J. Math. Anal. Appl. 402 (2013) 453–462

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Journal of Mathematical Analysis andApplications

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Massera problem for non-autonomous retardeddifferential equationsMohamed Zitane ∗, Charaf BensoudaUniversité Ibn Tofaïl, Faculté des Sciences, Département de Mathématiques, Laboratoire d’An. Maths et GNC, B.P. 133, Kénitra 1400, Maroc

a r t i c l e i n f o

Article history:Received 31 July 2012Available online 1 February 2013Submitted by Xu Zhang

Keywords:Evolution familyMild solutionPeriodic solutionsFixed point theoremMultivalued mapPoincaré map

a b s t r a c t

In this paper, under Acquistapace–Terreni conditions, we study the existence of periodicsolutions for some non-autonomous semilinear partial functional differential equationwith delay. We assume that the linear part is non-densely defined. The delayed part isassumed to be ω-periodic with respect to the first argument. Using a fixed point theoremfor multivalued mapping, some sufficient conditions are given to prove the existence ofperiodic solutions. An example is shown to illustrate our results.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

The aim of this work is to study the existence of periodic solutions for the following non-autonomous evolution equationwith delaydu(t)

dt= A(t)u(t)+ L(t, ut)+ G(t, ut) for t ≥ s

u(t) = ϕ(t − s), s − r ≤ t ≤ s,(1.1)

in a Banach space (X, ∥·∥), where A(t) : D ⊂ X → X is a family of closed linear operators defined on a common domain D,independent of t ∈ R. The history function ut : [−r, 0] → X defined by ut(θ) = u(t + θ); for each θ ∈ [−r, 0]; belongsto Cr := C ([−r, 0] ; X) the space of continuous functions endowed with the supremum norm. L : [0,+∞) × Cr → Xis continuous, linear with respect to the second argument, ω-periodic in t with values in Cb(Cr ,X); the Banach space ofbounded and strongly continuous linear operators mapping Cr into X,G : [0,+∞) × Cr → X is a nonlinear continuousfunction and ω-periodic in t .

The problem of finding periodic solutions to differential equations constitutes one of the most attractive topics inqualitative theory of functional differential equations due to possible applications. The existence of periodic solutions isstarted by Massera for ordinary differential equations. In [29], Massera explains the relationship between the boundednessof solutions and the existence of periodic solutions. In this context many results have been developed to investigate theperiodicity of the solution, among others, we cite [9,8,10,11,20,26,35,13]. Later on, these results have been extended toevolution equations, see [23,24,30]. For delayed non-autonomous differential equations we cite [4,5,17,25,27,28,19,12].More precisely, the authors in [4] analyzed Massera’s problem for Eq. (1.1) when A(·) satisfies the conditions introduced

∗ Corresponding author.E-mail addresses: zitanem@gmail.com (M. Zitane), bensoudach@yahoo.fr (C. Bensouda).

0022-247X/$ – see front matter© 2013 Elsevier Inc. All rights reserved.doi:10.1016/j.jmaa.2013.01.046

454 M. Zitane, C. Bensouda / J. Math. Anal. Appl. 402 (2013) 453–462

by Tanaka. However, to the best of our knowledge, the existence of periodic solutions to Eq. (1.1) under the Acquistapace–Terreni conditions is an untreated topic and this provides the main motivation of the present paper.

In many works, the main approach to prove the existence of periodic solutions is to consider the Poincaré map P whichis defined by

P(ϕ) = uω(·, ϕ)

where ω is a period of the system and u the unique mild solution determined by ϕ. Then a fixed point theorem can be usedto derive periodic solutions.

Thiswork is organized as follows: in Section 2,we recall somepreliminary results about the evolution family andwe showthe well-posedness of mild solutions of Eq. (1.1). In Section 3, by using the Chow and Hale fixed point theorem, we provethat the existence of bounded mild solution on R+ of some nonhomogeneous linear equation is equivalent to the existenceof periodic mild solution. In Section 4, we study the existence of periodic mild solutions of the nonlinear equation (1.1) byusing a fixed-point theorem for set-valued maps. Finally, in Section 5, we propose an application.

