EXISTENCE OF PERIODIC SOLUTIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

In this work we establish a result of existence of periodic solutions for quasi-linear partial neutral functional differential equations with unbounded delay on a phase space defined axiomatically.


Introduction
In this note we establish a result of existence of periodic solutions for partial neutral functional differential equations with unbounded delay that can be modelled in the form where A is the infinitesimal generator of a strongly continuous semigroup of linear operators on a Banach space X and both F as G are appropriate functions. These equations will be called abstract neutral functional differential equations (abbreviated, ANFDE) with unbounded delay.
We refer the reader to [6] for a brief historical review and for the basic qualitative properties of the ANFDE with unbounded delay. Next, for completeness, we collect the notions that will be needed in section 2.
Throughout this paper X will be a Banach space provided with a norm · and A : D(A) ⊆ X → X will be the infinitesimal generator of a strongly continuous semigroup of linear operators T (t) defined on X. For the theory of strongly continuous semigroups of linear operators we refer to Nagel [11] and Pazy [12]. We only recall here some notions and properties that will be essential for us. In particular, it is well known that there exist constantsM ≥ 1 and ω ∈ IR such that T (t) ≤M e ωt , t ≥ 0. (1.2) Moreover, if T is an uniformly bounded and analytic semigroup with infinitesimal generator A such that 0 ∈ ρ(A) (the resolvent set of A), then it is possible to define the fractional power (−A) α , for 0 < α ≤ 1, as a closed linear operator on its domain D((−A) α ). Furthermore, the subspace D((−A) α ) is dense in X and the expression defines a norm on D((−A) α ). Hereafter we represent by X α to the space D((−A) α ) endowed with the norm · α . The following properties are well known ( [12]). Lemma 1.1 : Suppose that the preceding conditions are satisfied. (a) Let 0 < α ≤ 1. Then X α is a Banach space. (b) If 0 < β < α ≤ 1 then X α → X β and the imbedding is compact whenever the resolvent operator of A is compact.
(c) For every a > 0, there exists a positive constant C a such that (d) For every a > 0, there exists a positive constant C a such that To study equation (1.1) we assume that the histories x t : (−∞, 0] → X, x t (θ) := x(t + θ), belong to some abstract phase space B, that is a phase space defined axiomatically.
In this work we will employ an axiomatic definition of the phase space B introduced by Hale and Kato [3]. To establish the axioms of space B we follow the terminology used in the book [8]. Thus, B will be a linear space of functions mapping (−∞, 0] into X endowed with a seminorm · B . We will assume that B satisfies the following axioms: (A) If x : (−∞, σ + a) → X, a > 0, is continuous on [σ, σ + a) and x σ ∈ B then for every t in [σ, σ + a) the following conditions hold: Where H ≥ 0 is a constant; K, M : [0, ∞) → [0, ∞), K is continuous and M is locally bounded and H, K and M are independent of x(·).
We will denote byB the quotient Banach space B/ · B and, if ϕ ∈ B we writeφ for the coset determined by ϕ.
The axiom (A-1) implies that the operator functions S(·) and W (t) given by and are strongly continuous semigroups of linear operators on B.
In this note we will establish existence of solutions and existence of periodic solutions of equation (1.1), assuming that T (·), F and G satisfy certain compactness conditions. Similar results, but based on the contraction mapping theorem have been established in [7].
Throughout this paper we always assume that B is a phase space. The terminology and notations are those generally used in operator theory. In particular, if X and Y denote Banach spaces, we indicate by L(X, Y ) the Banach space of bounded linear operators from X into Y and we abbreviate this notation to L(X) whenever X = Y . In addition, we denote by B r [x] the closed ball with center at x and radius r and we reserve the bold type α to represent the Kuratowski's measure of non-compactness. For the properties of the measure α see Deimling [1].

Existence of periodic solutions
We begin by studying existence of mild solutions of the abstract Cauchy problem where Ω is an open subset of B; F, G : [σ, a] × Ω → X are continuous functions and 0 ≤ σ < a.
