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Nonlinear impulsive partial functional differential inclusions with state-dependent delay and multivalued jumps
Nonlinear Analysis: Hybrid Systems 4 (2010) 791–803
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Nonlinear impulsive partial functional differential inclusions with state-dependent delay and multivalued jumps Nadjet Abada a , Mouffak Benchohra b,∗ , Hadda Hammouche c a
Département de Mathématiques, Université Mentouri de Constantine, Algérie
b
Laboratoire de Mathématiques, Université de Sidi Bel Abbès, BP 89, 22000 Sidi Bel Abbès, Algérie
c
Département de Mathématiques, Université Kasdi Merbah de Ouargla, Algérie
article
abstract
info
Article history: Received 29 January 2009 Accepted 19 May 2010
In this paper, we shall establish sufficient conditions for the existence of integral solutions for some nondensely defined impulsive semilinear functional differential inclusions with state-dependent delay in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Partial differential inclusions Impulses Multivalued jumps State-dependent delay Integral solution Semigroup
1. Introduction In this paper, we shall be concerned with the existence of integral solutions defined on a compact real interval for first order impulsive semilinear functional inclusions with state-dependent delay in a separable Banach space of the form: y0 (t ) ∈ Ay(t ) + F (t , yρ(t ,yt ) ),
∆y(ti ) ∈ Ik (ytk ), y(t ) = φ(t ),
t ∈ J = [0, b],
k = 1, 2, . . . , m,
t ∈ (−∞, 0],
(1.1) (1.2) (1.3)
where F : J × D → P (E ) is a given multivalued map with nonempty convex compact values, D is the phase space defined axiomatically (see Section 2) which contains the mappings from (−∞, 0] into E, φ ∈ D , 0 = t0 < t1 < · · · < tm < tm+1 = b, Ik : D → P (E ), k = 1, 2, . . . , m are bounded valued multivalued maps, P (E ) is the collection of all nonempty subsets of E , ρ : I × D → (−∞, b], A : D(A) ⊂ E → E is a nondensely defined closed linear operator on E, and E a real separable Banach space with norm |.|. For any function y defined on (−∞, b] \ {t1 , t2 , . . . , tm } and any t ∈ J, we denote by yt the element of D defined by yt (θ ) = y(t + θ ),
θ ∈ (−∞, 0].
In recent years, impulsive differential and partial differential equations have been the object of much investigation because they can describe various models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics. We refer the reader to the monographs of Bainov and Simeonov [1], Benchohra
∗
Corresponding author. Fax: +213 48 54 43 44. E-mail addresses: [email protected] (N. Abada), [email protected] (M. Benchohra), [email protected] (H. Hammouche).
1751-570X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2010.05.008
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et al. [2], Lakshmikantham et al. [3], and Samoilenko and Perestyuk [4] where numerous properties of their solutions are studied, and a detailed bibliography is given. Semilinear functional differential equations and inclusions with or without impulses have been extensively studied where the operator A generates a C0 -semigroup. Existence and uniqueness, among other things, are derived; see the books of Ahmed [5,6], Benchohra et al. [7], Heikkila and Lakshmikantam [8], Kamenski et al. [9] and the papers by Ahmed [10,11], Liu [12] and Rogovchenko [13,14]. In [15] Abada et al. have studied the controllability of a class of impulsive semilinear functional differential inclusions in Fréchet spaces by means of the extrapolation method [16,17], and in [18] the existence of mild and extremal mild solutions for first-order semilinear densely defined impulsive functional differential inclusions in separable Banach spaces with local and nonlocal conditions has been considered. To the best of our knowledge, there are very few results for impulsive evolution inclusions with multivalued jump operators; see [18–20]. The notion of the phase space D plays an important role in the study of both qualitative and quantitative theory. A usual choice is a semi-normed space satisfying suitable axioms, which was introduced by Hale and Kato [21] (see also [22,23]). For a detailed discussion on this topic we refer the reader to the book by Hino et al. [24]. For the case where the impulses are absent (i.e Ik = 0, k = 1, . . . , m), an extensive theory has been developed for the problem (1.1)–(1.3). We refer to Belmekki et al. [25], Corduneanu and Lakshmikantham [26], Hale and Kato [21], Hino et al. [24], Lakshmikantham et al. [27] and Shin [28]. The literature related to ordinary and partial functional differential equations with delay for which ρ(t , ψ) = t is very extensive; see for instance the books [29–32] and the papers therein. On the other hand, functional differential equations with state-dependent delay appear frequently in applications as model of equations and for this reason the study of this type of equations has received great attention in the last year, see, for instance, [33,34] and the references therein. The literature related to partial functional differential equations with state-dependent delay is limited (see [35–37]). This paper is organized as follows. In Section 2, we will recall briefly some basic definitions and preliminary facts which will be used throughout the following sections. In Section 3 we give some examples of operators with nondense domain. In Section 4, we prove existence of integral solutions for problem (1.1)–(1.3). Our approach will be based for the existence of integral solutions, on a fixed point theorem of Dhage [38] for the sum of a contraction map and a completely continuous map. In Section 5 we present some examples of phase spaces. Finally in Section 6 we give an example to illustrate the abstract theory. The results of the present paper extend to a nondensely defined operator some ones considered in the previous literature. 2. Preliminaries In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. For
ψ ∈ D the norm of ψ is defined by
kψkD = sup{|ψ(θ )| : θ ∈ (−∞, 0]}. Also B(E ) denotes the Banach space of bounded linear operators from E into E, with norm
kN kB(E ) = sup{|N (y)| : |y| = 1}. L1 (J , E ) denotes the Banach space of measurable functions y : J −→ E which are Bochner integrable normed by b
Z
|y(t )|dt .
kykL1 = 0
To consider the define the solution of problem (1.1)–(1.3), it is convenient to introduce some additional concepts and notations. Consider the following space
n
Bb = y : (−∞, b] → E , yk ∈ C (Jk , E ) and there exist y(tk− ), y(tk+ ) with y(tk ) = y(tk− ), y(t ) = φ(t ), t ≤ 0
o
where yk is the restriction of y to Jk = (tk , tk+1 ], k = 0, . . . , m. Let k · kb be the semi-norm in Bb defined by
kykb = ky0 kD + sup{|y(s)| : 0 ≤ s ≤ b},
y ∈ Bb .
In this work, we will employ an axiomatic definition for the phase space D which is similar to those introduced in [24]. Specifically, D will be a linear space of functions mapping (−∞, 0] into E endowed with a semi norm k.kD , and satisfies the following axioms introduced at first by Hale and Kato in [21]:
(A1 ) There exist a positive constant H and functions K (·), M (·) : R+ → R+ with K continuous and M locally bounded, such that for any b > 0, if y : (−∞, b] → E, y ∈ D , and y(·) is continuous on [0, b], then for every t ∈ [0, b] the following conditions hold: (i) yt is in D ; (ii) |y(t )| ≤ H kyt kD ; (iii) kyt kD ≤ K (t ) sup{|y(s)| : 0 ≤ s ≤ t } + M (t )ky0 kD , and H , K and M are independent of y(·). (A2 ) The space D is complete.
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In what follows we use the following notations Kb = sup{K (t ) : t ∈ J } and Mb = sup{M (t ) : t ∈ J }. Definition 2.1 ([39]). Let E be a Banach space. An integrated semigroup is a family of operators (S (t ))t ≥0 of bounded linear operators S (t ) on E with the following properties: (i) S (0) = 0; (ii) t → S (t ) isRstrongly continuous; s (iii) S (s)S (t ) = 0 (S (t + r ) − S (r ))dr , for all t , s ≥ 0. Definition 2.2. An integrated semigroup (S (t ))t ≥0 is called exponential bounded, if there exists constant M ≥ 0 and ω ∈ R such that
|S (t )| ≤ Meωt ,
for t ≥ 0.
Moreover (S (t ))t ≥0 is called nondegenerate if S (t )y = 0, for all t ≥ 0, implies y = 0. Definition 2.3. An operator A is called a generator of an integrated semigroup, if there exists ω ∈ R such that (ω, +∞) ⊂ ρ(A), and there exists a strongly R ∞continuous exponentially bounded family (S (t ))t ≥0 of linear bounded operators such that S (0) = 0 and (λI − A)−1 = λ 0 e−λt S (t )dt for all λ > ω. If A is the generator of an integrated semigroup (S (t ))t ≥0 which is locally Lipschitz, then from [39], S (·)y is continuously d differentiable if and only if y ∈ D(A). In particular S 0 (t )y := dt S (t )y defines a bounded operator on the set E1 := {y ∈ E : t → S (t )y is continuously differentiable on [0, ∞)} and (S 0 (t ))t ≥0 is a C0 semigroup on D(A). Here and hereafter, we assume that A satisfies the Hille–Yosida condition, that is, there exists M > 0 and ω ∈ R such that (ω, ∞) ⊂ ρ(A) and sup{(λI − ω)n |(λI − A)−n | : λ > ω, n ∈ N} ≤ M . Note that, since A satisfies the Hille–Yosida condition,
kS 0 (t )kB(E ) ≤ Meωt ,
t ≥ 0,
where M and ω are the constants considered in the Hille–Yosida condition (see [40]). Let (S (t ))t ≥0 , be the integrated semigroup generated by A. Consider the Cauchy Problem y0 (t ) = Ay(t ) + f (t ),
t ∈ [0, b],
y(0) = y0 ∈ E .
(2.1)
Then we have the following. Theorem 2.1 ([40]). Let f : [0, b] → E be a continuous function. Then for y0 ∈ D(A), there exists a unique continuous function y : [0, b] → E of the Cauchy Problem (2.1) such that (i)
Rt 0
y(s)ds ∈ D(A) for t ∈ [0, b],
Rt Rt y(s)ds + 0 f (s)ds, t ∈ [0, b], 0 Rt (iii) |y(t )| ≤ Meωt |y0 | + 0 e−ωs |f (s)|ds , t ∈ [0, b]. (ii) y(t ) = y0 + A
Moreover, from [39,40] y satisfies the variation of constants formula, d y(t ) = S 0 (t )y0 + dt
t
Z
S (t − s)f (s)ds,
t ≥ 0.
(2.2)
0
Let Bλ = λR(λ, A) := λ(λI − A)−1 . Then [40] for all y ∈ D(A), Bλ y → y as λ → ∞. Also from the Hille–Yosida condition (with n = 1) it easy to see that limλ→∞ |Bλ y| ≤ M |y|, since
|Bλ | = |λ(λI − A)−1 | ≤
Mλ . λ−ω
Thus limλ→∞ |Bλ | ≤ M. Also if y is given by (2.2), then y(t ) = S 0 (t )y0 + lim
Z
λ→∞ 0
t
S 0 (t − s)Bλ f (s)ds,
t ≥ 0.