2. Well-posedness and some preliminary results

Throughout this paper, we suppose that(H1) the family of closed linear operators A(t), for t ∈ R, onXwith domainD(A(t)) (possibly not densely defined) satisfy theso-calledAcquistapace–Terreni conditions; namely, there exist constantsλ0 ≥ 0, θ ∈ (π2 , π),M1,M2 ≥ 0, andα, β ∈ (0, 1]with α + β > 1 such that

Σθ ∪ 0 ⊂ ρ(A(t)− λ0), ∥R(λ, A(t)− λ0)∥ ≤M1

1 + |λ|(2.1)

and

∥(A(t)− λ0)R(λ, A(t)− λ0)[R(λ0, A(t))− R(λ0, A(s))]∥ ≤ M2|t − s|α|λ|−β (2.2)

for t, s ∈ R, λ ∈ Σθ := λ ∈ C − 0 : |argλ| ≤ θ.(H2) D(A(t)) = D is independent of t , and there are positive constants c1 > 0 and c2 > 0 such that

c1 ∥y∥D ≤ ∥y∥ + ∥A(t)∥ ≤ c2 ∥y∥D for y ∈ D and t ≥ 0.

(H3) The evolution family (U(t, s))t≥s generated by A(·) is exponentially stable with positive constants M ≥ 1 and δ suchthat

∥U(t, s)∥B(X) ≤ Me−δ(t−s) for t > s.

(H4) The family (U(t, s))t≥s is compact and ω-periodic.(H5) There existsM1 > 0, such that for φ, ϕ ∈ Cr and t ≥ 0 we have

∥G(t, φ)− G(t, ϕ)∥ ≤ M1∥φ − ϕ∥.

Let us mention that in the case when A(t) has a constant domain D(A(t)) = D, it is well-known [6,32] that Eq. (2.2) canbe replaced with the following

(H′

1) There exist constantM2 and 0 < α ≤ 1 such that

∥(A(t)− A(s))R(λ0, A(r))∥ ≤ M2|t − s|α, for s, t, r ∈ R. (2.3)

Let us also mention that (H1) was introduced in the literature by Acquistapace and Terreni in [3,2]. Among other things,from [1, Theorem 2.3]; see also [3,33,34]; it ensure that the system

u′(t) = A(t)u(t), t ≥ s, u(s) = ϕ ∈ X (2.4)

has a unique strongly continuous associated evolution family of operators (U(t, s))t≥s on X satisfying.

Theorem 2.1. The following properties hold,(1) U(t, s) = U(t, r)U(r, s) and U(t, t) = I for all t ≥ r ≥ s and t, r, s ∈ R.(2) the map (t, s) → U(t, s)x is continuous for all x ∈ X, t ≥ s and t, s ∈ R.(3) U(·, s) ∈ C1((s,∞),L(X)), dU

dt (t, s) = A(t)U(t, s) and ∥A(t)kU(t, s)∥ ≤ c(t − s)−k for 0 < t − s ≤ 1, k = 0, 1, and theconstant c depending only on the constants appearing in (H1).(4) ∂+

s U(t, s)x = −U(t, s)A(s)x, for t > s and x ∈ D(A(s)) with A(s)x ∈ D(A(s)).

Remark 2.2. There are some conditions on A(·) implying that U is exponentially stable. Such results are usually based on(H1) and its variants (cf. [31, Theorem 9.30, p. 494]; see also [22,15,18]).

M. Zitane, C. Bensouda / J. Math. Anal. Appl. 402 (2013) 453–462 455

Definition 2.3. Let ϕ ∈ Cr , a continuous function x : [−r,+∞) → X is called a mild solution of Eq. (1.1) if it satisfies thefollowing

x(t) :=

U(t, 0)ϕ(0)+

t

0U(t, σ )[L(σ , xσ )+ G(σ , xσ )]dσ t ≥ 0,

ϕ(t) −r ≤ t ≤ 0.(2.5)

Now, we state our first result.

Theorem 2.4. Assume that (H1)–(H3), (H5) hold and ϕ ∈ Cr such that ϕ(0) ∈ D. Then there exists a unique mild solutionu(·, ϕ, f ) on R+ for the Eq. (1.1)Moreover, the solution depends continuously on the initial data ϕ.