Henceforth we will assume that A is the infinitesimal generator of an analytic semigroup T (·) of bounded linear operators on X. In further, to avoid unnecessary notations, we suppose that 0 ∈ ρ(A) and that the semigroup T (·) is uniformly bounded, that is to say, T (t) ≤ M , for some constant M ≥ 1 and every t ≥ 0. Definition 2.1 : We will say that a function x : (−∞, σ + b) → X, b > 0, is a mild solution of the Cauchy problem (2.1)-(2.2) if x σ = ϕ; the restriction of x(·) to the interval [σ, σ + b) is continuous and for each σ ≤ t < σ + b the function AT (t − s)F (s, x s ), s ∈ [σ, t), is integrable and the integral equation is verified.
Initially we establish a result of existence of local solutions.
Proof. To simplify the notations we consider σ = 0. In view of (−A) β F and G are continuous functions and Ω is open in B we can assert that there exists 0 < r < r(ϕ) such that Since W (·)ϕ is continuous , we can choose δ > 0 such that for all 0 ≤ t ≤ δ . We set K := max 0≤t≤δ K(t) and r := min{r 0 , r /(2 K)}.
If x(·) satisfies the equation (2.3), we can decompose it as x(t) = u(t) + y(t, ϕ), t ≤ δ, such as was indicated in the observations preceding the statement of this theorem. It is clear that the function u(·) verifies the equation where we have abbreviated the notations by using y(·) instead of y(·, ϕ). This expression leads us to define the maps T , T 1 , and T 2 on C(δ, r) by means of In what follows we will show that T 1 and T 2 are completely continuous maps with values in C([0, δ]; X) and that T has compact range included in C(δ, r).
To prove these assertions, we observe initially that if Now, since G is a continuous function it is clear that T 2 is well defined on C(δ, r) and that (T 2 u)(·) is also a continuous function . On the other hand, since F is X β -valued and (−A) β F is continuous then both (−A) β F (s,ũ s + y s ) as F (s,ũ s + y s ) are continuous . In addition, in view of T (·) is an analytic semigroup ( see [12]), the operator function s → AT (t − s) is continuous in the uniform operator topology on [0, t) and thus AT (t − s)F (s,ũ s + y s ) is also continuous on [0, t). Applying the estimations established in Lemma 1.1 we obtain that This concludes the proof that T 1 is a well defined map with values in C([0, δ]; X).
and we can consider u ∈ C(δ 0 , r), it is follows from (2.5), (2.6) and (2.7) that Next we will prove that the range of T 1 is relatively compact. By Ascoli's theorem it is sufficient to show that the set R(T 1 ) is equicontinuous on [0, δ] and R(T 1 )(t) is relatively compact in X for each 0 ≤ t ≤ δ. We begin by showing this last assertion. Clearly, we may suppose that t > 0. Let 0 < η < t. Then From (a-2) we obtain that F (s,ũ s + y s ), 0 ≤ s ≤ δ, is included in a compact set and since (−A)T (η) is a bounded operator, by the mean value theorem for the Bochner integral ( [9]) we infer that the first term of the right hand side is also included in a compact set. Furthermore, since To prove the equicontinuity of R(T 1 ) at t 0 we take 0 ≤ t 0 < t < δ.
From the definition of T 1 it follows that From this expression and using both the compactness of for u ∈ C(δ, r), we obtain that R(T 1 ) is equicontinuous from the right at t 0 . Proceeding similarly we can prove that R(T 1 ) is equicontinuous at any t 0 ≥ 0. On the other hand, using hypotheses (a-2) and (b-1), and repeating the argument (see also [4] Finally, applying the Schauder's fixed point theorem we conclude the existence of a fixed point u( 2), which completes the proof of the theorem.
Proceeding as above we can also establish a result of existence of global solutions.
Proof. If we suppose that b < ∞, it follows from (a-3) that there exists lim t→b − x(t, ϕ). Hence we obtain that the extension of is also a solution of the problem (2.1) with initial condition x 0 = ϕ, which is contrary to our hypothesis.
