Periodic solutions for neutral evolution equations with delays

The aim is to study the periodic solution problem for neutral evolution equation $$(u(t)-G(t,u(t-\xi)))'+Au(t)=F(t,u(t),u(t-\tau)),\ \ \ \ t\in\R$$in Banach space $X$, where $A:D(A)\subset X\rightarrow X$ is a closed linear operator, and $-A$ generates a compact analytic operator semigroup $T(t)(t\geq0)$. With the aid of the analytic operator semigroup theories and some fixed point theorems, we obtain the existence and uniqueness of periodic mild solution for neutral evolution equations. The regularity of periodic mild solution for evolution equation with delay is studied, and some the existence results of the classical and strong solutions are obtained. In the end, we give an example to illustrate the applicability of abstract results. Our works greatly improve and generalize the relevant results of existing literatures.


Introduction
The theory of partial differential equations with delays has extensive physical background and realistic mathematical model, and it has undergone a rapid development in the last fifty years see [1,3] and references therein. More recently researchers have given special attentions to the study of equations in which the delay argument occurs in the derivative of the state variable as well as in the independent variable, so-called neutral differential equations. Neutral differential equations have many applications. It can model a lot of problems arising from engineering, such as population dynamics, transmission line, immune response or distribution of albumin in the blood etc.
In [3,4] the authors studied a the partial neutral functional differential-difference equations which is defined on a unit circle S 1 : where a, k, q are positive constants, g : R → R is continuously differentiable, τ ≥ 0 which denotes the time delay. Thereafter, more results on partial neutral functional differential equations are published, and we refer readers to [5,6,7,8,9,10].The idea of studying partial neutral functional differential equations with operators satisfying Hille-Yosida condition, begins with [11], where the authors studied the following class of equation in a Banach space X.
(u(t) − Bu(t − ξ)) ′ = A(u(t) − Bu(t − ξ)) + F (u(t), u(t − τ )), (1.2) where A satisfies the Hille-Yosida condition, B are bounded linear operators from X into X, F : X ×X → X is continuous, ξ, τ are poeitive constants which denote the time delays. It has been proved in particular, that the solutions generate a locally Lipschitz continuous integrated semigroup. The problems concerning periodic solutions of partial neutral functional differential equations with delay are an important area of investigation since they can take into account seasonal fluctuations occurring in the phenomena appearing in the models, and have been studied by some researchers in recent years. Specially, the existence of periodic solutions of neutral evolution equations with delay has been considered by several authors, see [12,13,14,15,16,17,18]. For the delayed evolution equations without neutral term, the existence of periodic solutions has been discussed by more authors, see [19,20,21,22,23] and references therein. Naturally, fixed point theorems play a significant role in the investigation of the existence of periodic solutions. It is well known that the Massera's approach (see [24]) on periodic partial functional differential equations explains the relationship between the existence of bounded solutions and the existence of periodic solutions. However, in many of the studies mentioned above, the key assumption of prior boundedness of solutions was employed and the most important feature is to show that Poincaré's mapping is condensing, where ω is a period of the system and u the unique mild solution determined by φ. Therefore, a fixed point theorem can be used to derive periodic solutions.
Recently, Zhu, Liu and Li in [25] investigated the existence of time periodic solutions for a class of one-dimensional parabolic evolution equation with delays. They obtained the existence of time periodic solutions by constructing some suitable Lyapunov functionals and establishing the prior bound for all possible periodic solutions. And, Li in [26] discussed the existence of the time periodic solution for the evolution equation with multiple delays in a Hilbert space H where A : D(A) ⊂ H → H is a positive definite selfadjoint operator, F : R×H n+1 → H is a nonlinear mapping which is ω-periodic in t, and τ 1 , τ 2 , · · · , τ n are positive constants which denote the time delays. By using periodic extension and Schauder fixed point theorem, the author presented essential conditions on the nonlinearity F to guarantee that the equation has ω-periodic solutions. Motivated by the papers mentioned above, the aim of this work is to study the existence of periodic solution for some the partial neutral functional differential equations. Our discussion will be made in a frame of abstract Banach spaces.
Throughout this paper, X is a Banach space provided with norm · and A : D(A) ⊂ X → X is a closed linear operator, and −A generates a compact analytic operator semigroup T (t)(t ≥ 0) in Banach space X. Let G, F be appropriate continuous functions which will be specified later, and G(t, ·), F (t, ·, ·) be ω-periodic in t.
Under the above assumptions we discuss the existence and uniqueness of ω-periodic solutions of the abstract neutral functional differential equations with delays in X of the form where ξ, τ are positive constants which denote the time delays. The purpose of the present note is to extend and develop the work in [25,26], that is, we will discuss the existence and regularity of periodic solutions for Eq. (1.4). The obtained results will also improve the main results in [13,17,18]. In this paper, it is worth mentioning that assumption of prior boundedness of solutions is not necessary. More precisely, the nonlinear term F only satisfies some growth conditions and the function G and F may not be defined on the whole space X. These conditions are much weaker than Lipschitz conditions. The paper is organized as follows. In Section 2, we collect some known notions and results on the fractional powers of the generator of an analytic semigroup and provide preliminary results to be used in theorems stated and proved in the paper. In Section 3, we apply the operator semigroup theory to find the periodic mild solutions for Eq. (1.4) and in Section 4, we investigate conditions for Eq.(1.4) to have the calssical and strong periodic solutions. In the last section, we give an example to illustrate the applicability of abstract results obtained in Section 3 and Section 4.