Proof. Let T > 0 and ϕ ∈ C([−r, 0]; X). We consider the set

Γϕ = u ∈ C([−r, T ]; X) : u(θ) = ϕ(θ) for θ ∈ [−r, 0]

Γϕ is a closed subset of C([−r, T ]; X) provided with the uniform norm topology. Let K be the operator defined onC([−r, T ]; X) by

K(u)(t) =

U(t, 0)ϕ(0)+

t

0U(t, σ )[L(σ , uσ )+ G(σ , uσ )]dσ , if t ∈ [0, T ]

ϕ(t), if t ∈ [−r, 0].

For the simplicity, we set: H(·, φ) = L(·, φ)+ G(·, φ).First, we have immediately that K(Γϕ) ⊆ Γϕ . Furthermore, for u, v ∈ Γϕ and t ∈ [0, T ], one has

∥K(u)(t)− K(v)(t)∥ ≤

t

0U(t, σ )[H(σ , uσ )− H(σ , vσ )]dσ

.Using (H3), (H5) and the boundedness of L, there exists a constant c > 0 such that

∥K(u)(t)− K(v)(t)∥ ≤ c t

0∥uσ − vσ∥ dσ

≤ ct ∥u − v∥ .

Then, by induction on n it follows thatK n(u)− K n(v) ≤

cntn

n!∥u − v∥ . (2.6)

For n large enough ( cntnn! ) < 1 and by a well known extension of the Banach contraction principle, K has a fixed point

u ∈ C([−r, T ]; X). This fixed point is the desired solution of Eq. (1.1).The uniqueness of a solution u and the Lipschitz continuity of the map ϕ → u(·, ϕ) are consequences of the followingargument. Let t ∈ [0, T ], and u(·, ψ) be a mild solution of (1.1) with initial data ψ , one has

∥u(t, ϕ)− u(t, ψ)∥ ≤ ∥U(t, 0)(ϕ(0)− ψ(0))∥ +

t

0U(t, σ ) ∥H(σ , uσ (·, ϕ))− H(σ , uσ (·, ψ))∥ dσ

≤ c ′∥ϕ − ψ∥ + c

t

0∥uσ (·, ϕ)− uσ (·, ψ)∥dσ .

Thus,

∥ut(·, ϕ)− ut(·, ψ)∥ = maxθ∈[−r,0]

∥u(t + θ, ϕ)− u(t + θ ·, ψ)∥

≤

∥ϕ(t + θ)− ψ(t + θ)∥ , if − r ≤ t + θ ≤ 0

c ′∥ϕ − ψ∥ + c

t

0∥uσ (·, ϕ)− uσ (·, ψ)∥ dσ , if 0 ≤ t + θ ≤ T .

Then, by Gronwall’s inequality we deduce that

∥ut(·, ϕ)− ut(·, ψ)∥ ≤ c ′ecT ∥ϕ − ψ∥

which yields both the uniqueness of u and the continuity of the map ϕ → ut(·, ϕ).

Proceeding inductively, we extend the solution uniquely and continuously to [−r,+∞[.

456 M. Zitane, C. Bensouda / J. Math. Anal. Appl. 402 (2013) 453–462

3. Existence of periodic solutions in the linear case

Consider the nonhomogeneous linear equation ddt

u(t) = Au(t)+ L(t, ut)+ f (t), t ≥ su(t) = ϕ(t − s), s − r ≤ t ≤ s

(3.1)

where, f : R → X is a continuous ω-periodic function.

A function u defined on R is said to be a mild solution of Eq. (3.1) if it satisfies the following

u(t) = U(t, s)u(s)+

t

sU(t, σ )[L(σ , uσ )+ f (σ )]dσ t ≥ s, (3.2)

we state now the following result which connects between bounded and periodic solution for the Eq. (3.1).

Theorem 3.1. Assume that (H1)–(H4) hold true. Then the following assertions are equivalent

(i) The Eq. (3.1) has a bounded mild solution on R+.

(ii) The Eq. (3.1) has an ω-periodic mild solution.

To prove Theorem 3.1, the following two lemmas are needed:

Lemma 3.2 ([14]). Let Y be a Banach space, P0 : Y → Y be a continuous linear operator and y ∈ Y. Consider the operatorP : Y → Y which is defined by

Px = P0x + y.

Suppose that there exists x0 ∈ Y such that Pnx0 : n ∈ N is relatively compact in Y, then P has a fixed point in Y.

