Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 9, 1-23; http://www.math.u-szeged.hu/ejqtde/

AN EXISTENCE RESULT FOR NEUTRAL FUNCTIONAL DIFFERENTIAL INCLUSIONS IN A

BANACH SPACE

⊤L.GUEDDA AND ^{♢}A. HALLOUZ

Abstract. In this paper we prove the existence of mild solutions for semilinear neutral functional differential inclusions with unbounded linear part generating a noncompact semigroup in a Banach space. This work generalizes the result given in [4].

1. Introduction

Semilinear neutral functional differential inclusion has been the object of many studies by many researchers in the recent years. We only mention the works of some authors ([1], [2], [6]). The method which consists in defining an integral multioperator for which fixed points set coincides with the solutions set of differential inclusion has been often applied to existence problems.

Our aim in this paper is to give an existence result for initial value problems for first order semilinear neutral functional differential inclusions in a separable Banach space E of the form:

(1.1) \frac{d}{dt} [x(t) − h(t, x_{t})] ∈ Ax(t) + F (t, x_{t}), t ∈ [0, T],
(1.2)  x(t) = φ(t), t ∈ [−r, 0],

where A : D(A) ⊂ E → E is the infinitesimal generator of an uniformly bounded analytic semigroup of linear operators {S(t)}_{t≥0} on a separable Banach space E; the multimap F : [0, T] × C([−r, 0] , E) → P (E) and h : [0, T] × C([−r, 0] , E) → E, are given functions, 0 < r < ∞, φ ∈ C([−r, 0] , E), where P (E) denotes the class of all nonempty subsets of E, and C([−r, 0] , E) denotes the space of continuous functions from [−r, 0] to E.

EJQTDE, 2008 No. 9, p. 1

2000 Mathematics Subject Classification. 34A60, 34K05.

Key words and phrases. Functional inclusions, condensing operators, semigroup of linear operators.

For any continuous function x defined on [−r, T] and any t ∈ [0, T], we denote by x_{t} the element of C([−r, 0] , E) defined by

x_{t}(θ) = x(t + θ), θ ∈ [−r, 0] .

For any u ∈ C([−r, 0] , E) the norm ∥.∥_{C([−r,0],E)} of u is defined by

∥u∥_{C([−r,0],E)} = sup{∥u(s)∥ : s ∈ [−r, 0]}.

The function x_{t}(.) represents the history of the state from time t−r, up the present time t.

In [8] using topological methods of multivalued analysis, existence results for semilinear inclusions with unbounded linear part generating a noncompact semigroup in Banach space were given. In this paper, using the method of fractional power of closed operators theory and by giving a special measure of noncompactness, we extend this line of attack to the problem (1.1)-(1.2). More precisely in section 3 we give the measure of noncompactness for which the integral multioperator is condensing, this will allow us to give an existence result for the problem (1.1)-(1.2), and by using the properties of fixed points set of condensing operators we deduce that the mild solutions set is compact.

2. Preliminaries

Along this work, E will be a separable Banach space provided with norm ∥.∥ , A : D(A) ⊂ E → E is the infinitesimal generator of an uniformly bounded analytic semigroup of linear operators {S(t)}_{t≥0} in E. We will assume that 0 ∈ ρ(A) and that ∥S(t)∥ ≤ M for all t ∈ [0, T]. Under these conditions it is possible to define the fractional power (−A)^{α}, 0 < α ≤ 1, as closed linear operator on its domain D(−A)^{α}. Furthermore, D(−A)^{α} is dense in E and the function ∥x∥_{α} = ∥(−A)^{α}x∥ defines a norm in D(−A)^{α}. If X_{α} is the space D(−A)^{α} endowed with the norm ∥.∥_{α}, then X_{α} is a Banach space and there exists c_{α} > 0 such that ∥(−A)^{α}S(t)∥ ≤ \frac{c_{α}}{t^{α}} , for t > 0. Also the inclusion X_{α} ↪ X_{β} for 0 < β ≤ α ≤ 1 is continuous.

For additional details respect of fractional power of a linear operator and semigroup theory, we refer the reader to [11] and [16] ..

Let X be a Banach space and (Y, ≥) a partially ordered set. A function Ψ : P (X) → Y is called a measure of noncompactness in X if Ψ(Ω) = Ψ(co^{−}Ω) for every Ω ∈ P (X), where co^{−}Ω denote the closed convex hull of Ω.

A measure of noncompactness Ψ is called:

EJQTDE, 2008 No. 9, p. 2 (i) nonsingular if Ψ({a} ∪ Ω) = Ψ(Ω) for every a ∈ X, Ω ∈ P (X); (ii) monotone if Ω_{0,}Ω_{1} ∈ P (X) and Ω_{0} ⊆ Ω_{1} imply Ψ(Ω_{0}) ≤ Ψ(Ω_{1}); (iii) real if Y = [0, ∞] with the natural ordering, and Ψ(Ω) < +∞

for every bounded set Ω ∈ P (X).

If Y is a cone in a Banach space we will say that the measure of noncompactness Ψ is regular if Ψ(Ω) = 0 is equivalent to the relative compactness of Ω.

One of most important examples of measure of noncompactness possessing all these properties, is the Hausdorff measure of noncompactness

χ(Ω) = inf{ε > 0; Ω has a finite ε-net in X}

Let K(X) denotes the class of compact subsets of X, Kv(X) denotes the class of compact convex subsets of X, and (Q, d) a metric space.

A multimap G : Z → K(X) is called Ψ−condensing if for every bounded set Ω ⊂ E, that is not relatively compact we have Ψ(G(Ω)) ⩾̸ Ψ(Ω), where Z ⊂ X.

A multivalued map G : X → K(Q) is u.s.c at a point x ∈ X, if for every ε > 0 there exists neighborhood V (x) such that G(x′) ⊂ W_{ε}(G(x)), for every x′ ∈ V (x). Here by W_{ε}(A) we denote the ε-neighborhood of a set A, i.e., W_{ε}(A) = {y ∈ Y : d(y, A) < ε}, where d(y, A) = inf_{x∈A} d(x, y).

A multimap G : X → P (Q) is said to be quasicompact if its restriction to every compact subset A ⊂ X is compact.

A multifunction 𝟋 : [0, T] → K(X) is said to be strongly measurable if there exists a sequence {𝟋_{n}}_{n=1}^{∞} of step multifunctions such that Haus (𝟋(t), 𝟋_{n}(t)) → 0 as n → ∞ for μ − a.e. t ∈ [0, T] where μ denotes a Lebesgue measure on [0, T] and Haus is the Hausdorff metric on K(X). Every strongly measurable multifunction 𝟋 admits a strongly measurable selection g : [0, T] → X, i.e., g(t) ∈ 𝟋(t) for a.e. t ∈ [0, T].

