DICHOTOMY AND PERIODIC SOLUTIONS TO PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS

. We establish a suﬃcient condition for existence and uniqueness of periodic solutions to partial functional diﬀerential equations of the form ˙ u = A ( t ) u + F ( t )( u t ) + g ( t,u t ) on a Banach space X where the operator-valued functions t (cid:55)→ A ( t ) and t (cid:55)→ F ( t ) are 1-periodic, the nonlinear operator g ( t,φ ) is 1-periodic with respect to t for each ﬁxed φ ∈ C := C ([ − r, 0] ,X ), and satisfying (cid:107) g ( t,φ 1 ) − g ( t,φ 2 ) (cid:107) (cid:54) ϕ ( t ) (cid:107) φ 1 − φ 2 (cid:107) C for φ 1 ,φ 2 ∈ C with ϕ being a positive function such that sup t (cid:62) 0 (cid:82) t +1 t ϕ ( τ ) dτ < ∞ . We then apply the results to study the existence, uniqueness, and conditional stability of periodic solutions to the above equation in the case that the family ( A ( t )) t ≥ 0 gener- ates an evolution family having an exponential dichotomy. We also prove the existence of a local stable manifold near the periodic solution in that case.

1. Introduction and preliminaries. Consider the abstract partial functional differential equationu = A(t)u + F (t)(u t ) + g(t, u t ), t ∈ R + , (1.1) where for each t ∈ R + , A(t) is a possibly unbounded operator on a Banach space X such that (A(t)) t≥0 generates an evolution family (U (t, s)) t≥s≥0 on X; F (t) ∈ L(C, X) with C := C([−r, 0], X), and g : R + × C → X is continuous and locally Lipschitz; u t is the history function defined by u t (θ) = u(t + θ) for θ ∈ [−r, 0]. One of the important research directions related to the asymptotic behavior of the solutions to the above equation is to find conditions for the existence and stability of a periodic solution to (1.1) in case that F and g are 1-periodic functions with respect to t. There are several approaches to handle this problem such as Tikhonov's fixed point method [21], Lyapunov functionals [24], Fredholm operator and translation 3128 NGUYEN THIEU HUY AND NGO QUY DANG formulae [17,23], and the most popular approach is the use of ultimate boundedness of solutions and the compactness of Poincaré map realized through some compact embeddings (see [1,11,20,21,22,24] and references therein). However, in some concrete applications, e.g., to partial differential equations in unbounded domains or to equations that have unbounded solutions, such compact embeddings are no longer valid, and the existence of bounded solutions is not easy to obtain since one has to carefully choose an appropriate initial vector (or conditions) to guarantee the boundedness of the solution emanating from that vector.
Recently, in [9], for the case of partial differential equations without delay we have proposed a new approach to overcome such difficulties. Namely, we start with the linear equationu = A(t)u + f (t), t ≥ 0 (1. 2) and use a Cesàro sum to prove the existence of a periodic solution through the existence of bounded solution whose sup-norm can be controlled by the sup-norm of the input function f . Then, we use the fixed point argument to prove counterpart results for the corresponding semi-linear problem. We refer to [7] for the use of an ergodic approach for the case of Stokes and Navier-Stokes equations around rotating obstacles. Such an approach has also been extended to more general fluid flow problems in [4].
In the present paper, we will consider the existence and uniqueness of periodic solutions to partial functional differential equations (PFDE) with a ϕ-Lipschitz nonlinear term g, i.e., g(t, φ 1 ) − g(t, φ 2 ) ϕ(t) φ 1 − φ 2 C for φ 1 , φ 2 ∈ C where ϕ is a real and positive function such that sup t 0 t+1 t ϕ(τ )dτ < ∞. The difficulties we face when passing to this case of PFDE are the following two features: Firstly, since the nonlinear delay g is ϕ-Lipschitz, the standard method for construction of bounded solutions relevant for uniform Lipschitz continuous functions is no longer valid. Secondly, the evolution family generated by (A(t)) t≥0 does not act on the same Banach space as that the initial functions belong to (in fact, the former acts on X, and the latter belong to C).