(2.3)
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We finish this section, with notations, definitions, and some results from multivalued analysis. Let (X , d) be a metric space. We use the notations: Pcl (X ) = {Y ∈ P (X ) : Y closed},
Pbd (X ) = {Y ∈ P (X ) : Y bounded}
Pc v (X ) = {Y ∈ P (X ) : Y convex},
Pcp (X ) = {Y ∈ P (X ) : Y compact}. S Consider Hd : P (X ) × P (X ) → R+ {∞} given by Hd (A, B) = max{sup d(a, B), sup d(A, b)}, a∈A
b∈B
where d(A, b) = infa∈A d(a, b), d(a, B) = infb∈B d(a, b). Then (Pbd,cl (X ), Hd ) is a metric space and (Pcl (X ), Hd ) is a generalized metric space (see [41]). A multivalued map N : J → Pcl (X ) is said to be measurable if, for each x ∈ X , the function Y : J → R defined by Y (t ) = d(x, N (t )) = inf{d(x, z ) : z ∈ N (t )}, is measurable. Definition 2.4. A measurable multivalued function F : J → Pbd,cl (X ) is said to be integrably bounded if there exists a function w ∈ L1 (J , R+ ) such that kvk ≤ w(t ) a.e. t ∈ J for all v ∈ F (t ). A multivalued S map F : X → P (X ) is convex (closed) valued if F (x) is convex (closed) for all x ∈ X . F is bounded on bounded sets if F (B) = x∈B F (x) is bounded in X for all B ∈ Pb (X ) i.e. supx∈B {sup{|y| : y ∈ F (x)}} < ∞. F is upper semi-continuous (u.s.c for short) on X if for each x0 ∈ X the set F (x0 ) is nonempty, closed subset of X , and for each open set U of X containing F (x0 ), there exists an open neighborhood V of x0 such that F (V ) ⊆ U.G is said to be completely continuous if F (B) is relatively compact for every B ∈ Pbd (X ). If the multivalued map F is completely continuous with nonempty compact valued, then G is u.s.c if and only if F has closed graph i.e. xn → x∗ , yn → y∗ , yn ∈ G(x∗ ) imply y∗ ∈ G(x∗ ). Definition 2.5. A multivalued map F : J × D → P (E ) is said to be Carathéodory if (i) t 7−→ F (t , u) is measurable for each u ∈ D (ii) u − 7 → F (t , u) is u.s.c. for almost each t ∈ J . Definition 2.6. A multivalued operator N : J → Pcl (X ) is called (a) contraction if and only if there exists γ > 0 such that Hd (N (x), N (y)) ≤ γ d(x, y),
for each x, y ∈ X ,
with γ < 1, (b) N has a fixed point if there exists x ∈ X such that x ∈ N (x). For more details on multivalued maps and the proof of the known results cited in this section we refer interested reader to the books of Deimling [42], Gorniewicz [43] and Hu and Papageorgiou [44]. 3. Examples of operators with nondense domain In this section we shall present examples of linear operators with nondense domain satisfying the Hille–Yosida estimate. More details can be found in the paper by Da Prato and Sinestrari [45]. Example 3.1. Let E = C ([0, 1], R) and the operator A : D(A) → E defined by Ay = y0 , where D(A) = {y ∈ C 1 ((0, 1), R) : y(0) = 0}. Then D(A) = {y ∈ C ((0, 1), R) : y(0) = 0} 6= E . Example 3.2. Let E = C ([0, 1], R) and the operator A : D(A) → E defined by Ay = y00 , where D(A) = {y ∈ C 2 ((0, 1), R) : y(0) = y(1) = 0}. Then D(A) = {y ∈ C ((0, 1), R) : y(0) = y(1) = 0} 6= E .
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Example 3.3. Let us set for some α ∈ (0, 1) E = C0α ([0, 1], R) =
y : [0, 1] → R : y(0) = 0 and
|y(t ) − y(s)| <∞ |t − s|α 0≤t
sup
and the operator A : D(A) → E defined by Ay = −y0 , where D(A) = {y ∈ C 1+α ((0, 1), R) : y(0) = y0 (0) = 0}. Then α
D(A) = h0 ((0, 1), R) =
y : [0, 1] → R : lim
|y(t ) − y(s)| = 0 6= E . | t − s| α
sup
δ→0 0<|t −s|≤δ
Here C 1+α ([0, 1], R) = {y : [0, 1] → R : y0 ∈ C α ([0, 1], R)}. The elements of hα ((0, 1), R) are called little Holder functions and it can be proved that the closure of C 1 ((0, 1), R) in C α ((0, 1), R) is hα ((0, 1), R) (see [46] Theorem 5.3). Example 3.4. Let Ω ⊂ Rn be a bounded open set with regular boundary Γ and define E = C (Ω , R) and the operator A : D(A) → E defined by Ay = ∆y, where D(A) = {y ∈ C (Ω , R) : y = 0 on Γ ; ∆y ∈ C (Ω , R)}. Here ∆ is the Laplacian in the sense of distributions on Ω . In this case we have D(A) = {y ∈ C (Ω , R) : y = 0 on Γ } 6= E . 4. Existence of integral solutions Now, we are able to state and prove our main theorem for the initial value problem (1.1)–(1.3). Before starting and proving this one, we give the definition of the integral solution. Definition 4.1. We say that y : (−∞, T ] → E is an integral solution of (1.1)–(1.3) if y(t ) = φ(t ) for all t ∈ (−∞, 0], the restriction of y(·) to the interval [0, b] is continuous, and there exist v(·) ∈ L1 (Jk , E ) and Ik ∈ Ik (ytk ), such that v(t ) ∈ F (t , yρ(t ,yt ) ) a.e t ∈ [0, b], and y satisfies the integral equation, (i) y(t ) = φ(0) + A (ii)
Rt 0
Rt 0
y(s)ds +
Rt 0
v(s)ds +
P
y(s)ds ∈ D(A) for t ∈ J.
0
Ik , t ∈ J .
From the definition it follows that y(t ) ∈ D(A), for each t ≥ 0, in particular φ(0) ∈ D(A). Moreover, from [39,40] y satisfies the following variation of constants formula: d y(t ) = S 0 (t )φ(0) + dt
t
Z
S (t − s)v(s)ds +
X
S 0 (t − tk ) Ik
t ≥ 0.
(4.1)
0 < tk < t
0
We notice also that, if y satisfies (4.1), then y(t ) = S 0 (t )φ(0) + lim
Z
λ→∞ 0
t
S 0 (t − s)Bλ v(s)ds +
X
S 0 (t − tk ) Ik ,
t ≥ 0.
0
The key tool in our approach is the following form of the fixed point theorem of Dhage [38,47]. Theorem 4.1. Let X be a Banach space, A : X → Pcl,c v,bd (X ) and B : X → Pcp,c v (X ), two multivalued operators satisfying (a) A is a contraction, and (b) B is completely continuous. Then either (i) The operator inclusion λx ∈ Ax + B x has a solution for λ = 1, or (ii) the set E = {u ∈ X |u ∈ λAu + λB u, 0 ≤ λ ≤ 1} is unbounded.
˜ be the space of functions in D which have values in D(A). We always assume that ρ : I × D → (−∞, b] is Set D continuous. Additionally, we introduce following hypotheses:
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(Hφ ) The function t → φt is continuous from R(ρ − ) = {ρ(s, ϕ) : (s, ϕ) ∈ J × D , ρ(s, ϕ) ≤ 0} into D and there exists a continuous and bounded function Lφ : R(ρ − ) → (0, ∞) such that kφt kD ≤ Lφ (t )kφkD for every t ∈ R(ρ − ). (H1) A satisfies Hille–Yosida condition; Pm (H2) There exist constants ck > 0, k = 1, . . . , m with Meωb Kb k=1 ck < 1 such that Hd (Ik (y), Ik (x)) ≤ ck |y − x|
˜. for each x, y ∈ D
(H3) The multivalued map F : J × D → Pcp,c v (E ) is Carathéodory; (H4) the operator S 0 (t ) is compact in D(A) wherever t > 0; (H5) There exist a function p ∈ L1 (J , R+ ) and a continuous nondecreasing function ψ : [0, ∞) → (0, ∞) such that
kF (t , x)kP ≤ p(t )ψ(kxkD ) for a.e. t ∈ J and each x ∈ D , R b −ωs with 0 e p(s)ds < ∞, (Mb + Lφ + MKb )kφkD + Kb u >1 lim sup Rb u→+∞ c ∗ + c ∗ ψ Kb u + (Mb + Lφ + MKb )kφkD e−ωs p(s)ds 1 2 0
(4.2)
where c ∗ eω b Kb
c1∗ =
m P
1 − Meωb Kb
+ Mb + Lφ + MKb kφkD ,
(4.3)
ck
k=1 m
X |Ik (0)| + ck Mb + Lφ + MKb kφkD ,
c∗ =
(4.4)
k=1
and MKb eωb
c2∗ =
1 − Meωb Kb
m P
.
(4.5)
ck
k=1
The next result is a consequence of the phase space axioms. Lemma 4.1 ([36], Lemma 2.1). If y : (−∞, b] → E is a function such that y0 = φ and y|J ∈ PC (J : D(A)), then
kys kD ≤ (Ma + Lφ )kφkD + Ka sup{ky(θ )k; θ ∈ [0, max{0, s}]},
s ∈ R(ρ − ) ∪ J ,
where Lφ = supt ∈R(ρ − ) Lφ (t ), Ma = supt ∈J M (t ) and Ka = supt ∈J K (t ). Remark 4.1. We remark that condition (Hφ ) is satisfied by functions which are continuous and bounded. In fact, if the space D satisfies axiom C2 in [24] then there exists a constant L > 0 such that kφkD ≤ L sup{kφ(θ )k : θ ∈ [−∞, 0]} for every φ ∈ D that is continuous and bounded, (see [24] Proposition 7.1.1) for details. Consequently sup kφ(θ )k
kφt kD ≤ L
θ≤0
kφkD
kφkD ,
for every φ ∈ D \ {0}.
Theorem 4.2. Assume that (Hφ ) and (H1)–(H5) hold. If φ(0) ∈ D(A), then the problem (1.1)–(1.3) has at least one integral solution on (−∞, b]. Proof. Transform the problem (1.1)–(1.3) into a fixed point problem. Set
Ω = PC (−∞, b], D(A) , and consider the multivalued operator N : Ω → P (Ω ) defined by N (y) = {h ∈ Ω } with h(t ) =
φ(t ),
if t ≤ 0, Z t X d S ( t )φ( 0 ) + S (t − s)v(s)ds + S 0 (t − tk )Ik , dt 0 0
k
v ∈ SF ,yρ(s,ys ) , Ik ∈ Ik (y(tk− )) if t ∈ J .
