Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 33, 1-24; http://www.math.u-szeged.hu/ejqtde/

Multivalued Evolution Equations with Infinite Delay in Fréchet Spaces

Selma Baghli and Mouffak Benchohra^{1}

Laboratoire de Mathématiques, Université de Sidi Bel-Abbès BP 89, 22000 Sidi Bel-Abbès, Algérie

e-mail : selma baghli@yahoo.fr & benchohra@univ-sba.dz

Abstract In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for two classes of first order semilinear functional and neutral functional differential evolution inclusions with infinite delay using a recent nonlinear alternative for contractive multivalued maps in Fréchet spaces due to Frigon, combined with semigroup theory.

Key words : Functional and neutral evolution inclusions, mild solution, fixed-point, semigroup theory, contractive multivalued maps, Fréchet spaces, infinite delay. AMS Subject Classification : 34A60, 34G25, 34K40, 49K24.

1 Introduction

In this paper, we consider the existence of mild solutions defined on a semi-infinite positive real interval J := [0, +∞), for two classes of first order semilinear functional and neutral functional differential evolution inclusions with infinite delay in a real Banach space (E, | · |). Firstly, in Section 3, we study the following evolution inclusion of the form

y′(t) ∈ A(t)y(t) + F (t, y_{t}), a.e. t ∈ J  (1)
y_{0} = ϕ ∈ B,  (2)

where F : J ×B → P(E) is a multivalued map with nonempty compact values, P(E) is the family of all subsets of E, ϕ ∈ B are given functions and {A(t)}_{0≤t<+∞} is a family of linear closed (not necessarily bounded) operators from E into E that generate an evolution system of operators {U(t, s)}_{(t,s)∈J×J} for 0 ≤ s ≤ t < +∞.

For any continuous function y and any t ≥ 0, we denote by y_{t} the element of B defined by y_{t}(θ) = y(t + θ) for θ ∈ (−∞, 0]. We assume that the histories y_{t} belongs to some abstract phase space B, to be specified later.

In Section 4, we consider the following neutral evolution inclusion of the form

\frac{d}{dt}[y(t) − g(t, y_{t})] ∈ A(t)y(t) + F (t, y_{t}), a.e. t ∈ J  (3)

1Corresponding author

y_{0} = ϕ ∈ B,  (4)

where A(·), F and ϕ are as in problem (1) − (2) and g : J × B → E is a given function. Finally in Section 5, two examples are provided illustrating the abstract theory.

Functional differential and partial differential equations arise in many areas of applied mathematics and such equations have received much attention in recent years. A good guide to the literature for functional differential equations is the books by Hale [25] and Hale and Verduyn Lunel [27], Kolmanovskii and Myshkis [36], Kuang [37] and Wu [41] and the references therein.

During the last decades, several authors considered the problem of existence of mild solutions for semilinear evolution equations with finite delay. Some results can be found in the books by Ahmed [1, 2], Heikkila and Lakshmikantham [28] and Pazy [38] and Wu [41] and the references therein. When the delay is infinite, the notion of the phase space B plays an important role in the study of both qualitative and quantitative theory. A usual choice is a seminormed space satisfying suitable axioms, which was introduced by Hale and Kato in [26], see also Corduneanu and Lakshmikantham [17], Hino et al. [31], Kappel and Schappacher [33], and Schumacher [39].

An extensive theory is developed for inclusion (1) with A(t) = A. We refer the reader to the books by Heikkila and Lakshmikantham [28], Kamenski et al [34] and the pioneer Hino and Murakami paper [30]. By means of fixed point arguments, Benchohra and his collaborators have studied many classes of first and second order functional differential inclusions with local and nonlocal conditions in [7, 8, 10, 11, 13, 14, 15, 23] on a bounded interval. Extension to the semiinfinite interval is given by Benchohra and Ntouyas in [9, 12] and by Henderson and Ouahab [29] with finite delay.

When A is depending on the time, Arara et al [3] considered control multivalued problem on a bounded interval [0, b] and very recently Baghli and Benchohra [4, 5] provided uniqueness results for some classes of partial and neutral functional differential evolution equations on the interval J = [0, +∞) when the delay is finite. The perturbed problem with infinite delay is studied in [6]. Our main purpose in this paper is to look for the multivalued version of these problems.

Sufficient conditions are established to get existence results of mild solutions which are fixed points of the appropriate operators of the semilinear functional and the neutral functional differential evolution problems by applying the nonlinear alternative of Leray-Schauder type due to Frigon [21] for contractive multivalued maps in Fréchet spaces, combined with the semigroup theory [1, 2, 38].

2 Preliminaries

We introduce here notations, definitions and preliminary facts from multivalued analysis which are used throughout this paper.

Let C([0, +∞); E) be the space of continuous functions from [0, +∞) into E and B(E) be the space of all bounded linear operators from E into E, with the norm

∥N∥_{B(E)} = sup { |N(y)| : |y| = 1 }.

A measurable function y : [0, +∞) → E is Bochner integrable if and only if |y| is Lebesgue integrable. (For the Bochner integral properties, see Yosida [42] for instance).

Let L^{1}([0, +∞), E) denotes the Banach space of measurable functions y : [0, +∞) → E which are Bochner integrable normed by

∥y∥_{L^{1}} = ∫_{0}^{+∞} |y(t)| dt.

Consider the following space

B_{+∞} = {y : (−∞, +∞) → E : y|_{J} ∈ C(J, E), y_{0} ∈ B} ,

where y|_{J} is the restriction of y to J = [0, +∞).

In this paper, we will employ an axiomatic definition of the phase space B introduced by Hale and Kato in [26] and follow the terminology used in [31]. Thus, (B, ∥ · ∥_{B}) will be a seminormed linear space of functions mapping (−∞, 0] into E, and satisfying the following axioms :

(A_{1}) If y : (−∞, b) → E, b > 0, is continuous on [0, b] and y_{0} ∈ B, then for every t ∈ [0, b) the following conditions hold :

(i) y_{t} ∈ B ;

(ii) There exists a positive constant H such that |y(t)| ≤ H∥y_{t}∥_{B} ;

(iii) There exist two functions K(·), M(·) : ℝ_{+} → ℝ_{+} independent of y(t) with K continuous and M locally bounded such that :

∥y_{t}∥_{B} ≤ K(t) sup{ |y(s)| : 0 ≤ s ≤ t} + M(t)∥y_{0}∥_{B}.

Denote K_{b} = sup{K(t) : t ∈ [0, b]} and M_{b} = sup{M(t) : t ∈ [0, b]}.

(A_{2}) For the function y(.) in (A_{1}), y_{t} is a B−valued continuous function on [0, b].

(A_{3}) The space B is complete.

Remark 2.1

1. (ii) is equivalent to |ϕ(0)| ≤ H∥ϕ∥_{B} for every ϕ ∈ B.

2. Since ∥·∥_{B} is a seminorm, two elements ϕ, ψ ∈ B can verify ∥ϕ−ψ∥_{B} = 0 without necessarily ϕ(θ) = ψ(θ) for all θ ≤ 0.

3. From the equivalence of (ii), we can see that for all ϕ, ψ ∈ B such that ∥ϕ−ψ∥_{B} = 0 : This implies necessarily that ϕ(0) = ψ(0).

Hereafter are some examples of phase spaces. For other details we refer, for instance to the book by Hino et al [31].

Example 2.2 The spaces BC, BUC, C^{∞} and C^{0}. Let :

BC the space of bounded continuous functions defined from (−∞, 0] to E;

BUC the space of bounded uniformly continuous functions defined from (−∞, 0] to E;

C^{∞} := {ϕ ∈ BC : lim_{θ→−∞} ϕ(θ) exist in E};

C^{0} := {ϕ ∈ BC : lim_{θ→−∞} ϕ(θ) = 0} , endowed with the uniform norm

∥ϕ∥ = sup{|ϕ(θ)| : θ ≤ 0}.