To overcome such difficulties, we combine the methods and results in [9] with the use of admissible spaces and appropriate choices of nonlinear operators to prove the existence and uniqueness of the periodic solution to Equation (1.1) without using the uniform boundedness and smallness of Lipschitz constants of the nonlinear terms. Instead, the smallness is now understood as the sufficient smallness of sup t≥0 t+1 t ϕ(τ )dτ . It is worth noting that our framework fits perfectly to the situation of exponentially dichotomic linear parts, i.e., the case when family (A(t)) t≥0 generates an evolution family (U (t, s)) t≥s≥0 having an exponential dichotomy (see Definition 4.1 below), since in this case we can choose the initial vector from that emanates a bounded solution. Moreover, we can also prove the conditional stability of periodic solutions as well as the existence of a local stable manifold around the periodic solution. Our main results are contained in Theorems 2.3 and 3.1. The applications of our abstract results to semi-linear delay PFDE with the exponentially dichotomic linear parts are given in Subsection 4.1. Finally, in Subsection 4.2. we also prove the existence of a local stable manifold around the periodic solution.
We now recall some notions for latter use. Firstly, as in [10] we denote by (b) For ϕ ∈ M and σ > 0 we define functions Λ σ ϕ and Λ σ ϕ by Then, Λ σ ϕ and Λ σ ϕ belong to M, and they are bounded. Moreover, denoted by · ∞ the esssup-norm, we have Next, in space M we consider the following subset consisting of 1-periodic functions denoted by Let now ϕ be a positive function belonging to P. Then, we have the following inequalities which are taken from [10, Ineq. (1.8)] (1.6) We now recall the cone-inequality theorem which will be used to prove the conditional stability of solutions. Given a cone K in a Banach space W (see [2,Chapt. I] for the notion of a cone), we will write x y if y − x ∈ K. The following coneinequality theorem is taken from ([2], Theorem I.9.3). Theorem 1.2 (Cone Inequality). Let K be a cone given in a Banach space W such that K is invariant under a bounded linear operator A ∈ L(W ) having spectral radius r < 1. If a vector x ∈ W satisfies the inequality x Ax + z for some given z ∈ W, then it also satisfies the estimate x y, where y ∈ W is the solution of the equation y = Ay + z.
Next, for a Banach space X (with a norm · ) and a given r > 0 we denote by C := C([−r, 0], X) the Banach space of all continuous functions from [−r, 0] into X, equipped with the norm φ C = sup θ∈[−r,0] φ(θ) for φ ∈ C. For a continuous function v : [−r, ∞) → X the history function v t ∈ C is defined by v t (θ) = v(t + θ) for all θ ∈ [−r, 0]. . Consider a positive function ϕ ∈ M and let B ρ be the ball with radius ρ in C i.e, B ρ := {φ ∈ C : φ C ρ}. A function g : [0, ∞)×B ρ → X is said to belong to the class (L, ϕ, ρ) for some positive constants L, ρ if g satisfies, Now, for the space M defined as in (1.3) we denote by endowed with the norm f M := f (·) M . Clearly M is a Banach Space. We also need the following space of bounded, continuous functions 2. Bounded and periodic solutions to linear evolution equations. Given a function f taking values in a Banach space X having a separable predual Y (i.e., X = Y for a separable Banach space Y ) we consider the non-homogeneous linear problem for the unknown function u(t) where the family of partial differential operators (A(t)) t≥0 is given such that the homogeneous Cauchy problem is well-posed. By this we mean that there exists an evolution family (U (t, s)) t≥s≥0 such that the solution of the Cauchy problem (2.2) is given by u(t) = U (t, s)u(s). For more details on the notion of evolution families, conditions for the existence of such families and applications to partial differential equations we refer the readers to Pazy [19] (see also Nagel and Nickel [18] for a detailed discussion of well-posedness for non-autonomous abstract Cauchy problems on the whole line R). We next give the precise concept of an evolution family in the following definition.
Definition 2.1. A family of bounded linear operators (U (t, s)) t≥s≥0 on a Banach space X is a (strongly continuous, exponentially bounded) evolution family if Ke α(t−s) x for all t ≥ s ≥ 0 and x ∈ X.