Let L^{1}([0, T], X) denotes the space of all Bochner summable functions

A multifunction 𝟋 : [0, T] → K(X) is said to be

(i) integrable provided it has a summable selection g ∈ L^{1}([0, T], X), (ii) integrably bounded if there exists a summable function q(.) ∈

L^{1}([0, T], X) such that ∥𝟋(t)∥ = sup {∥y∥ : y ∈ 𝟋(t)} ≤ q(t) for a.e. t ∈ [0, T].

A sequence {f_{n}}_{n=1}^{∞} ⊂ L^{1}([0, T], X) is semicompact if

(i) it is integrably bounded: ∥f_{n}(t)∥ ≤ q(t) for a.e. t ∈ [0, T] and for every n ≥ 1, where q(.) ∈ L^{1}([0, T], ℝ^{+})

EJQTDE, 2008 No. 9, p. 3 (ii) the set {f_{n}(t)}_{n=1}^{∞}is relatively compact for almost every t ∈ [0, T]. Any semicompact sequence in L^{1}([0, T], X) is weakly compact in

L^{1}([0, T], X).

For all this definitions see for example [8].

In the sequel, C([−r, T], E) denotes the space of continuous functions from [−r, T] to E endowed with the supremum norm. For any x ∈ C([−r, T], E),

∥x∥_{C([−r,T],E)} = sup {∥x(t)∥ : t ∈ [−r, T]} .

In section 3 we establish an existence result to the problem (1.1)-(1.2) using the following well known results. (See [8]).

Lemma 1. Let E be a Banach space and ϕ : E → E a bounded linear operator. Then for every bounded subset Ω ⊂ E

χ(ϕ(Ω)) ≤ ∥ϕ∥ χ(Ω).

Lemma 2. Let E be a separable Banach space and G : [0, T] → P (E) an integrable, integrably bounded multifunction such that

χ (G(t)) ≤ q(t)

for a.e. t ∈ [0, T] where q ∈ L_{+}^{1} ([0, T]) . Then for all t ∈ [0, T]

∫_{0}^{t} χ (G(s)) ds ≤ ∫_{0}^{t} q(s)ds.

Lemma 3. Let E be a separable Banach space and J an operator

J : L^{1}([0, T], E) → C([0, T], E)

which satisfies the following conditions:

J_{1}) There exists D > 0 such that

∥Jf(t) − J(g)(t)∥ ≤ D ∫_{0}^{t} ∥f(s) − g(s)∥ ds, 0 ≤ t ≤ T
for every f, g ∈ L^{1}([0, T], E).

J_{2}) For any compact K ⊂ E and sequence {f_{n}}_{n=1}^{∞} ⊂ L^{1}([0, T], E) such that {f_{n}(t)}_{n=1}^{∞} ⊂ K for a.e. t ∈ [0, T] the weak conver-

gence f_{n} →_{w} f_{0} implies J(f_{n}) → J(f_{0}).

Then, EJQTDE, 2008 No. 9, p. 4 (i) If the sequence of functions {f_{n}}_{n=1}^{∞} ⊂ L^{1}([0, T], E) is such that ∥f_{n}(t)∥ ≤ π(t) a.e t ∈ [0, T], for all n = 1, 2, …, and χ({f_{n}}_{n=1}^{∞}) ≤ ζ(t) a.e. t ∈ [0, T], where π, ζ ∈ L_{+}^{1}([0, T], then

χ( {J(f_{n})(t)}_{n=1}^{∞}) ≤ D ∫_{0}^{t} ζ(s)ds.

for all t ∈ [0, T].

(ii) For every semicompact sequence {f_{n}}_{n=1}^{∞} ⊂ L^{1}([0, T]; E) the sequence {J(f_{n})}_{n=1}^{∞} is relatively compact in C([0, T]; E), and; moreover, if f_{n} →_{w} f_{0} then J(f_{n}) →> J( f_{0}).

An example of this operator is the Cauchy operator J : L^{1}([0, T], E) → C([0, T], E) defined for every f ∈ L^{1}([0, T], E) by

J(f)(t) = ∫_{0}^{t} S(t − s)f(s)ds,

where {S(t)}_{t≥0} is a C_{0}−semigroup in E (see [3]).

Lemma 4. ([8]). If G is a convex closed subset of a Banach space E, and Γ : G → Kv(G) is closed Θ condensing, where Θ is nonsingular measure of noncompactness defined on subsets of G, then F ixΓ ≠ ∅.

Lemma 5. ([8]). Let Z be a closed subset of a Banach space E and F : Z → K(E) a closed multimap, which is α-condensing on every bounded subset of Z, where α is a monotone measure of noncompactness. If the fixed points set F ixF is bounded, then it is compact.

3. Existence Result

Let us define what we mean by a mild solution of the problem (1.1)- (1.2).

Definition 1. A function x ∈ C([−r, T], E) is said to be a mild solution of the problem (1.1)-(1.2) if the function s → AS(t−s)h(s, x_{s}) is integrable on [0, t) for each 0 ≤ t < T, and there exists f ∈ L^{1}([0, T], E), f(t) ∈ F (t, x_{t}) a.e. t ∈ [0, T], such that

x(t) = S(t)(φ(0) − h(0, φ)) + h(t, x_{t}) + ∫_{0}^{t} S(t − s)f(s)ds
+ ∫_{0}^{t} AS(t − s)h(s, x_{s})ds, t ∈ [0, T] ,

EJQTDE, 2008 No. 9, p. 5 and

x(t) = φ(t), t ∈ [−r, 0] .

To establish our result we consider the following conditions:

Suppose that the multimap F : [0, T] × C([−r, 0], E) → Kv(E) satisfies the following properties:

F 1) the multifunction F (·, u) has a strongly measurable selection

for every u ∈ C([−r, 0], E); F 2) the multimap F : (t, ·) → Kv(E) is upper semicontinuous for

e.a. t ∈ [0, T]; F 3) there exists a function β ∈ L^{1}([0, T], ℝ^{+}) such that, for every

u ∈ C([−r, 0], E),

∥F (t, u)∥ ≤ β(t)(1 + ∥u∥_{C([−r,0],E)}), a.e. t ∈ [0, T];

F 4) there exists a function κ ∈ L^{1}([0, T], ℝ^{+}) such that for every

Ω ⊂ C([−r, 0], E),

χ(F (t, Ω)) ≤ κ(t) sup_{s∈[−r,0]}χ(Ω(s)), a.e. t ∈ [0, T],

where, for s ∈ [−r, 0], Ω(s) = {u(s); u ∈ Ω} . Assume also that

H) there exist constants d_{1}, d_{2}, ω, θ ∈ ℝ^{+} and 0 < α < 1, such that

h is X_{α}-valued, and

(i) for every u ∈ C([−r, 0], E), and t ∈ [0, T] ∥(−A)^{α}h(t, u)∥ ≤ d_{1} ∥u∥_{C([−r,0],E)} + d_{2};

(ii) for every bounded set Ω ⊂ C([−r, 0], E) and t ∈ [0, T],

χ((−A)^{α}h(t, Ω)) ≤ ω sup_{s∈[−r,0]}χ(Ω(s));

(iii) for every u_{1}, u_{2} ∈ C([−r, 0], E) and t, s ∈ [0, T], ∥(−A)^{α}h(t, u_{1}) − (−A)^{α}h(s, u_{2})∥ ≤ θ ∥u_{1} − u_{2}∥_{C([−r,0],E)} + ϑ(|t − s|), where ϑ : [0, T] → ℝ^{+} is a continuous function, such that ϑ(0) = 0.