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For φ ∈ D define the function e φ : (−∞, b] → E such that:
φ(t ), e φ(t ) = 0 S (t )φ(0),
if t ≤ 0 if t ∈ J .
Then e φ0 = φ . For each x ∈ Bb with x(0) = 0, we denote by x the function defined by x(t ) =
0, x(t ),
t ∈ (−∞, 0], t ∈ J.
If y(.) satisfies (4.1), we can decompose it as y(t ) = e φ(t ) + x(t ), 0 ≤ t ≤ b, which implies yt = e φt + xt , for every 0 ≤ t ≤ b and the function x(.) satisfies x(t ) =
t
Z
d dt
Let
S 0 (t − tk )Ik
t ∈ J,
0
0
where v(s) ∈ SF ,x
X
S (t − s)v(s)ds +
φtk . and Ik ∈ Ik xtk + e
ρ(s,xs +e φs ) +φρ(s,xs +e φs )
e
Bb0 = {x ∈ Bb : x0 = 0 ∈ D }. For any x ∈ Bb0 we have
kxkb = kx0 kD + sup{|x(s)| : 0 ≤ s ≤ b} = sup{|x(s)| : 0 ≤ s ≤ b}. Thus (Bb0 , k · kb ) is a Banach space. We define the two multivalued operators A, B : Bb0 → P (Bb0 ) by A(x) := {h ∈ Bb0 }, B (x) := {h ∈ Bb0 } with h( t ) =
, 0X
S 0 (t − tk )Ik ,
I k ∈ I k x tk + e φ tk ,
if t ∈ (−∞, 0]; if t ∈ J ,
0
and
0, Z t d h( t ) = S (t − s)v(s)ds, dt
if t ∈ (−∞, 0];
v(s) ∈ SF ,x
0
ρ(s,xs +e φs ) +φρ(s,xs +e φs )
e
if t ∈ J .
Obviously to prove that the multivalued operator N has a fixed point is reduced that the operator inclusion x ∈ A(x) + B (x) has one, so it turns to show that the multivalued operators A and B satisfy all conditions of Theorem 4.1. For better readability, we break the proof into a sequence of steps. Step 1: A is a contraction. Let x1 , x2 ∈ Bb0 . Then for t ∈ J
! X
Hd (A(x1 ), A(x2 )) = Hd
0 < tk < t
≤ Meωb
S 0 (t − tk )Ik (x1tk + e φtk ),
X 0
S 0 (t − tk ) Ik (x2tk + e φtk )
X Ik (x1 ) − Ik (x2 ) tk
tk
0≤tk ≤t
≤ Meωb
m X
ck kx1tk − x2tk kD
k=1
≤ Meωb Kb
m X
ck kx1 − x2 kD .
k=1
Hence by (H2) A is a contraction. Step 2: B has compact, convex values, and it is completely continuous. This will be given in several claims. Claim 1: B has compact values. The operator B is equivalent to the composition L ◦ SF on L1 (J , E ), where L : L1 (J , E ) → Bb0 is the continuous operator defined by
L(v)(t ) =
d dt
Z 0
t
S (t − s)v(s)ds,
t ∈ J.
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Then, it suffices to show that L ◦ SF has compact values on Bb0 . Let x ∈ Bb0 arbitrary and vn a sequence such that e vn (t ) ∈ SF ,x , a.e. t ∈ J. Since F (t , xρ(t ,xt +e φ φt ) + φρ(t ,xt +e φt ) ) is compact, we may pass to a subsequence. Suppose e +e e ρ(t ,xt +φt )
ρ(t ,xt +φt )
e that vn → v in L1w (J , E ) (the space endowed with the weak topology), where v(t ) ∈ F (t , xρ(t ,xt +e φt ) + φρ(t ,xt +e φt ) ), a.e. t ∈ J. An application of Mazur’s theorem [48] implies that vn converges strongly to v and hence v(t ) ∈ SF ,x . From e + φ e e ρ(t ,xt +φt )
ρ(t ,xt +φt )
the continuity of L, it follows that Lvn (t ) → Lv(t ) pointwise on J as n → ∞. In order to show that the convergence is uniform, we first show that {Lvn } is an equicontinuous sequence. Let τ1 , τ2 ∈ J, then we have:
Z τ Z τ2 1 d d |L(vn (τ1 )) − L(vn (τ2 ))| = S (τ1 − s)vn (s)ds − S (τ2 − s)vn (s)ds dt 0 dt 0 Z τ1 Z ≤ lim [S 0 (τ1 − s) − S 0 (τ2 − s)]Bλ vn (s)ds + lim λ→∞ λ→∞ 0
τ2 τ1
S 0 (τ2 − s)Bλ vn (s)ds .
As τ1 → τ2 , the right hand-side of the above inequality tends to zero. Since S 0 (t ) is a strongly continuous operator and the compactness of S 0 (t ), t > 0, implies the uniform continuity (see [5,49]). Hence {Lvn } is equi-continuous, and an application of Arzéla–Ascoli theorem implies that there exists a subsequence which is uniformly convergent. Then we have Lvnj → Lv ∈ (L ◦ SF )(x) as j 7→ ∞, and so (L ◦ SF )(x) is compact. Therefore B is a compact valued multivalued operator on Bb0 . Claim 2: B (x) is convex for each z ∈ D0b . Let h1 , h2 ∈ B (x), then there exist v1 , v2 ∈ SF ,x such that, for each t ∈ J we have e ρ(t ,xt +e φt ) +φρ(t ,xt +e φt )
0, Z t d hi ( t ) = S (t − s)vi (s)ds dt
if t ∈ (−∞, 0],
if t ∈ J ,
0
,
i = 1, 2.
Let 0 ≤ δ ≤ 1. Then, for each t ∈ J, we have
0, Z (δ h1 + (1 − δ)h2 )(t ) = d t S (t − s)[δv1 (s) + (1 − δ)v2 (s)]ds dt
if t ∈ (−∞, 0],
if t ∈ J .
0
Since F has convex values, one has
δ h1 + (1 − δ)h2 ∈ B (x). Claim 3: B maps bounded sets into bounded sets in Bb0 . Let Bq = {x ∈ Bb0 : kxkb ≤ q}, q > 0 a bounded set in Bb0 . It is equivalent to show that there exists a positive constant l such that for each x ∈ Bq we have kB (x)kb ≤ l. So choose x ∈ Bq , then from Lemma 4.1 it follows that For each h ∈ B (x), and each x ∈ Bq , there exists v ∈ SF ,x such that φ e +e e ρ(t ,xt +φt )
h(t ) =
d dt
ρ(t ,xt +φt )
t
Z
S (t − s)v(s)ds. 0
From (A) we have
kxρ(t ,xt +eφt ) + e φρ(t ,xt +eφt ) kD ≤ Kb q + (Mb + Lφ )kφkD + Kb M |φ(0)| = q∗ . Then by (H6) we have ωb
|h(t )| ≤ Me ψ(q∗ )
Z
t
e−ωs p(s)ds := l.
0
This further, implies that
khkB 0 ≤ l. b
Hence B (B) is bounded. Claim 4: B maps bounded sets into equicontinuous sets. Let Bq be, as above, a bounded set and h ∈ B (x) for some x ∈ B. Then, there exists v ∈ SF ,x such that e ρ(t ,xt +e φt ) +φρ(t ,xt +e φt ) h(t ) =
d dt
t
Z
S (t − s)v(s)ds, 0
t ∈ J.
N. Abada et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 791–803
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Let τ1 , τ2 ∈ J \ {t1 , t2 , . . . , tm }, τ1 < τ2 . Thus if > 0, we have
Z τ1 − Z τ1 |h(τ2 ) − h(τ1 )| ≤ lim [S 0 (τ2 − s) − S 0 (τ1 − s)]Bλ v(s)ds + lim [S 0 (τ2 − s) − S 0 (τ1 − s)]Bλ v(s)ds λ→∞ 0 λ→∞ τ − 1 Z τ2 + lim S 0 (τ2 − s)Bλ v(s)ds λ→∞ τ 1 Z τ1 −
≤ ψ(q∗ )
0
+ ψ(q∗ )
kS 0 (τ2 − s) − S 0 (τ1 − s)kB(E ) p(s)ds
τ1
Z
τ1 −
kS 0 (τ2 − s) − S 0 (τ1 − s)kB(E ) p(s)ds + Meωb ψ(q∗ )
τ2
Z
e−ωs p(s)ds.
τ1
As τ1 → τ2 and becomes sufficiently small, the right-hand side of the above inequality tends to zero, since S 0 (t ) is a strongly continuous operator and the compactness of S 0 (t ) for t > 0 implies the uniform continuity. This proves the equicontinuity for the case where t 6= ti , i = 1, . . . , m + 1. It remains to examine the equicontinuity at t = ti . First we prove the equicontinuity at t = ti− , we have for some x ∈ Bq , there exists v ∈ SF ,x such that φ e +e e ρ(t ,xt +φt )
d
h( t ) =
dt
ρ(t ,xt +φt )
t
Z
S (t − s)v(s)ds,
t ∈ J.
0
Fix δ1 > 0 such that {tk , k 6= i} ∩ [ti − δ1 , ti + δ1 ] = ∅. For 0 < µ < δ1 , we have
|h(ti − µ) − h(ti )| ≤ lim
Z
ti −µ
λ→∞ 0
0
S (ti − µ − s) − S 0 (ti − s) Bλ v(s) ds + Meωb ψ(q∗ )
Z
ti
e−ωs p(s) ds;
ti −µ
which tends to zero as µ → 0. Define hˆ 0 (t ) = h(t ),
t ∈ [0, t1 ]
and hˆ i (t ) =
h(t ), h(ti+ ),
if t ∈ (ti , ti+1 ] if t = ti .
Next, we prove equicontinuity at t = ti+ . Fix δ2 > 0 such that {tk , k 6= i} ∩ [ti − δ2 , ti + δ2 ] = ∅. Then hˆ (ti ) =
ti
Z
T (ti − s)v(s)ds.
