© Hindawi Publishing Corp. THE KNESER PROPERTY FOR ABSTRACT RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

We establish existence of mild solutions for a class of 
semilinear first-order abstract retarded functional differential 
equations (ARFDEs) with infinite delay and we prove 
that the set consisting of mild solutions for this problem is 
connected in the space of continuous functions.

d(T (x), B) = 0, which in turn implies that T (x) ∈ B and x ∈ S. Consequently, x ∈ Fr(U ) ∩ S, which is absurd by the construction of U .
To apply this result to the set formed by the solutions of a certain equation, the following version is more convenient.Afterwards X denotes a Banach space, I ⊆ R is a compact interval, and C(I; X) stands for the space of continuous functions from I into X endowed with the norm of uniform convergence.
Corollary 1.2.Let -: C(I; X) → C(I; X) be continuous and S the set of fixed points of -.Assume that there is a compact set K ⊆ C(I; X) such that for each ε > 0, there is a set K ε ⊆ K with the following properties: (i) the sets K ε are connected; (ii) d(x, K ε ) < ε for all x ∈ S; (iii) y − -y ∞ < δ(ε) for all y ∈ K ε , where δ(ε) → 0, as ε → 0. Then S is connected.
Another abstract version of the Kneser property, which is known as the Krasnoselskii-Perov theorem (see [13]), is obtained using degree theory.
Let X be a Banach space endowed with a norm • .Henceforth, we always assume that A : D(A) → X is the infinitesimal generator of a strongly continuous semigroup of linear operators T (t) defined on X.In this paper, we are concerned with the initial value problem defined by the semilinear ARFDE with infinite delay x (t) = Ax(t) + f t, x t , 0 ≤ t ≤ τ, (1.1) with initial condition where Ꮾ is a space, called the phase space for the equation, f : [0,τ]× Ꮾ is an appropriated function, and x t represents the function defined from (−∞, 0] into X by x t (θ) = x(t + θ), −∞ < θ ≤ 0. Our purpose is to establish that the set of mild solutions of (1.1) and (1.2) is connected.
To study this problem in a general context, we will employ an axiomatic definition of the phase space Ꮾ introduced by Hale and Kato [4].To establish the axioms of space Ꮾ, we follow the terminology used in [8].Thus, Ꮾ will be a linear space of functions mapping (−∞, 0] into X endowed with a seminorm • Ꮾ .We will assume that Ꮾ satisfies the following axioms.(A) If x : (−∞,σ + a) → X, a > 0, is continuous on [σ , σ + a) and x σ ∈ Ꮾ, then for every t in [σ , σ + a), the following conditions hold: (i) x t is in Ꮾ; ), K is continuous and M is locally bounded and H, K, M are independent of x(•).
(B) The space Ꮾ is complete.Throughout this paper, we always assume that Ꮾ is a phase space.
2. Abstract retarded functional differential equations.Henceforth, T (t) is a strongly continuous semigroup of linear operators on X with infinitesimal generator A. We refer the reader to [3,12] and the references cited therein for the theory of strongly continuous semigroup and the associated abstract Cauchy problem (ACP).We only mention here a few results needed for our developments.
The existence and regularity of solutions of the first-order ACP where h : [0, ∞) → X is a locally integrable function, have been treated in several works (see [3,12]).We only recall here that the function is said to be a mild solution of (2.1).
Similarly, the existence of solutions of the semilinear ACP has been discussed in [1,11].Our interest in this section is to establish a Kneser's type property for the solutions of the functional semilinear first-order ACP (1.1) and (1.2).Next, we abbreviate the notation by writing I = [0,τ].
