PERIODIC SOLUTIONS OF NON-DENSELY DEFINED DELAY EVOLUTION EQUATIONS

where the linear operator is non-densely defined and satisfies the Hille-Yosida E condition. First, we obtain some properties of “integral solutions" for this case and prove the compactness of an operator determined by integral solutions. This allows us to apply Horn's fixed point theorem to prove the existence of periodic integral solutions when integral solutions are bounded and ultimately bounded. This extends the study of periodic solutions for densely defined operators to the non-densely defined operators. An example is given. Periodic Solutions, Non-Densely Defined Equations.


INTRODUCTION.
In this note, we consider finite delay evolution equation x (t) = Ax(t) + F (t, x t ), t ≥ 0, x 0 = ϕ ∈ C ([−r, 0] , E) , (1.1) where A is a non-densely defined linear operator in a Banach space E; C = C ([−r, 0] , E) is the space of continuous functions from [−r, 0] (r > 0 is a constant) to E endowed with the super-norm; and for every t ≥ 0, the function x t ∈ C is defined by and F (t, ϕ) is a continuous function from × C to E, and is ω−periodic in t.
It's well known that if A is the infinitesimal generator of a C 0 -semigroup of bounded linear operators, or equivalently (i).D(A) = E, (D means domain), (ii).there exist M ≥ 0 and τ ∈ such that ]τ, ∞[ ⊂ ρ(A), and sup where ρ(A) is the resolvent set of A, then the classical semigroup theory ensures the wellposedness of Eq. (1.1).( [12]).
In this work we investigate the case when the operator A satisfies only the assumption (ii) above, that is, when A is non-densely defined; and we are concerned with the existence of periodic solutions for Eq.(1.1).The related studies and examples concerning non-densely defined operators can be found in the references such as [1,3,4,7].For example, in [3], the authors proved that Eq.(1.1) is wellposed in the set The problem of finding periodic solutions is an important subject in the qualitative study of ordinary and functional differential equations.The famous Massera's theorem [9] on twodimensional periodic ordinary differential equations explains the relationship between the bounded solutions and periodic solutions.Using Browder's fixed point theorem, Yoshizawa proved in [13] that if the solutions of an n−dimensional periodic ordinary differential equation are either uniformly bounded or uniformly ultimately bounded, then the system has a periodic solution.Hale and Lopes [6] used Horn's fixed point theorem to obtain the same result for n−dimensional periodic ordinary and functional differential equations with finite delay.In [11], Show studied the Massera's theorem for some functional differential equations in finite dimensional spaces and proved that the existence of a bounded solution implies the existence of a periodic solution.
In many of those studies, the most important feature is to show that the operator is compact (continuous and takes a bounded set into a precompact set), where ω is the period of the system and x is the unique solution determined by φ.Then, some fixed point theorem can be used to derive periodic solutions.
For equations in general Banach spaces (infinite dimensional), showing the compactness of operator P is a very hard task, because the difficulty involving the abstract version of Arzela-Ascoli's theorem.As can be seen in [2,8,14], if this difficulty can be overcome, then other steps for equations in finite dimensional spaces can be carried over to show that operator P is compact, and hence to derive periodic solutions.For example, in [8], the author treated the evolution equation where A(t) is ω−periodic in t and generates an evolution system (U (t, s)) t≥s≥0 in a Banach space E, and f (t, ., .) is ω−periodic in t; and proved the compactness of P and hence the existence of periodic mild solutions using the boundedness and ultimate boundedness of mild solutions when ω > r.
When operator A in Eq.(1.1) satisfies only (ii) above, that is, when A is non-densely defined, the appropriate solutions to work with will be integral solutions; see Definition 2.1.To our knowledge, we have not seen results about periodic integral solutions, thus we would like to provide one here.Similar to the cases for densely defined operators mentioned above, the most important step here is to show that P φ = x ω (•, φ) is compact, where x is the unique integral solution determined by φ.After this, we are able to apply Horn's fixed point theorem and prove, under the boundedness and ultimate boundedness assumptions, the existence of ω−periodic integral solutions for Eq.(1.1) when ω > r.This way, we can extend the study of periodic solutions for densely defined operators to non-densely defined operators.Finally, we give an example of a partial functional differential equation in continuous functions space with the super-norm.