0
For 0 < µ < δ2 , we have
|hˆ (ti + µ) − hˆ (ti )| ≤ lim
Z
λ→∞ 0
ti
0
S (ti + µ − s) − S 0 (ti − s) Bλ v(s) ds + Meωb ψ(q∗ )
Z
ti +µ
e−ωs p(s) ds;
ti
The right hand-side tends to zero as µ → 0. The equicontinuity for the cases τ1 < τ2 ≤ 0 and τ1 ≤ 0 ≤ τ2 follows from the uniform continuity of φ on the interval (−∞, 0]. As a consequence of Claims 1–3 together with Arzelá–Ascoli theorem it suffices to show that B maps B into a precompact set in E. Let 0 < t < b be fixed and let be a real number satisfying 0 < < t. For x ∈ Bq , we define h (t ) = S 0 () lim
Z
λ→∞ 0
where v ∈ SF ,x
Z lim λ→∞
t −
S 0 (t − s − )Bλ v(s)ds,
ρ(t ,xt +e φt ) +φρ(t ,xt +e φt )
e
t −
. Since
Z ωb S (t − s − )Bλ v(s)ds ≤ Me ψ(q∗ )
t −
e−ωs p(s)ds,
0
0
0
the set
Z lim
λ→∞ 0
t −
S 0 (t − s − )Bλ v(s)ds : v ∈ SF ,x
.x e ρ(t ,xt +e φt ) +φρ(t ,xt +e φt )
is bounded. Since S 0 (t ) is a compact operator for t > 0, the set H (t ) = {h (t ) : h ∈ B (x)}
∈ Bq
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N. Abada et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 791–803
is precompact in E for every , 0 < < t. Moreover, for every h ∈ B (x) we have ωb
|h(t ) − h (t )| ≤ Me ψ(q∗ )
t −
Z
e−ωs p(s)ds. t
Therefore, there are precompact sets arbitrarily close to the set H (t ) = {h(t ) : h ∈ B (x)}. Hence the set H (t ) = {h(t ) : h ∈ B (Bq )} is precompact in E. Hence the operator B is totally bounded. Step 3: A priori bounds. Now it remains to show that the set
E = x ∈ Bb0 : x ∈ λA(x) + λB (x) for some 0 < λ < 1
φtk such that for each t ∈ J , is bounded. Let x ∈ E , then there exist v ∈ SF ,x and Ik ∈ Ik xtk + e e ρ(t ,xt +e φt ) +φρ(t ,xt +e φt )
x(t ) = λ
d
t
Z
dt
X
S (t − s)v(s) + λ
S 0 ( t − t k ) Ik .
0
0
This implies by (H2), (H5) that, for each t ∈ J, we have t
|x(t )| ≤ λMeωt
Z
≤ λMeωt
Z
ωt e e−ωs p(s)ψ(kxρ(s,xs +e φs ) + φρ(s,xs +e φs ) kD )ds + λMe
0
m X Ik x t + e φtk k k =1
t
e−ωs p(s)ψ Kb |x(s)| + (Mb + Lφ + MKb )kφkD ds
0
+ λMeωt
m m X X Ik xt + e + λMeωt |Ik (0)| φ − I ( 0 ) t k k k k=1
≤ λMe
ωt
Z
k=1
t
e
−ωs
φ
p(s)ψ Kb |x(s)| + (Mb + L + MKb )kφkD ds
0
+ λMe
ωt
m X
|Ik (0)| + λMeωt
k=1
∗ ωt
≤c e
+ Me
ωt
m X
ck Kb |x(s)| + (Mb + Lφ + MKb )kφkD
k=1
"Z
t
e
−ωs
φ
p(s)ψ Kb |x(s)| + (Mb + L + MKb )kφkD ds + Kb
0
m X
# ck |x(t )| .
k=1
Hence from (4.3)–(4.5) we have
(Mb + Lφ + MKb )kφkD + Kb |x(s)| ≤ c1∗ + c2∗
t
Z
e−ωs p(s)ψ Kb |x(t )| + (Mb + Lφ + MKb )kφkD ds.
0
Thus
(Mb + Lφ + MKb )kφkD + Kb kxkB 0 b R ≤ 1. b ∗ ∗ φ c1 + c2 ψ Kb kxkB 0 + (Mb + L + MKb )kφkD 0 e−ωs p(s)ds
(4.6)
b
From (4.2) it follows that there exists a constant R > 0 such that for each x ∈ E with kxkB 0 > R the condition (4.6) is b
violated. Hence kxkB 0 ≤ R for each x ∈ E , which means that the set E is bounded. As a consequence of Theorem 4.1, A + B b
has a fixed point x∗ on the interval (−∞, b], so y∗ = x∗ + e φ is a fixed point of the operator N which is the mild solution of problem (1.1)–(1.3). 5. Examples of phase spaces In this section we give some usual phase spaces. Let g : (−∞, 0] → [1, ∞) be a continuous, nondecreasing function with g (0) = 1, which satisfies the conditions (g-1), (g-2) of [24]. This means that the function G(t ) =
sup −∞<θ ≤−t
g (t + θ ) g (θ )
is locally bounded for t ≥ 0 and that limθ→−∞ g (θ ) = ∞. We said that φ : [−∞, 0] → E is normalized piecewise continuous, if φ is left continuous and the restriction of φ to any interval [−r , 0] is piecewise continuous.
N. Abada et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 791–803
801
Next we modify slightly the definition of the spaces Cg , Cg0 of [24]. We denote by P C g (E ) the space formed by the normalized piecewise continuous functions φ such that by the functions φ such that lim
θ→−∞
φ g
is bounded on (−∞, 0] and by P C 0g the subspace of PCg (E ) formed
φ(θ ) = 0. g (θ )
It is easy to see that D = P C g (E ) and D = P C 0g (E ) endowed with the norm
kφkD =
kφ(θ )k g (θ )
sup
θ∈(−∞,0]
are phase spaces. Moreover, in these cases K (s) = 1 for s ≥ 0. Let 1 ≤ p < ∞, 0 ≤ r < ∞ and g (·) is a Borel nonnegative measurable function on (−∞, r ) which satisfies the conditions (g-5)–(g-6) in the terminology of [24]. This means that g (·) is locally integrable on (−∞, −r ) and there exists a nonnegative and locally bounded function G on (−∞, 0] such that g (ξ +θ ) ≤ G(ξ )g (θ ) for all ξ ≤ 0 and θ ∈ (−∞, −r )\ Nξ , where Nξ ⊂ (−∞, −r ) is a set with Lebesgue measure 0. Let D := P C r × Lp (g , E ), r ≥ 0, p > 1, be the space formed of all classes of functions φ : (−∞, 0] → E such that φ|[−r ,0] ∈ PC ([−r , 0], E ), φ(·) is Lebesgue measurable on (−∞, −r ] and g |φ|p is Lebesgue integrable on (−∞, −r ]. The seminorm in k · kD is defined by
kφkD := sup kφ(θ )k +
Z
θ∈[−r ,0]
−r
g (θ )kφ(θ )kp dθ
1p
.
−∞
Proceeding as in the proof of ([24], Theorem 1.3.8), it follows that D is a phase space which satisfies Axioms (A1 ) and (A2 ). 1
Moreover, for r = 0 and p = 2 this space coincides (see [24]) with C0 × L2 (g , E ), H = 1, M (t ) = G(−t ) 2 and K (t ) = 1 +
0
Z
g (s)ds
21
,
for t ≥ 0.
−t
6. An example To apply our abstract results, we consider the partial functional differential equations with state dependent delay of the form
∂ ∂ v(t , ξ ) ∈ − v(t , ξ ) + m(t )a(t , v(t − σ (v(t , 0)), ξ )), ∂t ∂ξ
ξ ∈ [0, π], t ∈ [0, 1],
v(t , 0) = v(t , π ) = 0, t ∈ [0, 1], v(θ , ξ ) = v0 (θ , ξ ), ξ ∈ [0, π], θ ∈ (−∞, 0], Z ti γi (ti − s)[−|v(s, ξ )|, |v(s, ξ )|]ds, ∆v(ti )(ξ ) ∈
(6.1) (6.2) (6.3) (6.4)
−∞
where v0 : (−∞, 0] × [0, π] → R, γi : [0, ∞) → R are continuous functions, 0 < t1 < t2 < · · · < tn < 1, m : [0, 1] → R+ , a : [0, 1] × R → Pc v,cp (R), σ : R → R+ is continuous and we assume the existence of positive constants b1 , b2 such that
|a(t , u)| ≤ b1 |u| + b2 for every (t , u) ∈ [0, 1] × R. Let A be the operator defined on E = C ([0, π], R) by D(A) = {g ∈ C 1 ([0, π], R) : g (0) = 0}; Ag = g 0 . Then D(A) = C0 ([0, π], R) = {g ∈ C ([0, π], R) : g (0) = 0}. It is well known from [45] that A is sectorial, (0, +∞) ⊆ ρ(A) and for λ > 0
kR(λ, A)kB(E ) ≤
1
λ
.
It follows that A generates an integrated semigroup (S (t ))t ≥0 and that kS 0 (t )kB(E ) ≤ e−µt for t ≥ 0 for some constant µ > 0 and A satisfied the Hille–Yosida condition. Set γ > 0. For the phase space, we choose D to be defined by
D = PC γ = {φ ∈ PC ((−∞, 0], E ) : lim eγ θ φ(θ ) exists in E } θ→−∞
802
N. Abada et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 791–803
with norm
kφkγ =
sup
θ∈(−∞,0]
eγ θ |φ(θ )|,
φ ∈ PC γ .
Notice that the phase space D satisfies axioms (A1 ), (A2 ) (see [24] for more details). By making the following change of variables y(t )(ξ ) = v(t , ξ ), t ∈ [0, 1],
ξ ∈ [0, π], φ(θ )(ξ ) = v0 (θ , ξ ), θ ≤ 0, ξ ∈ [0, π], F (t , ϕ)(ξ ) = m(t )a(t , ϕ(0, ξ )), t ∈ [0, 1], ξ ∈ [0, π], φ ∈ Cγ ρ(t , ϕ) = t − σ (ϕ(0, 0)) Z 0 γk (−s)[−|v0 (s, ξ )|, |v0 (s, ξ )|]ds, Ik (ytk ) = −∞
the problem (6.1)–(6.4) takes the abstract form (1.1)–(1.3). Moreover, we have
kF (t , ϕ)kP ≤ m(t )(b1 kϕkD + b2 ) for all (t , ϕ) ∈ I × D with ∞
Z 1
ds
ψ(s)
Z = 1
∞
ds b1 s + b2
= +∞.