Lemma 3.3. Let u be a bounded solution of Eq. (3.1), then u is uniformly continuous with relatively compact rang u(t) : t > 0in X. Furthermore, the set ut : t ≥ 0 is relatively compact in Cr .

Proof of Lemma 3.3. Let u be a bounded mild solution of Eq. (3.1) and 0 < ε < 1, then

u(t) : t ≥ 0 = u(t) : 0 ≤ t ≤ ε ∪ u(t) : t ≥ ε.

Since u is continuous, then the first set on the right hand side is relatively compact in X. For notational simplicity we set:F(t;ϕ) = L(t;ϕ)+ f (t) and let t > ε, one has

u(t) = U(t, 0)u(0)+

t

0U(t, s)F(s, us)ds

= U(t, t − ε)

U(t − ε, 0)u(0)+

t−ε

0U(t − ε, s)F(s, us)ds

+

t

t−εU(t, s)F(s, us)ds

= U(t, t − ε)u(t − ε)+

t

t−εU(t, s)F(s, us)ds.

Firstly, we show that the set K := U(t, t − ε)u(t − ε); t > ε is relatively compact in X. To this end, let (yn)n be a sequencein K , then there exists a sequence (tn)n with values tn > ε such that

yn = U(tn, tn − ε)u(tn − ε).

So, there exists a unique qn ∈ N such that: tn = qn ∗ ω + rn + ε with 0 ≤ rn < ω. Using the ω-periodicity of the family(U(t, s))t≥s, we get

yn = U(rn + ε, rn)u(qn ∗ ω + rn).

Since (rn)n is a bounded sequence in [0, ω], then there exists a subsequence (rnk)k which converges to r0 ∈ [0, ω], i.e

limk→+∞

rnk = r0.

Since the family of operator (U(t, s))t≥s is compact, then it is continuous with respect to the operator norm. Hence

limk→+∞

∥U(rnk + ε, rnk)− U(r0 + ε, r0)∥ = 0.

M. Zitane, C. Bensouda / J. Math. Anal. Appl. 402 (2013) 453–462 457

From the compactness property of the operator U(r0 + ε, r0) and the boundedness of the solution u, it follows that,U(r0 + ε, r0)u(qn ∗ ω + rn); n ∈ N is relatively compact in X, then, there exists a subsequence of (rnk)k which we denotesimilarly (rnk)k such that the sequence U(r0 + ε, r0)u(qnk ∗ ω + rnk)k converges to y∗ in X, i.e

limk→+∞

U(r0 + ε, r0)u(qnk ∗ ω + rnk) = y∗.

Then, one hasU(rnk + ε, rnk)u(qnk ∗ ω + rnk)− y∗ ≤ ∥U(rnk + ε, rnk)u(qnk ∗ ω + rnk)− U(r0 + ε, r0)u(qnk ∗ ω + rnk)∥

+U(r0 + ε, r0)u(qnk ∗ ω + rnk)− y∗

≤

U(rnk + ε, rnk)− U(r0 + ε, r0) u(qnk ∗ ω + rnk)

+

U(r0 + ε, r0)u(qnk ∗ ω + rnk)− y∗ .

Letting k → +∞, we have

limk→+∞

U(rnk + ε, rnk)u(qnk ∗ ω + rnk) = y∗.

So, we conclude that the set K := U(t, t − ε)u(t − ε); t > ε is relatively compact in X. Secondly, from the boundednessof F , there exists a positive constant c2 such that t

t−εU(t, s)F(s, us)ds

≤ c2ε.

Hence, u(t) : t ≥ 0 is relatively compact in X.To prove the uniform continuity of uwe let t > s ≥ 0, then

u(t)− u(s) = (U(t, 0)− U(s, 0))u(0)+

t

0U(t, σ )F(σ , uσ )dσ −

s

0U(s, σ )F(σ , uσ )ds

= (U(t, s)− I)U(s, 0)u(0)+ (U(t, s)− I) s

0U(s, σ )F(σ , uσ )dσ +

t

sU(t, σ )F(σ , uσ )dσ

= (U(t, s)− I)U(s, 0)u(0)+

s

0U(s, σ )F(σ , uσ )dσ

+

t

sU(t, σ )F(σ , uσ )dσ

= (U(t, s)− I)u(s)+

t

sU(t, σ )F(σ , uσ )dσ .