Preliminaries
Throughout this paper, we assume that X is a Banach space with norm · , that A : D(A) ⊂ X → X is a closed linear operator and −A generates a compact analytic operator semigroup T (t)(t ≥ 0) in Banach space X. For the theory of semigroups of linear operators we refer to [27].
We only recall here some notions and properties that are essential for us. For a general C 0 -semigroup T (t)(t ≥ 0), there exist M ≥ 1 and ν ∈ R such that (see [27]) then ν 0 is called the growth exponent of the semigroup T (t)(t ≥ 0). Furthermore, ν 0 can be also obtained by the following formula If C 0 -semigroup T (t) is analytic on (0, +∞), it is well known that ν 0 can also be determined by σ(A) (see [27,28]), where −A is the infinitesimal generator of C 0 -semigroup T (t)(t ≥ 0). We know that T (t)(t ≥ 0) is continuous in the uniform operator topology for t > 0 if T (t)(t ≥ 0) is compact semigroup or analytic semigroup(see [29]). In particular, if T (t)(t ≥ 0) is analytic semigroup with infinitesimal generator A satisfying 0 ∈ ρ(A)(ρ(A) is the resolvent set of A). Then for any α > 0, we can define A −α by 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. Furthermore, the subspace D(A α ) is dense in X and the expression defines a norm on D(A α ). Hereafter we respresent X α to the space D(A α ) endowed with the norm · α and denote by C α the operator norm of A −α , i.e., C α := A −α . The following preperties are well known( [27]).
, and the embedding is continuous. and the embedding X β ֒→ X α is compact whenever the resolvent operator of A is compact.
Observe by Lemma 2.1 (iii) and (iv) that the restriction T α (t) of T (t) to X α is exactly the part of T (t) in X α . Moreover, for any x ∈ X α , we have and it follows that T α (t)(t ≥ 0) is a strongly continuous semigroup on X α and T α (t) α ≤ T (t) for all t ≥ 0. To prove our main results, we need the following lemmas.
) If X is reflexive, then X α is also reflexive. Now, recall some basic facts on abstract linear evolutions, which are needed to prove our main results.
Let J denote the infinite interval [0, ∞) and h : J → X, consider the initial value problem of the linear evolution equation It is well known, when x 0 ∈ D(A) and h ∈ C 1 (J, X), the initial value problem (2.6) has a unique classical solution u ∈ C 1 (J, X) ∩ C(J, X 1 ) expressed by Generally, for x 0 and h ∈ C(J, X), the function u given by (2.7) belongs to C(J, X) and it is called a mild solution of the linear evolution equation (2.6). A mild solution u of Eq. (2.6) is called a strong solution if u is continuously differentiable a.e. on J, u ′ ∈ L 1 loc (J, X) and satisfies Eq. (2.6). Furthermore, we have the following results.
Given h ∈ C ω (R, X), we consider the existence of ω-periodic mild solution of linear evolution equation , the linear evolution equation (2.8) exists a unique ω-periodic mild solution u, which can be expressed by

