Now, according to Lemma 9, it remains to prove that 𝑄 is a 𝛽-contraction in 𝑊_𝑟. From condition (H2), we get that 𝑄₁∶𝑊_𝑟→𝐶([0,𝑏],𝑋) is Lipschitz continuous with constant 𝑀𝑘. In fact, for 𝑢,𝑣∈𝑊_𝑟, by (H2), we have 𝑄₁𝑢−𝑄₁𝑣≤𝑀𝑔(𝑢)−𝑔(𝑣)≤𝑀𝑘‖𝑢−𝑣‖. (35) Thus, it follows from Lemma 8-(30) that 𝛽(𝑄₁𝑊_𝑟)≤𝑀𝑘𝛽(𝑊_𝑟).
For operator 𝑄₂∶𝑊_𝑟→𝐶([0,𝑏],𝑋), we have 𝑄₂𝑢=𝐺𝑓_𝑢, where 𝑓_𝑢(⋅)=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(𝑄₂𝑊_𝑟)=0. Consequently, 𝛽𝑄𝑊_𝑟≤𝛽𝑄₁𝑊_𝑟+𝛽𝑄₂𝑊_𝑟≤𝑀𝑘𝛽𝑊_𝑟. (36) Since 𝑀𝑘<1, the mapping 𝑄 is a 𝛽-contraction on 𝑊_𝑟. By Darbo-Sadovskii’s fixed point theorem, the operator 𝑄 has a fixed point in 𝑊_𝑟, which is just the mild solution of nonlocal Cauchy problem (1).

Now, we give the following technical condition on function 𝑔.(H)𝑔∶𝐶([0,𝑏],𝑋)→𝑋 is continuous, and the set 𝑔(conv𝑄𝑊_𝑟) is precompact, where conv𝐵 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 of 𝑔. 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 that 𝑄 has a fixed point by using Schauder’s fixed point theorem. According to the proof of Theorem 12, we have proven that 𝑄∶𝑊_𝑟→𝑊_𝑟 is continuous. Next, we will prove that there exists a set 𝑊⊆𝑊_𝑟 such that 𝑄∶𝑊→𝑊 is compact.
For this purpose, let 0<𝑡≤𝑏 and 0<𝛿<𝑡. It is easy to see that the set 𝑆_𝛼(𝛿)𝑆_𝛼(𝑡−𝛿)𝑢₀−𝑔(𝑢)∶𝑢∈𝑊_𝑟 (37) is precompact since 𝑆_𝛼(𝛿) is compact. On the other hand, by Lemma 10(iii), we obtain lim_𝛿→0𝑆_𝛼(𝑡)𝑢₀−𝑔(𝑢)−𝑆_𝛼(𝛿)𝑆_𝛼(𝑡−𝛿)𝑢₀−𝑔(𝑢)=0,∀𝑢∈𝑊_𝑟, (38) which implies that 𝑄₁𝑊_𝑟(𝑡) is precompact in 𝑋 by using the total boundedness. Next, we claim 𝑄₁𝑊_𝑟 is equicontinuous on [𝜂,𝑏] for any small positive number 𝜂. In fact, for 𝑢∈𝑊_𝑟 and 𝜂≤𝑡₁≤𝑡₂≤𝑏, we have 𝑄₁𝑢𝑡₂−𝑄₁𝑢𝑡₁≤𝑆_𝛼𝑡₂−𝑆_𝛼𝑡₁𝑢₀+𝑔(𝑢). (39) By Lemma 10(i), 𝑆_𝛼(𝑡) is operator norm continuous for 𝑡>0. Thus 𝑆_𝛼(𝑡) is operator norm continuous uniformly for 𝑡∈[𝜂,𝑏], and hence lim_{|𝑡2−𝑡1|→0}𝑄₁𝑢𝑡₂−𝑄₁𝑢𝑡₁=0,uniformlyfor𝑢∈𝑊_𝑟, (40) that is, 𝑄₁𝑊_𝑟 is equicontinuous on [𝜂,𝑏]. Note that 𝑄₂ is compact by Lemma 11. Therefore, we have proven that 𝑄𝑊_𝑟(𝑡) is precompact for every 𝑡∈(0,𝑏] and 𝑄𝑊_𝑟 is equicontinuous on [𝜂,𝑏] for any small positive number 𝜂.
Now, let 𝑊=conv𝑄𝑊_𝑟, 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(conv𝑄𝑊_𝑟) is precompact due to condition (H).
Thus, we can now claim that 𝑄∶𝑊→𝑊 is compact. In fact, it is easy to see that 𝑄₁𝑊(0)=𝑢₀−𝑔(𝑊) is precompact since 𝑔(𝑊)=𝑔(conv𝑄𝑊_𝑟) 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 𝑄₁𝑢𝑡₂−𝑄₁𝑢𝑡₁≤𝑆_𝛼𝑡₂−𝑆_𝛼𝑡₁𝑢₀−𝑔(𝑢). (41) In view of the compactness of 𝑔(𝑊) and the strong continuity of 𝑆_𝛼(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 Theorem 14.