The next results are consequence of Theorem 4.2 and Remark 4.1. Theorem 6.1. Let φ ∈ B be such that condition (Hφ ) holds, then problem (6.1)–(6.4) has a mild solution. Corollary 6.1. Let φ ∈ B be continuous and bounded, then problem (6.1)–(6.4) has a mild solution. Acknowledgement The authors are grateful to the referee for his/her remarks. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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Contents lists available at ScienceDirect
Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Nonlinear impulsive partial functional differential inclusions with state-dependent delay and multivalued jumps Nadjet Abada a , Mouffak Benchohra b,∗ , Hadda Hammouche c a
Département de Mathématiques, Université Mentouri de Constantine, Algérie
b
Laboratoire de Mathématiques, Université de Sidi Bel Abbès, BP 89, 22000 Sidi Bel Abbès, Algérie
c
Département de Mathématiques, Université Kasdi Merbah de Ouargla, Algérie
article
abstract
info
Article history: Received 29 January 2009 Accepted 19 May 2010
In this paper, we shall establish sufficient conditions for the existence of integral solutions for some nondensely defined impulsive semilinear functional differential inclusions with state-dependent delay in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Partial differential inclusions Impulses Multivalued jumps State-dependent delay Integral solution Semigroup
1. Introduction In this paper, we shall be concerned with the existence of integral solutions defined on a compact real interval for first order impulsive semilinear functional inclusions with state-dependent delay in a separable Banach space of the form: y0 (t ) ∈ Ay(t ) + F (t , yρ(t ,yt ) ),
∆y(ti ) ∈ Ik (ytk ), y(t ) = φ(t ),
t ∈ J = [0, b],
k = 1, 2, . . . , m,
t ∈ (−∞, 0],
(1.1) (1.2) (1.3)
where F : J × D → P (E ) is a given multivalued map with nonempty convex compact values, D is the phase space defined axiomatically (see Section 2) which contains the mappings from (−∞, 0] into E, φ ∈ D , 0 = t0 < t1 < · · · < tm < tm+1 = b, Ik : D → P (E ), k = 1, 2, . . . , m are bounded valued multivalued maps, P (E ) is the collection of all nonempty subsets of E , ρ : I × D → (−∞, b], A : D(A) ⊂ E → E is a nondensely defined closed linear operator on E, and E a real separable Banach space with norm |.|. For any function y defined on (−∞, b] \ {t1 , t2 , . . . , tm } and any t ∈ J, we denote by yt the element of D defined by yt (θ ) = y(t + θ ),
θ ∈ (−∞, 0].
In recent years, impulsive differential and partial differential equations have been the object of much investigation because they can describe various models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics. We refer the reader to the monographs of Bainov and Simeonov [1], Benchohra
∗
Corresponding author. Fax: +213 48 54 43 44. E-mail addresses: [email protected] (N. Abada), [email protected] (M. Benchohra), [email protected] (H. Hammouche).
1751-570X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2010.05.008
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et al. [2], Lakshmikantham et al. [3], and Samoilenko and Perestyuk [4] where numerous properties of their solutions are studied, and a detailed bibliography is given. Semilinear functional differential equations and inclusions with or without impulses have been extensively studied where the operator A generates a C0 -semigroup. Existence and uniqueness, among other things, are derived; see the books of Ahmed [5,6], Benchohra et al. [7], Heikkila and Lakshmikantam [8], Kamenski et al. [9] and the papers by Ahmed [10,11], Liu [12] and Rogovchenko [13,14]. In [15] Abada et al. have studied the controllability of a class of impulsive semilinear functional differential inclusions in Fréchet spaces by means of the extrapolation method [16,17], and in [18] the existence of mild and extremal mild solutions for first-order semilinear densely defined impulsive functional differential inclusions in separable Banach spaces with local and nonlocal conditions has been considered. To the best of our knowledge, there are very few results for impulsive evolution inclusions with multivalued jump operators; see [18–20]. The notion of the phase space D plays an important role in the study of both qualitative and quantitative theory. A usual choice is a semi-normed space satisfying suitable axioms, which was introduced by Hale and Kato [21] (see also [22,23]). For a detailed discussion on this topic we refer the reader to the book by Hino et al. [24]. For the case where the impulses are absent (i.e Ik = 0, k = 1, . . . , m), an extensive theory has been developed for the problem (1.1)–(1.3). We refer to Belmekki et al. [25], Corduneanu and Lakshmikantham [26], Hale and Kato [21], Hino et al. [24], Lakshmikantham et al. [27] and Shin [28]. The literature related to ordinary and partial functional differential equations with delay for which ρ(t , ψ) = t is very extensive; see for instance the books [29–32] and the papers therein. On the other hand, functional differential equations with state-dependent delay appear frequently in applications as model of equations and for this reason the study of this type of equations has received great attention in the last year, see, for instance, [33,34] and the references therein. The literature related to partial functional differential equations with state-dependent delay is limited (see [35–37]). This paper is organized as follows. In Section 2, we will recall briefly some basic definitions and preliminary facts which will be used throughout the following sections. In Section 3 we give some examples of operators with nondense domain. In Section 4, we prove existence of integral solutions for problem (1.1)–(1.3). Our approach will be based for the existence of integral solutions, on a fixed point theorem of Dhage [38] for the sum of a contraction map and a completely continuous map. In Section 5 we present some examples of phase spaces. Finally in Section 6 we give an example to illustrate the abstract theory. The results of the present paper extend to a nondensely defined operator some ones considered in the previous literature. 2. Preliminaries In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. For
ψ ∈ D the norm of ψ is defined by
kψkD = sup{|ψ(θ )| : θ ∈ (−∞, 0]}. Also B(E ) denotes the Banach space of bounded linear operators from E into E, with norm
kN kB(E ) = sup{|N (y)| : |y| = 1}. L1 (J , E ) denotes the Banach space of measurable functions y : J −→ E which are Bochner integrable normed by b
Z
|y(t )|dt .
kykL1 = 0
To consider the define the solution of problem (1.1)–(1.3), it is convenient to introduce some additional concepts and notations. Consider the following space
n
Bb = y : (−∞, b] → E , yk ∈ C (Jk , E ) and there exist y(tk− ), y(tk+ ) with y(tk ) = y(tk− ), y(t ) = φ(t ), t ≤ 0
o
where yk is the restriction of y to Jk = (tk , tk+1 ], k = 0, . . . , m. Let k · kb be the semi-norm in Bb defined by
kykb = ky0 kD + sup{|y(s)| : 0 ≤ s ≤ b},
y ∈ Bb .
In this work, we will employ an axiomatic definition for the phase space D which is similar to those introduced in [24]. Specifically, D will be a linear space of functions mapping (−∞, 0] into E endowed with a semi norm k.kD , and satisfies the following axioms introduced at first by Hale and Kato in [21]:
(A1 ) There exist a positive constant H and functions K (·), M (·) : R+ → R+ with K continuous and M locally bounded, such that for any b > 0, if y : (−∞, b] → E, y ∈ D , and y(·) is continuous on [0, b], then for every t ∈ [0, b] the following conditions hold: (i) yt is in D ; (ii) |y(t )| ≤ H kyt kD ; (iii) kyt kD ≤ K (t ) sup{|y(s)| : 0 ≤ s ≤ t } + M (t )ky0 kD , and H , K and M are independent of y(·). (A2 ) The space D is complete.
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In what follows we use the following notations Kb = sup{K (t ) : t ∈ J } and Mb = sup{M (t ) : t ∈ J }. Definition 2.1 ([39]). Let E be a Banach space. An integrated semigroup is a family of operators (S (t ))t ≥0 of bounded linear operators S (t ) on E with the following properties: (i) S (0) = 0; (ii) t → S (t ) isRstrongly continuous; s (iii) S (s)S (t ) = 0 (S (t + r ) − S (r ))dr , for all t , s ≥ 0. Definition 2.2. An integrated semigroup (S (t ))t ≥0 is called exponential bounded, if there exists constant M ≥ 0 and ω ∈ R such that
|S (t )| ≤ Meωt ,
for t ≥ 0.
Moreover (S (t ))t ≥0 is called nondegenerate if S (t )y = 0, for all t ≥ 0, implies y = 0. Definition 2.3. An operator A is called a generator of an integrated semigroup, if there exists ω ∈ R such that (ω, +∞) ⊂ ρ(A), and there exists a strongly R ∞continuous exponentially bounded family (S (t ))t ≥0 of linear bounded operators such that S (0) = 0 and (λI − A)−1 = λ 0 e−λt S (t )dt for all λ > ω. If A is the generator of an integrated semigroup (S (t ))t ≥0 which is locally Lipschitz, then from [39], S (·)y is continuously d differentiable if and only if y ∈ D(A). In particular S 0 (t )y := dt S (t )y defines a bounded operator on the set E1 := {y ∈ E : t → S (t )y is continuously differentiable on [0, ∞)} and (S 0 (t ))t ≥0 is a C0 semigroup on D(A). Here and hereafter, we assume that A satisfies the Hille–Yosida condition, that is, there exists M > 0 and ω ∈ R such that (ω, ∞) ⊂ ρ(A) and sup{(λI − ω)n |(λI − A)−n | : λ > ω, n ∈ N} ≤ M . Note that, since A satisfies the Hille–Yosida condition,
kS 0 (t )kB(E ) ≤ Meωt ,
t ≥ 0,
where M and ω are the constants considered in the Hille–Yosida condition (see [40]). Let (S (t ))t ≥0 , be the integrated semigroup generated by A. Consider the Cauchy Problem y0 (t ) = Ay(t ) + f (t ),
t ∈ [0, b],
y(0) = y0 ∈ E .
(2.1)
Then we have the following. Theorem 2.1 ([40]). Let f : [0, b] → E be a continuous function. Then for y0 ∈ D(A), there exists a unique continuous function y : [0, b] → E of the Cauchy Problem (2.1) such that (i)
Rt 0
y(s)ds ∈ D(A) for t ∈ [0, b],
Rt Rt y(s)ds + 0 f (s)ds, t ∈ [0, b], 0 Rt (iii) |y(t )| ≤ Meωt |y0 | + 0 e−ωs |f (s)|ds , t ∈ [0, b]. (ii) y(t ) = y0 + A
Moreover, from [39,40] y satisfies the variation of constants formula, d y(t ) = S 0 (t )y0 + dt
t
Z
S (t − s)f (s)ds,
t ≥ 0.
(2.2)
0
Let Bλ = λR(λ, A) := λ(λI − A)−1 . Then [40] for all y ∈ D(A), Bλ y → y as λ → ∞. Also from the Hille–Yosida condition (with n = 1) it easy to see that limλ→∞ |Bλ y| ≤ M |y|, since
|Bλ | = |λ(λI − A)−1 | ≤
Mλ . λ−ω
Thus limλ→∞ |Bλ | ≤ M. Also if y is given by (2.2), then y(t ) = S 0 (t )y0 + lim
Z
λ→∞ 0
t
S 0 (t − s)Bλ f (s)ds,
t ≥ 0.