We have that the spaces BUC, C^{∞} and C^{0} satisfy conditions (A_{1})−(A_{3}). BC satisfies (A_{1}), (A_{3}) but (A_{2}) is not satisfied.

Example 2.3 The spaces C_{g}, UC_{g}, C_{g}^{∞} and C_{g}^{0}. Let g be a positive continuous function on (−∞, 0]. We define :

C_{g} := \{ϕ ∈ C((−∞, 0], E) : \frac{ϕ(θ)}{g(θ)} is bounded on (−∞, 0]\};

C_{g}^{0} := \{ϕ ∈ C_{g} : lim_{θ→−∞} \frac{ϕ(θ)}{g(θ)} = 0\} , endowed with the uniform norm

∥ϕ∥ = sup \{\frac{|ϕ(θ)|}{g(θ)} : θ ≤ 0\} .

We consider the following condition on the function g.

(g_{1}) For all a > 0, sup_{0≤t≤a} sup \{\frac{g(t + θ)}{g(θ)} : −∞ < θ ≤ −t\} < ∞.

Then we have that the spaces C_{g} and C_{g}^{0} satisfy conditions (A_{3}). They satisfy conditions (A_{1}) and (A_{2}) if (g_{1}) holds.

Example 2.4 The space C_{γ}. For any real constant γ, we define the functional space C_{γ} by

C_{γ} := {ϕ ∈ C((−∞, 0], E) : lim_{θ→−∞} e^{γθ}ϕ(θ) exist in E}

endowed with the following norm

∥ϕ∥ = sup{e^{γθ}|ϕ(θ)| : θ ≤ 0}.

Then in the space C_{γ} the axioms (A_{1}) − (A_{3}) are satisfied.

In what follows, we assume that {A(t), t ≥ 0} is a family of closed densely defined linear unbounded operators on the Banach space E and with domain D(A(t)) independent of t.

Definition 2.5 We say that a family {A(t)}_{t≥0} generates a unique linear evolution system {U(t, s)}_{(t,s)∈Δ} for Δ := {(t, s) ∈ J × J : 0 ≤ s ≤ t < +∞} satisfying the following properties :

1. U(t, t) = I where I is the identity operator in E,

2. U(t, s) U(s, τ) = U(t, τ) for 0 ≤ τ ≤ s ≤ t < +∞,

3. U(t, s) ∈ B(E) the space of bounded linear operators on E, where for every (t, s) ∈ Δ and for each y ∈ E, the mapping (t, s) → U(t, s)y is continuous.

More details on evolution systems and their properties could be found on the books of Ahmed [1], Engel and Nagel [19] and Pazy [38].

Let X be a Fréchet space with a family of semi-norms {∥ · ∥_{n}}_{n∈IN}. Let Y ⊂ X, we say that F is bounded if for every n ∈ IN, there exists \overline{M}_{n} > 0 such that

∥y∥_{n} ≤ \overline{M}_{n} for all y ∈ Y.

To X we associate a sequence of Banach spaces {(X^{n}, ∥ · ∥_{n})} as follows : For every n ∈ IN, we consider the equivalence relation ∼_{n} defined by : x ∼_{n} y if and only if ∥x − y∥_{n} = 0 for all x, y ∈ X. We denote X^{n} = (X|_{∼_{n}}, ∥ · ∥_{n}) the quotient space, the completion of X^{n} with respect to ∥ · ∥_{n}. To every Y ⊂ X, we associate a sequence the {Y^{n}} of subsets Y^{n} ⊂ X^{n} as follows : For every x ∈ X, we denote [x]_{n} the equivalence class of x of subset X^{n} and we defined Y^{n} = \{[x]_{n} : x ∈ Y \}. We denote \overline{Y^{n}}, int_{n}(Y^{n}) and ∂_{n}Y^{n}, respectively, the closure, the interior and the boundary of Y^{n} with respect to ∥ · ∥ in X^{n}. We assume that the family of semi-norms {∥ · ∥_{n}} verifies :

∥x∥_{1} ≤ ∥x∥_{2} ≤ ∥x∥_{3} ≤ ... for every x ∈ X.

Let (X, d) be a metric space. We use the following notations :

P_{cl}(X) = {Y ∈ P(X) : Y closed}, P_{b}(X) = {Y ∈ P(X) : Y bounded},

P_{cv}(X) = {Y ∈ P(X) : Y convexe}, P_{cp}(X) = {Y ∈ P(X) : Y compact}.

Consider H_{d} : P(X) × P(X) ⟶ ℝ_{+} ∪ {∞}, given by

H_{d}(A, B) = max { sup_{a∈A} d(a, B) , sup_{b∈B} d(A, b) } ,

where d(A, b) = inf_{a∈A} d(a, b), d(a, B) = inf_{b∈B} d(a, b). Then (P_{b,cl}(X), H_{d}) is a metric space

and (P_{cl}(X), H_{d}) is a generalized (complete) metric space (see [35]).

Definition 2.6 A multivalued map G : J → P_{cl}(X) is said to be measurable if for each x ∈ E, the function Y : J → X defined by

Y (t) = d(x, G(t)) = inf{|x − z| : z ∈ G(t)}

is measurable where d is the metric induced by the normed Banach space X.

Definition 2.7 A function F : J × B ⟶ P(X) is said to be an L_{loc}^{1}-Carathéodory multivalued map if it satisfies :

(i) x ↦ F (t, y) is continuous (with respect to the metric H_{d}) for almost all t ∈ J;

(ii) t ↦ F (t, y) is measurable for each y ∈ B;

(iii) for every positive constant k there exists h_{k} ∈ L_{loc}^{1}(J; ℝ^{+}) such that

∥F (t, y)∥ ≤ h_{k}(t) for all ∥y∥_{B} ≤ k and for almost all t ∈ J.

Let (X, ∥ · ∥) be a Banach space. A multivalued map G : X → P(X) has convex (closed) values if G(x) is convex (closed) for all x ∈ X. We say that G is bounded on bounded sets if G(B) is bounded in X for each bounded set B of X, i.e.,

sup_{x∈B} {sup{ ∥y∥ : y ∈ G(x)}} < ∞.

Finally, we say that G has a fixed point if there exists x ∈ X such that x ∈ G(x).

For each y ∈ B_{+∞} let the set S_{F,y} known as the set of selectors from F defined by

S_{F,y} = {v ∈ L^{1}(J; E) : v(t) ∈ F (t, y_{t}) , a.e. t ∈ J}.

For more details on multivalued maps we refer to the books of Deimling [18], Górniewicz [24], Hu and Papageorgiou [32] and Tolstonogov [40].

Definition 2.8 A multivalued map F : X → P(X) is called an admissible contraction with constant {k_{n}}_{n∈ℕ} if for each n ∈ ℕ there exists k_{n} ∈ (0, 1) such that

i) H_{d}(F (x), F (y)) ≤ k_{n} ∥x − y∥_{n} for all x, y ∈ X.

ii) For every x ∈ X and every ϵ ∈ (0, ∞)^{n}, there exists y ∈ F (x) such that

∥x − y∥_{n} ≤ ∥x − F (x)∥_{n} + ϵ_{n} for every n ∈ ℕ

Theorem 2.9 (Nonlinear Alternative of Frigon, [21, 22]). Let X be a Fréchet space and U an open neighborhood of the origin in X and let N : \overline{U} → P(X) be an admissible multivalued contraction. Assume that N is bounded. Then one of the following statements holds :

(C1) N has a fixed point;

(C2) There exists λ ∈ [0, 1) and x ∈ ∂U such that x ∈ λ N (x).

3 Semilinear Evolution Inclusions

The main result of this section concerns the semilinear evolution problem (1) − (2). Before stating and proving this one, we give first the definition of the mild solution.

