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

and Applied Analysis 3 iii For every t > 0, AS t is bounded in X and there existsMα > 0 such that ‖AS t ‖ ≤ Mαt−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,Xα the Banach space of all continuous functions from J into Xα with supnorm given by ‖u‖C supt∈J‖u t ‖α for u ∈ C J,Xα . From Lemma 2.1 iv , since A−α is a bounded linear operator for 0 ≤ α ≤ 1, there exists a constant Cα such that ‖A−α‖ ≤ Cα for 0 ≤ α ≤ 1. For any t ≥ 0, denote by Sα t the restriction of S t to Xα. From Lemma 2.1 i and ii , for any x ∈ Xα, we have ‖S t x‖α ‖A · S t x‖ ‖S t ·Aαx‖ ≤ ‖S t ‖ · ‖Ax‖ ‖S t ‖ · ‖x‖α, ‖S t x − xα‖ ‖A · S t x −Aαx‖ ‖S t ·Aαx −Aαx‖ −→ 0 2.3 as t → 0. Therefore, S t t ≥ 0 is a strongly continuous semigroup in Xα, and ‖Sα t ‖α ≤ ‖S t ‖ for all t ≥ 0. To prove our main results, the following lemma is also needed. Lemma 2.2 see 27 . Sα t t ≥ 0 is an immediately compact semigroup in Xα, and hence it is immediately norm-continuous. Let us recall the following known definitions in fractional calculus. For more details, see 16–20, 23 . Definition 2.3. The fractional integral of order σ > 0 with the lower limits zero for a function f is defined by


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: 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 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 H 1 and H 2 .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.

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

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 β x A α A β x A β A α x for every α, β ∈ R and x ∈ D A μ with μ : max{α, β, α β}, where A 0 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 α x 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 X 0 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 α S t x S t A α x for each x ∈ D A α and t ≥ 0.

Abstract and Applied Analysis 3
iii For every t > 0, A α S t is bounded in X and there exists M α > 0 such that 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 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 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 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

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
For x ∈ X, we define two families {U t } t≥0 and {V t } t≥0 of operators by where η q is a probability density function defined on 0, ∞ , which has properties η q θ ≥ 0 for all θ ∈ 0, ∞ and 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 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 normcontinuous.
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 for all t ∈ J.

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.
iii for any r > 0, there exists a function g r ∈ L ∞ J, R such that and there is a constant γ > 0 such that 1 has at least one mild solution.
Proof.Define an operator Q by 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 H 1 iii , we see that g r s ds.

3.4
Dividing on both sides by r and taking the lower limit as r → ∞, we have 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 From the assumption H 1 ii , for each s ∈ J, we have f s, 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 as n → ∞, which implies that Q : B r → B r is continuous.
Abstract and Applied Analysis 7 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 } {x 0 } is compact in X α .Hence it is only necessary to consider the case of t > 0. For each t ∈ 0, T , ∈ 0, t , and any δ > 0, we define a set Q ,δ B r t by where θη q θ S t − s q θ − q δ dθ • f s, u s , Gu s ds .

3.9
Then the set Q ,δ B r t is relatively compact in X α since by Lemma 2.2, the operator S α q δ is compact in X α .For any u ∈ B r and t ∈ 0, T , from the following inequality: Γ q 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.

3.11
Hence it is only necessary to consider the case of t > 0. For 0 < t 1 < t 2 ≤ T , by Lemma 2.1 and Lemma 2.6 i , we have Abstract and Applied Analysis 9

3.12
From Lemma 2.6 v , we see that I 1 → 0 as t 2 → t 1 independently of u ∈ B r .From the expressions of I 2 and I 3 , it is clear that I 2 → 0 and For any ∈ 0, t 1 , we have Γ q 1 q .

3.13
It follows from Lemma 2.6 v that I 4 → 0 as t 2 → t 1 and → 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 H 1 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 f t, x, y β ≤ a f 1 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 f T q 1 K * T > 0 is the constant in 3.2 .
More generally, if f satisfies the following condition: ii for any r > 0, there exists a function ∈ L ∞ J, R such that 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 H
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 H 2 ii , we have where b t f t, 0, 0 β .Therefore, the condition H 1 iii is satisfied with γ 1 K * T T q L ∞ /q.By Theorem 3.1, 1.1 has at least one mild solution u * ∈ B r .Let u 1 , u 2 ∈ B r be the solutions of 1.1 .We show that u 1 ≡ u 2 .Since u 1 t Qu 1 t and u 2 t Qu 2 t for all t ∈ J, we have 3.17 By using the Gronwall-Bellman inequality see 14, Theorem 1 , we can deduce that u 1 t − u 2 t α 0 for all t ∈ J, which implies that u 1 ≡ u 2 .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 H 2 , and an existence and uniqueness result is obtained.If f t, u, v ≡ f t, u : J × X α → X, then the assumption H 2 deletes the linear growth condition 3 of assumption Hf in 12 .Therefore, the Theorem 3.3 extends and improves the main result in 12 .

An Example
Assume that X L 2 0, π equipped with its natural norm and inner product defined, respectively, for all u, v ∈ L 2 0, π , by Consider the following fractional partial differential equation: where T > 0 is a constant.Let the operator A : D A ⊂ X → X be defined by It is well known that A has a discrete spectrum with eigenvalues of the form n 2 , 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 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 for every u ∈ D A .
v There exist the functions 1 , 0 ∈ L ∞ 0, T , R such that

4.11
This shows that f satisfies the condition H 2 .Hence by Theorem 3.3, the mild solution of 4.2 is unique.