We begin with a result that ensures existence of solutions under quite general hypotheses which are suitable for our purposes.To study this initial value problem, we assume that the function f : I × Ꮾ → X satisfies the following Carathéodory conditions: In what follows, to study problem (1.1) and (1.2) and in order to avoid some cumbersome notations, if x : I → X is a function such that x(0) = ϕ(0), we identify x with its extension x : In what follows, we denote by B R the closed ball with center at 0 and radius R in an appropriated space.Since the strongly continuous semigroups are uniformly bounded on bounded intervals, we represent by M a constant such that T (t) ≤ M for all 0 ≤ t ≤ τ.Moreover, we abbreviate our notations by writing K = sup 0≤t≤τ K(t) and M = sup 0≤t≤τ M(t) and we denote by co(B) the closed convex hull of a set B. Theorem 2.2.Assume that the following conditions hold: (H1) for each R > 0, there is a positive integrable function γ R ∈ ᏸ 1 (I) such that sup{ f (t,ψ) : Then there is a mild solution of (1.1) and (1.2).In further, if the following condition is fulfilled: (H4) lim sup R→∞ ( MK/R)( τ 0 γ R (s)ds) < 1, then the set formed by the mild solutions of (1.1) and (1.2) is compact in C(I; X) and the set H() is compact in C(I; Ꮾ).

Proof.
Let C ϕ = {x ∈ C(I; X) : x(0) = ϕ(0)}.We define the map -: Clearly,is well defined and the Lebesgue's dominated convergence theorem implies thatis continuous.We affirm that there exists n ∈ N such that -: In fact, if we assume that the assertion is false, we can select an increasing sequence R j such that as well as a sequence (n j ) j in N and a sequence x j ∈ B n j such that q j = Kn j + M ϕ Ꮾ ≤ R j ≤ K(n j + 1) + M ϕ Ꮾ and -(x j ) > n j .Therefore, for each t ∈ I, we have that x t j Ꮾ ≤ q j from which it follows that Hence, we obtain that which contradicts (H3).
Next, using (H1) and (H2), we establish thatis completely continuous.Sinceis continuous, it only remains to prove thattakes bounded sets into relatively compact sets in the space C(I; X).From the Ascoli-Arzela theorem, it is sufficient to prove that for each R ≥ 0, the set {-0 (x)(t) : x ∞ ≤ R} is relatively compact in X for all 0 ≤ t ≤ τ and that the set {-0 (x) : x ∞ ≤ R} is equicontinuous, where we have denoted (2.8) We begin by establishing the first assertion.Let t > 0 and we take ε > 0 small enough.We can write (2.9) Applying the mean value theorem for the Bochner integral (see [10]), it follows that where Q = KR + M ϕ Ꮾ .By (H2), the set on the right-hand side is compact so that the first term on the right-hand side of (2.9) is included in a compact set which does not depend on the function x(•).
On the other hand, for the second term on the right-hand side of (2.9), we obtain the estimation which shows that this term converges towards zero as ε → 0 since γ R is integrable.Hence, we can assure that the set To establish the second assertion, we first observe that (2.12) Note that the first term on the right-hand side is included in (T (h)−I)V (t) so that this term converges to zero as h → 0, independent of x(•) ∈ B R .Similarly, for the second term, we have which implies that this term tends towards zero as h → 0.
Applying now the fixed-point theorem of Schauder, we infer thathas a fixed-point x in B n .Clearly, x is a mild solution of (1.1) and (1.2).Moreover, the continuity ofimplies that the set consisting of mild solutions is closed.
On the other hand, if condition (H4) holds, then is bounded.In fact, if we assume that is not bounded, then there exists a sequence of functions x j ∈ such that q j = x j ∞ ≥ j.Let R j = sup 0≤t≤τ x j t Ꮾ ≤ Kq j + M ϕ .Hence, which is absurd by (H4).Finally, using thatis completely continuous, we deduce that is compact.The last assertion is a direct consequence of the next property (see [8]).Thus, the proof is complete.
We refer to [6,7] for another existence results.Next, for future reference, we strengthen a property established in the proof of Theorem 2.2.