Using the continuity of (U(t, s) − I) and the compactness of u(t); t > 0 in X, one has (U(t, s) − I)u(s) is uniformlycontinuous in u(t); t > 0. So

limt→s+

∥(U(t, s)− I)u(s)∥ = 0. (3.3)

Moreover, we know that (U(t, s))t≥s and F(t; ut) are continuous in X, then

limt→s+

t

sU(t, σ )F(σ , uσ )dσ

= 0. (3.4)

From (3.3) and (3.4) we deduce that

limt→s+

∥u(t)− u(s)∥ = 0. (3.5)

Using a similar argument, one can also show that

limt→s−

∥u(t)− u(s)∥ = 0. (3.6)

From (3.5) and (3.6)we conclude that the solutionu is uniformly continuous inX.Moreover, ut : t ≥ 0 is an equicontinuousfamily in Cr , so the range of u is relatively compact in X. Hence, by Arzela–Ascoli theorem, we determine that ut : t ≥ 0 isrelatively compact in Cr .

Proof of Theorem 3.1. As usual, define the Poincaré map P(ϕ) = xω(·, ϕ, f ) on the phase space C0 := ϕ ∈ Cr;ϕ(0) ∈ D,where x(·, ϕ, f ) is the mild solution of Eq. (3.1) through ϕ.Because of the uniqueness property, it is enough to show that P has a fixed point to get an ω-periodic mild solution of Eq.(3.1). Also, the uniqueness property of the solution with respect to ϕ allows the Poincaré map P to be decomposed as

P(ϕ) = xω(·, ϕ, 0)+ xω(·, 0, f )

where xω(·, ϕ, 0) is the mild solution of Eq. (3.1) with f = 0, and xω(·, 0, f ) is the mild solution of Eq. (3.1) with ϕ = 0.

458 M. Zitane, C. Bensouda / J. Math. Anal. Appl. 402 (2013) 453–462

Let u be the bounded solution of Eq. (3.1) on R+ with u0 = ϕ. Then the uniqueness of the solution of Eq. (3.1) implies that

Pn(ϕ) = unω(·, ϕ, f ) for n ∈ N. (3.7)

By Lemma 3.3, the set

Pn(ϕ) : n ∈ N = unω(·, ϕ, f ); n ∈ N

is relatively compact in Cr , and by Lemma 3.2, we conclude that the mapping P has a fixed point in C0. Hence, the Eq. (3.1)has an ω-periodic mild solution.

4. Periodic solutions of the nonlinear equation

To get the aim of this section, we need to recall the following definition and theorem below.

Definition 4.1 ([36, Definition 9.3]). Let Γ : M → 2M be a multivalued map, where M is a subset of a Banach space and 2M

is the power set ofM .(i) ForΩ ⊂ M , the inverse Γ −1(Ω) is the set of all x ∈ M such that Γ (x) ∩Ω = ∅.(ii) The map Γ is called upper semi-continuous if Γ −1(Ω) is closed for all closed setΩ in M .

Theorem 4.2 ([36, Corollary 9.8]). Let Γ : M → 2M be a multivalued map, where M is a nonempty convex set in a Banach spaceY such that:(i) The set Γ (x) is nonempty, closed and convex for all x ∈ M.(ii) The set Γ (M) is relatively compact in Y.(iii) The map Γ is upper semi-continuous.Then the map Γ has a fixed point in the sense that there exists x ∈ M such that x ∈ Γ (x).

Let Bω be the space of all continuous ω-periodic functions from R into X, endowed with the uniform norm topology.We state now the second main result.

Theorem 4.3. Assume that (H1)–(H5) hold. Further, assume that there exists a positive constant ρ such that for any y ∈ Sρ =

v ∈ Bω : ∥v∥ ≤ ρ, the equation

ddt

u(t) = A(t)u(t)+ L(t, ut)+ G(t, yt); for t ≥ 0 (4.1)

has an ω-periodic mild solution in Sρ . Then, Eq. (1.1) has at least an ω-periodic mild solution.