The existence of the evolution family (U (t, s)) t≥s≥0 allows us to define a notion of mild solutions as follows. By the mild solution to (2.1) we mean a function u satisfying the following integral equation We refer the reader to Pazy [19] for more detailed treatments on the relations between classical and mild solutions of evolution equations of the form (2.1). We now state an assumption that will be used in the rest of the paper.
Assumption 2.2. We assume that A(t) is 1-periodic, i.e., A(t + 1) = A(t) for all t ∈ R + . Then (U (t, s)) t≥s≥0 becomes 1 -periodic in the sense that We also assume that the space Y considered as a subspace of Y (through the canonical embedding) is invariant under the operator U (1, 0) which is the dual of U (1, 0).
We now recall a Massera type's Theorem for existence and uniqueness of a periodic solution whose proof can be found in [ .7). For the Banach space X possessing a separable predual Y let the following condition holds true: For Then, under the Assumption 2.2, if f is 1-periodic, then Equation (2.1) has an 1-periodic mild solutionû satisfying Furthermore, if the evolution family U (t, s) t≥s≥0 satisfies: then the 1-periodic mild solution of (2.1) is unique.
3. Bounded and periodic solutions to semi-linear problems. For a Banach space X with a separable predual Y as in the previous section, we now consider the following partial functional differential equation where the linear operators A(t), t ≥ 0, act on X and satisfy the hypotheses of Theorem 2.3, and the linear term F : [0, ∞) → L(C, X) and the nonlinear term g : [0, ∞) × C → X satisfies: (2) the map t → F (t) belongs to P, (3) g belongs to class (L, ϕ, ρ) for some L, ρ > 0 and 0 < ϕ ∈ P, Furthermore, by the mild solution to (3.1) we mean the function u satisfying the following equation We then come to our next result on the existence and uniqueness of the periodic mild solution to Equation (3.1).
Theorem 3.1. Assume that there exists a constant M such that for each f ∈ M there is a mild solution u of (2.1)satisfying u ∈ C b (R + , X) and and that the evolution family U (t, s) t≥s≥0 satisfies: Let F and g satisfy the conditions in (3.2). Then, if γ := F (·) M + ϕ M is small enough, Equation (3.1) has one and only one 1-periodic mild solutionû in We then define the following transformation Φ given as follows: Consider the equation for given v ∈ C b ([−r, ∞), X) with u being the solution: where u ∈ C b (R + , X) is the unique 1-periodic solution to (3.5) (the existence and uniqueness of such a u is guaranteed by Theorem 2.3), andũ(t), −r t < 0, is the 1-periodic extension of u on the interval [−r, 0). We will prove that if γ is small enough, then the transformation Φ acts from B 1 ρ into itself and is a contraction. To do this, fixing any v ∈ B 1 ρ , since F and g satisfy the conditions in (3.2) we have (3.7) Applying Theorem 2.3 for the right-hand side F (τ )(v τ ) + g(τ, v τ ) instead of f (τ ) (in formula of the mild solution) we obtain that for v ∈ B 1 ρ there exists a unique 1-periodic solution u to (3.5) satisfying Here, the first equality in the above formulas holds true since Φ(v)(t), t ∈ [−r, 0), is the 1-periodic extension of u to the interval [−r, 0). Therefore, we obtain that if γ is small enough, then the map Φ acts from B 1 ρ into itself. Now, by Formula (3.5) we have the following representation of Φ where, as above, the functionũ(t) is the 1-periodic extension to interval [−r, 0) of the periodic function Since u(t), t ≥ 0, is 1-periodic, and for −r t < 0 the functionũ(t) is an 1-periodic extension of u to interval [−r, 0), we have that Thus, from Theorem 2.3 and the fact that g belongs to class (L, ϕ, ρ) we arrive at We thus obtain that if γ = F (·) M + ϕ M is small enough, then Φ : is a contraction. Therefore, for such a γ there exists a unique fixed pointû in B 1 ρ of Φ, and by definition of Φ, this functionû is the unique 1-periodic mild solution to Equation (3.1).

4.
Periodic solutions in the case of dichotomic evolution families.

4.1.