We note that from assumptions (F 1) − (F 3) it follows that the superposition multioperator

sel_{F} : C([−r, T], E) → P (L^{1}([0, T], E))

defined for x ∈ C([−r, T], E) by:

sel_{F} (x) = {f ∈ L^{1}([0, T], E), f(t) ∈ F (t, x_{t}), a.e. t ∈ [0, T]} EJQTDE, 2008 No. 9, p. 6 is correctly defined (see [8]) and is weakly closed in the following sense: if the sequences {x^{n}}_{n=1}^{∞} ⊂ C([−r, T], E), {f_{n}}_{n=1}^{∞} ⊂ L^{1}([0, T], E),

f_{n}(t) ∈ F (t, x_{t}^{n}), a.e. t ∈ [0, T], n ≥ 1 are such that x^{n} → x^{0}, f_{n} →_{w} f_{0}.,

then f_{0}(t) ∈ F (t, x_{t}^{0}) a.e. t ∈ [0, T] (see [8]). Since the family {S(t)}_{t≥0} is an analytic semigroup [16], the operator function s → AS(t − s) is continuous in the uniform operator topology on [0, t) which from the estimate

∥(−A)S(t − s)h(s, x_{s})∥ = ∥(−A)^{1−α}S(t − s)(−A)^{α}h(s, x_{s})∥
≤ \frac{C_{1−α}}{(t − s)^{1−α}} (d_{1} ∥x_{s}∥_{C([−r,0],E)} + d_{2})
≤ \frac{C_{1−α}}{(t − s)^{1−α}} (d_{1} ∥x∥_{C([−r,T],E)} + d_{2})

and the Bochner’s theorem implies that AS(t − s)h(s, x_{s}) is integrable on [0, t).

Now we shall prove our main result.

Theorem 1. Let the assumptions (F 1) − (F 4) and (H) be satisfied. If

∥(−A)^{−α}∥ max {ω, θ, d_{1}} < 1

then the mild solutions set of the problem (1.1)-(1.2) is a nonempty compact subset of the space C([−r, T], E).

Proof. In the space C([−r, T], E), Let define the operator Γ : C([−r, T], E) → P (C([−r, T], E)) in the following way:

Γ(x)(t) = {y ∈ C([−r, T], E) : y(t) = φ(t), t ∈ [−r, 0] and
y(t) = Υ(f)(t) + h(t, x_{t}) + ∫_{0}^{t} AS(t − s)h(s, x_{s})ds; for t ∈ [0, T] }

where f ∈ sel_{F} (x), and the operator Υ : L^{1}([0, T], E) → C([0, T], E) is defined by

Υ(f)(t) = S(t)x_{0} + ∫_{0}^{t} S(t − s)f(s)ds, t ∈ [0, T]

where x_{0} = φ(0) − h(0, φ).

Remark 1. It is clear that the operator Γ is well defined, and the fixed points of Γ are mild solutions of the problem (1.1)-(1.2).

EJQTDE, 2008 No. 9, p. 7 The proof will be given in four steps.

Step 1. The multivalued operator Γ is closed.

The multivalued operator Γ can be written in the form Γ = ∑^{3} ; 1Γ_{i}

where the operators Γ_{i}, i = 1, 2, 3 are defined as follows: the multivalued operator Γ_{1} : C([−r, T], E) → P (C([−r, T], E)) by

Γ_{1}(x)(t) = \{ \begin{matrix}φ(t) − h(0, φ), & t ∈ [−r, 0], \\ Υ(f)(t), & t ∈ [0, T] \end{matrix} \}

where f ∈ Sel_{F} (x), the operator Γ_{2} : C([−r, T], E) → C([−r, T], E) by

Γ_{2}(x)(t) = \{ \begin{matrix}h(0, φ), & t ∈ [−r, 0], \\ h(t, x_{t}), & t ∈ [0, T] \end{matrix}

and the operator Γ_{3} : C([−r, T], E) → C([−r, T], E) by

Γ_{3}(x)(t) = \{ \begin{matrix}0, & t ∈ [−r, 0] \\ ∫_{0}^{t} AS(t − s)h(s, x_{s})ds, & t ∈ [0, T]. \end{matrix}

Let {x^{n}}_{n=1}^{∞}, {z^{n}}_{n=1}^{∞}, x^{n} → x^{0}, z^{n} ∈ Γ((x^{n}), n ≥ 1, and z^{n} → z^{0}. Let {f_{n}}_{n=1}^{∞} ⊂ L^{1}([0, T], E) an arbitrary sequence such that, for n ≥ 1

f_{n}(t) ∈ F (t, x_{t}^{n}), a.e. t ∈ [0, T],

and

z^{n} = \{ \begin{matrix}φ(t), & t ∈ [−r, 0], \\ Υ(f_{n})(t) + h(t, x_{t}^{n}) + ∫_{0}^{t} AS(t − s)h(s, x_{s}^{n})ds, & t ∈ [0, T]. \end{matrix}

Since {S(t)}_{t≥0} is a strongly continuous semigroup (see [3]), the operator Υ satisfies the properties (J_{1}) and (J_{2}) of Lemma 3, by using hypothesis (F 3) we have that sequence {f_{n}}_{n=1}^{∞} is integrably bounded. Hypothesis (F 4) implies that

χ({f_{n}(t)}_{n=1}^{∞}) ≤ κ(t)χ({x^{n}(t)}_{n=1}^{∞} = 0

for a.e. t ∈ [0, T], i.e., the set {f_{n}(t)}_{n=1}^{∞} is relatively compact for a.e. t ∈ [0, T], thus {f_{n}}_{n=1}^{∞} is semicompact sequence. Consequently {f_{n}}_{n=1}^{∞} is weakly compact in L^{1}([0, T], E) so we can assume without loss of generality, that f_{n} →_{w} f_{0}.

By applying Lemma 3 , Υ(f_{n}) → Υ(f_{0}) in C([0, T], E). Moreover, by using the fact that the operator sel_{F} is closed, we have f_{0} ∈ sel_{F} (x^{0}).

