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

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.

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: 𝐷^𝑞𝑢(𝑡)(𝑡)((𝑡)(𝑡[],𝑢(0)+𝐴𝑢=𝑓𝑡,𝑢,𝐺𝑢)),𝑡∈𝐽=0,𝑇=𝑥₀, (1.1) where 𝐷^𝑞 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 𝐺𝑢(𝑡)=^𝑡₀𝐾(𝑡,𝑠)𝑢(𝑠)𝑑𝑠 (1.2) 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)∈Δ}𝐾(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 Section 2, 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.

Let 𝑋 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 𝐴^−𝛼1∶=Γ(𝛼)^∞₀𝑡^(𝛼−1)𝑆(𝑡)𝑑𝑡. (2.1)

It follows that each 𝐴^−𝛼 is an injective continuous endomorphism of 𝑋. Hence we can define 𝐴^𝛼 by 𝐴^𝛼∶=(𝐴^−𝛼)⁻¹, which is a closed bijective linear operator in 𝑋. It can be shown that each 𝐴^𝛼 has dense domain and that 𝐷(𝐴^𝛽)⊂𝐷(𝐴^𝛼) for 0≤𝛼≤𝛽. Moreover, 𝐴^𝛼+𝛽𝑥=𝐴^𝛼𝐴^𝛽𝑥=𝐴^𝛽𝐴^𝛼𝑥 for every 𝛼,𝛽∈ℝ and 𝑥∈𝐷(𝐴^𝜇) 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 by 𝑋_𝛼 the Banach space of 𝐷(𝐴^𝛼) equipped with norm ‖𝑥‖_𝛼=‖𝐴^𝛼𝑥‖ for 𝑥∈𝐷(𝐴^𝛼), which is equivalent to the graph norm of 𝐴^𝛼. Then we have 𝑋_𝛽↪𝑋_𝛼 for 0≤𝛼≤𝛽≤1 (with 𝑋₀=𝑋), and the embedding is continuous. Moreover, 𝐴^𝛼 has the following basic properties.

Lemma 2.1 (see [24]). 𝐴^𝛼 has the following properties.(i)𝑆(𝑡)∶𝑋→𝑋_𝛼 for each 𝑡>0 and 𝛼≥0. (ii)𝐴^𝛼𝑆(𝑡)𝑥=𝑆(𝑡)𝐴^𝛼𝑥 for each 𝑥∈𝐷(𝐴^𝛼) and 𝑡≥0. (iii)For every 𝑡>0,𝐴^𝛼𝑆(𝑡) is bounded in 𝑋 and there exists 𝑀_𝛼>0 such that 𝐴^𝛼𝑆(𝑡)≤𝑀_𝛼𝑡^−𝛼. (2.2)(iv)𝐴^−𝛼 is a bounded linear operator for 0≤𝛼≤1 in 𝑋.

In the following, we denote by 𝐶(𝐽,𝑋_𝛼) the Banach space of all continuous functions from 𝐽 into 𝑋_𝛼 with supnorm given by ‖𝑢‖_𝐶=sup_𝑡∈𝐽‖𝑢(𝑡)‖_𝛼 for 𝑢∈𝐶(𝐽,𝑋_𝛼). From Lemma 2.1(iv), since 𝐴^−𝛼 is a bounded linear operator for 0≤𝛼≤1, there exists a constant 𝐶_𝛼 such that ‖𝐴^−𝛼‖≤𝐶_𝛼 for 0≤𝛼≤1.

For any 𝑡≥0, denote by 𝑆_𝛼(𝑡) the restriction of 𝑆(𝑡) to 𝑋_𝛼. From Lemma 2.1(i) and (ii), for any 𝑥∈𝑋_𝛼, we have ‖𝑆(𝑡)𝑥‖_𝛼=𝐴^𝛼⋅𝑆(𝑡)𝑥=𝑆(𝑡)⋅𝐴^𝛼𝑥≤‖𝑆(𝑡)‖⋅𝐴^𝛼𝑥=‖𝑆(𝑡)‖⋅‖𝑥‖_𝛼,𝑆(𝑡)𝑥−𝑥_𝛼=𝐴^𝛼⋅𝑆(𝑡)𝑥−𝐴^𝛼𝑥=𝑆(𝑡)⋅𝐴^𝛼𝑥−𝐴^𝛼𝑥→0 (2.3) as 𝑡→0. Therefore, 𝑆(𝑡)(𝑡≥0) is a strongly continuous semigroup in 𝑋_𝛼, and ‖𝑆_𝛼(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 𝑋_𝛼, 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 order 𝜎>0 with the lower limits zero for a function 𝑓 is defined by 𝐼^𝜎𝑓(𝑡)=1Γ(𝜎)^𝑡₀(𝑡−𝑠)^𝜎−1𝑓(𝑠)𝑑𝑠,𝑡>0, (2.4) where Γ is the gamma function.
The Riemann-Liouville fractional derivative of order 𝑛−1<𝜎<𝑛 with the lower limits zero for a function 𝑓 can be written as ^𝐿𝐷^𝜎𝑓(𝑡)=1Γ()𝑑𝑛−𝜎𝑑𝑡^𝑛𝑛𝑡₀(𝑡−𝑠)^{𝑛−𝜎−1}𝑓(𝑠)𝑑𝑠,𝑡>0,𝑛∈ℕ. (2.5) Also the Caputo fractional derivative of order 𝑛−1<𝜎<𝑛 with the lower limits zero for a function 𝑓∈𝐶^𝑛[0,∞) can be written as 𝐷^𝜎𝑓(𝑡)=1Γ(𝑛−𝜎)^𝑡₀(𝑡−𝑠)^{𝑛−𝜎−1}𝑓^(𝑛)(𝑠)𝑑𝑠,𝑡>0,𝑛∈ℕ. (2.6)