Lemma 2.4. Assume that conditions (H1) and (H2) hold. Then for each
Proof.Proceeding as in the proof of Theorem 2.2, we infer that the set is relatively compact for each 0 ≤ t ≤ τ.We put V = 0≤t≤τ V (t) and we prove that this set is relatively compact.To this finality, we first observe that the following property holds: for each ε > 0, there is δ > 0 such that (2.17) In fact, we chose δ > 0 such that Since the first term on the right-hand side is included in T (h)V (t) and the norm of the second term is less than or equal to M t+h t γ Q (s)ds, we establish the assertion.Now, assuming that δ = τ/n for some n ∈ N large enough, we have that which implies that V is relatively compact.
Next, we assume that problem (1.1) and (1.2) has mild solutions and we denote by the set consisting of such solutions.For a set Ᏺ ⊆ C(I; Y ), we denote by Ᏺ(I) the set {f (t) : f ∈ Ᏺ, t ∈ I}.Theorem 2.5.Assume that f is a continuous function and that (H1) and (H2) are fulfilled.If in further the following conditions hold: (H5) the set is compact, Proof.For the remainder of this proof, we abbreviate our notations by writing h(t) = T (t)ϕ(0).On the other hand, applying (H5) and (H6), we can select a constant R > 0 large enough such that x ∞ ≤ R for all x ∈ and where Q = KR + M ϕ .This implies that Let V be the set constructed in the proof of Lemma 2.4 for the constant Q.
Without loss of generality, we can assume that V is absolutely convex.We put U = 2V and U 1 = 3V .We denote N 1 = 2 M τ 0 γ Q (s)ds.We divide the proof into several steps.
Step 1.For a division d of I formed by the points 0 for all i = 1,...,n.Clearly, z is a continuous function.Next, for a fixed z given by (2.22), we denote by Ψ (•) and u(•) the step functions defined by ..,n.Thus, we can rewrite the definition of z as (2.23) We will show that z(t) ≤ R for 0 ≤ t ≤ τ, independent of the division d and the choice of points u i .In fact, from (2.20), we easily obtain that z(t) ≤ R for 0 < t ≤ t 1 .We assume that the assertion is true for t ∈ [0,t i−1 ] and we show that z(t) ≤ R, for t i−1 < t ≤ t i .In fact, it is clear from the axioms of phase space that z t Ꮾ ≤ Q for 0 ≤ t ≤ t i−1 , and since which establishes our assertion.
Step 2. To simplify the construction, we consider the points t k equally spaced with δ = t k − t k−1 and, in addition to the conditions considered in Step 1, we suppose that for all 1 ≤ i + 1 ≤ j ≤ n.We prove that the set W formed by the functions z defined by (2.22) is relatively compact in C(I; X).In fact, since h is a fixed function if we denote z = z − h, we must prove that W 0 = {z : z ∈ W } is relatively compact.
First, turning to apply (2.23) and Lemma 2.4, we infer that for every z ∈ W and t ∈ I. Now, we prove that W 0 is equicontinuous.Let 0 ≤ t ≤ t ≤ τ.From (2.23), we can write (2.27) In view of that V is a compact set and z(t ) − t 0 u(s)ds ∈ V , in order to prove the assertion, it is sufficient to show that t t u(s)ds converges towards zero as t − t → 0, independent of the construction of z.
Initially, we assume that t and t coincide with some points of the division.Thus, we suppose that t = t i and t = t j .In this case, we have that and applying (2.25), we obtain the estimation (2.30) Consequently, since Regarding the relative location of points t k , we analyze three possible situations.In first case, we assume that there is no point t k between t and t.Hence, there is an index i such that t i < t < t ≤ t i+1 and t t u(s)ds = (t − t )u i+1 .From (2.25), we have that Hence, if δ ≤ η 0 , using that t −t ≤ δ from the above expression, it follows that while if δ ≥ η 0 , we obtain that which establishes the assertion in this case.Now, we assume that there is an index i such that t i−1 < t < t i < t < t i+1 .Therefore, from our definitions, it follows that (2.34) Since t i − t < t − t and t − t i < t − t , we can argue as in the preceding case.