EJQTDE, 2008 No. 9, p. 8 Consequently

z_{1}^{n}(t) = \{ \begin{matrix}φ(t) − h(0, φ), & t ∈ [−r, 0], \\ Υ(f_{n})(t), & t ∈ [0, T]. \end{matrix}
(3.1)  →_{n→∞} z_{1}^{0}(t) = \{ \begin{matrix}φ(t) − h(0, φ), & t ∈ [−r, 0], \\ Υ(f_{0})(t), & t ∈ [0, T]. \end{matrix}

in the space C([−r, T], E), with f_{0} ∈ sel_{F} (x^{0}). On the other hand, using (H) − (iii), for t ∈ [0, T] we get

∥h(t, x_{t}^{n}) − h(t, x_{t}^{0})∥ ≤ ∥(−A)^{−α}∥ ∥(−A)^{α}h(t, x_{t}^{n}) − (−A)^{α}h(t, x_{t}^{0})∥
≤ θ ∥(−A)^{−α}∥ ∥x_{s}^{n} − x_{s}^{0}∥_{C([−r,0],E)}
≤ θ ∥(−A)^{−α}∥ ∥x^{n} − x^{0}∥_{C([−r,T],E)} .

It results that

(3.2) ∥Γ_{2}(x^{0}) − Γ_{2}(x^{n})∥_{C([−r,T],E)} ≤ θ ∥(−A)^{−α}∥ ∥x^{n} − x^{0}∥_{C([−r,T],E)}

Using hypothesis (H)−(ii) and the estimate in the family {(−A)^{1−α}S(t)}_{t>0}, for any t ∈ [0, T] we have

∥∫_{0}^{t} [AS(t − s)h(s, x_{s}^{n}) − AS(t − s)h(s, x_{s}^{0})] ds∥
≤ ∫_{0}^{t} ∥AS(t − s)h(s, x_{s}^{n}) − AS(t − s)h(s, x_{s}^{0})∥ ds
≤ θ ∥(−A)^{−α}∥ ∥x^{n} − x^{0}∥_{C([−r,T],E)} ∫_{0}^{t} ∥(−A)^{1−α}S(t − s)∥ ds
≤ θ ∥(−A)^{−α}∥ ∥x^{n} − x^{0}∥_{C([−r,T],E)} ∫_{0}^{t} \frac{C_{1−α}}{(t−s)^{1−α}} ds
≤ θ ∥(−A)^{−α}∥ \frac{C_{1−α}T^{α}}{α} ∥x^{n} − x^{0}∥_{C([−r,T],E)}

Then

(3.3) ∥Γ_{3}(x^{0}) − Γ_{3}(x^{n})∥_{C([−r,T],E)}

≤ θ ∥(−A)^{−α}∥ \frac{C_{1−α}T^{α}}{α} ∥x^{n} − x^{0}∥_{C([−r,T],E)} .

From the inequalities (3.1)-(3.3) follows immediately that z^{n} → z^{0} in the space C([−r, T], E), with

z^{0}(t) = \{ \begin{matrix}φ(t), & t ∈ [−r, 0], \\ Υ(f_{0})(t) + h(t, x_{t}^{0}) + ∫_{0}^{t} AS(t − s)h(s, x_{s}^{0})ds, & t ∈ [0, T]. \end{matrix} \}

EJQTDE, 2008 No. 9, p. 9 and f_{0} ∈ sel_{F} (x^{0}). Thus z^{0} ∈ Γ(x^{0}) and hence Γ is closed. Now in the space C([−r, T], E) we consider the measure of noncompactness Θ defined in the following way: for every bounded subset Ω ⊂ C([−r, T], E)

Θ(Ω) = (χ(Ω([−r, 0]), Ψ(Ω), mod_{c}Ω)

where

Ψ(Ω) = sup_{t∈[0,T]} (e^{−Lt} sup_{s∈[0,t]}χ(Ω(s)))

and mod_{c}Ω is the module of equicontinuity of the set Ω ⊂ C([−r, T], E) given by:

mod_{c}Ω = lim_{δ→0}sup_{x∈Ω} max_{|t_{1}−t_{2}|≤δ} ∥x(t_{1}) − x(t_{2})∥

and L > 0 is chosen so that

M sup_{t∈[0,T]} ∫_{0}^{t} e^{−L(t−s)}κ(s)ds ≤ q_{1} < 1

ω sup_{t∈[0,T]} ∫_{0}^{t} \frac{C_{1−α}}{(t − s)^{1−α}} e^{−L(t−s)}ds ≤ q_{2} < 1

d_{1} sup_{t∈[0,T]} ∫_{0}^{t} \frac{e^{−L(t−s)}}{(t − s)^{1−α}} c_{1−α}ds ≤ q_{3} < 1

M sup_{t∈[0,T]} ∫_{0}^{t} e^{−L(t−s)}β(s)ds ≤ q_{4} < 1

where M is the constant from the estimation in the family {S(t)}_{t≥0}, the constants d_{1}, d_{2} from (H) − (i), the constant ω from (H) − (ii), the function β from the hypothesis (F 3) and the function κ from the hypothesis (F 4). From the Arzelá-Ascoli theorem, the measure Θ give a nonsingular and regular measure of noncompactness in C([−r, T], E).

Remark 2. If ξ ∈ L^{1}([0, T], E), it is clear that

sup_{t∈[0,T]} ∫_{0}^{t} e^{−L(t−s)}ξ(s)ds →_{L→+∞} 0.

Step 2. The miltioperator Γ is Θ condensing on every bounded subset of C([−r, T], E).

Let Ω ⊂ C([−r, T], E) be a bounded subset such that

(3.4)  Θ(Γ(Ω)) ≥ Θ(Ω),

EJQTDE, 2008 No. 9, p. 10 where the inequality is taking in the sense of the order in ℝ^{3} induced by the positive cone ℝ_{+}^{3}. We will show that (3.4) implies that Ω is relatively compact in C([−r, T], E). From the inequality (3.4) follows immediately that

(3.5)  χ(Ω([−r, 0])) = 0.

Indeed, we have

χ(ΓΩ)[−r, 0]) = χ {φ([−r, 0])} = 0 ≥ χ(Ω[−r, 0]) ≥ 0.

Remark that from (3.5) it follows that

sup_{α∈[−r,0]}χ(Ω(α)) = 0.

Therefore, for s ∈ [0, T]

sup_{α∈[s−r,s]}χ(Ω(α)) ≤ sup_{α∈[−r,0]}χ(Ω(α)) + sup_{α∈[0,s]}χ(Ω(α))
(3.6)  ≤ sup_{α∈[0,s]}χ(Ω(α)).

We give now an upper estimate for χ({f(s), f ∈ sel_{F} (Ω)}, for s ∈ [0, t], t ≤ T. By using (3.6) and the assumption (F 4) we have

χ({f(s), f ∈ sel_{F} (Ω)} ≤ χ(F (s, Ω_{s}))
≤ e^{Ls}κ(s)e^{−Ls} sup_{α∈[−r,0]}χ(Ω_{s}(α))
≤ e^{Ls}κ(s)e^{−Ls} sup_{α∈[s−r,s]}χ(Ω(α))
≤ e^{Ls}κ(s)e^{−Ls} sup_{α∈[0,s]}χ(Ω(α))
≤ e^{Ls}κ(s) sup_{s∈[0,t]}e^{−Ls} sup_{α∈[0,s]}χ(Ω(α)).
≤ e^{Ls}κ(s) sup_{s∈[0,T]}e^{−Ls} sup_{α∈[0,s]}χ(Ω(α))
≤ e^{Ls}κ(s)Ψ(Ω)

Using Lemma 2 with D = M, we get

χ({Υ(f)(s), f ∈ sel_{F} (Ω)} ≤ MΨ(Ω) ∫_{0}^{s} e^{Lλ}κ(λ)dλ.