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 function ℎ∶𝐽→𝑋 is Bochner integrable if ‖ℎ‖ is Lebesgue integrable.

For 𝑥∈𝑋, we define two families {𝑈(𝑡)}_𝑡≥0 and {𝑉(𝑡)}_𝑡≥0 of operators by 𝑈(𝑡)𝑥=^∞₀𝜂_𝑞(𝜃)𝑆𝑡^𝑞𝜃𝑥𝑑𝜃,𝑉(𝑡)𝑥=𝑞^∞₀𝜃𝜂_𝑞(𝜃)𝑆𝑡^𝑞𝜃𝑥𝑑𝜃,0<𝑞<1, (2.7) Where 𝜂_𝑞(𝜃)=1𝑞𝜃^{−1−(1/𝑞)}𝜌_𝑞𝜃^−(1/𝑞),𝜌_𝑞(𝜃)=1𝜋^∞_𝑛=1(−1)^𝑛−1𝜃^{−𝑞𝑛−1}Γ𝑛𝑞+1,(),𝑛!sin𝑛𝜋𝑞𝜃∈0,∞ (2.8)𝜂_𝑞 is a probability density function defined on (0,∞), which has properties 𝜂_𝑞(𝜃)≥0 for all 𝜃∈(0,∞) and ^∞₀𝜂_𝑞(𝜃)𝑑𝜃=1,^∞₀𝜃𝜂_𝑞(𝜃)1𝑑𝜃=Γ.𝑞+1 (2.9) The following lemma follows from the results in [7, 11–13].

Lemma 2.6. The operators U and V have the following properties.(i)For fixed 𝑡≥0 and any 𝑥∈𝑋_𝛼, we have ‖𝑈(𝑡)𝑥‖_𝛼≤𝑀‖𝑥‖_𝛼,‖𝑉(𝑡)𝑥‖_𝛼≤𝑞𝑀Γ‖𝑥‖1+𝑞^𝛼=𝑀Γ𝑞‖𝑥‖_𝛼. (2.10)(ii)The operators 𝑈(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 every 𝑡>0, the restriction of 𝑈(𝑡) to 𝑋_𝛼 and the restriction of 𝑉(𝑡) to 𝑋_𝛼 are norm-continuous.(vi)For every 𝑡>0, the restriction of 𝑈(𝑡) to 𝑋_𝛼 and the restriction of 𝑉(𝑡) to 𝑋_𝛼 are compact operators in 𝑋_𝛼.

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,𝑋_𝛼) satisfying 𝑢(𝑡)=𝑈(𝑡)𝑥₀+^𝑡₀(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))𝑑𝑠 (2.11) 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 on 𝑓 is satisfied.(H₁) There exists a constant 𝛽∈[𝛼,1] such that 𝑓∶𝐽×𝑋_𝛼×𝑋_𝛼→𝑋_𝛽 satisfies:(i) for each (𝑥,𝑦)∈𝑋_𝛼×𝑋_𝛼, the function 𝑓(⋅,𝑥,𝑦)∶𝐽→𝑋_𝛽 is measurable;(ii) for each 𝑡∈𝐽, the function 𝑓(𝑡,⋅,⋅)∶𝑋_𝛼×𝑋_𝛼→𝑋_𝛽 is continuous;(iii) for any 𝑟>0, there exists a function 𝑔_𝑟∈𝐿^∞(𝐽,ℝ⁺) such that sup_{‖𝑥‖𝛼𝑦≤𝑟,𝛼≤𝐾∗𝑇𝑟}𝑓𝑡,𝑥,𝑦_𝛽≤𝑔_𝑟(𝑡),𝑡∈𝐽, (3.1)and there is a constant 𝛾>0 such that liminf_𝑟→+∞1𝑟^𝑡₀𝑔_𝑟(𝑠)(𝑡−𝑠)^1−𝑞𝑑𝑠≤𝛾<+∞. (3.2) If 𝑥₀∈𝑋_𝛼 and 𝑀𝐶_𝛽−𝛼𝛾<Γ(𝑞), then (1.1) has at least one mild solution.