In connection with these results it should be noted that if (−A) β F (t, ϕ) and G(t, ϕ) satisfy further certain local Lipschitz or yet Hölder conditions on ϕ then the mild solution of (2.1)-(2.2) is unique. Next, for completeness, we state a pair of results of this type. For the sake of brevity we omit their proofs.
Lemma 2.1 : Assume that for each ϕ ∈ Ω and each σ ≥ 0 there exist positive constants r, δ, C 1 and C 2 such that the following conditions hold: Then the mild solution of problem (2.1)-(2.2) is unique. Lemma 2.2 : Assume that for each ϕ ∈ Ω and each σ ≥ 0 there exist constants C, δ > 0, 0 < ν < 1 and continuous functions k 1 , k 2 : [0, ∞) → [0, ∞) such that the following conditions hold: In the sequel we assume that the functions F, G and the semigroup T (·) satisfy appropriate conditions to assure existence and unique-ness of mild solutions of equation In these conditions we refer to (2.11) as an ANFDE(F,G) system. Furthermore, we will say that the ANFDE(F,G) system is ω-periodic if F (t, ϕ) and G(t, ϕ) are ω-periodic at t. In the rest of this section we use ω to represent a fixed strictly positive constant.
Definition 2.2 : We will say that a function x : IR → X is an ω-periodic solution of equation (2.11) if x(·) is a mild solution of (2.11) with initial condition x 0 ∈ Ω and It is clear that if x : IR → X is a function such that x 0 = ϕ ∈ B, the restriction of x(·) on [0, ω) is continuous and x ω = ϕ then ϕ is ω-periodic on (−∞, 0]. In addition, if the ANFDE(F,G) system is ω-periodic and x(·, ϕ) is a mild solution of (2.11)-(2.12) then the condition x ω = ϕ is sufficient to guarantee that x(·, ϕ) is an ωperiodic solution of (2.11). Since this is the essential property in order to obtain existence of periodic solutions we shall state it formally. Proposition 2.1 : Assume that the ANFDE(F,G) system is ωperiodic and that the mild solution of (2.11) with initial condition x 0 = ϕ is defined on IR. If x ω (·, ϕ) = ϕ then x(·) is an ω-periodic solution.
Let E be a non empty closed subset of Ω such that the mild solution x(·, ϕ) of (2.11)-(2.12) is unique and defined on [0, ω], for each ϕ ∈ E. In this case we represent by P ω the map E → B, ϕ → x ω (·, ϕ).
If the ANFDE(F,G) system is ω-periodic, it is clear from the previous Proposition that a sufficient condition for the existence of a periodic solution of (2.11) is the existence of a fixed point for the map P ω . In order to establish the existence of a fixed point of P ω we will apply the Sadovskii's Theorem. Nevertheless, an essential condition needed to apply this result is that the domain of the respective map it will be bounded, closed and convex. For this reason we introduce the following assumption.
Later we will exhibit a class of ANFDE(F,G) systems for which this assumption is valid. Since one of the hypotheses of the Sadovkii's theorem is the continuity of the respective map, our next result establish a criteria to assure the continuity of P ω . Theorem 2.2 : Suppose that assumption (F,G) holds. If we assume further that: (a-4) There exists β ∈ (0, 1) such that F is X β -valued and the function (−A) β F is continuous and takes closed and bounded sets into bounded sets.
(b-2) The map G takes closed and bounded sets into bounded sets and for each closed and bounded set B ⊆ Ω and each t > 0 there exists a compact subset W t of X such that T (t)G(s, ψ) ∈ W t , for all ψ ∈ B and every 0 ≤ s ≤ ω; then the map P ω : E → B, ϕ → x ω (·, ϕ) is continuous.