(2.3)
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We finish this section, with notations, definitions, and some results from multivalued analysis. Let (X , d) be a metric space. We use the notations: Pcl (X ) = {Y ∈ P (X ) : Y closed},
Pbd (X ) = {Y ∈ P (X ) : Y bounded}
Pc v (X ) = {Y ∈ P (X ) : Y convex},
Pcp (X ) = {Y ∈ P (X ) : Y compact}. S Consider Hd : P (X ) × P (X ) → R+ {∞} given by Hd (A, B) = max{sup d(a, B), sup d(A, b)}, a∈A
b∈B
where d(A, b) = infa∈A d(a, b), d(a, B) = infb∈B d(a, b). Then (Pbd,cl (X ), Hd ) is a metric space and (Pcl (X ), Hd ) is a generalized metric space (see [41]). A multivalued map N : J → Pcl (X ) is said to be measurable if, for each x ∈ X , the function Y : J → R defined by Y (t ) = d(x, N (t )) = inf{d(x, z ) : z ∈ N (t )}, is measurable. Definition 2.4. A measurable multivalued function F : J → Pbd,cl (X ) is said to be integrably bounded if there exists a function w ∈ L1 (J , R+ ) such that kvk ≤ w(t ) a.e. t ∈ J for all v ∈ F (t ). A multivalued S map F : X → P (X ) is convex (closed) valued if F (x) is convex (closed) for all x ∈ X . F is bounded on bounded sets if F (B) = x∈B F (x) is bounded in X for all B ∈ Pb (X ) i.e. supx∈B {sup{|y| : y ∈ F (x)}} < ∞. F is upper semi-continuous (u.s.c for short) on X if for each x0 ∈ X the set F (x0 ) is nonempty, closed subset of X , and for each open set U of X containing F (x0 ), there exists an open neighborhood V of x0 such that F (V ) ⊆ U.G is said to be completely continuous if F (B) is relatively compact for every B ∈ Pbd (X ). If the multivalued map F is completely continuous with nonempty compact valued, then G is u.s.c if and only if F has closed graph i.e. xn → x∗ , yn → y∗ , yn ∈ G(x∗ ) imply y∗ ∈ G(x∗ ). Definition 2.5. A multivalued map F : J × D → P (E ) is said to be Carathéodory if (i) t 7−→ F (t , u) is measurable for each u ∈ D (ii) u − 7 → F (t , u) is u.s.c. for almost each t ∈ J . Definition 2.6. A multivalued operator N : J → Pcl (X ) is called (a) contraction if and only if there exists γ > 0 such that Hd (N (x), N (y)) ≤ γ d(x, y),
for each x, y ∈ X ,
with γ < 1, (b) N has a fixed point if there exists x ∈ X such that x ∈ N (x). For more details on multivalued maps and the proof of the known results cited in this section we refer interested reader to the books of Deimling [42], Gorniewicz [43] and Hu and Papageorgiou [44]. 3. Examples of operators with nondense domain In this section we shall present examples of linear operators with nondense domain satisfying the Hille–Yosida estimate. More details can be found in the paper by Da Prato and Sinestrari [45]. Example 3.1. Let E = C ([0, 1], R) and the operator A : D(A) → E defined by Ay = y0 , where D(A) = {y ∈ C 1 ((0, 1), R) : y(0) = 0}. Then D(A) = {y ∈ C ((0, 1), R) : y(0) = 0} 6= E . Example 3.2. Let E = C ([0, 1], R) and the operator A : D(A) → E defined by Ay = y00 , where D(A) = {y ∈ C 2 ((0, 1), R) : y(0) = y(1) = 0}. Then D(A) = {y ∈ C ((0, 1), R) : y(0) = y(1) = 0} 6= E .
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Example 3.3. Let us set for some α ∈ (0, 1) E = C0α ([0, 1], R) =
y : [0, 1] → R : y(0) = 0 and
|y(t ) − y(s)| <∞ |t − s|α 0≤t
sup
and the operator A : D(A) → E defined by Ay = −y0 , where D(A) = {y ∈ C 1+α ((0, 1), R) : y(0) = y0 (0) = 0}. Then α
D(A) = h0 ((0, 1), R) =
y : [0, 1] → R : lim
|y(t ) − y(s)| = 0 6= E . | t − s| α
sup
δ→0 0<|t −s|≤δ
Here C 1+α ([0, 1], R) = {y : [0, 1] → R : y0 ∈ C α ([0, 1], R)}. The elements of hα ((0, 1), R) are called little Holder functions and it can be proved that the closure of C 1 ((0, 1), R) in C α ((0, 1), R) is hα ((0, 1), R) (see [46] Theorem 5.3). Example 3.4. Let Ω ⊂ Rn be a bounded open set with regular boundary Γ and define E = C (Ω , R) and the operator A : D(A) → E defined by Ay = ∆y, where D(A) = {y ∈ C (Ω , R) : y = 0 on Γ ; ∆y ∈ C (Ω , R)}. Here ∆ is the Laplacian in the sense of distributions on Ω . In this case we have D(A) = {y ∈ C (Ω , R) : y = 0 on Γ } 6= E . 4. Existence of integral solutions Now, we are able to state and prove our main theorem for the initial value problem (1.1)–(1.3). Before starting and proving this one, we give the definition of the integral solution. Definition 4.1. We say that y : (−∞, T ] → E is an integral solution of (1.1)–(1.3) if y(t ) = φ(t ) for all t ∈ (−∞, 0], the restriction of y(·) to the interval [0, b] is continuous, and there exist v(·) ∈ L1 (Jk , E ) and Ik ∈ Ik (ytk ), such that v(t ) ∈ F (t , yρ(t ,yt ) ) a.e t ∈ [0, b], and y satisfies the integral equation, (i) y(t ) = φ(0) + A (ii)
Rt 0
Rt 0
y(s)ds +
Rt 0
v(s)ds +
P
y(s)ds ∈ D(A) for t ∈ J.
0
Ik , t ∈ J .
From the definition it follows that y(t ) ∈ D(A), for each t ≥ 0, in particular φ(0) ∈ D(A). Moreover, from [39,40] y satisfies the following variation of constants formula: d y(t ) = S 0 (t )φ(0) + dt
t
Z
S (t − s)v(s)ds +
X
S 0 (t − tk ) Ik
t ≥ 0.
(4.1)
0 < tk < t
0
We notice also that, if y satisfies (4.1), then y(t ) = S 0 (t )φ(0) + lim
Z
λ→∞ 0
t
S 0 (t − s)Bλ v(s)ds +
X
S 0 (t − tk ) Ik ,
t ≥ 0.
0
The key tool in our approach is the following form of the fixed point theorem of Dhage [38,47]. Theorem 4.1. Let X be a Banach space, A : X → Pcl,c v,bd (X ) and B : X → Pcp,c v (X ), two multivalued operators satisfying (a) A is a contraction, and (b) B is completely continuous. Then either (i) The operator inclusion λx ∈ Ax + B x has a solution for λ = 1, or (ii) the set E = {u ∈ X |u ∈ λAu + λB u, 0 ≤ λ ≤ 1} is unbounded.
˜ be the space of functions in D which have values in D(A). We always assume that ρ : I × D → (−∞, b] is Set D continuous. Additionally, we introduce following hypotheses:
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(Hφ ) The function t → φt is continuous from R(ρ − ) = {ρ(s, ϕ) : (s, ϕ) ∈ J × D , ρ(s, ϕ) ≤ 0} into D and there exists a continuous and bounded function Lφ : R(ρ − ) → (0, ∞) such that kφt kD ≤ Lφ (t )kφkD for every t ∈ R(ρ − ). (H1) A satisfies Hille–Yosida condition; Pm (H2) There exist constants ck > 0, k = 1, . . . , m with Meωb Kb k=1 ck < 1 such that Hd (Ik (y), Ik (x)) ≤ ck |y − x|
˜. for each x, y ∈ D
(H3) The multivalued map F : J × D → Pcp,c v (E ) is Carathéodory; (H4) the operator S 0 (t ) is compact in D(A) wherever t > 0; (H5) There exist a function p ∈ L1 (J , R+ ) and a continuous nondecreasing function ψ : [0, ∞) → (0, ∞) such that
kF (t , x)kP ≤ p(t )ψ(kxkD ) for a.e. t ∈ J and each x ∈ D , R b −ωs with 0 e p(s)ds < ∞, (Mb + Lφ + MKb )kφkD + Kb u >1 lim sup Rb u→+∞ c ∗ + c ∗ ψ Kb u + (Mb + Lφ + MKb )kφkD e−ωs p(s)ds 1 2 0
(4.2)
where c ∗ eω b Kb
c1∗ =
m P
1 − Meωb Kb
+ Mb + Lφ + MKb kφkD ,
(4.3)
ck
k=1 m
X |Ik (0)| + ck Mb + Lφ + MKb kφkD ,
c∗ =
(4.4)
k=1
and MKb eωb
c2∗ =
1 − Meωb Kb
m P
.
(4.5)
ck
k=1
The next result is a consequence of the phase space axioms. Lemma 4.1 ([36], Lemma 2.1). If y : (−∞, b] → E is a function such that y0 = φ and y|J ∈ PC (J : D(A)), then
kys kD ≤ (Ma + Lφ )kφkD + Ka sup{ky(θ )k; θ ∈ [0, max{0, s}]},
s ∈ R(ρ − ) ∪ J ,
where Lφ = supt ∈R(ρ − ) Lφ (t ), Ma = supt ∈J M (t ) and Ka = supt ∈J K (t ). Remark 4.1. We remark that condition (Hφ ) is satisfied by functions which are continuous and bounded. In fact, if the space D satisfies axiom C2 in [24] then there exists a constant L > 0 such that kφkD ≤ L sup{kφ(θ )k : θ ∈ [−∞, 0]} for every φ ∈ D that is continuous and bounded, (see [24] Proposition 7.1.1) for details. Consequently sup kφ(θ )k
kφt kD ≤ L
θ≤0
kφkD
kφkD ,
for every φ ∈ D \ {0}.
Theorem 4.2. Assume that (Hφ ) and (H1)–(H5) hold. If φ(0) ∈ D(A), then the problem (1.1)–(1.3) has at least one integral solution on (−∞, b]. Proof. Transform the problem (1.1)–(1.3) into a fixed point problem. Set
Ω = PC (−∞, b], D(A) , and consider the multivalued operator N : Ω → P (Ω ) defined by N (y) = {h ∈ Ω } with h(t ) =
φ(t ),
if t ≤ 0, Z t X d S ( t )φ( 0 ) + S (t − s)v(s)ds + S 0 (t − tk )Ik , dt 0 0
k
v ∈ SF ,yρ(s,ys ) , Ik ∈ Ik (y(tk− )) if t ∈ J .