Proof. Define an operator 𝑄 by (𝑄𝑢)(𝑡)=𝑈(𝑡)𝑥₀+^𝑡₀(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))𝑑𝑠,𝑡∈𝐽. (3.3) It is not difficult to verify that 𝑄∶𝐶(𝐽,𝑋_𝛼)→𝐶(𝐽,𝑋_𝛼). We will use Schauder fixed point theorem to prove that 𝑄 has fixed points in 𝐶(𝐽,𝑋_𝛼).
For any 𝑟>0, let 𝐵_𝑟∶={𝑢∈𝐶(𝐽,𝑋_𝛼)∶‖𝑢(𝑡)‖_𝛼≤𝑟,𝑡∈𝐽}. We first show that there is a positive number 𝑟 such that 𝑄(𝐵_𝑟)⊂𝐵_𝑟. If this were not the case, then for each 𝑟>0, there would exist 𝑢_𝑟∈𝐵_𝑟 and 𝑡_𝑟∈𝐽 such that ‖(𝑄𝑢_𝑟)(𝑡_𝑟)‖_𝛼>𝑟. Thus, from Lemma 2.6(i) and (H₁)(iii), we see that 𝑟<𝑄𝑢_𝑟𝑡_𝑟𝛼≤𝑈𝑡_𝑟𝑥_0𝛼+^𝑡𝑟₀𝑡_𝑟−𝑠^𝑞−1𝑉𝑡_𝑟−𝑠𝑓𝑠,𝑢_𝑟(𝑠),𝐺𝑢_𝑟(𝑠)_𝛼𝑥𝑑𝑠≤𝑀_0𝛼+^𝑡𝑟₀𝑡_𝑟−𝑠^𝑞−1𝐴^𝛼−𝛽𝑉𝑡_𝑟−𝑠⋅𝐴^𝛽𝑓𝑠,𝑢_𝑟(𝑠),𝐺𝑢_𝑟(𝑠)𝑥𝑑𝑠≤𝑀_0𝛼+𝑀𝐶_𝛽−𝛼Γ𝑞^𝑡𝑟₀𝑡_𝑟−𝑠^𝑞−1𝑔_𝑟(𝑠)𝑑𝑠. (3.4) Dividing on both sides by 𝑟 and taking the lower limit as 𝑟→+∞, we have 𝑀𝐶_𝛽−𝛼𝛾≥Γ𝑞, (3.5) which is a contradiction. Hence 𝑄(𝐵_𝑟)⊂𝐵_𝑟 for some 𝑟>0.
To complete the proof, we separate the rest of proof into the following three steps.
Step 1. 𝑄∶𝐵_𝑟→𝐵_𝑟 is continuous.
Let {𝑢_𝑛}⊂𝐵_𝑟 with 𝑢_𝑛→𝑢∈𝐵_𝑟 as 𝑛→∞. From the assumption (H₁)(ii), for each 𝑠∈𝐽, we have 𝑓𝑠,𝑢_𝑛(𝑠),𝐺𝑢_𝑛(𝑠)→𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠)) (3.6) as 𝑛→∞. Since ‖𝑓(𝑠,𝑢_𝑛(𝑠),𝐺𝑢_𝑛(𝑠))−𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))‖_𝛽≤2𝑔_𝑟(𝑠), by the Lebesgue dominated convergence theorem, for each 𝑡∈𝐽, we have 𝑄𝑢_𝑛(𝑡)−(𝑄𝑢)(𝑡)_𝛼≤^𝑡₀(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝑓𝑠,𝑢_𝑛(𝑠),𝐺𝑢_𝑛(𝑠)−𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))_𝛼≤𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝐴^𝛼−𝛽𝑉(𝑡−𝑠)⋅𝐴^𝛽𝑓𝑠,𝑢_𝑛(𝑠),𝐺𝑢_𝑛(𝑠)((𝑠)(𝑠≤−𝑓𝑠,𝑢,𝐺𝑢))𝑑𝑠𝑀𝐶_𝛽−𝛼Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑓𝑠,𝑢_𝑛(𝑠),𝐺𝑢_𝑛(𝑠)−𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))_𝛽𝑑𝑠→0 (3.7) as 𝑛→∞, which implies that 𝑄∶𝐵_𝑟→𝐵_𝑟 is continuous.
Step 2. (𝑄𝐵_𝑟)(𝑡)∶={(𝑄𝑢)(𝑡)∶𝑢∈𝐵_𝑟} is relatively compact in 𝑋_𝛼 for all 𝑡∈𝐽.
It follows from (2.9) and (3.3) that (𝑄𝐵_𝑟)(0)={(𝑄𝑢)(0)∶𝑢∈𝐵_𝑟}={𝑥₀} is compact in 𝑋_𝛼. 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 (𝑄_𝜖,𝛿𝐵_𝑟)(𝑡) by 𝑄_𝜖,𝛿𝐵_𝑟(𝑡)∶=𝑄_𝜖,𝛿𝑢(𝑡)∶𝑢∈𝐵_𝑟, (3.8) where 𝑄_𝜖,𝛿𝑢(𝑡)=^∞_𝛿𝜂_𝑞(𝜃)𝑆𝑡^𝑞𝜃𝑑𝜃𝑥₀+𝑞₀^𝑡−𝜖(𝑡−𝑠)^𝑞−1∞_𝛿𝜃𝜂_𝑞(𝜃)𝑆(𝑡−𝑠)^𝑞𝜃((𝑠)(𝑠𝜖𝑑𝜃⋅𝑓𝑠,𝑢,𝐺𝑢))𝑑𝑠=𝑆^𝑞𝛿^∞_𝛿𝜂_𝑞(𝜃)𝑆𝑡^𝑞𝜃−𝜖^𝑞𝛿𝑑𝜃𝑥₀+𝑞₀^𝑡−𝜖(𝑡−𝑠)^𝑞−1∞_𝛿𝜃𝜂_𝑞(𝜃)𝑆(𝑡−𝑠)^𝑞𝜃−𝜖^𝑞𝛿𝑑𝜃⋅𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))𝑑𝑠. (3.9)