Proof. We begin by showing that for each relatively compact subset B of B the set ∪ ϕ∈B R( F ϕ ) is relatively compact in C([0, ω]; X). In fact, from the continuity of F and the compactness of the interval [0, ω] we obtain that for each ε > 0 and each ϕ ∈ Ω there exists δ > 0 such that sup for all 0 ≤ t ≤ ω and certain constant C 1 > 0, it follows that there is δ(ϕ) > 0 such that for all u ∈ C(ω, r) and ϕ − ψ B ≤ δ(ϕ). In view of the fact that B is relatively compact , we can assert the existence of ϕ 1 , ϕ 2 , · · · , ϕ n such Applying (a-5) we infer that the first set on the right hand side of the above inclusion is relatively compact . Since ε was chosen arbitrarily this proves our asertion. Let now (ϕ n ) n be a sequence in E which converges to ϕ. We are going to prove that P ω ϕ n converges to P ω ϕ. Let x n := x(·, ϕ n ). First we will show that the set {x n : n ∈ IN } is relatively compact in C([0, ω]; X). In fact, since x n (t) = z n (t) − F (t, x n t ), where and by the Assumption (F, G) the set {x n t : 0 ≤ t ≤ ω, n ∈ IN } is bounded, we may proceed as in the proof of Theorem 2.1 to show that {z n (·) : n ∈ IN } is relatively compact in C([0, ω]; X). Furthermore, turning to use the boundedness of the set {x n t : 0 ≤ t ≤ ω, n ∈ IN } we obtain that there is r > 0 for which which implies the set {F (t, x n t ) : n ∈ IN } is relatively compact in C([0, ω]; X). This completes the proof of this asertion.
We are now in conditions to establish the main result of this section.
Then the equation (2.11) has an ω-periodic solution.
Proof. Since T (·) is a compact semigroup and G(·) takes bounded and closed sets into bounded sets, it follows from Theorem 2.2 that the map P ω ( in short, P ) : E → E, ϕ → x ω (·, ϕ) is continuous . Consequently, there exists an induced mapP :Ê →Ê which satisfies the conditionP (φ) = P (ϕ), for everyφ ∈Ê and every ϕ ∈φ. Our objective is to show thatP is a condensing map. Since for each subset C ofÊ there is D ⊆ E such that C =D and α(C) = α(D), we restrict our attention to estimate the value α(P (D)), for each D ⊆ E with α(D) > 0.
Proceeding as in the proof of Theorem 2.2 we obtain that D 2 [σ 1 , σ 2 ] is a relatively compact subset of C([σ 1 , σ 2 ]; X) so that Furthermore, since the semigroup T is compact , α(D 1 [σ 1 , σ 2 ]) = 0, for σ 1 > 0 and On the other hand, combining the previous estimations with the The-orem 2.1 in [14] we obtain that for each 0 < σ ≤ ω which, by condition (d), implies thatP is a condensing map. Finally, by the Sadovskii's fixed point theorem ( [13]), we infer that P has a fixed point in E and, based in our previous discussion, we can assert that there exists an ω-periodic solution of (2.11).
In practical applications the property K(·) bounded and M (t) convergent towards zero when t goes to infinity is frequently encountered (see [8]). For this reason, next we present a pair of consequences of Theorem 2.3 in the context of phase spaces verifying such property. In this case, and without any danger of confussion, we will employ the same symbol K to denote sup t≥0 K(t). First we present a class of systems which satisfy the Assumption (F,G). and G(t, ϕ) ≤ N 3 ϕ B + N 4 (2.15) then for N 1 and N 3 µ small enough and for every ϕ ∈ B the mild solution x(·, ϕ) is defined and bounded on IR and the Assumption (F, G) is fulfilled.
Proof. Let x = x(·, ϕ) be the mild solution of (2.1-2.2) corresponding to σ = 0. From Theorem 2.1 we obtain that x is defined on (−∞, b), for some b > 0. Applying (2.3) we infer the existence of positive constants C 1 and C 2 which are independent of ϕ such that for 0 ≤ t < b. This estimation and Theorem 2.2 show that x is defined and bounded on IR. Furthermore, it is clear from this estimation that if we choose R > C 2 K and ω large enough then for 0 ≤ s ≤ t < b and for every function x : (−∞, b) → X which is continuous on [0, b) and x t ∈ Ω, for 0 ≤ t < b, then F satisfies condition (a-3). If, in further, F (t, ·) is completely continuous for each 0 ≤ t ≤ ω, then F also verifies (a-5). Working in concrete phase spaces it is not difficult to present examples of ANFDE(F, G) systems which verify the conditions considered in the previous results.