Definition 3.1 We say that the function y(·) : (−∞, +∞) → E is a mild solution of the evolution system (1) − (2) if y(t) = ϕ(t) for all t ∈ (−∞, 0] and the restriction of y(·) to the interval J is continuous and there exists f(·) ∈ L^{1}(J; E) : f(t) ∈ F (t, y_{t}) a.e. in J such that y satisfies the following integral equation :

y(t) = U(t, 0) ϕ(0) + ∫_{0}^{t} U(t, s) f(s) ds for each t ∈ [0, +∞).  (5)

We will need to introduce the following hypothesis which are assumed hereafter :

(H1) There exists a constant M̂ ≥ 1 such that :

∥U(t, s)∥_{B(E)} ≤ M̂ for every (t, s) ∈ Δ.

(H2) The multifunction F : J × B ⟶ P(E) is L_{loc}^{1}-Carathéodory with compact and convex values for each u ∈ B and there exist a function p ∈ L_{loc}^{1}(J; ℝ_{+}) and a continuous nondecreasing function ψ : J → (0, ∞) such that:

∥F (t, u)∥_{P(E)} ≤ p(t) ψ(∥u∥_{B}) for a.e. t ∈ J and each u ∈ B.

(H3) For all R > 0, there exists l_{R} ∈ L_{loc}^{1}(J; ℝ_{+}) such that :

H_{d}(F (t, u) − F (t, v)) ≤ l_{R}(t) ∥u − v∥_{B}

for each t ∈ J and for all u, v ∈ B with ∥u∥_{B} ≤ R and ∥v∥_{B} ≤ R and

d(0, F (t, 0)) ≤ l_{R}(t) a.e. t ∈ J.

For every n ∈ IN, we define in B_{+∞} the family of semi-norms by :

∥y∥_{n} := sup { e^{−τ L_{n}^{∗}(t)} |y(t)| : t ∈ [0, n] }

where : L_{n}^{∗}(t) = ∫_{0}^{t} \overline{l}_{n}(s) ds , \overline{l}_{n}(t) = M̂K_{n}l_{n}(t) and l_{n} is the function from (H3).

Then B_{+∞} is a Fréchet space with the family of semi-norms ∥ · ∥_{n∈ℕ}. In what follows we will choose τ > 1.

Theorem 3.2 Suppose that hypotheses (H1) − (H3) are satisfied and moreover

∫_{c_{n}}^{+∞} \frac{ds}{ψ(s)} > K_{n}M̂∫_{0}^{n} p(s) ds for each n ∈ IN  (6)

with c_{n} = (K_{n}M̂H + M_{n})∥ϕ∥_{B}. Then evolution problem (1) − (2) has a mild solution.

Proof. Transform the problem (1) − (2) into a fixed-point problem. Consider the multivalued operator N : B_{+∞} → P(B_{+∞}) defined by :

N(y) = \{h ∈ B_{+∞} : h(t) = \{ \begin{matrix}ϕ(t), & if t ≤ 0; \\ U(t, 0) ϕ(0) + ∫_{0}^{t} U(t, s) f(s) ds, & if t ≥ 0, \end{matrix} \}

where f ∈ S_{F,y} = {v ∈ L^{1}(J, E) : v(t) ∈ F (t, y_{t}) for a.e. t ∈ J}.

Clearly, the fixed points of the operator N are mild solutions of the problem (1)−(2). We remark also that, for each y ∈ B_{+∞}, the set S_{F,y} is nonempty since, by (H2), F has a measurable selection (see [16], Theorem III.6).

For ϕ ∈ B, we will define the function x(.) : (−∞, +∞) → E by

x(t) = \{ \begin{matrix}ϕ(t), & if t ∈ (−∞, 0]; \\ U(t, 0) ϕ(0), & if t ∈ J. \end{matrix}

Then x_{0} = ϕ. For each function z ∈ B_{+∞}, set

y(t) = z(t) + x(t).  (7)

It is obvious that y satisfies (5) if and only if z satisfies z_{0} = 0 and

z(t) = ∫_{0}^{t} U(t, s) f(s) ds for t ∈ J.

where f(t) ∈ F (t, z_{t} + x_{t}) a.e. t ∈ J.

Let

B_{+∞}^{0} = {z ∈ B_{+∞} : z_{0} = 0} .

Define in B_{+∞}^{0}, the multivalued operator F : B_{+∞}^{0} → P(B_{+∞}^{0}) by :

F(z) = {h ∈ B_{+∞}^{0} : h(t) = ∫_{0}^{t} U(t, s) f(s) ds, t ∈ J} ,

where f ∈ S_{F,z} = {v ∈ L^{1}(J, E) : v(t) ∈ F (t, z_{t} + x_{t}) for a.e. t ∈ J}.

Obviously the operator inclusion N has a fixed point is equivalent to the operator inclusion F has one, so it turns to prove that F has a fixed point.

Let z ∈ B_{+∞}^{0} be a possible fixed point of the operator F. Given n ∈ ℕ, then z should be solution of the inclusion z ∈ λ F(z) for some λ ∈ (0, 1) and there exists f ∈ S_{F,z} ⇔ f(t) ∈ F (t, z_{t} + x_{t}) such that, for each t ∈ [0, n], we have

|z(t)| ≤ ∫_{0}^{t} ∥U(t, s)∥_{B(E)} |f(s)| ds
≤ M̂ ∫_{0}^{t} p(s) ψ (∥z_{s} + x_{s}∥_{B}) ds.

Assumption (A_{1}) gives

∥z_{s} + x_{s}∥_{B} ≤ ∥z_{s}∥_{B} + ∥x_{s}∥_{B}
≤ K(s)|z(s)| + M(s)∥z_{0}∥_{B} + K(s)|x(s)| + M(s)∥x_{0}∥_{B}
≤ K_{n}|z(s)| + K_{n}∥U(s, 0)∥_{B(E)}|ϕ(0)| + M_{n}∥ϕ∥_{B}
≤ K_{n}|z(s)| + K_{n}M̂|ϕ(0)| + M_{n}∥ϕ∥_{B}
≤ K_{n}|z(s)| + K_{n}M̂H∥ϕ∥_{B} + M_{n}∥ϕ∥_{B}
≤ K_{n}|z(s)| + (K_{n}M̂H + M_{n})∥ϕ∥_{B}.

Set c_{n} := (K_{n}M̂H + M_{n})∥ϕ∥_{B}, then we have

∥z_{s} + x_{s}∥_{B} ≤ K_{n}|z(s)| + c_{n}  (8)

Using the nondecreasing character of ψ, we get

|z(t)| ≤ M̂ ∫_{0}^{t} p(s) ψ (K_{n}|z(s)| + c_{n}) ds.

Then

K_{n}|z(t)| + c_{n} ≤ K_{n}M̂∫_{0}^{t} p(s)ψ(K_{n}|z(s)| + c_{n})ds + c_{n}.

We consider the function μ defined by

μ(t) := sup { K_{n}|z(s)| + c_{n} : 0 ≤ s ≤ t }, 0 ≤ t < +∞.

Let t^{⋆} ∈ [0, t] be such that

μ(t) = K_{n}|z(t^{⋆})| + c_{n}.

By the previous inequality, we have

μ(t) ≤ K_{n}M̂∫_{0}^{t} p(s) ψ(μ(s)) ds + c_{n} for t ∈ [0, n].

Let us take the right-hand side of the above inequality as v(t). Then, we have

μ(t) ≤ v(t) for all t ∈ [0, n].

From the definition of v, we have

v(0) = c_{n} and v′(t) = K_{n}M̂p(t) ψ(μ(t)) a.e. t ∈ [0, n].