Then the set (𝑄_𝜖,𝛿𝐵_𝑟)(𝑡) is relatively compact in 𝑋_𝛼 since by Lemma 2.2, the operator 𝑆_𝛼(𝜖^𝑞𝛿) is compact in 𝑋_𝛼. For any 𝑢∈𝐵_𝑟 and 𝑡∈(0,𝑇], from the following inequality:(𝑄𝑢)(𝑡)−𝑄_𝜖,𝛿𝑢(𝑡)_𝛼≤^𝛿₀𝜂_𝑞(𝜃)𝑆𝑡^𝑞𝜃𝑑𝜃𝑥_0𝛼+𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝛿₀𝜃𝜂_𝑞(𝜃)𝑆(𝑡−𝑠)^𝑞𝜃((𝑠)(𝑠𝑑𝜃⋅𝑓𝑠,𝑢,𝐺𝑢))𝑑𝑠_𝛼+𝑞^𝑡₀(𝑡−𝑠)^𝑞−1∞_𝛿𝜃𝜂_𝑞(𝜃)𝑆(𝑡−𝑠)^𝑞𝜃𝑑𝜃⋅𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))𝑑𝑠−𝑞₀^𝑡−𝜖(𝑡−𝑠)^𝑞−1∞_𝛿𝜃𝜂_𝑞(𝜃)𝑆(𝑡−𝑠)^𝑞𝜃((𝑠)(𝑠𝑑𝜃⋅𝑓𝑠,𝑢,𝐺𝑢))𝑑𝑠_𝛼≤𝑀𝑥_0𝛼^𝛿₀𝜂_𝑞(𝜃)𝑑𝜃+𝑞𝑀𝐶_𝛽−𝛼𝑔_𝑟𝐿∞^𝑡₀(𝑡−𝑠)^𝑞−1𝑑𝑠^𝛿₀𝜃𝜂_𝑞(𝜃)𝑑𝜃+𝑞𝑀𝐶_𝛽−𝛼𝑔_𝑟𝐿∞^𝑡_𝑡−𝜖(𝑡−𝑠)^𝑞−1𝑑𝑠^∞₀𝜃𝜂_𝑞(𝜃)𝑥𝑑𝜃≤𝑀_0𝛼^𝛿₀𝜂_𝑞(𝜃)𝑑𝜃+𝑀𝐶_𝛽−𝛼𝑇^𝑞𝑔_𝑟𝐿∞×^𝛿₀𝜃𝜂_𝑞(𝜃)𝑑𝜃+𝑀𝐶_𝛽−𝛼𝑔_𝑟𝐿∞Γ𝜖𝑞+1^𝑞. (3.10) One can obtain that the set (𝑄𝐵_𝑟)(𝑡) is relatively compact in 𝑋_𝛼 for all 𝑡∈(0,𝑇). And since it is compact at 𝑡=0, we have the relatively compactness of (𝑄𝐵_𝑟)(𝑡) in 𝑋_𝛼 for all 𝑡∈𝐽.
Step 3. 𝑄𝐵_𝑟∶={𝑄𝑢∈𝐶(𝐽,𝑋_𝛼)∶𝑢∈𝐵_𝑟} is equicontinuous.
For 𝜏∈[0,𝑇), by (3.3), we have ‖(𝑄𝑢)(𝜏)−(𝑄𝑢)(0)‖_𝛼≤𝑈(𝜏)𝑥₀−𝑥_0𝛼+^𝜏₀(𝜏−𝑠)^𝑞−1𝑉(𝜏−𝑠)𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))𝑑𝑠_𝛼≤‖𝑈(𝜏)−𝐼‖⋅𝑥_0𝛼+𝑀𝐶_𝛽−𝛼𝑔_𝑟𝐿∞Γ𝜏𝑞+1^𝑞. (3.11) 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 (𝑄𝑢)𝑡₂−(𝑄𝑢)𝑡_1𝛼≤𝑈𝑡₂𝑥₀−𝑈𝑡₁𝑥_0𝛼+^𝑡2₀𝑡₂−𝑠^𝑞−1𝑉𝑡₂−𝑠𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))𝑑𝑠−^𝑡1₀𝑡₁−𝑠^𝑞−1𝑉𝑡₁−𝑠𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))𝑑𝑠_𝛼≤𝑈𝑡₂−𝑈𝑡₁⋅𝑥_0𝛼+^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝑉𝑡₂−𝑠𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))𝑑𝑠_𝛼+^𝑡1₀𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−1𝑉𝑡₂−𝑠𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))𝑑𝑠_𝛼+^𝑡1₀𝑡₁−𝑠^𝑞−1𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))𝑉𝑡₂−𝑠−𝑉𝑡₁−𝑠𝑑𝑠_𝛼≤𝑈𝑡₂−𝑈𝑡₁⋅𝑥_0𝛼+𝑀𝐶_𝛽−𝛼𝑔_𝑟𝐿∞Γ𝑡𝑞+1²−𝑡₁^𝑞+𝑀𝐶_𝛽−𝛼𝑔_𝑟𝐿∞Γ𝑡𝑞+1^𝑞2−𝑡^𝑞₁−𝑡₂−𝑡₁^𝑞+^𝑡1₀𝑡₁−𝑠^𝑞−1𝑉𝑡₂−𝑠−𝑉𝑡₁−𝑠⋅𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))𝑑𝑠_𝛼≜𝐼₁+𝐼₂+𝐼₃+𝐼₄. (3.12) From Lemma 2.6(v), we see that 𝐼₁→0 as 𝑡₂→𝑡₁ independently of 𝑢∈𝐵_𝑟. From the expressions of 𝐼₂ and 𝐼₃, it is clear that 𝐼₂→0 and 𝐼₃→0as 𝑡₂→𝑡₁ independently of 𝑢∈𝐵_𝑟. For any 𝜖∈(0,𝑡₁), we have 𝐼₄≤^𝑡1₀^−𝜖𝑡₁−𝑠^𝑞−1𝑉𝑡₂−𝑠−𝑉𝑡₁−𝑠⋅𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))𝑑𝑠_𝛼+^𝑡1_𝑡1−𝜖𝑡₁−𝑠^𝑞−1𝑉𝑡₂−𝑠−𝑉𝑡₁−𝑠⋅𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))𝑑𝑠_𝛼≤1𝑞𝐶_𝛽−𝛼𝑔_𝑟𝐿∞𝑇^𝑞+𝜖^𝑞sup_{0≤𝑠≤𝑡1−𝜖}𝑉𝑡₂−𝑠−𝑉𝑡₁−𝑠+2𝑀𝐶_𝛽−𝛼𝑔_𝑟𝐿∞Γ𝜖𝑞+1^𝑞. (3.13)
It follows from Lemma 2.6(v) that 𝐼₄→0 as 𝑡₂→𝑡₁ and 𝜖→0 independently of 𝑢∈𝐵_𝑟. Therefore, we prove that 𝑄𝐵_𝑟 is equicontinuous.
Thus, the Arzela-Ascoli theorem guarantees that 𝑄 is a compact operator. By the Schauder fixed point theorem, the operator 𝑄 has at least one fixed point 𝑢^∗ in 𝐵_𝑟, which is a mild solution of (1.1). This completes the proof.