9)
and the solution operator P : Proof. For any ν ∈ (0, |ν 0 |), there exists M > 0 such that In X, define the equivalent norm | · | by , then for t ≥ 0, it is easy to obtain that |T (t)| < e −νt . Hence, (I − T (ω)) has bounded inverse operator and its norm satisfies (2.10) then the mild solution u(t) of the linear initial value problem (2.6) given by (2.7) satisfies the periodic boundary condition u(0) = u(ω) = x 0 . For t ∈ R + , by (2.7) and the properties of the semigroup T (t)(t ≥ 0), we have Therefore, the ω-periodic extension of u on R is a unique ω-periodic mild solution of Eq.(2.8). By (2.7) and (2.11), the ω-periodic mild solution can be expressed by where C := (I − T (ω)) −1 , which implies that P is bounded. This completes the proof of Lemma 2.6. ✷ To prove our main results, we also need the following lemma.
Lemma 2.7.( [33]) Assume that Q is a condensing operator on a Banach space X, i.e.,Q is continuous and takes bounded sets into bounded sets, and α(Q(D)) < α(D) for every bounded set D of X with α(D) > 0. If Q(Ω) ⊂ Ω for a convex, closed, and bounded set Ω of X, then Q has a fixed point in Ω (where α(·) denotes the kuratowski measure of non-compactness).
Remark 2.8. It is easy to see that, if Q = Q 1 + Q 2 with Q 1 a completely continuous operator and Q 2 a contractive one, then Q is a condensing operator on X.

Existence of Mild solution
Now, we are in a position to state and prove our main results of this section.
If the following conditions (H1) for any r > 0, there exists a positive value function h r : R → R + such that (H2) G(t, θ) = θ for t ∈ R, and there is a constant L ≥ 0 such that Proof From the assumption, we know that G(t, u(t − ξ)) ∈ D(A) for every u ∈ C ω (R, X α ), thus, we can rewrite Eq.(1.4) as following For any r > 0, let Note that Ω r is a closed ball in C ω (R, X α ) with centre θ and radius r. Moreover, by the condition (H2), it follows that, Hence, we can define the operator Q on C ω (R, X α ) by From Lemma 2.6, it is sufficient to prove that Q has a fixed point. Now, we show that there is a positive constant r such that Q(Ω r ) ⊂ Ω r . If this were not case, then for any r > 0, there exist u r ∈ Ω r and t r ∈ R such that Qu r (t r ) α > r. Thus, we see by (H1),(H2) and (H3) that Dividing on both sides by r and taking the lower limit as r → ∞, we have which contradicts (H3). Hence, there is a positive constant r such that Q(Ω r ) ⊂ Ω r .
To show that the operator Q has a fixed point on Ω r , we also introduce the decom- Then we will prove that Q 1 is a compact operator and Q 2 is a contraction. Firstly, we prove that Q 1 is a compact operator. Let {u n } ⊂ Ω r with u n → u in Ω r , then by the continuity of F , we have then the dominated convergence theorem ensure that It is easy to see that Q 1 maps Ω r into a bounded set in C ω (R, X α ). Now, we demonstrate that Q(Ω r ) is equicontinuous. For every u ∈ Ω r , by the periodicity of u, we only consider it on [0, ω]. Set 0 ≤ t 1 < t 2 ≤ ω, we get that It is clear that Thus, we only need to check I i α tend to 0 independently of u ∈ Ω r when t 2 − t 1 → 0, i = 1, 2, 3. From the continuity of t → T (t) for t > 0 and the condition (H1), we can easily see It remains to show that (Q 1 Ω r )(t) is relatively compact in X α for all t ∈ R. To do this, we define a set (Q ε Ω r )(t) by From Lemma 2.2, the operator T α (ε) is compact in X α , it is follows that the set (Q ε Ω r )(t) is relatively compact in X α . For any u ∈ Ω r and t ∈ R, from the following inequality one can obtain that the set (Q 1 Ω r )(t) is relatively compact in X α for all t ∈ R.
Thus, the Arzela-Ascoli theorem guarantees that Q 1 is a compact operator. Secondly, we prove that Q 2 is a contraction. Let u, v ∈ Ω r , by the condition (H2), Lemma 2.1(vi) and Lemma 2.6, we have therefore, By Lemma 2.7, we know that Q has a fixed point u ∈ Ω r , that is, Eq (1.4) has a ω-periodic mild solution. The proof is completed.