Finally, we consider that there are indices i < j such that t i−1 < t < t i < t j < t < t j+1 .Clearly, (2.35) where we have abbreviated v(t) = t 0 u(s)ds.In view of that t j − t i ≤ t − t , from our initial remark, we obtain that v(t j ) − v(t i ) ≤ ε.Thus, this case is reduced to estimate the first and third term on the right-hand side of the above expression.For the first term, we observe that v(t) is a convex combination of v(t j ) and v(t j+1 ).Therefore, (2.36) and since δ = t j+1 −t j ≤ t −t , we can repeat the previous argument.The third term is estimated similarly.
From the Ascoli-Arzela theorem, it follows that W 0 is relatively compact and so is W = h + W 0 .
Step 3. Now let ε > 0 be fixed.Without loss of generality, we assume also that ε ≤ min{τ/2, 2N 1 } and we take ε 1 = ε/2 Mτ.Using the compactness of sets and W as well as the continuity of f , we infer the existence of 0 for all s ∈ I and for every Similarly, there is δ 2 > 0 such that In the sequel, we consider the division d defined by t i = iδ, i = 0,...,n.Let W ε be the set formed by the functions defined by (2.22) with (u 1 ,...,u n ) ∈ Z ε , where Z ε is the set consisting of points (u 1 ,...,u n ) ∈ ((2/δ)U ) n that satisfy the following conditions: (i) We notice that condition (ii) implies that δ i k=1 u k ≤ N 1 .Next, we establish some properties of W ε .
Step 4. The set W ε is connected.This assertion is an easy consequence of the fact that the functions z ∈ W ε depend continuously on the choice of (u 1 ,...,u n ) ∈ Z ε and Z ε is convex by our construction.
Step 5.In this step, we show that the solutions of (2.3) can be approximated by the elements in W ε .Let x ∈ be fixed.We proceed to define z ∈ W ε so that x − z ∞ ≤ ε.We define z(•) inductively on the intervals [t i−1 ,t i ].
Step 6.In this step, we prove that the elements of W ε are approximate solutions of (2.3).Specifically, we show that for all t ∈ I and z ∈ W ε .In fact, for t i−1 < t ≤ t i , using (2.22), we have and employing now (2.37), (2.38), and the choice of δ, we can establish the estimation which shows our assertion.
Since is the set of fixed points of the mapgiven by (2.4), gathering Step 1 and Step 6 and applying Corollary 1.2, we obtain that is connected in the space C(I; X).Finally, to complete the proof, we observe that the map Λ : C ϕ → C(I; Ꮾ), x(•) → (t → x t ), is continuous and that H() = Λ().

Application.
The equations studied in this work are widely used to model concrete problems.We consider, as an example, the equation where d is a positive constant and q(•) and h are continuous functions that satisfy some conditions that will be specified later, which is a simplification of the predator-prey model proposed by Cohen et al. [2] to represent some inherent time delay in the population model.
With these definitions, problem (3.1) can be set in the form (1.1) and (1.2) and we can verify easily that the hypotheses of Theorem 2.2 hold.In fact, it is not difficult to see that f satisfies the Carathéodory conditions.Furthermore, which shows that condition (H1) holds with γ R (t) = C 1 C α R α , and, in turn, this definition shows that (H4) and (H6) are fulfilled for all τ > 0 when α < 1, while (H4) and (H6) are verified for τ > 0 small enough when α = 1.Finally, condition (H2) follows from the fact that T is a compact semigroup .Consequently, the hypotheses of Theorems 2.2 and 2.5 are satisfied and we can assert that problem (3.1) has mild solutions and that the set formed by these solutions is connected.
It is worth to point out that we have obtained these properties without assuming that h satisfies a local Lipschitz condition.