Remark 3.2. In assumption (H₁)(iii), if the function 𝑔_𝑟(𝑡) is independent of 𝑡, then we can easily obtain a constant 𝛾>0 satisfying (3.2). For example, if there is a constant 𝑎_𝑓>0 such that 𝑓𝑡,𝑥,𝑦_𝛽≤𝑎_𝑓1+‖𝑥‖_𝛼+𝑦_𝛼 (3.14) for all 𝑥,𝑦∈𝑋_𝛼 and 𝑡∈𝐽, then for any 𝑟>0, 𝑥,𝑦∈𝑋_𝛼 with ‖𝑥‖_𝛼≤𝑟,‖𝑦‖_𝛼≤𝐾^∗𝑇𝑟, we have ‖𝑓(𝑡,𝑥,𝑦)‖_𝛽≤𝑎_𝑓+𝑎_𝑓(1+𝐾^∗𝑇)𝑟≜𝑔_𝑟(𝑡), where 𝑔_𝑟(𝑡) is independent of 𝑡. Thus, 𝛾∶=(1/𝑞)𝑎_𝑓𝑇^𝑞(1+𝐾^∗𝑇)>0 is the constant in (3.2).
More generally, if 𝑓 satisfies the following condition:(H2) there is a constant 𝛽∈[𝛼,1] such that 𝑓∶𝐽×𝑋_𝛼×𝑋_𝛼→𝑋_𝛽 satisfies:(i) for each (𝑥,𝑦)∈𝑋_𝛼×𝑋_𝛼, the function 𝑓(⋅,𝑥,𝑦)∶𝐽→𝑋_𝛽 is measurable,(ii) for any 𝑟>0, there exists a function ℓ∈𝐿^∞(𝐽,ℝ⁺) such that 𝑓𝑡,𝑥₁,𝑦₁−𝑓𝑡,𝑥₂,𝑦_2𝛽≤ℓ(𝑡)𝑥₁−𝑥_2𝛼+𝑦₁−𝑦_2𝛼 (3.15)for any 𝑥_𝑖,𝑦_𝑖∈𝑋_𝛼 with ‖𝑥_𝑖‖_𝛼≤𝑟,‖𝑦_𝑖‖_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∈𝑋_𝛼 and 𝑀𝐶_𝛽−𝛼𝑇^𝑞(1+𝐾^∗𝑇)‖ℓ‖_𝐿^∞<Γ(𝑞+1), then (1.1) has a unique mild solution.