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For φ ∈ D define the function e φ : (−∞, b] → E such that:
φ(t ), e φ(t ) = 0 S (t )φ(0),
if t ≤ 0 if t ∈ J .
Then e φ0 = φ . For each x ∈ Bb with x(0) = 0, we denote by x the function defined by x(t ) =
0, x(t ),
t ∈ (−∞, 0], t ∈ J.
If y(.) satisfies (4.1), we can decompose it as y(t ) = e φ(t ) + x(t ), 0 ≤ t ≤ b, which implies yt = e φt + xt , for every 0 ≤ t ≤ b and the function x(.) satisfies x(t ) =
t
Z
d dt
Let
S 0 (t − tk )Ik
t ∈ J,
0
0
where v(s) ∈ SF ,x
X
S (t − s)v(s)ds +
φtk . and Ik ∈ Ik xtk + e
ρ(s,xs +e φs ) +φρ(s,xs +e φs )
e
Bb0 = {x ∈ Bb : x0 = 0 ∈ D }. For any x ∈ Bb0 we have
kxkb = kx0 kD + sup{|x(s)| : 0 ≤ s ≤ b} = sup{|x(s)| : 0 ≤ s ≤ b}. Thus (Bb0 , k · kb ) is a Banach space. We define the two multivalued operators A, B : Bb0 → P (Bb0 ) by A(x) := {h ∈ Bb0 }, B (x) := {h ∈ Bb0 } with h( t ) =
, 0X
S 0 (t − tk )Ik ,
I k ∈ I k x tk + e φ tk ,
if t ∈ (−∞, 0]; if t ∈ J ,
0
and
0, Z t d h( t ) = S (t − s)v(s)ds, dt
if t ∈ (−∞, 0];
v(s) ∈ SF ,x
0
ρ(s,xs +e φs ) +φρ(s,xs +e φs )
e
if t ∈ J .
Obviously to prove that the multivalued operator N has a fixed point is reduced that the operator inclusion x ∈ A(x) + B (x) has one, so it turns to show that the multivalued operators A and B satisfy all conditions of Theorem 4.1. For better readability, we break the proof into a sequence of steps. Step 1: A is a contraction. Let x1 , x2 ∈ Bb0 . Then for t ∈ J
! X
Hd (A(x1 ), A(x2 )) = Hd
0 < tk < t
≤ Meωb
S 0 (t − tk )Ik (x1tk + e φtk ),
X 0
S 0 (t − tk ) Ik (x2tk + e φtk )
X Ik (x1 ) − Ik (x2 ) tk
tk
0≤tk ≤t
≤ Meωb
m X
ck kx1tk − x2tk kD
k=1
≤ Meωb Kb
m X
ck kx1 − x2 kD .
k=1
Hence by (H2) A is a contraction. Step 2: B has compact, convex values, and it is completely continuous. This will be given in several claims. Claim 1: B has compact values. The operator B is equivalent to the composition L ◦ SF on L1 (J , E ), where L : L1 (J , E ) → Bb0 is the continuous operator defined by
L(v)(t ) =
d dt
Z 0
t
S (t − s)v(s)ds,
t ∈ J.
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Then, it suffices to show that L ◦ SF has compact values on Bb0 . Let x ∈ Bb0 arbitrary and vn a sequence such that e vn (t ) ∈ SF ,x , a.e. t ∈ J. Since F (t , xρ(t ,xt +e φ φt ) + φρ(t ,xt +e φt ) ) is compact, we may pass to a subsequence. Suppose e +e e ρ(t ,xt +φt )
ρ(t ,xt +φt )
e that vn → v in L1w (J , E ) (the space endowed with the weak topology), where v(t ) ∈ F (t , xρ(t ,xt +e φt ) + φρ(t ,xt +e φt ) ), a.e. t ∈ J. An application of Mazur’s theorem [48] implies that vn converges strongly to v and hence v(t ) ∈ SF ,x . From e + φ e e ρ(t ,xt +φt )
ρ(t ,xt +φt )
the continuity of L, it follows that Lvn (t ) → Lv(t ) pointwise on J as n → ∞. In order to show that the convergence is uniform, we first show that {Lvn } is an equicontinuous sequence. Let τ1 , τ2 ∈ J, then we have:
Z τ Z τ2 1 d d |L(vn (τ1 )) − L(vn (τ2 ))| = S (τ1 − s)vn (s)ds − S (τ2 − s)vn (s)ds dt 0 dt 0 Z τ1 Z ≤ lim [S 0 (τ1 − s) − S 0 (τ2 − s)]Bλ vn (s)ds + lim λ→∞ λ→∞ 0
τ2 τ1
S 0 (τ2 − s)Bλ vn (s)ds .
As τ1 → τ2 , the right hand-side of the above inequality tends to zero. Since S 0 (t ) is a strongly continuous operator and the compactness of S 0 (t ), t > 0, implies the uniform continuity (see [5,49]). Hence {Lvn } is equi-continuous, and an application of Arzéla–Ascoli theorem implies that there exists a subsequence which is uniformly convergent. Then we have Lvnj → Lv ∈ (L ◦ SF )(x) as j 7→ ∞, and so (L ◦ SF )(x) is compact. Therefore B is a compact valued multivalued operator on Bb0 . Claim 2: B (x) is convex for each z ∈ D0b . Let h1 , h2 ∈ B (x), then there exist v1 , v2 ∈ SF ,x such that, for each t ∈ J we have e ρ(t ,xt +e φt ) +φρ(t ,xt +e φt )
0, Z t d hi ( t ) = S (t − s)vi (s)ds dt
if t ∈ (−∞, 0],
if t ∈ J ,
0
,
i = 1, 2.
Let 0 ≤ δ ≤ 1. Then, for each t ∈ J, we have
0, Z (δ h1 + (1 − δ)h2 )(t ) = d t S (t − s)[δv1 (s) + (1 − δ)v2 (s)]ds dt
if t ∈ (−∞, 0],
if t ∈ J .
0
Since F has convex values, one has
δ h1 + (1 − δ)h2 ∈ B (x). Claim 3: B maps bounded sets into bounded sets in Bb0 . Let Bq = {x ∈ Bb0 : kxkb ≤ q}, q > 0 a bounded set in Bb0 . It is equivalent to show that there exists a positive constant l such that for each x ∈ Bq we have kB (x)kb ≤ l. So choose x ∈ Bq , then from Lemma 4.1 it follows that For each h ∈ B (x), and each x ∈ Bq , there exists v ∈ SF ,x such that φ e +e e ρ(t ,xt +φt )
h(t ) =
d dt
ρ(t ,xt +φt )
t
Z
S (t − s)v(s)ds. 0
From (A) we have
kxρ(t ,xt +eφt ) + e φρ(t ,xt +eφt ) kD ≤ Kb q + (Mb + Lφ )kφkD + Kb M |φ(0)| = q∗ . Then by (H6) we have ωb
|h(t )| ≤ Me ψ(q∗ )
Z
t
e−ωs p(s)ds := l.
0
This further, implies that
khkB 0 ≤ l. b
Hence B (B) is bounded. Claim 4: B maps bounded sets into equicontinuous sets. Let Bq be, as above, a bounded set and h ∈ B (x) for some x ∈ B. Then, there exists v ∈ SF ,x such that e ρ(t ,xt +e φt ) +φρ(t ,xt +e φt ) h(t ) =
d dt
t
Z
S (t − s)v(s)ds, 0
t ∈ J.
N. Abada et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 791–803
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Let τ1 , τ2 ∈ J \ {t1 , t2 , . . . , tm }, τ1 < τ2 . Thus if > 0, we have
Z τ1 − Z τ1 |h(τ2 ) − h(τ1 )| ≤ lim [S 0 (τ2 − s) − S 0 (τ1 − s)]Bλ v(s)ds + lim [S 0 (τ2 − s) − S 0 (τ1 − s)]Bλ v(s)ds λ→∞ 0 λ→∞ τ − 1 Z τ2 + lim S 0 (τ2 − s)Bλ v(s)ds λ→∞ τ 1 Z τ1 −
≤ ψ(q∗ )
0
+ ψ(q∗ )
kS 0 (τ2 − s) − S 0 (τ1 − s)kB(E ) p(s)ds
τ1
Z
τ1 −
kS 0 (τ2 − s) − S 0 (τ1 − s)kB(E ) p(s)ds + Meωb ψ(q∗ )
τ2
Z
e−ωs p(s)ds.
τ1
As τ1 → τ2 and becomes sufficiently small, the right-hand side of the above inequality tends to zero, since S 0 (t ) is a strongly continuous operator and the compactness of S 0 (t ) for t > 0 implies the uniform continuity. This proves the equicontinuity for the case where t 6= ti , i = 1, . . . , m + 1. It remains to examine the equicontinuity at t = ti . First we prove the equicontinuity at t = ti− , we have for some x ∈ Bq , there exists v ∈ SF ,x such that φ e +e e ρ(t ,xt +φt )
d
h( t ) =
dt
ρ(t ,xt +φt )
t
Z
S (t − s)v(s)ds,
t ∈ J.
0
Fix δ1 > 0 such that {tk , k 6= i} ∩ [ti − δ1 , ti + δ1 ] = ∅. For 0 < µ < δ1 , we have
|h(ti − µ) − h(ti )| ≤ lim
Z
ti −µ
λ→∞ 0
0
S (ti − µ − s) − S 0 (ti − s) Bλ v(s) ds + Meωb ψ(q∗ )
Z
ti
e−ωs p(s) ds;
ti −µ
which tends to zero as µ → 0. Define hˆ 0 (t ) = h(t ),
t ∈ [0, t1 ]
and hˆ i (t ) =
h(t ), h(ti+ ),
if t ∈ (ti , ti+1 ] if t = ti .
Next, we prove equicontinuity at t = ti+ . Fix δ2 > 0 such that {tk , k 6= i} ∩ [ti − δ2 , ti + δ2 ] = ∅. Then hˆ (ti ) =
ti
Z
T (ti − s)v(s)ds.