Theorem 15. Assume that conditions (H1) and (H) are satisfied for each 𝑟>0. If 𝑔(𝑢)‖𝑢‖‖𝑢‖𝑓(⟶0,⟶∞,𝑡,𝑥)‖𝑥‖⟶0,‖𝑥‖⟶∞ (42) 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 constants 𝐿₁, 𝐿₂>0 and 𝛾₁, 𝛾₂∈[0,1) such that 𝑔(𝑢)≤𝐿₁(1+‖𝑢‖)^𝛾1,𝑓(𝑡,𝑥)≤𝐿₂(1+‖𝑥‖)^𝛾2 (43) for 𝑡∈[0,𝑏], then conditions (42) are satisfied.

Next, we will give special types of nonlocal item 𝑔 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 𝑊_𝑟 into a bounded set, and there is a 𝛿=𝛿(𝑟)∈(0,𝑏) such that 𝑔(𝑢)=𝑔(𝑣) for any 𝑢,𝑣∈𝑊_𝑟 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 𝑄𝑊_𝑟𝛿={])(𝑡)(𝑡)[],𝑢(𝑡)(𝛿)[)𝑢∈𝐶([0,𝑏,𝑋;𝑢=𝑣for𝑡∈𝛿,𝑏=𝑢for𝑡∈0,𝛿,where𝑣∈𝑄𝑊_𝑟. (44) From the proof of Theorem 14, we know that (𝑄𝑊_𝑟)_𝛿 is precompact in 𝐶([0,𝑏],𝑋). Moreover, by condition (H3), 𝑔(conv𝑄𝑊_𝑟)=𝑔(conv(𝑄𝑊_𝑟)_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 that 𝑔(𝑢)=∑^𝑝_𝑗=1𝑐_𝑗𝑢(𝑡_𝑗), where 𝑐_𝑗 are given positive constants, and 0<𝑡₁<𝑡₂<⋯<𝑡_𝑝≤𝑏. Then the nonlocal problem (1) has at least one mild solution on [0,𝑏] provided that 𝑀𝑢₀+^𝑝_𝑗=1𝑐_𝑗𝑟+𝑏sup_{[]𝑠∈0,𝑏,𝑢∈𝑊𝑟}𝑓(𝑠,𝑢(𝑠))≤𝑟. (45)

Proof. It is easy to see that the mapping 𝑔 with 𝑔(𝑢)=∑^𝑝_𝑗=1𝑐_𝑗𝑢(𝑡_𝑗) 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 Theorem 14, it is sufficient to prove that the hypothesis (H) is satisfied. For arbitrary 𝜖>0, there exists 0<𝛿<𝑏 such that ∫^𝛿₀‖𝑢(𝑠)‖d𝑠<𝜖 for all 𝑢∈𝑄𝑊_𝑟. Let 𝑄𝑊_𝑟𝛿={])(𝑡)(𝑡)[],𝑢(𝑡)(𝛿)[)𝑢∈𝐶([0,𝑏,𝑋;𝑢=𝑣for𝑡∈𝛿,𝑏=𝑢for𝑡∈0,𝛿,where𝑣∈𝑄𝑊_𝑟. (46) From the proof of Theorem 14, we know that (𝑄𝑊_𝑟)_𝛿 is precompact in 𝐶([0,𝑏],𝑋), which implies that (𝑄𝑊_𝑟)_𝛿 is precompact in 𝐿¹([0,𝑏],𝑋). Thus, 𝑄𝑊_𝑟 is precompact in 𝐿¹([0,𝑏],𝑋) as it has an 𝜖-net (𝑄𝑊_𝑟)_𝛿. By condition 𝑔∶(𝐶([0,𝑏],𝑋),‖⋅‖_𝐿¹)→𝑋 is continuous and conv𝑄𝑊_𝑟⊆(𝐿)conv𝑄𝑊_𝑟, it follows that condition (H) is satisfied, where (𝐿)conv𝐵 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 in ℝ: 𝐷^𝛼_𝑡𝑤()=𝜕𝑡,𝑥₂𝑤(𝑡,𝑥)𝜕𝑥²+𝐽_𝑡^1−𝛼𝐹((𝑤()()𝑡,𝑤𝑡,𝑥)),0<𝑥<1,0<𝑡<1,𝑡,0=𝑤𝑡,1,𝑤^′_𝑥(𝑡,0)=𝑤^′_𝑥()𝑤()𝑡,1,0≤𝑡≤1,0,𝑥=𝑤₀(𝑥)−^𝑝_𝑗=1𝑐_𝑗𝑤𝑡_𝑗,𝑥,0<𝑥<1,0<𝑡_𝑗≤1, (47) where 0<𝛼<1 and 𝐷^𝛼_𝑡 denotes the Caputo fractional derivative.
We introduce the abstract frame as follows. Let 𝑋={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 𝑆_𝛼(𝑡) for 𝑡>0.
Assume that 𝑓∶[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)−∑^𝑝_𝑗=1𝑐_𝑗𝑢(𝑡_𝑗)(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 𝑀𝑤₀+^𝑝_𝑗=1𝑐_𝑗𝑟+sup_{[]𝑠∈0,1,𝑢∈𝑊𝑟}𝑓(𝑠,𝑢(𝑠))≤𝑟 (48) holds for some constant 𝑟>0, there exists at least one mild solution for fractional equation (47) in view of Corollary 18.

Acknowledgments

This work was completed when the first author visited the Pennsylvania State University. The author is grateful to Professor Alberto Bressan and the Department of Mathematics for their hospitality and providing good working conditions. The work was supported by the NSF of China (11001034) and Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-aged Teachers and Presidents.