Therefore,

sup_{s∈[0,t]}χ({Υ(f)(s), f ∈ sel_{F} (Ω)} ≤ MΨ(Ω) ∫_{0}^{t} e^{Lλ}κ(λ)dλ.

By multiplying both sides with e^{−Lt} and bearing in mind the definition of q_{1}, we get

sup_{t∈[0,T]}e^{−Lt} sup_{s∈[0,t]}χ({Υ(f)(s), f ∈ sel_{F} (Ω)})
= Ψ({Υ(f), f ∈ sel_{F} (Ω)})
≤ Ψ(Ω)M sup_{t∈[0,T]} ∫_{0}^{t} e^{−L(t−λ)}κ(λ)dλ
(3.7)  ≤ q_{1}Ψ(Ω)

Since the measure χ is monotone, by using (H_{1}) − (iii) and Lemma 1, we obtain for s ∈ [0, t], t ≤ T

χ(h(s, Ω_{s})) ≤ e^{Lt}e^{−Lt}χ ((A)^{−α}(−A)^{α}h(s, Ω_{s}))
≤ e^{Lt}ω ∥(−A)^{−α}∥ e^{−Lt} sup_{α∈[0,s]}χ(Ω(α))
≤ e^{Lt}ω ∥(−A)^{−α}∥ e^{−Lt} sup_{α∈[0,t]}χ(Ω(α))
≤ e^{Lt}ω ∥(−A)^{−α}∥ sup_{t∈[0,T]}e^{−Lt} sup_{α∈[0,t]}χ(Ω(α))
≤ e^{Lt}ω ∥(−A)^{−α}∥ Ψ(Ω).

Then,

sup_{s∈[0,t]}χ(h(s, Ω_{s})) ≤ .e^{Lt}ω ∥(−A)^{−α}∥ Ψ(Ω)

By multiplying both sides with e^{−Lt}, we have

(3.8)  sup_{t∈[0,T]}e^{−Lt} sup_{s∈[0,t]}χ(h(s, Ω_{s})) ≤ .ω ∥(−A)^{−α}∥ Ψ(Ω).

The multifunction G : s → AS(t − s)h(s, Ω_{s}), s ∈ [0, t) is integrable and integrably bounded. Indeed for any x ∈ Ω we have:

∥(−A)S(t − s)h(s, x_{s})∥ = ∥(−A)^{1−α}S(t − s)(−A)^{α})h(s, x_{s})∥
≤ \frac{C_{1−α}}{(t − s)^{1−α}} (d_{1} ∥x_{s}∥_{C([−r,0],E)} + d_{2})
≤ \frac{C_{1−α}}{(t − s)^{1−α}} (d_{1} ∥x∥_{C([−r,T],E)} + d_{2})
≤ \frac{C_{1−α}}{(t − s)^{1−α}} (d_{1}sup_{x∈Ω} ∥x∥_{C([−r,T],E)} + d_{2})

EJQTDE, 2008 No. 9, p. 12 Using the assumption (H) − (ii) and Lemma 1, we get

χ (AS(t − s)h(s, x_{s})) = χ ((−A)^{1−α}(−A)^{α}S(t − s)h(s, x_{s}))
≤ ∥(−A)^{1−α}S(t − s)∥ χ ((−A)^{α}h(s, x_{s}))
≤ \frac{C_{1−α}}{(t − s)^{1−α}} ω sup_{λ∈[0,s]}χ(Ω(λ))
≤ \frac{ωC_{1−α}}{(t − s)^{1−α}} e^{Ls} sup_{s∈[0,T]}e^{−Ls} sup_{λ∈[0,s]}χ(Ω(λ))
≤ \frac{ωC_{1−α}}{(t − s)^{1−α}} e^{Ls}Ψ(Ω)

By lemma 2, we get for every s ∈ [0, t]

χ (∫_{0}^{s}(−A)S(t − λ)h(λ, x_{λ})dλ) ≤ Ψ(Ω) ∫_{0}^{s} \frac{ωC_{1−α}}{(t − λ)^{1−α}} e^{Lλ}dλ
≤ Ψ(Ω) ∫_{0}^{t} \frac{ωC_{1−α}}{(t − λ)^{1−α}} e^{Lλ}dλ.

Thus,

sup_{s∈[0,t]}χ (∫_{0}^{s}(−A)S(t − λ)h(λ, x_{λ})dλ) ≤ Ψ(Ω) ∫_{0}^{t} \frac{ωC_{1−α}}{(t − λ)^{1−α}} e^{Lλ}dλ

By multiplying both sides with e^{−Lt} and bearing in mind the definition of q_{2}, we get

sup_{t∈[0,T]}e^{−Lt} sup_{s∈[0,t]}χ (∫_{0}^{s}(−A)S(t − λ)h(λ, x_{λ})dλ)

≤ Ψ(Ω) sup_{t∈[0,T]} ∫_{0}^{t} \frac{ωC_{1−α}}{(t − λ)^{1−α}} e^{−L(t−λ)}dλ
(3.9)  ≤ q_{2}Ψ(Ω).

EJQTDE, 2008 No. 9, p. 13 From the inequalities (3.7)-(3.9), remark 2 and the fact that ω ∥(−A)^{α}∥ < 1, we get

Ψ(Γ(Ω)) = sup_{t∈[0,T]}e^{−Lt} sup_{s∈[0,t]}χ {Υ(f)(s) + h(s, Ω_{s}) + ∫_{0}^{s} AS(t − s)h(λ, Ω_{λ})dλ}

≤ Ψ(Ω[ sup_{t∈[0,T]}M ∫_{0}^{t} e^{−L(t−s)}κ(s)ds + ∥(−A)^{α}∥ ω

+ sup_{t∈[0,T]} ∫_{0}^{t} \frac{ωC_{1−α}}{(t − s)^{1−α}} e^{−L(t−S)}ds]

≤ Ψ(Ω) [q_{1} + q_{2} + ω ∥(−A)^{α}∥]
< Ψ(Ω).

Using the inequality (3.4), the last inequality implies that

(3.10)  Ψ(Ω) = 0.

We shall give now an upper estimate for mod_{c}Γ(Ω). We have shown that χ {Υ(f)(t), f ∈ sel_{F} (x), x ∈ Ω} = 0, for any t ∈ [0, T]. From the conditions (F 3) and (F 4) follows that the sequence {f ∈ sel_{F} (x), x ∈ Ω} is semicompact in L^{1}([0, T], E), and hence the set

{y; y(t) = Υf(t), t ∈ [0, T], f ∈ sel_{F} (x), x ∈ Ω}

is relatively compact in C([0, T], E) (see [3]). Therefore, the set

Γ_{1}(Ω) = {y(t) = φ(t) − h(0, φ), t ∈ [−r, 0];
y(t) = Υ(f)(t), t ∈ [0, T], f ∈ sel_{F} (x), x ∈ Ω}

is relatively compact in C([−r, T], E). Consequently

(3.11)  mod_{c}Γ_{1}(Ω) = 0.