Using the nondecreasing character of ψ, we get

v′(t) ≤ K_{n}M̂ p(t) ψ(v(t)) a.e. t ∈ [0, n].

This implies that for each t ∈ [0, n] and using the condition (6), we get

∫_{c_{n}}^{v(t)} \frac{ds}{ψ(s)} ≤ K_{n}M̂ ∫_{0}^{t} p(s) ds ≤ K_{n}M̂ ∫_{0}^{n} p(s) ds < ∫_{c_{n}}^{+∞} \frac{ds}{ψ(s)}.

Thus, for every t ∈ [0, n], there exists a constant Λ_{n} such that v(t) ≤ N_{n} and hence μ(t) ≤ Λ_{n}. Since ∥z∥_{n} ≤ μ(t), we have ∥z∥_{n} ≤ Λ_{n}. Set

U = { z ∈ B_{+∞}^{0} : sup{ |z(t)| : 0 ≤ t ≤ n} < Λ_{n} + 1 for all n ∈ IN}.

Clearly, U is an open subset of B_{+∞}^{0}.

We shall show that F : \overline{U} → P(B_{+∞}^{0}) is a contraction and an admissible operator.

First, we prove that F is a contraction ; Let z, \overline{z} ∈ B_{+∞}^{0} and h ∈ F(z). Then there exists f(t) ∈ F (t, z_{t} + x_{t}) such that for each t ∈ [0, n]

h(t) = ∫_{0}^{t} U(t, s) f(s) ds.

From (H3) it follows that

H_{d}(F (t, z_{t} + x_{t}), F (t, \overline{z}_{t} + x_{t})) ≤ l_{n}(t) ∥z_{t} − \overline{z}_{t}∥_{B}.

Hence, there is ρ ∈ F (t, \overline{z}_{t} + x_{t}) such that

|f(t) − ρ| ≤ l_{n}(t) ∥z_{t} − \overline{z}_{t}∥_{B} t ∈ [0, n].

Consider U_{⋆} : [0, n] → P(E), given by

U_{⋆} = \{ρ ∈ E : |f(t) − ρ| ≤ l_{n}(t) ∥z_{t} − \overline{z}_{t}∥_{B}\}.

Since the multivalued operator V(t) = U_{⋆}(t) ∩ F (t, \overline{z}_{t} + x_{t}) is measurable (in [16], see Proposition III.4), there exists a function \overline{f}(t), which is a measurable selection for V. So, \overline{f}(t) ∈ F (t, \overline{z}_{t} + x_{t}) and using (A_{1}), we obtain for each t ∈ [0, n]

|f(t) − \overline{f}(t)| ≤ l_{n}(t) ∥z_{t} − \overline{z}_{t}∥_{B}
≤ l_{n}(t) [K(t) |z(t) − \overline{z}(t)| + M(t) ∥z_{0} − \overline{z}_{0}∥_{B}]
≤ l_{n}(t) K_{n} |z(t) − \overline{z}(t)|

Let us define, for each t ∈ [0, n]

\overline{h}(t) = ∫_{0}^{t} U(t, s) \overline{f}(s) ds.

Then we have

|h(t) − \overline{h}(t)| ≤ ∫_{0}^{t} ∥U(t, s)∥_{B(E)} |f(s) − \overline{f}(s)| ds
≤ ∫_{0}^{t} M̂ K_{n} l_{n}(s) |z(s) − \overline{z}(s)| ds
≤ ∫_{0}^{t} [\overline{l}_{n}(s) e^{τ L_{n}^{∗}(s)} ] [e^{−τ L_{n}^{∗}(s)} |z(s) − \overline{z}(s)|] ds
≤ ∫_{0}^{t} [\frac{e^{τ L_{n}^{∗}(s)}}{τ} ]′ ds ∥z − \overline{z}∥_{n}
≤ \frac{1}{τ} e^{τ L_{n}^{∗}(t)} ∥z − \overline{z}∥_{n}.

Therefore,

∥h − \overline{h}∥_{n} ≤ \frac{1}{τ}∥z − \overline{z}∥_{n}.

By an analogous relation, obtained by interchanging the roles of z and \overline{z}, it follows that

H_{d}(F(z), F(\overline{z})) ≤ \frac{1}{τ}∥z − \overline{z}∥\overline{B}.

So, F is a contraction for all n ∈ ℕ.

Now we shall show that F is an admissible operator. Let z ∈ B_{+∞}^{0}. Set, for every n ∈ ℕ, the space

B_{n}^{0} := {y : (−∞, n] → E : y|_{[0,n]} ∈ C([0, n], E), y_{0} ∈ B} ,

and let us consider the multivalued operator F : B_{n}^{0} → P_{cl}(B_{n}^{0}) defined by :

F(z) = {h ∈ B_{n}^{0} : h(t) = ∫_{0}^{t} U(t, s) f(s) ds, t ∈ [0, n]}

where f ∈ S_{F,y}^{n} = {v ∈ L^{1}([0, n], E) : v(t) ∈ F (t, y_{t}) for a.e. t ∈ [0, n]}.

From (H1) − (H3) and since F is a multivalued map with compact values, we can prove that for every z ∈ B_{n}^{0}, F(z) ∈ P_{cl}(B_{n}^{0}) and there exists z_{⋆} ∈ B_{n}^{0} such that z_{⋆} ∈ F(z_{⋆}). Let h ∈ B_{n}^{0}, \overline{y} ∈ \overline{U} and ϵ > 0. Assume that z_{⋆} ∈ F(\overline{z}), then we have

|\overline{z}(t) − z_{⋆}(t)| ≤ |\overline{z}(t) − h(t)| + |z_{⋆}(t) − h(t)|
≤ e^{τ L_{n}^{∗}(t)} ∥\overline{z} − F(\overline{z})∥_{n} + ∥z_{⋆} − h∥.

Since h is arbitrary, we may suppose that h ∈ B(z_{⋆}, ϵ) = {h ∈ B_{n}^{0} : ∥h − z_{⋆}∥_{n} ≤ ϵ}. Therefore,

∥\overline{z} − z_{⋆}∥_{n} ≤ ∥\overline{z} − F(\overline{z})∥_{n} + ϵ.

If z is not in F(\overline{z}), then ∥z_{⋆} − F(\overline{z})∥ ≠ 0. Since F(\overline{z}) is compact, there exists x ∈ F(\overline{z}) such that ∥z_{⋆} − F(\overline{z})∥ = ∥z_{⋆} − x∥. Then we have

|\overline{z}(t) − z_{⋆}(t)| ≤ |\overline{z}(t) − h(t)| + |x(t) − h(t)|
≤ e^{τ L_{n}^{∗}(t)} ∥\overline{z} − F(\overline{z})∥_{n} + |x(t) − h(t)|.

Thus,

∥\overline{z} − x∥_{n} ≤ ∥\overline{z} − F(\overline{z})∥_{n} + ϵ.

So, F is an admissible operator contraction. From the choice of U there is no z ∈ ∂U such that z = λ F(z) for some λ ∈ (0, 1). Then the statement (C2) in Theorem 2.9 does not hold. We deduce that the operator F has a fixed point z^{⋆}. Then y^{⋆}(t) = z^{⋆}(t) + x(t), t ∈ (−∞, +∞) is a fixed point of the operator N, which is a mild solution of the evolution inclusion problem (1) − (2).

4 Semilinear Neutral Evolution Inclusions

In this section, we give existence results for the neutral functional differential evolution problem with infinite delay (3) − (4). Firstly we define the mild solution.