Existence, uniqueness and conditional stability. In this subsection, we will consider equations (2.3) and (3.3) in the case that the evolution family (U (t, s)) t≥s≥0 has an exponential dichotomy. In this case, the existence of bounded solutions to (2.3) (i.e., bounded mild solutions to (2.1)) is convenient to prove. Therefore, the existence and uniqueness of periodic solutions to (2.3) and hence to (3.3) easily follow. Moreover, using the cone-inequality Theorem 1.2, we will show the conditional stability of such periodic solutions. To do so, we start with the definitions of exponential dichotomy and stability of an evolution family.
The projections P (t), t ≥ 0, are called the dichotomy projections, and the constants N, ν -the dichotomy constants.
(2) The evolution family U is called exponentially stable if it has an exponential dichotomy with the dichotomy projections P (t) = Id for all t ≥ 0. In other words, U is exponentially stable if there exist positive constants N and ν such that We remark that properties (a)-(d) of dichotomy projections P (t) already imply that i) H := sup t≥0 P (t) < ∞, ii) t → P (t) is strongly continuous (see [16,Lemm. 4.2]). We refer the reader to [5] for characterizations of exponential dichotomies of evolution families in general admissible spaces.
If (U (t, s)) t≥s≥0 has an exponential dichotomy with dichotomy projections (P (t)) t≥0 and constants N, ν > 0, then we can define the Green's function on a half-line as follows: Also, G(t, τ ) satisfies the estimate Using the projections (P (t), t ≥ 0 on X, we can define the family of operators P (t), t ≥ 0 on C as follows.
Lemma 4.2. Let the evolution family (U (t, s)) t≥s≥0 have an exponential dichotomy with the corresponding dichotomy projections (P (t)) t≥0 and dichotomy constants N, ν > 0. Let f ∈ M, and let F and g satisfy conditions given in (3.2). Then, the following assertions hold true.
(a) Let v ∈ C b (R + , X) be the solution to equation (2.3). Then, v can be rewritten in the form where G(t, τ ) is the Green's function defined by equality (4.2).
ρ for a fixed ρ > 0. Then, for t ≥ 0 this function u(t) can be rewritten in the form for some η = P (0)φ(0) ∈ X 0 where G and X 0 are determined as in Item (a). Proof.
We now prove the conditional stability of periodic solutions to (3.3). To do this, for a}, respectively. Let B ρ (0) be the ball containingû as in Assertion (b) of Theorem 4.4.
Suppose further that there exists a positive functionφ ∈ P such that: for some positive constants C µ and µ independent of u,û and ρ.
Proof. Putting w = u −û we have that u is a solution to Equation (3.3) in B ρ (û) with u 0 = ζ if and only if w is the solution in B ρ (0) of the equation for t 0. (4.11) We now prove that Equation (4.11) has a unique solution in B ρ (0). To do this, puttingg(t, w t ) = g(t, w t +û t ) − g(t,û t ) we obtain thatg(t, 0) = 0 andg belongs to class (ρ,φ, 2ρ). Setting ξ = P (0)ζ(0) − P (0)û(0) we prove that the transformation K defined by for − r t 0 acts from B ρ (0) into itself and is a contraction. In fact, we have Thus, we obtain that if ( F (·) M + φ M ) is small enough, then the transformation K acts form B ρ (0) into B ρ (0).
(4.13) We will use the cone-inequality Theorem 1.2 applying to Banach space W := L ∞ (R + ) which is the space of real-valued functions defined and essentially bounded on R + (endowed with the esssup-norm denoted by · ∞ ) with the cone K being the set of all (a.e.) nonnegative functions. We then consider the linear operator B defined for u ∈ W by (Bu)(t) = (1 + H)N e νr ∞ 0 e −ν|t−τ | ( F (τ ) +φ(τ ))u(τ )dτ for t 0.
By inequalities in (1.6) we have that Therefore, B ∈ L(W ) and B Obviously, B leaves the cone K invariant. The inequality (4.13) can now be rewritten as φ Bφ + z for z(t) = ρ 2 e νr e −νt , t 0.