EJQTDE, 2008 No. 9, p. 14 Let δ > 0, and t, t′ ∈ [0, T], such that For 0 ≤ t′ − t < δ, and let x ∈ Ω, we have

∥Γ_{2}(x)(t) − Γ_{2}(x)(t′)∥ = ∥h(t, x_{t}) − h(t′, x_{t′})∥ ≤
≤ ∥(−A)^{−α}(−A)^{α}h(t, x_{t}) − (−A)^{−α}(−A)^{α}h(t′, x_{t′})∥
≤ ∥(−A)^{−α}∥ [θ ∥x_{t} − x_{t′}∥_{C([−r,0],E)} + sup_{t′−t<δ}ϑ(t′ − t)]

≤ ∥(−A)^{−α}∥ [θ sup_{α∈[−r,0]}_{t′−t<δ} ∥x(t + α) − x(t′ + α)∥ + sup_{t′−t<δ}ϑ(t′ − t)]

≤ ∥(−A)^{−α}∥ [θ sup_{s,s′∈[t−r,t′]}_{|s−s′|<δ} ∥x(s) − x(s′)∥ + sup_{t′−t<δ}ϑ(t′ − t)]

≤ ∥(−A)^{−α}∥ [θ sup_{s,s′∈[−r,T]}_{|s−s′|<δ} ∥x(s) − x(s′)∥ + sup_{t′−t<δ}ϑ(t′ − t)]

Since

lim_{δ→0} sup_{t′−t<δ}ϑ(t′ − t) = ϑ(0) = 0

It results that

(3.12)  mod_{c}Γ_{2}(Ω) ≤ θ ∥(−A)^{−α}∥ mod_{c}Ω

Now we will show that the set

Γ_{3}(Ω) = \{y; y(t) = \{ \begin{matrix}0,  t ∈ [−r, 0], \\ ∫_{0}^{t} Ae^{A(t−s)}h(s, x_{s})ds,  t ∈ [0, T], \end{matrix} \}

EJQTDE, 2008 No. 9, p. 15 where x ∈ Ω, is equicontinuous on C([−r, T], E). Let 0 ≤ t ≤ t′ ≤ T, and x ∈ Ω. We have

∥∫_{0}^{t′} [AS(t′ − s)h(s, x_{s}) − ∫_{0}^{t} AS(t − s)h(s, x_{s})] ds∥

≤ ∥(S(t′ − t) − I) ∫_{0}^{t} AS(t − s)h(s, x_{s})ds∥ + ∥∫_{t}^{t′} AS(t′ − s)h(s, x_{s})ds∥

≤ ∥(S(t′ − t) − I) ∫_{0}^{t} AS(t − s)h(s, x_{s})ds∥

+C_{1−α}(d_{1}sup_{x∈Ω} ∥x∥_{C([−r,T],E)} + d_{2})\frac{(t′ − t)^{α}}{α}

Since χ (∫_{0}^{t} AS(t − s)h(s, Ω_{s})ds) = 0, i.e., the set {∫_{0}^{t} AS(t − s)h(s, Ω_{s})ds} is relatively compact for every t ∈ [0, T], the first term on the right hand side converge to zero when t′ → t uniformly on x ∈ Ω. As consequence we get

(3.13)  mod_{c}Γ_{3}(Ω) = 0.

Since

mod_{c}Γ(Ω) ≤ 3 ; ∑_{i=1}mod_{c}Γ_{i}(Ω)

From the inequalities (3.11)-(3.13), we obtain

mod_{c}Γ(Ω) ≤ θ ∥(−A)^{−α}∥ mod_{c}Ω.

Since θ ∥(−A)^{−α}∥ < 1, from the inequality (3.4) follows

(3.14)  mod_{c}(Ω) = 0.

Finally from the inequalities (3.5), (3.10) and (3.14) we get

Θ(Ω) = (0, 0, 0).

This shows that the subset Ω is relatively compact, concluding the proof of Step 2.

Now in the space C([−r, T], E) we introduce the equivalent norm, given by

∥x∥_{∗} = sup_{t∈[−r,0]} ∥x(t)∥ + sup_{t∈[0,T]}e^{−Lt} sup_{s∈[0,t]} ∥x(s)∥

Consider the ball

B_{r}(0) = {x ∈ C([−r, T], E); ∥x∥_{∗} ≤ r}

where r is a constant chosen so that

r ≥ \frac{∥φ∥_{C([−r,0],E)} + ∥h(0, φ)∥ + M (∥x_{0}∥ + ∥β∥_{L^{1}}) + d_{2}C_{1−α} \frac{T^{α}}{α}}{1 − d_{1} ∥(−A)^{−α}∥}

where x_{0} = φ(0) − h(0, φ). Since d_{1} ∥(−A)^{−α}∥ < 1, the last inequality implies

d_{1} ∥(−A)^{−α}∥ r+∥φ∥_{C([−r,0],E)}+∥h(0, φ)∥+M (∥x_{0}∥ + ∥β∥_{L^{1}})+d_{2}C_{1−α} \frac{T^{α}}{α} ≤ r.

Step 3. The multioperator Γ maps the ball B_{r}(0) into itself.

Let x ∈ B_{r}(0) and y ∈ Γ(x),

y(t) = S(t)x_{0} + h(t, x_{t}) + ∫_{0}^{t} S(t − s)f(s)ds
+ ∫_{0}^{t} AS(t − s)h(s, x_{s})ds, t ∈ [0, T]
y(t) = φ(t), t ∈ [−r, 0]

where f ∈ sel_{F} (x). Remark first that

y = y_{1} + y_{2} + y_{3}

where

y_{1}(t) = \{ \begin{matrix}φ(t) − h(0, φ), & t ∈ [−r, 0] \\ y(t) = Υf(t), & t ∈ [0, T] \end{matrix} \} ,

y_{2}(t) = \{ \begin{matrix}h(0, φ), & t ∈ [−r, 0] \\ h(t, x_{t}) & t ∈ [0, T] \end{matrix} ,

and

y_{3}(t) = \{ \begin{matrix}0, & t ∈ [−r, 0], \\ ∫_{0}^{t} AS(t − s)h(s, x_{s})ds, & t ∈ [0, T]. \end{matrix} .

Therefore,

∥y∥_{⋆} ≤ ∥y_{1}∥_{⋆} + ∥y_{2}∥_{⋆} + ∥y_{3}∥_{⋆}

Let give an upper estimate for each ∥y_{i}∥_{⋆}; i = 1, 2, 3. For s ∈ [−r, 0], we have (3.15)

sup_{s∈[−r,0]} ∥y_{1}(s)∥ = sup_{s∈[−r,0]} ∥φ(t) − h(0, φ)∥ ≤ ∥φ∥_{C([−r,0],E)} + ∥h(0, φ)∥ .