0
For 0 < µ < δ2 , we have
|hˆ (ti + µ) − hˆ (ti )| ≤ lim
Z
λ→∞ 0
ti
0
S (ti + µ − s) − S 0 (ti − s) Bλ v(s) ds + Meωb ψ(q∗ )
Z
ti +µ
e−ωs p(s) ds;
ti
The right hand-side tends to zero as µ → 0. The equicontinuity for the cases τ1 < τ2 ≤ 0 and τ1 ≤ 0 ≤ τ2 follows from the uniform continuity of φ on the interval (−∞, 0]. As a consequence of Claims 1–3 together with Arzelá–Ascoli theorem it suffices to show that B maps B into a precompact set in E. Let 0 < t < b be fixed and let be a real number satisfying 0 < < t. For x ∈ Bq , we define h (t ) = S 0 () lim
Z
λ→∞ 0
where v ∈ SF ,x
Z lim λ→∞
t −
S 0 (t − s − )Bλ v(s)ds,
ρ(t ,xt +e φt ) +φρ(t ,xt +e φt )
e
t −
. Since
Z ωb S (t − s − )Bλ v(s)ds ≤ Me ψ(q∗ )
t −
e−ωs p(s)ds,
0
0
0
the set
Z lim
λ→∞ 0
t −
S 0 (t − s − )Bλ v(s)ds : v ∈ SF ,x
.x e ρ(t ,xt +e φt ) +φρ(t ,xt +e φt )
is bounded. Since S 0 (t ) is a compact operator for t > 0, the set H (t ) = {h (t ) : h ∈ B (x)}
∈ Bq
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N. Abada et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 791–803
is precompact in E for every , 0 < < t. Moreover, for every h ∈ B (x) we have ωb
|h(t ) − h (t )| ≤ Me ψ(q∗ )
t −
Z
e−ωs p(s)ds. t
Therefore, there are precompact sets arbitrarily close to the set H (t ) = {h(t ) : h ∈ B (x)}. Hence the set H (t ) = {h(t ) : h ∈ B (Bq )} is precompact in E. Hence the operator B is totally bounded. Step 3: A priori bounds. Now it remains to show that the set
E = x ∈ Bb0 : x ∈ λA(x) + λB (x) for some 0 < λ < 1
φtk such that for each t ∈ J , is bounded. Let x ∈ E , then there exist v ∈ SF ,x and Ik ∈ Ik xtk + e e ρ(t ,xt +e φt ) +φρ(t ,xt +e φt )
x(t ) = λ
d
t
Z
dt
X
S (t − s)v(s) + λ
S 0 ( t − t k ) Ik .
0
0
This implies by (H2), (H5) that, for each t ∈ J, we have t
|x(t )| ≤ λMeωt
Z
≤ λMeωt
Z
ωt e e−ωs p(s)ψ(kxρ(s,xs +e φs ) + φρ(s,xs +e φs ) kD )ds + λMe
0
m X Ik x t + e φtk k k =1
t
e−ωs p(s)ψ Kb |x(s)| + (Mb + Lφ + MKb )kφkD ds
0
+ λMeωt
m m X X Ik xt + e + λMeωt |Ik (0)| φ − I ( 0 ) t k k k k=1
≤ λMe
ωt
Z
k=1
t
e
−ωs
φ
p(s)ψ Kb |x(s)| + (Mb + L + MKb )kφkD ds
0
+ λMe
ωt
m X
|Ik (0)| + λMeωt
k=1
∗ ωt
≤c e
+ Me
ωt
m X
ck Kb |x(s)| + (Mb + Lφ + MKb )kφkD
k=1
"Z
t
e
−ωs
φ
p(s)ψ Kb |x(s)| + (Mb + L + MKb )kφkD ds + Kb
0
m X
# ck |x(t )| .
k=1
Hence from (4.3)–(4.5) we have
(Mb + Lφ + MKb )kφkD + Kb |x(s)| ≤ c1∗ + c2∗
t
Z
e−ωs p(s)ψ Kb |x(t )| + (Mb + Lφ + MKb )kφkD ds.
0
Thus
(Mb + Lφ + MKb )kφkD + Kb kxkB 0 b R ≤ 1. b ∗ ∗ φ c1 + c2 ψ Kb kxkB 0 + (Mb + L + MKb )kφkD 0 e−ωs p(s)ds
(4.6)
b
From (4.2) it follows that there exists a constant R > 0 such that for each x ∈ E with kxkB 0 > R the condition (4.6) is b
violated. Hence kxkB 0 ≤ R for each x ∈ E , which means that the set E is bounded. As a consequence of Theorem 4.1, A + B b
has a fixed point x∗ on the interval (−∞, b], so y∗ = x∗ + e φ is a fixed point of the operator N which is the mild solution of problem (1.1)–(1.3). 5. Examples of phase spaces In this section we give some usual phase spaces. Let g : (−∞, 0] → [1, ∞) be a continuous, nondecreasing function with g (0) = 1, which satisfies the conditions (g-1), (g-2) of [24]. This means that the function G(t ) =
sup −∞<θ ≤−t
g (t + θ ) g (θ )
is locally bounded for t ≥ 0 and that limθ→−∞ g (θ ) = ∞. We said that φ : [−∞, 0] → E is normalized piecewise continuous, if φ is left continuous and the restriction of φ to any interval [−r , 0] is piecewise continuous.
N. Abada et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 791–803
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Next we modify slightly the definition of the spaces Cg , Cg0 of [24]. We denote by P C g (E ) the space formed by the normalized piecewise continuous functions φ such that by the functions φ such that lim
θ→−∞
φ g
is bounded on (−∞, 0] and by P C 0g the subspace of PCg (E ) formed
φ(θ ) = 0. g (θ )
It is easy to see that D = P C g (E ) and D = P C 0g (E ) endowed with the norm
kφkD =
kφ(θ )k g (θ )
sup
θ∈(−∞,0]
are phase spaces. Moreover, in these cases K (s) = 1 for s ≥ 0. Let 1 ≤ p < ∞, 0 ≤ r < ∞ and g (·) is a Borel nonnegative measurable function on (−∞, r ) which satisfies the conditions (g-5)–(g-6) in the terminology of [24]. This means that g (·) is locally integrable on (−∞, −r ) and there exists a nonnegative and locally bounded function G on (−∞, 0] such that g (ξ +θ ) ≤ G(ξ )g (θ ) for all ξ ≤ 0 and θ ∈ (−∞, −r )\ Nξ , where Nξ ⊂ (−∞, −r ) is a set with Lebesgue measure 0. Let D := P C r × Lp (g , E ), r ≥ 0, p > 1, be the space formed of all classes of functions φ : (−∞, 0] → E such that φ|[−r ,0] ∈ PC ([−r , 0], E ), φ(·) is Lebesgue measurable on (−∞, −r ] and g |φ|p is Lebesgue integrable on (−∞, −r ]. The seminorm in k · kD is defined by
kφkD := sup kφ(θ )k +
Z
θ∈[−r ,0]
−r
g (θ )kφ(θ )kp dθ
1p
.
−∞
Proceeding as in the proof of ([24], Theorem 1.3.8), it follows that D is a phase space which satisfies Axioms (A1 ) and (A2 ). 1
Moreover, for r = 0 and p = 2 this space coincides (see [24]) with C0 × L2 (g , E ), H = 1, M (t ) = G(−t ) 2 and K (t ) = 1 +
0
Z
g (s)ds
21
,
for t ≥ 0.
−t
6. An example To apply our abstract results, we consider the partial functional differential equations with state dependent delay of the form
∂ ∂ v(t , ξ ) ∈ − v(t , ξ ) + m(t )a(t , v(t − σ (v(t , 0)), ξ )), ∂t ∂ξ
ξ ∈ [0, π], t ∈ [0, 1],
v(t , 0) = v(t , π ) = 0, t ∈ [0, 1], v(θ , ξ ) = v0 (θ , ξ ), ξ ∈ [0, π], θ ∈ (−∞, 0], Z ti γi (ti − s)[−|v(s, ξ )|, |v(s, ξ )|]ds, ∆v(ti )(ξ ) ∈
(6.1) (6.2) (6.3) (6.4)
−∞
where v0 : (−∞, 0] × [0, π] → R, γi : [0, ∞) → R are continuous functions, 0 < t1 < t2 < · · · < tn < 1, m : [0, 1] → R+ , a : [0, 1] × R → Pc v,cp (R), σ : R → R+ is continuous and we assume the existence of positive constants b1 , b2 such that
|a(t , u)| ≤ b1 |u| + b2 for every (t , u) ∈ [0, 1] × R. Let A be the operator defined on E = C ([0, π], R) by D(A) = {g ∈ C 1 ([0, π], R) : g (0) = 0}; Ag = g 0 . Then D(A) = C0 ([0, π], R) = {g ∈ C ([0, π], R) : g (0) = 0}. It is well known from [45] that A is sectorial, (0, +∞) ⊆ ρ(A) and for λ > 0
kR(λ, A)kB(E ) ≤
1
λ
.
It follows that A generates an integrated semigroup (S (t ))t ≥0 and that kS 0 (t )kB(E ) ≤ e−µt for t ≥ 0 for some constant µ > 0 and A satisfied the Hille–Yosida condition. Set γ > 0. For the phase space, we choose D to be defined by
D = PC γ = {φ ∈ PC ((−∞, 0], E ) : lim eγ θ φ(θ ) exists in E } θ→−∞
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N. Abada et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 791–803
with norm
kφkγ =
sup
θ∈(−∞,0]
eγ θ |φ(θ )|,
φ ∈ PC γ .
Notice that the phase space D satisfies axioms (A1 ), (A2 ) (see [24] for more details). By making the following change of variables y(t )(ξ ) = v(t , ξ ), t ∈ [0, 1],
ξ ∈ [0, π], φ(θ )(ξ ) = v0 (θ , ξ ), θ ≤ 0, ξ ∈ [0, π], F (t , ϕ)(ξ ) = m(t )a(t , ϕ(0, ξ )), t ∈ [0, 1], ξ ∈ [0, π], φ ∈ Cγ ρ(t , ϕ) = t − σ (ϕ(0, 0)) Z 0 γk (−s)[−|v0 (s, ξ )|, |v0 (s, ξ )|]ds, Ik (ytk ) = −∞
the problem (6.1)–(6.4) takes the abstract form (1.1)–(1.3). Moreover, we have
kF (t , ϕ)kP ≤ m(t )(b1 kϕkD + b2 ) for all (t , ϕ) ∈ I × D with ∞
Z 1
ds
ψ(s)
Z = 1
∞
ds b1 s + b2
= +∞.
The next results are consequence of Theorem 4.2 and Remark 4.1. Theorem 6.1. Let φ ∈ B be such that condition (Hφ ) holds, then problem (6.1)–(6.4) has a mild solution. Corollary 6.1. Let φ ∈ B be continuous and bounded, then problem (6.1)–(6.4) has a mild solution. Acknowledgement The authors are grateful to the referee for his/her remarks. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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