THE COMPACTNESS OF OPERATOR P.
In this section we make the following assumptions.
(H 1 ).The operator A is a Hille-Yosida operator.That means: there exist M > 0 and τ ∈ such that ]τ, ∞[ ⊂ ρ(A), and sup (H 2 ).The function F is continuous in t and is locally Lipschitz in the second variable in the sense that for each h > 0 there exists a constant k 1 (h) > 0 such that We now list the following definitions.
From the closedness of operator A, we see that if an integral solution x of Eq.(1.1) is continuously differentiable, then x is a strict solution of Eq.(1.1).
We can use methods similar to those in [3] for autonomous equations to obtain the following existence and uniqueness results concerning integral solutions and strict solutions for non-autonomous equations.Details are omitted here.with its maximal interval of existence [−r, t ϕ [ , t ϕ > 0, and either t ϕ = ∞, or lim sup Moreover x(., ϕ) is a continuous function of ϕ in the sense that, for ϕ ∈ C 0 and t ∈ [0, t ϕ [, there exist positive constants L and ε such that for any ψ ∈ C 0 with |ϕ − ψ| < ε, we have Theorem 2.2.Assume that (H 1 ) and (H 2 ) hold.Furthermore assume that F is continuously differentiable, and for each h > 0, there exist positive constants k 2 (h) and k 3 (h) such that Then the integral solution x(ϕ) of Eq.(1.1) is a strict solution in [0, t ϕ [ .Remark 2.2.If we assume that function F (t, ϕ) is continuous and Lipschitz with respect to the second argument, then the integral solution of Eq.(1.1) exists for all t ≥ 0. Moreover the solutions x(t, ϕ) are locally bounded in t and ϕ.In the following, we're concerned with periodic integral solutions, so we will assume that all integral solutions exist on [0, ∞).

Now, we introduce the part A 0 of A in D(A), defined by
Then it's well known ( [1,7]) that the part A 0 of A generates a strongly continuous semigroup T 0 (•) on D(A); and for ϕ ∈ C 0 , the integral solution of Eq.(1.1) is given by where In the sequel, we need to prove the compactness of operator P , so we assume (H 3 ).The semigroup (T 0 (t)) t≥0 is compact on D(A).That means for each t > 0, operator T 0 (t) is compact on D(A).
Now, we are ready to prove our main result in this paper, that is, the compactness of operator P .Theorem 2.3.Assume that (H 1 ), (H 2 ), (H 3 ) hold and that all integral solutions of Eq.(1.1) exist for t ≥ 0 and are locally bounded (that is, for any t > 0, the integral solutions of Eq.(1.1) are bounded on [0, t] by a constant if their initial functions are bounded by a constant).Let ω > r be fixed.Then the operator P ϕ = x ω (., ϕ) on C 0 is compact (continuous and takes a bounded set into a precompact set).