Proof of Theorem 4.3. Define the set-valued mapping Γ : Sρ → 2Sρ , for y ∈ Sρ by

Γ (y) =

x ∈ Sρ : x(t) = U(t, 0)x(0)+

t

0U(t, σ ) [L(σ , xσ )+ G(σ , yσ )] dσ ; t ≥ 0

. (4.2)

We will show that the mapping Γ satisfies the conditions of Theorem 4.2.(i) By assumption, Γ (y) is nonempty for all y ∈ Sρ .Let y ∈ Sρ, x1, x2 ∈ Sρ , and λ ∈ [0; 1], then λx1 + (1 − λ)x2 ∈ Sρ which implies that Γ (y) is convex.The continuity of L and G implies that Γ (y) is a closed set.(ii) Using the Arzela–Ascoli theorem, we show that the set Γ (Sρ) is relatively compact in Bω .To this end, we consider the functions in Γ (Sρ) on an interval of length the period ω, say [0, ω].Let x ∈ Γ (Sρ), then there exists y ∈ Sρ such that

x(t) = U(t, 0)x(0)+

t

0U(t, σ )[L(σ , xσ )+ G(σ , yσ )]dσ .

Firstly, we show that the set x(t); x ∈ Γ (Sρ) is relatively compact in X.Let t > 0 and ε > 0 such that t > ε, then

x(t) = U(t, 0)x(0)+ U(t, t − ε)

t−ε

0U(t − ε, σ ) [L(σ , xσ )+ G(σ , yσ )] dσ

+

t

t−εU(t, σ ) [L(σ , xσ )+ G(σ , yσ )] dσ .

M. Zitane, C. Bensouda / J. Math. Anal. Appl. 402 (2013) 453–462 459

From the compactness of the operator U(t, 0) and the boundedness of L and G, we deduce that the setU(t, t − ε)

t−ε

0U(t − ε, σ ) [L(σ , xσ )+ G(σ , yσ )] dσ ; x ∈ Γ (Sρ)

is relatively compact in X. On the other hand, for some positive constant c we have t

t−εU(t, σ ) [L(σ , xσ )+ G(σ , yσ )] dσ

≤ cε.

Thus, the set x(t); x ∈ Γ (Sρ) is relatively compact in X for t ∈ [0, ω].For the equicontinuity of Γ (Sρ), one has, for t, τ ∈ [0, ω] such that t > τ :

∥x(t)− x(τ )∥ ≤

(U(t, τ )− I)U(τ , 0)x(0)+

τ

0U(τ , σ ) [L(σ , xσ )+ G(σ , yσ )] dσ

+

t

τ

U(t, σ ) [L(σ , xσ )+ G(σ , yσ )] dσ

≤ ∥(U(t, τ )− I)x(τ )∥ +

t

τ

U(t, σ ) [L(σ , xσ )+ G(σ , yσ )] dσ .

Since the family (U(t, s))t>s is compact, then it is continuous with respect to the operator norm. Hence

limt→τ+

(U(t, τ )− I) = 0.

From the assumption (H5) and the continuity of L and the boundedness of x(τ ) independently on y ∈ Sρ , there exists apositive constant c such that t

τ

U(t, σ ) [L(σ , xσ )+ G(σ , yσ )] dσ ≤ c(t − τ); uniformly for x, y in Sρ .

Then,

limt→τ+

supx∈Γ (Sρ )

∥x(t)− x(τ )∥ = 0.

Similarly, one can also prove that

limt→τ−

supx∈Γ (Sρ )

sup ∥x(t)− x(τ )∥ = 0.

Therefore, we deduce that Γ (Sρ) is relatively compact in Bω .(iii) To prove that Γ is upper semi-continuous, it is enough to show that Γ is closed.Let (yn)n≥0 and (zn)n≥0 be sequences, respectively, in Sρ and Γ (Sρ) such that for; n ≥ 0; zn ∈ Γ (yn)with

yn → y and zn → z, as n → +∞.

Then

zn(t) = U(t, 0)zn(0)+

t

0U(t, σ )

L(σ , znσ )+ G(σ , ynσ )

dσ ; t ≥ 0.

Letting n → +∞, by continuity argument, we obtain

z(t) = U(t, 0)z(0)+

t

0U(t, σ ) [L(σ , zσ )+ G(σ , yσ )] dσ ; t ≥ 0.