Definition 4.1 We say that the function y(·) : (−∞, +∞) → E is a mild solution of the neutral evolution system (3) − (4) if y(t) = ϕ(t) for all t ∈ (−∞, 0], the restriction of y(·) to the interval J is continuous and there exists f(·) ∈ L^{1}(J; E) : f(t) ∈ F (t, y_{t}) a.e. in J such that y satisfies the following integral equation

\begin{aligned}y(t) & = U(t, 0)[ϕ(0) − g(0, ϕ)] + g(t, y_{t}) + ∫_{0}^{t} U(t, s)A(s)g(s, y_{s})ds \\ & + ∫_{0}^{t} U(t, s)f(s) ds for each t ∈ [0, +∞). \end{aligned}  (9)

We consider the hypotheses (H1) − (H3) and we will need the following assumptions :

(H4) There exists a constant \overline{M}_{0} > 0 such that :

∥A^{−1}(t)∥_{B(E)} ≤ \overline{M}_{0} for all t ∈ J.

(H5) There exists a constant 0 < L < \frac{1}{\overline{M}_{0}K_{n}} such that :

|A(t) g(t, ϕ)| ≤ L (∥ϕ∥_{B} + 1) for all t ∈ J and ϕ ∈ B.

(H6) There exists a constant L_{⋆} > 0 such that :

|A(s) g(s, ϕ) − A(\overline{s}) g(\overline{s}, \overline{ϕ})| ≤ L_{⋆} (|s − \overline{s}| + ∥ϕ − \overline{ϕ}∥_{B})

for all s, \overline{s} ∈ J and ϕ, \overline{ϕ} ∈ B.

For every n ∈ IN, let us take here \overline{l}_{n}(t) = M̂K_{n}[L_{⋆} + l_{n}(t)] for the family of seminorm {∥ · ∥_{n}}_{n∈IN} defined in Section 3. In what follows we fix τ > 1 and assume

[\overline{M}_{0}L_{⋆}K_{n} + \frac{1}{τ}] < 1.

Theorem 4.2 Suppose that hypotheses (H1) − (H6) are satisfied and moreover

∫_{δ_{n}}^{+∞} \frac{ds}{s + ψ(s)} > \frac{M̂K_{n}}{1 − \overline{M}_{0}LK_{n}} ∫_{0}^{n} max(L, p(s))ds for each n ∈ IN (10)

with

δ_{n} := (K_{n}M̂H + M_{n})∥ϕ∥_{B} +\frac{K_{n}}{1 − \overline{M}_{0}LK_{n}} [(M̂ + 1)\overline{M}_{0}L + M̂Ln
+\overline{M}_{0}L [M̂(K_{n}H + 1) + M_{n}] ∥ϕ∥_{B}] .

Then the neutral evolution problem (3) − (4) has a mild solution.

Proof. Transform the neutral evolution problem (3) − (4) into a fixed-point problem. Consider the multivalued operator Ñ : B_{+∞} → P(B_{+∞}) defined by :

Ñ(y) = \{h ∈ B_{+∞} : h(t) = \{ \begin{matrix}ϕ(t), & if t ≤ 0; \\ U(t, 0) [ϕ(0) − g(0, ϕ)] + g(t, y_{t}) \\ + ∫_{0}^{t} U(t, s)A(s)g(s, y_{s})ds \\ + ∫_{0}^{t} U(t, s)f(s)ds, & if t ∈ J, \end{matrix} \}

where f ∈ S_{F,y} = {v ∈ L^{1}(J, E) : v(t) ∈ F (t, y_{t}) for a.e. t ∈ J}.

Clearly, the fixed points of the operator Ñ are mild solutions of the problem (3)−(4). We remark also that, for each y ∈ B_{+∞}, the set S_{F,y} is nonempty since, by (H2), F has a measurable selection (see [16], Theorem III.6).

For ϕ ∈ B, we will define the function x(.) : (−∞, +∞) → E by

x(t) = \{ \begin{matrix}ϕ(t), & if t ∈ (−∞, 0]; \\ U(t, 0) ϕ(0), & if t ∈ J. \end{matrix}

Then x_{0} = ϕ. For each function z ∈ B_{+∞}, set

y(t) = z(t) + x(t).  (11)

It is obvious that y satisfies (9) if and only if z satisfies z_{0} = 0 and

z(t) = g(t, z_{t} + x_{t}) − U(t, 0)g(0, ϕ)
+ ∫_{0}^{t} U(t, s)A(s)g(s, z_{s} + x_{s})ds + ∫_{0}^{t} U(t, s)f(s)ds.

where f(t) ∈ F (t, z_{t} + x_{t}) a.e. t ∈ J.

Let

B_{+∞}^{0} = {z ∈ B_{+∞} : z_{0} = 0} .

Define in B_{+∞}^{0}, the multivalued operator F̃ : B_{+∞}^{0} → P(B_{+∞}^{0}) by :

F̃(z) = {h ∈ B_{+∞}^{0} : h(t) = g(t, z_{t} + x_{t}) − U(t, 0)g(0, ϕ)
+ ∫_{0}^{t} U(t, s)A(s)g(s, z_{s} + x_{s})ds
+ ∫_{0}^{t} U(t, s)f(s)ds, t ∈ J}

where f ∈ S_{F,z} = {v ∈ L^{1}(J, E) : v(t) ∈ F (t, z_{t} + x_{t}) for a.e. t ∈ J}.

Obviously the operator inclusion Ñ has a fixed point is equivalent to the operator inclusion F̃ has one, so it turns to prove that F̃ has a fixed point.

Let z ∈ B_{+∞}^{0} be a possible fixed point of the operator F̃. Given n ∈ ℕ, then z should be solution of the inclusion z ∈ λ F̃(z) for some λ ∈ (0, 1) and there exists f ∈ S_{F,z} ⇔ f(t) ∈ F (t, z_{t} + x_{t}) such that, for each t ∈ [0, n], we have

|z(t)| ≤ ∥A^{−1}(t)∥_{B(E)}|A(t)g(t, z_{t} + x_{t})| + ∥U(t, 0)∥_{B(E)}∥A^{−1}(0)∥_{B(E)}|A(0) g(0, ϕ)|
+ ∫_{0}^{t} ∥U(t, s)∥_{B(E)}|A(s)g(s, z_{s} + x_{s})|ds + ∫_{0}^{t} ∥U(t, s)∥_{B(E)}|f(s)|ds
≤ \overline{M}_{0}L(∥z_{t} + x_{t}∥_{B} + 1) + M̂\overline{M}_{0}L(∥ϕ∥_{B} + 1)
+ M̂∫_{0}^{t} L(∥z_{s} + x_{s}∥_{B} + 1)ds + M̂∫_{0}^{t} p(s)ψ(∥z_{s} + x_{s}∥_{B})ds
≤ \overline{M}_{0}L∥z_{t} + x_{t}∥_{B} + \overline{M}_{0}L(1 + M̂) + M̂Ln + M̂\overline{M}_{0}L∥ϕ∥_{B}
+ M̂∫_{0}^{t} L∥z_{s} + x_{s}∥_{B}ds + M̂∫_{0}^{t} p(s)ψ(∥z_{s} + x_{s}∥_{B})ds.

Using the inequality (8) and the nondecreasing character of ψ, we obtain

|z(t)| ≤ \overline{M}_{0}L(K_{n}|z(t)| + c_{n}) + \overline{M}_{0}L(1 + M̂) + M̂Ln + M̂\overline{M}_{0}L∥ϕ∥_{B}
+ M̂∫_{0}^{t} L(K_{n}|z(s)| + c_{n})ds + M̂∫_{0}^{t} p(s)ψ(K_{n}|z(s)| + c_{n})ds
≤ \overline{M}_{0}LK_{n}|z(t)| + \overline{M}_{0}L(1 + M̂) + M̂Ln + \overline{M}_{0}Lc_{n} + M̂\overline{M}_{0}L∥ϕ∥_{B}
+ M̂[∫_{0}^{t} L(K_{n}|z(s)| + c_{n})ds + ∫_{0}^{t} p(s)ψ(K_{n}|z(s)| + c_{n})ds] .