Existence of Classical and Strong Solutions
In this section, we discuss the regularity properties of the ω-periodic mild solution of Eq. (1.4), and present essential conditions on the nonlinearity F and G to guarantee that Eq. (1.4) has ω-periodic classical and strong solutions. Now, we are in a position to state and prove the main result of this section: Theorem 4.1. Let A : D(A) ⊂ X → X be a closed linear operator, and −A generate a compact and exponentially stable analytic operator semigroup T (t)(t ≥ 0) in Banach space X. For α ∈ [0, 1), we assume that G : R × X α → X 1 and F : R × X 2 α → X are continuous functions, and for every x, x 0 , x 1 ∈ X α , G(t, x), F (t, x 0 , x 1 ) are ω-periodic in t. If the following conditions (H4) there exist L 1 and µ 1 ∈ (0, 1) such that for each t 1 , t 2 ∈ R and x 0 , x 1 , y 0 , y 1 ∈ X α , (H5) G(t, θ) = θ for t ∈ R, there exist L 2 and µ 2 ∈ (0, 1) such that for each t 1 , t 2 ∈ R and x, y ∈ X α , (H6) CM α (2L 1 + L 2 ) ω 1−α 1−α + C 1−α L 2 < 1, where C = (I − T (ω)) −1 , hold, then Eq.(1.1) has an ω-periodic classical solution.
By the arbitrariness of ε, we claim that Therefore, u 0 is ω-periodic classical solution of Eq.(1.4) and satisfies The proof is completed.
Theorem 4.2. Let X be a reflexive Banach space, A : D(A) ⊂ X → X is a closed linear operator and −A generates an exponentially stable and compact analytic semigroup For α ∈ [0, 1), we assume that G : R×X α → X 1 and F : R×X 2 α → X are continuous functions, and for every x, x 0 , x 1 ∈ X α , G(t, x), F (t, x 0 , x 1 ) are ωperiodic in t. If the conditions (H4 ′ ) there exists a constant L 1 > 0 such that for any t 1 , t 2 ∈ R and x 0 , x 1 , y 0 , y 1 ∈ X α , (H5 ′ ) G(t, θ) = θ for t ∈ R, there exist L 2 and such that for each t 1 , t 2 ∈ R and x, y ∈ X α , and (H6) hold, then Eq.(1.1) has an ω-periodic strong solution u.
Proof Let Q be the operator defined by (3.3) in the proof of Theorem 3.1. For a given r > 0, let Ω r ⊂ C ω (R, X α ) is defined by (3.2). By the conditions (H4 ′ -H6 ′ ), one can use the same argument as in the proof of Theorem 3.1 to obtain that (QΩ r ) ⊂ Ω r .
For this r, consider the set for some L * large enough. It is clear that Ω is convex closed and nonempty set. We shall prove that Q has a fixed point on Ω. Obviously, from the proof of Theorem 3.1, it is sufficient to show that for any u ∈ Ω In fact, by the definition of Q, the condition (H4 ′ ),(H5 ′ ) and (4.3), we have whenever L * ≥ K 0 1−K * . Therefore, Q has a fixed point u which is an ω-periodic mild solution of Eq.(1.4).
By the above calculation, we see that for this u(·), all the following functions are Lipschitz continuous, respectively. Since the u is Lipschitz continuous on R and the space X α is reflexive by the assumption and Lemma 2.3, then a result of [34] asserts that u(·) is a.e. differentiable on R and u ′ (·) ∈ L 1 loc (R, X α ). Furthermore, by a stantdard arguement as Theorem 4.2.4 in [27], we can obtain that Hence, for almost every t ∈ R a.e t ∈ R. (4.6) This shows that u is a strong solution for Eq.(1.4) and the proof is completed.

Application
In this section, we present one example, which indicates how our abstract results can be applied to concrete problems.
To treat this system in the abstract form (1.4), we choose the space X = L 2 ([0, 1], R), equipped with the L 2 -norm · L 2 , thus, X is reflexive.
Then −A generates an exponentially stable compact analytic semigroup T (t)(t ≥ 0) in X. It is well known that 0 ∈ ρ(A) and so the fractional powers of A are well defined. Moveover, A has a discrete spectrum with eigenvalues of the form n 2 π 2 , n ∈ N, and the associated normalized eigenfunctions are given by e n (x) = √ 2 sin(nπx) for x ∈ [0, 1], the associated semigroup T (t)(t ≥ 0) is explicitly given by e −n 2 π 2 t (u, e n )e n , t ≥ 0, u ∈ X, (5 .3) where (·, ·) is an inner product on X, and it is not difficult to verify that T (t) ≤ e −π 2 t for all t ≥ 0. Hence, we take M = 1, The proof of the following lemma can be found in [35].
According to Lemma 5.1, we define the Banach space X 1 It is clear that G : R×X 1 2 → X 1 and F : R×X 1 2 ×X 1 2 → X. Then the partial differential equation with delays (5.1) can be rewritten into the abstract evolution equation with delays (1.4).
Proof. Let φ, ϕ ∈ X 1 2 , from the condition (F1), we can get thus, the condition (H1 ′ ) in Section 3 holds. Let φ, ϕ ∈ X 1 2 , from the condition (F2), we have thus, the condition (H2) in Section 3 holds. Finally, by (F3), we can easily to prove that the condition (H3 ′ ) holds in Section 3. Therefore, from Corollary 3.2, it follows that the neutral partial differential equation with delays (5.1) has at least one time ω-periodic mild solution. The proof is completed.