Hence, z ∈ Γ (y), which implies that Γ is closed.Now, let D be a closed set in Sρ and take a sequence (xn)n≥0 in Γ −1(D) such that xn → x, as n → +∞, then there existsyn ∈ D such that yn ∈ Γ (xn). Moreover, Γ (Sρ) is relatively compact, thus, there exists a subsequence (ynk)k of (yn)n suchthat ynk → y, as k → +∞. Therefore, Γ is closed and it follows that y ∈ Γ (x) and x ∈ Γ −1(y). Consequently, Γ is uppersemi-continuous.All assumptions of Theorem 4.2 are satisfied. Hence, there exists x ∈ Sρ such that x ∈ Γ (x), which implies the existence ofan ω-periodic mild solution of Eq. (1.1).

To prove that Eq. (4.1) has a periodic mild solution in Sρ , by Theorem 3.1, it suffices to show that it has a mild solutionwhich is bounded by ρ.

460 M. Zitane, C. Bensouda / J. Math. Anal. Appl. 402 (2013) 453–462

Corollary 4.4. Assume that (H1)–(H5) hold. If there exists a positive constant ρ such that for any y ∈ Sρ = v ∈ Bω : ∥v∥ ≤ ρ,the nonhomogeneous linear equation (4.1) has a mild solution that is bounded by ρ . Then, the Eq. (1.1) has anω-periodic solutionon R+.

Proof of Corollary 4.4. Let u be a bounded mild solution of Eq. (4.1) such that u0 = ϕ. Following the proof of [21, Theorem2.5], the Poincaré map P has a fixed point in coPnϕ : n ≥ 0, where co denotes the closure of the convex hull. Let ψ bethe fixed point of P and x(·, ψ,G(·, y·)) the associated mild solution of Eq. (4.1) through ψ . By virtue of the linearity of theoperators (U(t, s))t≥s≥0 and (L(t; .))t≥0 together with the continuous dependence on the initial data, we can see that if y isbounded by ρ then x(·, ψ, f ) is also bounded by ρ.

5. Application

In order to apply the abstract result of the previous section, we consider the following periodic reaction–diffusionequation with delay

∂

∂tu(t, x) = −a(t, x)

∂2

∂x2u(t, x)+ b1(t)u(t − r, x)+ b2(t)h(u(t − r, x))+ g(t, u(t, x)),

u(t, 0) = u(t, π) = 0, for t ∈ R+,u(θ, x) = φ(θ, x) for θ ∈ [−r, 0] and x ∈ [0, π],

(5.1)

where b1, b2 : R+→ R are continuous and ω-periodic, h : R → R is continuous and there exists a constant k > 0 such

that

|h(x)| ≤ k|x|, x ∈ R (5.2)

g : R+× R → R is continuous and ω-periodic in t , φ : [−r, 0] × [0, π] → R is continuous and a : R+

× [0, π] → Ris jointly continuous, x → a(t, x) is differentiable for all t ∈ R+, t → a(t, x) is ω-periodic and the following assumptionshold(H6) inft∈R,x∈[0,π ] a(t, x) = m0 > 0.(H7) there exists d > 0 and 0 < µ ≤ 1 such that |a(t, x)− a(s, x)| ≤ d|t − s|µ for all t, s ∈ R uniformly in x ∈ [0, π].

Let X = C ([0, π ] ; R) be the space of continuous functions from [0, π ] to R endowed with the uniform norm topology,and∆ the Laplacian operator on [0, π ] with domain

D(∆) = z ∈ C ([0, π ] ; R) : ∆z ∈ C ([0, π ] ; R) : z(0) = z(π) = 0 .

By Da Prato [16], the operator∆ satisfies the Hille–Yosida condition on X

(0,+∞) ⊂ ρ(∆) and |R (λ;∆)| ≤1λ

for λ > 0. (5.3)

Moreover,

D(∆) = z ∈ C ([0, π ] ; R) : z(0) = z(π) = 0 := C0 ([0, π ] ; R) .