Proof:
The continuity of P follows from Theorem 2.1.Let B be a bounded set in C 0 .Then, as integral solutions are locally bounded, it follows that the set is bounded in C 0 .Next, we will show that W is precompact in C 0 by using Arzela-Ascoli's theorem.For that end we first prove that for every fixed θ ∈ [−r, 0], is precompact in D(A).From (2.1), we have Since ω > r is fixed, we can choose ε > 0 such that ω + θ − ε > 0. Then and Therefore, by Assumption (H 3 ) and the local boundedness assumption on integral solutions, we see that is precompact in D(A) because T 0 (ε) is applied to a bounded set.Next, for some positive constant a, one has This implies that the set is totally bounded, and therefore the set {x(ω + θ, ϕ) : ϕ ∈ B} is precompact in D(A).
It remains to prove the equicontinuity of functions in set W .Let θ, θ 0 ∈ [−r, 0] with θ > θ 0 .Then ω + θ > ω + θ 0 > 0, and From assumption (H 3 ) and a result in [10], semigroup (T 0 (t)) t≥0 is uniformly continuous for t > 0, which implies that lim We also have lim λ→∞ As in (2.3), we obtain that there is a compact set K in D(A) such that lim Thus, from Banach-Stenhauss's theorem, we have lim The proof for θ < θ 0 is similar.This completes the proof. 2 3 BOUNDEDNESS AND PERIODICITY.
After showing the compactness of operator P , we can follow [2,8,14] and derive periodic integral solutions for Eq.(1.1).In the sequel we make the following assumption.
We also need to bound certain terms involving integral solutions, so we define the following.Definition 3.1.We say that integral solutions of Eq.(1.1) are bounded if for each Definition 3.2.We say that integral solutions of Eq.(1.1) are ultimate bounded if there is a bound B > 0, such that for each We also list the following result as a reference.Lemma 3.1.[6] (Horn's fixed point Theorem).Let E 0 ⊂ E 1 ⊂ E 2 be convex subsets of Banach space Z, with E 0 and E 2 compact subsets and E 1 open relative to E 2 .Let P : E 2 → Z be a continuous map such that for some integer m one has With these preparations, we can prove Theorem 3.1.Assume that (H 1 ), (H 2 ), (H 3 ) and (H 4 ) hold.Furthermore suppose that integral solutions of Eq.(1.1) are bounded and ultimate bounded.Then Eq.(1.1) has an ω−periodic integral solution.
Proof: From Theorem 2.3, we know that operator P φ = x ω (•, φ) on C 0 is compact.Let B be the bound in the definition of ultimate boundedness, then by boundedness, there is a Let x(t) be an integral solution of Eq.(1.1) and define y(t) = x(t + ω).Then, for t ≥ 0, This implies that y(t) = x(t+ω) is also an integral solution of Eq.(1.1).Then the uniqueness implies that P m ϕ = x mω (., ϕ) for ϕ ∈ C 0 .Thus, we have the following where cov.(F ) is the convex hull of the set F .Then we see that E 0 , E 1 and E 2 are convex subsets of C 0 with E 0 and E 2 compact subsets and E 1 open relative to E 2 , and Consequently, from Horn's fixed point theorem we know that operator P has a fixed point φ.Then, for the integral solution x(•) = x(•, φ) with x 0 (•, φ) = φ, we see from (3.1) that y(t) = x(t + ω, φ) is also an integral solution with y 0 = x ω (φ) = P φ = φ.Thus the uniqueness implies x(t, φ) = y(t) = x(t + ω, φ); therefore x(t, φ) is an ω−periodic solution.This completes the proof.
Next, we give a criteria as in [8] to guarantee the boundedness and ultimate boundedness of integral solutions of Eq.(1.1).The proof can be found in [8].Theorem 3.2.Assume that there exist functions W i , i = 1, 2, 3 with W i : [0, ∞[ → [0, ∞[ , W i (0) = 0, W i strictly increasing, and W 1 (t) → ∞, as t → ∞.Further assume that there exists a Liapunov function V : E → such that for some M > 0, when x is an integral solution of Eq.(1.1) with |x(t)| ≥ M, then Then integral solutions of Eq.(1.1) are bounded and ultimate bounded.

AN EXAMPLE.
In the following, we apply our results here to the partial functional differential equation where u t (•, x) (not a partial derivative) is defined by u We study Eq.(4.1) in E = C[0, 1], the space of all continuous functions on [0, 1] with the super-norm, and define Then the closure of D(A) is thus A is not densely defined on E. Now, (10.2) in [4] is true, and it implies (1.1) in [4].Thus Hille-Yosida condition (H 1 ) in our paper is satisfied.
Next, similar to the argument in [5], we have where q = arg λ and B(E 0 ) is the Banach space of all bounded linear operators on E 0 .Now from [4] or [10], A 0 generates an analytic semigroup on E 0 .Then similar to [10], the analytic semigroup is continuous in the uniform operator topology for t > 0; and (λ − A 0 ) −1 , λ / ∈ (−∞, 0], maps bounded set in E 0 into bounded set in E 0 with a uniform bound on their first derivatives.It now follows from Arzela-Ascoli's theorem that (λ − A 0 ) −1 , λ / ∈ (−∞, 0], is a compact operator.Therefore the semigroup in compact according to [10]. 2 For this example, conditions (H 1 )-(H 4 ) can be met, so that results here can be applied.Details are left to the reader.