Hence, by cone-inequality Theorem 1.2 we obtain that φ ψ, where ψ is a solution in W of the equation ψ = Bψ + z which can be rewritten as (4.14) We now estimate ψ. To that purpose, for we set h(t) = e µt ψ(t) for t 0. Then, by (4.14) we obtain that Therefore, D ∈ L(W ) and D Since µ < ν + ln(1 − (1 + H)N e νr (N 1 + N 2 )( F (·) M + φ M )) we obtain that Therefore, the equation h = Dh +z is uniquely solvable in L ∞ (W ), and its solution is h = (I − D) −1z . Hence, we obtain that . Therefore, .
This yields that h(t) C µ ρ for t 0.
For an exponentially stable evolution family (see Definition 4.1 (2)) we have the following corollary which is a direct consequence of Theorem 4.5.
Corollary 4.7. Let the assumptions of the Theorem 4.4 hold, and letû be the periodic solution of (3.3) obtained in assertion (b) of Theorem 4.4. Let further the evolution family (U (t, s)) t≥s≥0 be exponentially stable. Then, the periodic solution u is exponentially stable in the sense that for any other solution u ∈ C b ([−r, ∞), X) of (3.3) such that u 0 −û 0 C is small enough we have for some positive constants C and µ independent of u andû.
Proof. We just apply Theorem 4.5 for P (t) = Id for all t ≥ 0 to obtain the assertion of the theorem.

4.2.
Local stable manifold around the periodic solution. In this subsection, under the same hypotheses as the previous subsection, we will prove the existence of a local stable manifold for Equation (3.3) near its periodic solution. As in the previous subsection, we denote by B r (φ) the ball in C centered at φ with radius r. We then give the definition of a local stable manifold for Equation (3.3) around its periodic solution. Then, a set S ⊂ R + × C is said to be a local stable manifold for the Equation (3.3) aroundû if for every t ∈ R + the phase space C splits into a direct sum C = X 0 (t) ⊕ X 1 (t) with corresponding projections P (t) (i.e X 0 (t) =Im P (t) and X 1 (t) =Ker P (t)) such that sup t≥0 P (t) < ∞ and if there exist positive constants ρ, ρ 0 , ρ 1 and a family of Lipschitz continuous mappings , t ∈ R + with the Lipschitz constants being independent of t such that and we denote by to each ψ ∈ S t0 there corresponds one and only one solution u(t) of Equation (3.3) on [t 0 − r, ∞) satisfying conditions u t0 = ψ and sup t≥t0 u t C ρ. Moreover, every solution u(t) on the manifold S is exponentially attracted tô u(t) in the sense that, there exist positive constants µ and C µ independent of t 0 ≥ 0 such that (4.17) Note that, if we identify X 0 (t) ⊕ X 1 (t) with X 0 (t) × X 1 (t), then we can write S t = graph(h t ) where graph(h t ) is denoted for the graph of the mapping h t .
We now state and prove our last result on the existence of a stable manifold for solutions to Equation  Proof. We will apply our result obtained in [8,Theorem 3.7]. To this purpose, let u be a solution to Equation (3.3) and put w = u −û. Then, u satisfies Equation Putting nowg(t, w t ) = g(t, w t +û t )−g(t,û t ) we obtain thatg(t, 0) = 0 andg belongs to class (2ρ,φ, 2ρ) since g satisfies the assumption of Theorem 4.5. Therefore, by a similar way as in the proof of [8,Theorem 3.7] we obtain that if F (·) M + φ M is small enough, then there exists a local stable manifold S (near 0) for Equation (4.18). Returning to the solution u of (3.3) by replacing w by u −û, we obtain that, this manifold S is the local stable manifold for Equation (3.3) near the solution u.
We finally illustrate our results by the following example.
Therefore, F and g satisfies the hypotheses of Theorems 4.4 and 4.5 with ρ = a, L := ρ + γ 2ρ , ϕ(t) = 2ρψ(t) andφ(t) = 4ρψ(t). By Theorem 4.4 and 4.5 we obtain that, if c is large enough (consequently, F (·) M + ϕ M and F (·) M + φ M are small enough), then Equation (4.19) has one and only one 1-periodic mild solution u ∈ B ρ (0) and this solutionû is conditionally stable in the sense of Remark 4.6. Moreover, by Theorem 4.9, there exists a local stable manifold for mild solutions to Equation (4.19) around the periodic solutionû.