Proof. For any 𝑟>0, if 𝑥,𝑦∈𝑋_𝛼 with ‖𝑥‖_𝛼≤𝑟,‖𝑦‖_𝛼≤𝐾^∗𝑇𝑟, then from (H₂)(ii), we have 𝑓𝑡,𝑥,𝑦_𝛽≤ℓ(𝑡)1+𝐾^∗𝑇𝑟+𝑏(𝑡)≜𝑔_𝑟(𝑡), (3.16) where 𝑏(𝑡)=‖𝑓(𝑡,0,0)‖_𝛽. Therefore, the condition (H₁)(iii) is satisfied with 𝛾=((1+𝐾^∗𝑇)𝑇^𝑞‖ℓ‖_𝐿^∞)/𝑞. By Theorem 3.1, (1.1) has at least one mild solution 𝑢^∗∈𝐵_𝑟.
Let 𝑢₁,𝑢₂∈𝐵_𝑟 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 𝑢₁(𝑡)−𝑢₂(𝑡)_𝛼=𝑄𝑢₁(𝑡)−𝑄𝑢₂(𝑡)_𝛼≤^𝑡₀(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝑓𝑠,𝑢₁(𝑠),𝐺𝑢₁(𝑠)−𝑓𝑠,𝑢₂(𝑠),𝐺𝑢₂(𝑠)_𝛼=𝑑𝑠_𝑡0(𝑡−𝑠)^𝑞−1𝐴^𝛼−𝛽𝑉(𝑡−𝑠)⋅𝐴^𝛽𝑓𝑠,𝑢₁(𝑠),𝐺𝑢₁(𝑠)−𝑓𝑠,𝑢₂(𝑠),𝐺𝑢₂(𝑠)≤𝑑𝑠𝑀𝐶_𝛽−𝛼Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1⋅𝑓𝑠,𝑢₁(𝑠),𝐺𝑢₁(𝑠)−𝑓𝑠,𝑢₂(𝑠),𝐺𝑢₂(𝑠)_𝛽≤𝑑𝑠𝑀𝐶_𝛽−𝛼Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1⋅ℓ(𝑠)𝑢₁(𝑠)−𝑢₂(𝑠)_𝛼+𝐺𝑢₁(𝑠)−𝐺𝑢₂(𝑠)_𝛼≤𝑑𝑠𝑀𝐶_𝛽−𝛼‖ℓ‖_𝐿^∞Γ𝑞1+𝐾^∗𝑇^𝑡₀(𝑡−𝑠)^𝑞−1⋅𝑢₁(𝑠)−𝑢₂(𝑠)_𝛼𝑑𝑠. (3.17) 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^∗∈𝐵_𝑟. This completes the proof.