Next we study a linear operator in the space B := C r × L p (g; X), with r = 0, defined in the Example 1.1.
Example 2.2. Let B = C r × L p (g; X), with r = 0 and p > 1, be the space defined in Example 1.1. We set where C(t, θ) ∈ L(X) is a strongly measurable map defined on [0, ∞)× (−∞, 0] which satisfies the following conditions: (i) For each t, θ, C(t, θ) is a compact linear operator and this property is verified locally uniformly at θ. This means that for every s > 0 the set where q denotes the conjugate exponent of p. Then Λ satisfies conditions (a-3) and (a-5). In fact, condition (ii-1) implies that Λ is well defined and Λ(t, ·) is a bounded linear map from B into X. Moreover, Λ(t, ·) is a compact operator . In fact, setting, for each s > 0, the same argument already used shows that Λ s (t, ·) is a bounded linear operator from B into X and Theorem 1 in [5] implies that Λ s (t, ·) is compact . Since Λ s (t, ·) converges uniformly to Λ(t, ·) as s → ∞ it follows that Λ(t, ·) is compact .
On the other hand, if x(·) ∈ S(ϕ, b, r), then it is clear that where we have denoted and h := t − s Thus, (ii-2) and the integrability of g show that condition (a-6) is verified so that the assertion is consequence of previous example. Related with this example and Proposition 2.2 it is worth to point out that if g satisfies (g-6) and (g-7) in the terminolgy of [8] and γ(−t) → 0, as t → ∞, then K(·) is a bounded function and M (t) → 0 as t → ∞ ([8], Theorem 1.3.7 and Example 7.1.8). In the example that follows we suppose g verifies these conditions as well as that ln g is uniformly continuous .

Example 2.3
We conclude this section with an application of our results to discuss the existence of periodic solutions of the boundary value problem (2.18) where the functions a 0 , a, a 1 , b, q and φ satisfy appropriate conditions. To represent this problem as the Cauchy problem (2.1)-(2.2) we shall take X := L 2 ([0, π]) and define x(t) := u(t, ·). The operator A is given by It is well known that A generates a strongly continuous semigroup T (·) which is compact, analytic, self-adjoint and uniformly stable. Specifically, T (t) ≤ e −t , t ≥ 0. Furthermore, A has discrete spectrum, the eigenvalues are −n 2 , n ∈ IN, with corresponding normalized eigenvectors z n (ξ) := 2 π 1/2 sin(nξ). The operator (−A) 1/2 is defined by n=1 n < f, z n > z n ∈ X}. Let B denote the space C r × L 2 (g; X), with r = 0, defined in Example 1.1. In this case (see [8] where µ is the measure µ(θ, ξ) = g(θ) dθ dξ. Next we assume that the following conditions hold: dη dθ dξ < ∞.

g(θ)
we obtain that the right hand side of the above inequality converges to zero as h → 0. From Example 2.1 and 2.2 we derive that Λ 1 verifies conditions (a-3) and (a-5). On the other hand, Q is a substitution operator which is continuous and takes bounded sets into bounded sets ( [10]). Moreover, the Lipschits or Hölder continuity of q implies that Q has the same property, respectively. Since F is a Hammerstein operator formed by the composition of the linear operator Λ 1 and the operator Q we obtain that F satisfies the properties already established for Λ 1 . Thus, the system (2.16-2.17-2.18) verifies conditions (a-3) and (a-5). In addition, it is not difficult to see that F satisfies the hypotheses considered in Lemma 2.1 or Lemma 2.2 so that for each φ there is a unique mild solution x(·, φ) defined on (−∞, b), for some b > 0. Furthermore, if C 1 Λ 1 and Λ 2 are small enough and h is bounded then the solution x(·, φ) is · 2 -bounded on [0, ∞).
In connection with the existence of periodic solutions, if h is ωperiodic, from Proposition 2.2 and Corollary 2.2 we obtain that for C 1 Λ 1 and Λ 2 enough small there exists an mω-periodic mild solution.