For s ∈ [0, t], t ≤ T, using the hypothesis (F_{3}), we have

∥y_{1}(s)∥ ≤ ∥S(s)x_{0}∥ + ∫_{0}^{s} ∥S(s − τ)∥ ∥f(τ)∥ dτ
≤ M ∥x_{0}∥ + M ∥β∥_{L^{1}} + M ∫_{0}^{s} β(τ) ∥x_{τ}∥_{2} dτ
≤ M ∥x_{0}∥ + M ∥β∥_{L^{1}} + M ∫_{0}^{s} β(τ) sup_{μ∈[τ−r,τ]} ∥x(μ)∥ dτ
≤ M ∥x_{0}∥ + M ∥β∥_{L^{1}}
+ M ∫_{0}^{s} e^{Lτ}β(τ) [e^{−Lτ} sup_{μ∈[−r,0]} ∥x(μ)∥ + e^{−Lτ} sup_{μ∈[0,τ]} ∥x(μ)∥] dτ
≤ M ∥x_{0}∥ + M ∥β∥_{L^{1}}
+ M ∫_{0}^{s} e^{Lτ}β(τ) [ sup_{μ∈[−r,0]} ∥x(μ)∥ + sup_{τ∈[0,T]}e^{−Lτ} sup_{μ∈[0,τ]} ∥x(μ)∥] dτ

≤ M (∥x_{0}∥ + ∥β∥_{L^{1}}) + M ∥x∥_{⋆} ∫_{0}^{s} e^{Lτ}β(τ)dτ

Thus,

sup_{s∈[0,t]} ∥y_{1}(s)∥ ≤ M (∥x_{0}∥ + ∥β∥_{L^{1}}) + M ∥x∥_{⋆} ∫_{0}^{t} e^{Lτ}β(τ)dτ

By multiplying both sides with e^{−Lt} and bearing in mind the definition of q_{4}, we obtain

sup_{t∈[0,T]}e^{−Lt} sup_{s∈[0,t]} ∥y_{1}(s)∥ ≤ M (∥x_{0}∥ + ∥β∥_{L^{1}}) + ∥x∥_{⋆} M sup_{t∈[0,T]} ∫_{0}^{t} e^{−L(t−τ)}β(τ)dτ
(3.16)  ≤ M (∥x_{0}∥ + ∥β∥_{L^{1}}) + q_{4} ∥x∥_{⋆} .

From inequalities (3.15) and (3.16), we get

(3.17) ∥y_{1}∥_{⋆} ≤ ∥φ∥_{C([−r,0],E)} + ∥h(0, φ)∥ + M (∥x_{0}∥ + ∥β∥_{L^{1}}) + q_{4} ∥x∥_{⋆}

Let now give an upper estimate fore ∥y_{2}∥_{⋆}. For s ∈ [−r, 0], we have

(3.18)  sup_{s∈[−r,0]} ∥y_{2}(s)∥_{⋆} = sup_{s∈[−r,0]} ∥h(0, φ)∥ = ∥h(0, φ)∥

For s ∈ [0, t], t ≤ T, using the hypothesis (H) − (i), we have

∥y_{2}(s)∥ ≤ ∥h(t, x_{t})∥
≤ d_{2} ∥(−A)^{−α}∥ + d_{1} ∥(−A)^{−α}∥ ∥x_{s}∥_{C([−r,0],E)}
≤ d_{2} ∥(−A)^{−α}∥ + d_{1} ∥(−A)^{−α}∥ [ sup_{μ∈[−r,0]} ∥x(μ)∥ + sup_{μ∈[0,s]} ∥x(μ)∥]
≤ d_{2} ∥(−A)^{−α}∥ + d_{1} ∥(−A)^{−α}∥ e^{Lt} [ sup_{μ∈[−r,0]} ∥x(μ)∥ + e^{−Lt} sup_{μ∈[0,t]} ∥x(μ)∥]
≤ d_{2} ∥(−A)^{−α}∥ + d_{1} ∥(−A)^{−α}∥ e^{Lt} [ sup_{μ∈[−r,0]} ∥x(μ)∥ + sup_{t∈[0,T]}e^{−Lt} sup_{μ∈[0,t]} ∥x(μ)∥]
≤ d_{2} ∥(−A)^{−α}∥ + d_{1} ∥(−A)^{−α}∥ e^{Lt} ∥x∥_{⋆}

It follows that

sup_{s∈[0,t]} ∥y_{2}(s)∥ ≤ d_{2} ∥(−A)^{−α}∥ + e^{Lt}d_{1} ∥(−A)^{−α}∥ ∥x∥_{⋆}

By multiplying both sides with e^{−Lt}, we get

(3.19) sup_{t∈[0,T]}e^{−Lt} sup_{s∈[0,t]} ∥y_{2}(s)∥ ≤ d_{2} ∥(−A)^{−α}∥ + d_{1} ∥(−A)^{−α}∥ ∥x∥_{⋆}

From the inequalities (3.18) and (3.19), it follows that

∥y_{2}∥_{⋆} ≤ ∥h(0, φ)∥ + d_{2} ∥(−A)^{−α}∥

+d_{1} ∥(−A)^{−α}∥ ∥x∥_{⋆} (3.20) It remains to give an upper estimate for ∥y_{3}∥_{⋆}. For s ∈ [−r, 0], we have

(3.21) sup_{s∈[−r,0]} ∥y_{3}(s)∥ = 0.

EJQTDE, 2008 No. 9, p. 19 For s ∈ [0, t], t ≤ T, by using (H) − (i), we have

∥y_{3}(s)∥ = ∥∫_{0}^{s} AS(s − τ)h(τ, x_{τ})dτ∥
≤ d_{1} ∫_{0}^{s} \frac{C_{1−α}}{(s − τ)^{1−α}} sup_{μ∈[τ−r,τ]} ∥x(μ)∥ dτ + d_{2} ∫_{0}^{s} \frac{C_{1−α}}{(s − τ)^{1−α}} dτ

≤ d_{1} ∫_{0}^{s} \frac{C_{1−α}}{(s − τ)^{1−α}} [ sup_{μ∈[−r,0]} ∥x(μ)∥ + sup_{μ∈[0,τ]} ∥x(μ)∥] dτ + d_{2} ∫_{0}^{T} \frac{C_{1−α}}{(T − τ)^{1−α}} dτ

≤ d_{1} ∫_{0}^{t} \frac{C_{1−α}}{(t − s)^{1−α}} e^{Lτ}dτ [ sup_{μ∈[−r,0]} ∥x(μ)∥ + sup_{τ∈[0,T]}e^{−Lτ} sup_{μ∈[0,τ]} ∥x(μ)∥] + d_{2}C_{1−α} \frac{T^{α}}{α}

≤ d_{1} ∫_{0}^{s} \frac{C_{1−α}}{(t − s)^{1−α}} e^{Lτ}dτ ∥x∥_{⋆} + d_{2}C_{1−α} \frac{T^{α}}{α}

Therefore

sup_{s∈[0,t]} ∥y_{3}(s)∥ ≤ ∥x∥_{⋆} d_{1} ∫_{0}^{t} \frac{C_{1−α}}{(t − τ)^{1−α}} e^{Lτ}dτ + d_{2}C_{1−α} \frac{T^{α}}{α} .