Remark 3.4. In Theorem 3.3, 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×𝑋_𝛼→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 that 𝑋=𝐿²[0,𝜋] equipped with its natural norm and inner product defined, respectively, for all 𝑢,𝑣∈𝐿²[0,𝜋], by ‖𝑢‖_𝑋=^𝜋₀|𝑢(𝑥)|²𝑑𝑥^1/2,⟨𝑢,𝑣⟩=^𝜋₀𝑢(𝑥)𝑣(𝑥)𝑑𝑥. (4.1) Consider the following fractional partial differential equation: 𝜕^1/2𝜕𝑡^1/2𝑢()−𝜕𝑥,𝑡²𝜕𝑥²𝑢(𝑥,𝑡)=𝑔𝑥,𝑡,𝑢(𝑥,𝑡),∫^𝑡₀𝐾()𝑢()[][]𝑢()()[],𝑢()𝑡,𝑠𝑥,𝑠𝑑𝑠,𝑡∈0,𝑇,𝑥∈0,𝜋0,𝑡=𝑢𝜋,𝑡=0,𝑡∈0,𝑇𝑥,0=𝑢₀(𝑥),𝑥∈[0,𝜋], (4.2) where 𝑇>0 is a constant.

Let the operator 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 be defined by 𝐷(𝐴)∶=𝑣∈𝑋∶𝑣^′′(0)(𝜋)𝜕∈𝑋,𝑣=𝑣=0,𝐴𝑢=−²𝑢𝜕𝑥². (4.3) It is well known that 𝐴 has a discrete spectrum with eigenvalues of the form 𝑛²,𝑛∈ℕ, and corresponding normalized eigenfunctions given by 𝑧_𝑛=(2/𝜋)sin(𝑛𝑥). Moreover, −𝐴 generates a compact analytic semigroup 𝑆(𝑡)(𝑡≥0) in X, and 𝑆(𝑡)𝑢=^∞_𝑛=1𝑒^−𝑛2𝑡𝑢,𝑧_𝑛𝑧_𝑛. (4.4) 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 operator 𝐴 can be written as 𝐴𝑢=^∞_𝑛=1𝑛²𝑢,𝑧_𝑛𝑧_𝑛 (4.5)for every 𝑢∈𝐷(𝐴).(II)The operator 𝐴^1/2 is given by 𝐴^1/2𝑢=^∞_𝑛=1𝑛𝑢,𝑧_𝑛𝑧_𝑛 (4.6)for each 𝑢∈𝐷(𝐴^1/2)∶={𝑣∈𝑋∶∑^∞_𝑛=1𝑛<𝑣,𝑧_𝑛>𝑧_𝑛∈𝑋} and ‖𝐴^−(1/2)‖_ℒ(𝑋)=1.

Lemma 4.1 (see [28]). If 𝑚∈𝐷(𝐴^1/2), then 𝑚 is absolutely continuous, 𝑚^′∈𝑋 and ‖𝑚^′‖_𝑋=‖𝐴^1/2𝑚‖_𝑋.

Let 𝑋_1/2=(𝐷(𝐴^1/2),‖⋅‖_1/2), where ‖𝑥‖_1/2∶=‖𝐴^1/2𝑥‖_𝑋 for all 𝑥∈𝐷(𝐴^1/2). 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 each 𝑡∈[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 functions ℓ₁,ℓ₀∈𝐿^∞([0,𝑇],ℝ⁺) such that 𝜕𝑔𝜕𝑥𝑥,𝑡,𝜉,𝜂≤ℓ₁(𝑡)𝜉+𝜂+ℓ₀(𝑡) (4.7)for all (𝑥,𝑡,𝜉,𝜂)∈[0,𝜋]×[0,𝑇]×ℝ×ℝ.

Define 𝑓(𝑡,𝑢(𝑡),𝐺𝑢(𝑡))(𝑥)=𝑔(𝑥,𝑡,𝑢(𝑥,𝑡),∫^𝑡₀𝐾(𝑡,𝑠)𝑢(𝑥,𝑠)𝑑𝑠). Then, for each 𝜙∈𝑋_1/2, from assumptions (iii) and (iv), we have 𝑓𝑡,𝜙,𝐺𝜙,𝑧_𝑛=^𝜋₀𝑔𝑥,𝑡,𝜙(𝑥,𝑡),^𝑡₀𝐾(𝑡,𝑠)𝜙(𝑥,𝑠)𝑑𝑠⋅2𝜋()=1sin𝑛𝑥𝑑𝑥𝑛^𝜋₀𝜕𝑔(),𝜕𝑥𝑥,𝑡,𝜙𝑥,𝑡^𝑡₀𝐾(𝑡,𝑠)𝜙(𝑥,𝑠)𝑑𝑠⋅2𝜋cos(𝑛𝑥)𝑑𝑥. (4.8)

This implies from (II) that 𝑓∶[0,𝑇]×𝑋_1/2×𝑋_1/2→𝑋_1/2. Moreover, for any 𝑟>0, by Minkowski inequality, assumption (v) and Lemma 4.1, we have sup_{𝜙1/2≤𝑟}𝑓𝑡,𝜙,𝐺𝜙_1/2=sup_{𝜙1/2≤𝑟}𝜕𝑔(),𝜕𝑥𝑥,𝑡,𝜙𝑥,𝑡^𝑡₀𝐾(𝑡,𝑠)𝜙(𝑥,𝑠)𝑑𝑠_𝑋=sup_{𝜙1/2≤𝑟}^𝜋₀𝜕𝑔(),𝜕𝑥𝑥,𝑡,𝜙𝑥,𝑡^𝑡₀𝐾(𝑡,𝑠)𝜙(𝑥,𝑠)𝑑𝑠²𝑑𝑥^1/2≤sup_{𝜙1/2≤𝑟}^𝜋₀ℓ₁(𝑡)𝜙(𝑥,𝑡)+^𝑡₀𝐾(𝑡,𝑠)𝜙(𝑥,𝑠)𝑑𝑠+ℓ₀(𝑡)²𝑑𝑥^1/2≤sup_{𝜙1/2≤𝑟}ℓ₁(𝑡)𝜙_𝑋+𝐾^∗𝑇𝜙_𝑋+ℓ₀(𝑡)=sup_{𝜙1/2≤𝑟}1+𝐾^∗𝑇ℓ₁(𝑡)𝐴^−(1/2)⋅𝐴^1/2𝜙_𝑋+ℓ₀(𝑡≤)1+𝐾^∗𝑇𝑟ℓ₁(𝑡)+ℓ₀(𝑡)≜𝑔_𝑟(𝑡). (4.9)