Then

(1 − \overline{M}_{0}LK_{n})|z(t)| ≤ (M̂ + 1)\overline{M}_{0}L + M̂Ln + \overline{M}_{0}Lc_{n} + M̂\overline{M}_{0}L∥ϕ∥_{B}
+ M̂[∫_{0}^{t} L(K_{n}|z(s)| + c_{n})ds + ∫_{0}^{t} p(s)ψ(K_{n}|z(s)| + c_{n})ds] .

Set δ_{n} := c_{n} + \frac{K_{n}}{1 − \overline{M}_{0}LK_{n}} [(M̂ + 1)\overline{M}_{0}L + M̂Ln + \overline{M}_{0}Lc_{n} + M̂\overline{M}_{0}L∥ϕ∥_{B}]. Thus

K_{n}|z(t)| + c_{n} ≤ δ_{n}
+ \frac{M̂K_{n}}{1 − \overline{M}_{0}LK_{n}} [∫_{0}^{t} L(K_{n}|z(s)| + c_{n})ds + ∫_{0}^{t} p(s)ψ(K_{n}|z(s)| + c_{n})ds] .

We consider the function μ defined by

μ(t) := sup { K_{n}|z(s)| + c_{n} : 0 ≤ s ≤ t }, 0 ≤ t < +∞.

Let t^{⋆} ∈ [0, t] be such that μ(t) = K_{n}|z(t^{⋆})| + c_{n}. By the previous inequality, we have

μ(t) ≤ δ_{n} + \frac{M̂K_{n}}{1 − \overline{M}_{0}LK_{n}} [∫_{0}^{t} Lμ(s)ds + ∫_{0}^{t} p(s)ψ(μ(s))ds] for t ∈ [0, n].

Let us take the right-hand side of the above inequality as v(t). Then, we have

μ(t) ≤ v(t) for all t ∈ [0, n].

From the definition of v, we have v(0) = δ_{n} and

v′(t) = \frac{M̂K_{n}}{1 − \overline{M}_{0}LK_{n}} [Lμ(t) + p(t)ψ(μ(t))] a.e. t ∈ [0, n].

Using the nondecreasing character of ψ, we get

v′(t) ≤ \frac{M̂K_{n}}{1 − \overline{M}_{0}LK_{n}} [Lv(t) + p(t)ψ(v(t))] a.e. t ∈ [0, n].

This implies that for each t ∈ [0, n] and using the condition (10), we get

∫_{δ_{n}}^{v(t)} \frac{ds}{s + ψ(s)} ≤ \frac{M̂K_{n}}{1 − \overline{M}_{0}LK_{n}} ∫_{0}^{t} max(L, p(s))ds
≤ \frac{M̂K_{n}}{1 − \overline{M}_{0}LK_{n}} ∫_{0}^{n} max(L, p(s))ds
< ∫_{δ_{n}}^{+∞} \frac{ds}{s + ψ(s)}.

Thus, for every t ∈ [0, n], there exists a constant Λ_{n} such that v(t) ≤ Λ_{n} and hence μ(t) ≤ Λ_{n}. Since ∥z∥_{n} ≤ μ(t), we have ∥z∥_{n} ≤ Λ_{n}.

We can show as in Section 3 that F̃ is an admissible operator and we shall prove now that F̃ : \overline{U} → P(B_{+∞}^{0}) is a contraction.

Let z, \overline{z} ∈ B_{+∞}^{0} and h ∈ F̃(z). Then there exists f(t) ∈ F (t, z_{t} + x_{t}) such that for each t ∈ [0, n]

h(t) = g(t, z_{t} + x_{t}) − U(t, 0)g(0, ϕ) + ∫_{0}^{t} U(t, s)A(s)g(s, z_{s} + x_{s})ds + ∫_{0}^{t} U(t, s)f(s)ds

From (H3) it follows that

H_{d}(F (t, z_{t} + x_{t}), F (t, \overline{z}_{t} + x_{t})) ≤ l_{n}(t) ∥z_{t} − \overline{z}_{t}∥_{B}.

Hence, there is ρ ∈ F (t, \overline{z}_{t} + x_{t}) such that

|f(t) − ρ| ≤ l_{n}(t) ∥z_{t} − \overline{z}_{t}∥_{B} t ∈ [0, n].

Consider U_{⋆} : [0, n] → P(E), given by

U_{⋆} = \{ρ ∈ E : |f(t) − ρ| ≤ l_{n}(t) ∥z_{t} − \overline{z}_{t}∥_{B}\}.

Since the multivalued operator V(t) = U_{⋆}(t) ∩ F (t, \overline{z}_{t} + x_{t}) is measurable (in [16], see Proposition III.4), there exists a function \overline{f}(t), which is a measurable selection for V. So, \overline{f}(t) ∈ F (t, \overline{z}_{t} + x_{t}) and using (A_{1}), we obtain for each t ∈ [0, n]

\begin{aligned}|f(t) − \overline{f}(t)| & ≤ & l_{n}(t) ∥z_{t} − \overline{z}_{t}∥_{B} \\ & ≤ & l_{n}(t) [K(t) |z(t) − \overline{z}(t)| + M(t) ∥z_{0} − \overline{z}_{0}∥_{B}] \\ & ≤ & l_{n}(t) K_{n} |z(t) − \overline{z}(t)|. \end{aligned}  (12)

Let us define, for each t ∈ [0, n]

\overline{h}(t) = g(t, \overline{z}_{t} + x_{t}) − U(t, 0)g(0, ϕ) + ∫_{0}^{t} U(t, s)A(s)g(s, \overline{z}_{s} + x_{s})ds + ∫_{0}^{t} U(t, s)\overline{f}(s)ds Then, for each t ∈ [0, n] and n ∈ IN and using (H1) and (H3) to (H6), we get

|h(t) − \overline{h}(t)| ≤ |g(t, z_{t} + x_{t}) − g(t, \overline{z}_{t} + x_{t})|
+ ∫_{0}^{t} |U(t, s)A(s)[g(s, z_{s} + x_{s}) − g(s, \overline{z}_{s} + x_{s})]| ds
+ ∫_{0}^{t} |U(t, s)[f(s) − \overline{f}(s)]| ds
≤ ∥A^{−1}(t)∥_{B(E)} |A(t)g(t, z_{t} + x_{t}) − A(t)g(t, \overline{z}_{t} + x_{t})|
+ ∫_{0}^{t} ∥U(t, s)∥_{B(E)}|A(s)g(s, z_{s} + x_{s}) − A(s)g(s, \overline{z}_{s} + x_{s})| ds
+ ∫_{0}^{t} ∥U(t, s)∥_{B(E)}|f(s) − \overline{f}(s)| ds
≤ \overline{M}_{0}L_{⋆}∥z_{t} − \overline{z}_{t}∥_{B} + ∫_{0}^{t} M̂L_{⋆}∥z_{s} − \overline{z}_{s}∥_{B} ds + ∫_{0}^{t} M̂|f(s) − \overline{f}(s)| ds.