By multiplying both sides with e^{−Lt} and bearing in mind the definition of q_{3}, we get

sup_{t∈[0,T]}e^{−Lt} sup_{s∈[0,t]} ∥y_{3}(s)∥

≤ ∥x∥_{⋆} (d_{1} sup_{t∈[0,T]} ∫_{0}^{t} \frac{C_{1−α}}{(t − τ)^{1−α}} e^{−L(t−τ)}dτ) + d_{2}C_{1−α} \frac{T^{α}}{α}

(3.22) ≤ q_{3} ∥x∥_{⋆} + d_{2}C_{1−α} \frac{T^{α}}{α}

From the inequalities (3.21) and (3.22), it follows that

(3.23)  ∥y_{3}∥_{⋆} ≤ q_{3} ∥x∥_{⋆} + d_{2}C_{1−α} \frac{T^{α}}{α}

Finally from (3.17), (3.20) , (3.23) and Remark 2, we get

∥y∥_{⋆} ≤ ∥y_{1}∥_{⋆} + ∥y_{2}∥_{⋆} + ∥y_{3}∥_{⋆}

≤ ∥φ∥_{C([−r,0],E)} + ∥h(0, φ)∥ + M (∥x_{0}∥ + ∥β∥_{L^{1}}) + d_{2}C_{1−α} \frac{T^{α}}{α}

+ [d_{1} ∥(−A)^{−α}∥ + q_{4} + q_{3}] ∥x∥_{⋆} EJQTDE, 2008 No. 9, p. 20

≤ ∥φ∥_{C([−r,0],E)} + ∥h(0, φ)∥ + M (∥x_{0}∥ + ∥β∥_{L^{1}}) + d_{2}C_{1−α} \frac{T^{α}}{α}
+ [d_{1} ∥(−A)^{−α}∥ + q_{4} + q_{3}] r
≤ r

According to Lemma 4, the problem (1.1)-(1.2) has at least one mild solution.

Step 4. The solutions set is compact.

The solution set is a priori bounded. In fact, if x is a mild solution of the problem (1.1)-(1.2), and the function υ(.) : [0, T] → ℝ^{+} is such that υ(t) = sup_{μ∈[0,t]} ∥x(μ)∥ then as above for t ∈ [0, T] we have

υ(t) = sup_{μ∈[0,t]} ∥x(μ)∥

≤ M (∥x_{0}∥ + ∥β∥_{L^{1}}) + M ∫_{0}^{t} β(τ) [ sup_{μ∈[−r,0]} ∥x(μ)∥ + sup_{μ∈[0,τ]} ∥x(μ)∥] dτ

+d_{2} ∥(−A)^{−α}∥ + d_{1} ∥(−A)^{−α}∥ [ sup_{μ∈[−r,0]} ∥x(μ)∥ + sup_{μ∈[0,t]} ∥x(μ)∥]

+d_{1} ∫_{0}^{t} \frac{C_{1−α}}{(t − s)^{1−α}} dτ [ sup_{μ∈[−r,0]} ∥x(μ)∥ + sup_{μ∈[0,τ]} ∥x(μ)∥] + d_{2} ∫_{0}^{t} \frac{C_{1−α}}{(t − s)^{1−α}} dτ

≤ M (∥x_{0}∥ + ∥β∥_{L^{1}}) + d_{2} ∥(−A)^{−α}∥
+ [M ∥β∥_{L^{1}} + d_{1} ∥(−A)^{−α}∥ + d_{2}C_{1−α} \frac{T^{α}}{α} ] ∥φ∥_{C([−r,0],E)}

+d_{1} ∥(−A)^{−α}∥ sup_{s∈[0,t]} ∥x(s)∥ + ∫_{0}^{t} [Mβ(τ) + d_{1} \frac{C_{1−α}}{(t − s)^{1−α}} ] sup_{μ∈[0,τ]} ∥x(μ)∥ dτ

≤ M (∥x_{0}∥ + ∥β∥_{L^{1}}) + d_{2} ∥(−A)^{−α}∥

+ [M ∥β∥_{L^{1}} + d_{1} ∥(−A)^{−α}∥ + d_{2}C_{1−α} \frac{T^{α}}{α} ] ∥φ∥_{C([−r,0],E)}

+d_{1} ∥(−A)^{−α}∥ υ(t) + ∫_{0}^{t} [Mβ(τ) + d_{1} \frac{C_{1−α}}{(t − s)^{1−α}} ] υ(τ)dτ

it results that

υ(t) ≤ \frac{1}{d_{1} ∥(−A)^{−α}∥} [ξ + ∫_{0}^{t} [Mβ(τ) + d_{1} \frac{C_{1−α}}{(t − s)^{1−α}} ] υ(τ)dτ]

EJQTDE, 2008 No. 9, p. 21 where

\begin{aligned}ξ & = M (∥x_{0}∥ + ∥β∥_{L^{1}}) + d_{2} ∥(−A)^{−α}∥ \\ & + [M ∥β∥_{L^{1}} + d_{1} ∥(−A)^{−α}∥ + d_{2}C_{1−α} \frac{T^{α}}{α} ] ∥φ∥_{C([−r,0],E)} \end{aligned}

Applying Gromwall-Bellmann type inequality, we get

∥υ(t)∥ ≤ \frac{ξ}{1 − d_{1} ∥(−A)^{−α}∥}e^{γ}

where,

γ = \frac{1}{1 − d_{1} ∥(−A)^{−α}∥} [M ∥B∥_{L^{1}} + T^{α} \frac{d_{1}C_{1−α}}{α} ] .

therefore,

υ(T) = sup_{μ∈[0,T]} ∥x(μ)∥ ≤ \frac{ξ}{1 − d_{1} ∥(−A)^{−α}∥}e^{γ}

Consequently

∥x∥_{C([−r,T],E)} ≤ sup_{μ∈[−r,0]} ∥x(μ)∥ + sup_{μ∈[0,T]} ∥x(μ)∥

≤ ∥φ∥_{C([−r,0],E)} + \frac{ξ}{1 − d_{1} ∥(−A)^{−α}∥}e^{γ}

To complete the proof it remains to apply Lemma 5.

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(Received November 19, 2007)

⊤University of Tiaret, BP 78, Algeria.

E-mail address: lahcene guedda@yahoo.fr

♢University of Tiaret, BP 78, Algeria.

E-mail address: ahmedhallouz@yahoo.fr