Let∆0 be the part of∆ in D(∆), by Arendt et al. [7] the domain of∆0 is given by

D(∆0) = z ∈ C0 ([0, π ] ; R) : ∆z ∈ C0 ([0, π ] ; R)

and∆0 generates a strongly continuous compact semigroup (T0(t))t≥0 on D(∆), and

∥T0(t)∥ ≤ e−t for t ≥ 0. (5.4)

Let (A(t))t≥0 be the operator defined by A(t)u = a(t, x)∆uwith domain D(A(t)) = D(∆) := D, thus, D = C ([0, π ] ; R).By using (5.3), one has; for every λ > 0

R (λ, a(·, x)∆) =1

a(·, x)R

λ

a(·, x),∆

.

Under previous assumptions, it is clear that the part of operators A(·) in D are invertible and satisfy Acquistapace–Terreniconditions. Clearly, the system

u′(t) = A(t)u(t), t ≥ s, u(s) = ϕ ∈ X

has an associated evolution family of operators (U(t, s))t≥s which satisfies: there exist some constantsM, δ > 0 such that

∥U(t, s)∥B(X) ≤ Me−δ(t−s) (5.5)

for all s, t ∈ R with t ≥ s.Moreover, since A(t + ω) = A(t) for all t ∈ R, it follows that the family (U(t, s))t≥s is ω-periodic.

M. Zitane, C. Bensouda / J. Math. Anal. Appl. 402 (2013) 453–462 461

Let L,G : R × C([−r, 0];D) → D be defined, for t ∈ R+, ϕ ∈ C([−r, 0];D), and x ∈ [0, π], by

(L(t, ϕ))(x) = b1(t)ϕ(−r)(x) (5.6)(G(t, ϕ))(x) = b2(t)h(ϕ(−r)(x))+ g(t, x). (5.7)

The initial data ϕ ∈ C([−r, 0];D) is given by

ϕ(θ)(x) = φ(θ, x) for θ ∈ [−r, 0] and x ∈ [0, π]. (5.8)

Then, the system (5.1) takes the abstract formdut

dt= A(t)ut + L(t, ut)+ G(t, ut), t ≥ 0

u(t) = ϕ(t), −r ≤ t ≤ 0.(5.9)

Hence, the assumptions (H1)–(H5) are satisfied, and we have the following proposition.

Proposition 5.1. Assume that there exists d ∈ (max0; 1 − m0, 1) such that

|b1(t)| + k|b2(t)| ≤ 1 − d, for t ∈ [0, ω]. (5.10)

Then, (5.9) has an ω-periodic mild solution.

Proof. Letm = max|g(t, x)|; x ∈ [0, π], t ∈ [0, 1] and ρ =m+d

m0+d−1 , then

m − ρ(d − 1) = ρm0 − d. (5.11)

We claim that if y is a continuous ω-periodic function such that ∥y∥ ≤ ρ, then for all ϕ with ∥ϕ∥ ≤ ρ, the solution u ofdut

dt= A(t)ut + L(t, ut)+ G(t, yt) for t ≥ 0

u(t) = ϕ(t), −r ≤ t ≤ 0,(5.12)

satisfies ∥u(t)∥ ≤ ρ, for all t ≥ 0. Proceeding by contradiction, suppose that there exists t1 such that ∥u(t1)∥ > ρ and let

t0 = inft ≥ 0; ∥u(t)∥ > ρ.

If t0 < ∞, by continuity, we get ∥u(t0)∥ = ρ and there exists α > 0 such that ∥u(t)∥ > ρ, for t ∈ (t0, t0 + α). By using(2.5), one has

u(t0) = U(t0, 0)ϕ(0)+

t0

0U(t0, σ )[L(σ , uσ )+ G(σ , yσ )]dσ , t ≥ 0.

By (5.5)–(5.7), we get that

∥u(t0)∥ ≤ ρe−δt0 +1δ[(|b1| + k|b2|)ρ + m](1 − e−δt0)

and by (5.10), (5.11), we obtain

∥u(t0)∥ ≤ ρe−δt0 +1δ[(1 − d)ρ + m](1 − e−δt0)

≤ ρe−δt0 +1δ(ρδ − d)(1 − e−δt0)

≤ ρ −dδ(1 − e−δt0)

which gives that ∥u(t0)∥ < ρ. This contradicts the definition of t0. Consequently, ∥u(t)∥ ≤ ρ for all t ≥ 0, and byCorollary 4.4, the Eq. (5.9) has an ω-periodic mild solution in Sρ . Thus, the Eq. (5.1) has an ω-periodic mild solution.

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