Using (A_{1}) and (12), we obtain

|h(t) − \overline{h}(t)| ≤ \overline{M}_{0}L_{⋆}K(t)|z(t) − \overline{z}(t)| + ∫_{0}^{t} M̂L_{⋆}K(s)|z(s) − \overline{z}(s)| ds
+ ∫_{0}^{t} M̂l_{n}(s)K_{n}|z(s) − \overline{z}(s)| ds
≤ \overline{M}_{0}L_{⋆}K_{n}|z(t) − \overline{z}(t)| + ∫_{0}^{t} M̂K_{n}[L_{⋆} + l_{n}(s)]|z(s) − \overline{z}(s)| ds
≤ \overline{M}_{0}L_{⋆}K_{n}|z(t) − \overline{z}(t)| + ∫_{0}^{t} \overline{l}_{n}(s)|z(s) − \overline{z}(s)| ds
≤ \overline{M}_{0}L_{⋆}K_{n}[e^{τ L_{n}^{∗}(t)} ] [e^{−τ L_{t}^{∗}(t)} |z(t) − \overline{z}(t)|]
+ ∫_{0}^{t} [\overline{l}_{n}(s) e^{τ L_{n}^{∗}(s)}] [e^{−τ L_{n}^{∗}(s)} |z(s) − \overline{z}(s)|] ds
≤ \overline{M}_{0}L_{⋆}K_{n} e^{τ L_{n}^{∗}(t)} ∥z − \overline{z}∥_{n} + ∫_{0}^{t} [\frac{e^{τ L_{n}^{∗}(s)}}{τ} ]′ ds ∥z − \overline{z}∥_{n}
≤ \overline{M}_{0}L_{⋆}K_{n} e^{τ L_{n}^{∗}(t)} ∥z − \overline{z}∥_{n} + \frac{1}{τ} e^{τ L_{n}^{∗}(t)} ∥z − \overline{z}∥_{n}
≤ [\overline{M}_{0}L_{⋆}K_{n} + \frac{1}{τ}] e^{τ L_{n}^{∗}(t)} ∥z − \overline{z}∥_{n}.

Therefore,

∥h − \overline{h}∥_{n} ≤ [\overline{M}_{0}L_{⋆}K_{n} + \frac{1}{τ}] ∥z − \overline{z}∥_{n}.

By an analogous relation, obtained by interchanging the roles of z and \overline{z}, it follows that

H_{d}(F̃(z), F̃(\overline{z})) ≤ [\overline{M}_{0}L_{⋆}K_{n} + \frac{1}{τ}] ∥z − \overline{z}∥\overline{B}.

So the operator F̃ is a contraction for all n ∈ ℕ and an admissible operator. From the choice of U there is no z ∈ ∂U such that z = λ F̃(z) for some λ ∈ (0, 1). Then the statement (C2) in Theorem 2.9 does not hold. This implies that the operator F̃ has a fixed point z^{⋆}. Then y^{⋆}(t) = z^{⋆}(t) + x(t), t ∈ (−∞, +∞) is a fixed point of the operator Ñ, which is a mild solution of the problem (3) − (4).

5 Applications

To illustrate the previous results, we give in this section two applications:

Example 5.1 Consider the following model

\{ \begin{matrix}\frac{∂v}{∂t} (t, ξ) & ∈ a(t, ξ) \frac{∂^{2}v}{∂ξ^{2}} (t, ξ) \\ & + ∫_{−∞}^{0} P (θ)R(t, v(t + θ, ξ))dθ & ξ ∈ [0, π] \\ v(t, 0) & = v(t, π) = 0 & t ∈ [0, +∞) \\ v(θ, ξ) & = v_{0}(θ, ξ) & −∞ < θ ≤ 0, & ξ ∈ [0, π], \end{matrix}  (13)

where a(t, ξ) is a continuous function and is uniformly Hölder continuous in t ; P : (−∞, 0] → ℝ and v_{0} : (−∞, 0] × [0, π] → ℝ are continuous functions and R : [0, +∞) × ℝ → P(ℝ) is a multivalued map with compact convex values.

Consider E = L^{2}([0, π], ℝ) and define A(t) by A(t)w = a(t, ξ)w′′ with domain

D(A) = { w ∈ E : w, w′ are absolutely continuous, w′′ ∈ E, w(0) = w(π) = 0 }

Then A(t) generates an evolution system U(t, s) satisfying assumption (H1) (see [20]).

For the phase space B, we choose the well known space BUC(ℝ^{−}, E) : the space of uniformly bounded continuous functions endowed with the following norm

∥φ∥ = sup_{θ≤0} |φ(θ)| for φ ∈ B.

If we put for φ ∈ BUC(ℝ^{−}, E) and ξ ∈ [0, π]

y(t)(ξ) = v(t, ξ), t ∈ [0, +∞), ξ ∈ [0, π],

ϕ(θ)(ξ) = v_{0}(θ, ξ), −∞ < θ ≤ 0, ξ ∈ [0, π],

and

F (t, φ)(ξ) = ∫_{−∞}^{0} P (θ)R(t, φ(θ)(ξ))dθ, −∞ < θ ≤ 0, ξ ∈ [0, π].

Then, the problem (13) takes the abstract semilinear functional evolution inclusion form (1) − (2). In order to prove the existence of mild solutions of problem (13), we suppose the following assumptions :

- There exist p ∈ L^{1}(J, ℝ^{+}) and a nondecreasing continuous function ψ : [0, +∞) → (0, +∞) such that

|R(t, u)| ≤ p(t)ψ(|u|), for ∈ J, and u ∈ ℝ.

- P is integrable on (−∞, 0].

By the dominated convergence theorem, one can show that f ∈ S_{F,y} is a continuous function from B to E. On the other hand, we have for φ ∈ B and ξ ∈ [0, π]

|F (t, φ)(ξ)| ≤ ∫_{−∞}^{0} |p(t)P (θ)| ψ(|(φ(θ))(ξ)|)dθ.

Since the function ψ is nondecreasing, it follows that

∥F (t, φ)∥_{P(E)} ≤ p(t) ∫_{−∞}^{0} |P (θ)| dθψ(|φ|), for φ ∈ B.

Proposition 5.2 Under the above assumptions, if we assume that condition (6) in Theorem 3.2 is true, φ ∈ B, then the problem (13) has a mild solution which is defined in (−∞, +∞).

Example 5.3 Consider the following model

\{ \begin{matrix}\frac{∂}{∂t} & [v(t, ξ) − ∫_{−∞}^{0} T (θ)u(t, v(t + θ, ξ))dθ] \\ & ∈ a(t, ξ) \frac{∂^{2}v}{∂ξ^{2}} (t, ξ) \\ & + ∫_{−∞}^{0} P (θ)R(t, v(t + θ, ξ))dθ & t ∈ [0, +∞), & ξ ∈ [0, π] \\ v(t, 0) & = v(t, π) = 0 & t ∈ [0, +∞) \\ v(θ, ξ) & = v_{0}(θ, ξ) & −∞ < θ ≤ 0, & ξ ∈ [0, π], \end{matrix} (14)

where a(t, ξ) is a continuous function and is uniformly Hölder continuous in t ; T, P : (−∞, 0] → ℝ ; u : (−∞, 0] × ℝ → ℝ and v_{0} : (−∞, 0] × [0, π] → ℝ are continuous functions and R : [0, +∞) × ℝ → P(ℝ) is a multivalued map with compact convex values.

Consider E = L^{2}([0, π], ℝ) and define A(t) by A(t)w = a(t, ξ)w′′ with domain

D(A) = { w ∈ E : w, w′ are absolutely continuous, w′′ ∈ E, w(0) = w(π) = 0 }

Then A(t) generates an evolution system U(t, s) satisfying assumption (H1) (see [20]).

For the phase space B, we choose the well known space BUC(ℝ^{−}, E) : the space of uniformly bounded continuous functions endowed with the following norm

∥φ∥ = sup_{θ≤0} |φ(θ)| for φ ∈ B.

If we put for φ ∈ BUC(ℝ^{−}, E) and ξ ∈ [0, π]

y(t)(ξ) = v(t, ξ), t ∈ [0, +∞), ξ ∈ [0, π],
ϕ(θ)(ξ) = v_{0}(θ, ξ), −∞ < θ ≤ 0, ξ ∈ [0, π],
g(t, φ)(ξ) = ∫_{−∞}^{0} T (θ)u(t, φ(θ)(ξ))dθ, −∞ < θ ≤ 0, ξ ∈ [0, π],

and

F (t, φ)(ξ) = ∫_{−∞}^{0} P (θ)R(t, φ(θ)(ξ))dθ, −∞ < θ ≤ 0, ξ ∈ [0, π].