Therefore, 𝑓 satisfies the condition (H₁) with 𝛾=2𝑇^1/2(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 function 𝑔 satisfies the following:(vi) for any 𝑟>0, there exists a function ℓ₂∈𝐿^∞([0,𝑇],ℝ⁺) such that 𝜕𝑔𝜕𝑥𝑥,𝑡,𝜉₁,𝜂₁−𝜕𝑔𝜕𝑥𝑥,𝑡,𝜉₂,𝜂₂≤ℓ₂(𝑡)𝜉₁−𝜉₂+𝜂₁−𝜂₂ (4.10)for (𝑥,𝑡,𝜉₁,𝜂₁),(𝑥,𝑡,𝜉₂,𝜂₂)∈[0,𝜋]×[0,𝑇]×ℝ×ℝ with |𝜉_𝑖|≤𝑟 and |𝜂_𝑖|≤𝐾^∗𝑇𝑟,𝑖=1,2.

Then for each 𝜙₁,𝜙₂∈𝑋_1/2, by Lemma 4.1, we have 𝑓𝑡,𝜙₁,𝐺𝜙₁−𝑓𝑡,𝜙₂,𝐺𝜙_21/2=𝐴^1/2𝑓𝑡,𝜙₁,𝐺𝜙₁−𝑓𝑡,𝜙₂,𝐺𝜙_2𝑋=𝜕𝑔𝜕𝑥𝑥,𝑡,𝜙₁(𝑥,𝑡),^𝑡₀𝐾(𝑡,𝑠)𝜙₁()−𝜕𝑥,𝑠𝑑𝑠𝑔𝜕𝑥𝑥,𝑡,𝜙₂(𝑥,𝑡),^𝑡₀𝐾(𝑡,𝑠)𝜙₂(𝑥,𝑠)𝑑𝑠_𝑋=^𝜋₀𝜕𝑔𝜕𝑥𝑥,𝑡,𝜙₁(𝑥,𝑡),^𝑡₀𝐾(𝑡,𝑠)𝜙₁()−𝜕𝑥,𝑠𝑑𝑠𝑔𝜕𝑥𝑥,𝑡,𝜙₂(𝑥,𝑡),^𝑡₀𝐾(𝑡,𝑠)𝜙₂(𝑥,𝑠)𝑑𝑠²𝑑𝑥^1/2≤^𝜋₀ℓ₂(𝑡)𝜙₁(𝑥,𝑡)−𝜙₂(+𝑥,𝑡)_𝑡0𝐾(𝑡,𝑠)𝜙₁()−𝑥,𝑠𝑑𝑠_𝑡0𝐾(𝑡,𝑠)𝜙₂(𝑥,𝑠)𝑑𝑠²𝑑𝑥^1/2≤ℓ₂(𝑡)^𝜋₀𝜙₁(𝑥,𝑡)−𝜙₂(𝑥,𝑡)²𝑑𝑥^1/2+^𝜋₀^𝑡₀𝐾(𝑡,𝑠)𝜙₁()−𝑥,𝑠𝑑𝑠_𝑡0𝐾(𝑡,𝑠)𝜙₂(𝑥,𝑠)𝑑𝑠²𝑑𝑥^1/2=ℓ₂(𝑡)𝜙₁−𝜙_2𝑋+𝐺𝜙₁−𝐺𝜙_2𝑋=ℓ₂(𝑡)𝐴^−(1/2)⋅𝐴^1/2𝜙₁−𝜙_2𝑋+𝐴^−(1/2)⋅𝐴^1/2𝐺𝜙₁−𝐺𝜙_2𝑋≤ℓ₂(𝑡)𝐴^1/2𝜙₁−𝜙_2𝑋+𝐴^1/2𝐺𝜙₁−𝐺𝜙_2𝑋=ℓ₂(𝑡)𝜙₁−𝜙_21/2+𝐺𝜙₁−𝐺𝜙_21/2. (4.11) This shows that 𝑓 satisfies the condition (H₂). Hence by Theorem 3.3, the mild solution of (4.2) is unique.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant no. 11261051), the Fundamental Research Funds for the Gansu Universities and the Project of NWNU-LKQN-11-3.