Then, the problem (14) takes the abstract neutral functional evolution inclusion form (3) − (4). In order to prove the existence of mild solutions of problem (14), we suppose the following assumptions :

- u is Lipschitz with respect to its second argument. Let lip(u) denotes the Lipschitz constant of u.

- There exist p ∈ L^{1}(J, ℝ^{+}) and a nondecreasing continuous function ψ : [0, +∞) → (0, +∞) such that

|R(t, x)| ≤ p(t)ψ(|x|), for ∈ J, and x ∈ ℝ.

- T, P are integrable on (−∞, 0].

By the dominated convergence theorem, one can show that f ∈ S_{F,y} is a continuous function from B to E. Moreover the mapping g is Lipschitz continuous in its second argument, in fact, we have

|g(t, φ_{1}) − g(t, φ_{2})| ≤ \overline{M}_{0}L_{∗}lip(u) ∫_{−∞}^{0} |T (θ)| dθ |φ_{1} − φ_{2}| , for φ_{1}, φ_{2} ∈ B.

On the other hand, we have for φ ∈ B and ξ ∈ [0, π]

|F (t, φ)(ξ)| ≤ ∫_{−∞}^{0} |p(t)P (θ)| ψ(|(φ(θ))(ξ)|)dθ.

Since the function ψ is nondecreasing, it follows that

∥F (t, φ)∥_{P(E)} ≤ p(t) ∫_{−∞}^{0} |P (θ)| dθψ(|φ|), for φ ∈ B.

Proposition 5.4 Under the above assumptions, if we assume that condition (10) in Theorem 4.2 is true, φ ∈ B, then the problem (14) has a mild solution which is defined in (−∞, +∞).

References

[1] N.U. Ahmed, Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series, 246. Longman Scientific & Technical, Harlow John Wiley & Sons, Inc., New York, 1991.

[2] N.U. Ahmed, Dynamic Systems and Control with Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.

[3] A. Arara, M. Benchohra, L. Górniewicz and A. Ouahab, Controllability results for semilinear functional differential inclusions with unbounded delay, Math. Bulletin, 3 (2006), 157-183.

[4] S. Baghli and M. Benchohra, Uniqueness results for partial functional differential equations in Fréchet spaces, Fixed Point Theory 9 (2) (2008), 395-406.

[5] S. Baghli and M. Benchohra, Existence results for semilinear neutral functional differential equations involving evolution operators in Fréchet spaces, Georgian Math. J. (to appear).

[6] S. Baghli and M. Benchohra, Perturbed functional and neutral functional evolution equations with infinite delay in Fréchet spaces, Elec. J. Diff. Equa. 2008 (2008), No. 69, pp. 1–19.

[7] M. Benchohra, E. P. Gatsori and S. K. Ntouyas, Multivalued semilinear neutral functional differential equations with nonconvex-valued right-hand side, Abst. Appl. Anal. 2004 (6) (2004), 525-541.

[8] M. Benchohra, E. P. Gastori and S. K. Ntouyas, Existence results for semi-linear integrodifferential inclusions with nonlocal conditions, Rocky Mountain J. Math., 34 (3) (2004), 833–848.

[9] M. Benchohra and S. K. Ntouyas, Existence of mild solutions on semiinfinite interval for first order differential equation with nonlocal condition, Comment. Math. Univ. Carolinae, 41 (3) (2000), 485-491.

[10] M. Benchohra and S. K. Ntouyas, Existence of mild solutions for certain delay semilinear evolution inclusions with nonlocal condition, Dynam. Systems Appl. 9 (3) (2000), 405–412.

[11] M. Benchohra and S. K. Ntouyas, Existence results for neutral functional differential and integrodifferential inclusions in Banach spaces, Elect. J. Diff. Equa. 2000 (2000), No. 20, 1-15.

[12] M. Benchohra and S. K. Ntouyas, Existence results on infinite intervals for neutral functional differential and integrodifferential inclusions in Banach spaces, Georgian Math. J. 7 (2000), No. 4, 609-625.

[13] M. Benchohra and S. K. Ntouyas, Existence of mild solutions of semilinear evolution inclusions with nonlocal conditions, Georgian Math. J. 7 (2000), No. 2, 221-230.

[14] M. Benchohra and S. K. Ntouyas, Existence results for functional differential inclusions, Elect. J. Diff. Equa. 2001 (2001), No. 41, 1-8.

[15] M. Benchohra and S. K. Ntouyas, Existence results for multivalued semilinear functional differential equations, Extracta Math., 18 (1) (2003), 1-12.

[16] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 580, Springer-Verlag, New York, 1977.

[17] C. Corduneanu and V. Lakshmikantham, Equations with unbounded delay, Nonlinear Anal., 4 (1980), 831-877.

[18] K. Deimling, Multivalued Differential Equations, Walter de Gruyter, Berlin-New York, 1992.

[19] K.J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.

[20] A. Freidman, Partial Differential Equations, Holt, Rinehat and Winston, New York, 1969.

[21] M. Frigon, Fixed point results for multivalued contractions on gauge spaces. Set valued mappings with applications in nonlinear analysis, 175–181, Ser. Math. Anal. Appl., 4, Taylor & Francis, London, 2002.

[22] M. Frigon, Fixed point and continuation results for contractions in metric and gauge spaces. Fixed point theory and its applications, 89–114, Banach Center Publ., 77, Polish Acad. Sci., Warsaw, 2007.

[23] E. Gastori, S. K. Ntouyas and Y. G. Sficas, On a nonlocal cauchy problem for differential inclusions, Abst. Appl. Anal. 2004 (5) (2004), 425-434.

[24] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its Applications, 495, Kluwer Academic Publishers, Dordrecht, 1999.

[25] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.

[26] J. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.

[27] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences 99, Springer-Verlag, New York, 1993.

[28] S. Heikkila and V. Lakshmikantham, Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations, Marcel Dekker Inc., New York, 1994.

[29] J. Henderson and A. Ouahab, Existence results for nondensely defined semilinear functional differential inclusions in Fréchet spaces, Elect. J. Qual. Theor. Differ. Equat. 17 (2005), 1-17.

[30] Y. Hino and S. Murakami, Total stability in abstract functional differential equations with infinite delay, Proceedings of the 6th Colloquium on the Qualitative Theory of Differential Equations (Szeged, 1999), No. 13, 9 pp. (electronic) Proc. Colloq. Qual. Theory Differ. Equ., Electron. J. Qual. Theory Differ. Equ., Szeged, 2000.

[31] Y. Hino, S. Murakami, and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Math., 1473, Springer-Verlag, Berlin, 1991.

[32] Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory, Kluwer, Dordrecht, Boston, London, 1997.

[33] F. Kappel and W. Schappacher, Some considerations to the fundamental theory of infinite delay equations, J. Differential Equations, 37 (1980), 141-183.

[34] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, Walter de Gruyter & Co., Berlin, 2001.

[35] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, 1991.

[36] V. Kolmanovskii, and A. Myshkis, Introduction to the Theory and Applications of Functional-Differential Equations. Mathematics and its Applications, 463. Kluwer Academic Publishers, Dordrecht, 1999.

[37] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics. Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993.

[38] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.

[39] K. Schumacher, Existence and continuous dependence for differential equations with unbounded delay, Arch. Rational Mech. Anal., 64 (1978), 315-335.

[40] A.A. Tolstonogov, Differential Inclusions in a Banach Space, Kluwer Academic Publishers, Dordrecht, 2000.

[41] J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences 119, Springer-Verlag, New York, 1996.

[42] K. Yosida, Functional Analysis, 6^{th} edn. Springer-Verlag, Berlin, 1980.

(Received July 5, 2008)