Perron’s theorem for nondensely defined partial functional differential equations

Perron’s theorem for nondensely defined partial functional differential equations

Nadia Drisi, Brahim Es-sebbar

and Khalil Ezzinbi

Département de Mathématiques, Faculté des Sciences Semlalia Université Cadi Ayyad, B.P. 2390

Marrakesh, Morocco

Received 12 July 2017, appeared 21 November 2017 Communicated by László Simon

Abstract. The aim of this work is to establish a Perron type theorem for some non- densely defined partial functional differential equations with infinite delay. More specifically, we show that if the nonlinear delayed part is “small” (in a sense to be made precise below), then the asymptotic behavior of solutions can be described in terms of the dynamic of the unperturbed linear part of the equation.

Keywords: functional differential equations, asymptotic behavior, Perron’s theorem.

2010 Mathematics Subject Classification: 35B40, 35R10.

1 Introduction

The aim of this work is to study the asymptotic behavior of solutions for the following partial functional differential equation





 d

dtx(t) = Ax(t) +L(x_t) + f(t,x_t) fort≥0, x₀= φ∈ B,

(1.1) where A is a linear operator on a Banach space X satisfying the well-known Hille–Yosida condition, the domain is not necessarily dense, namely, we suppose that:

(H0) There exist M₀≥1 andω ∈_Rsuch that(ω,∞)⊂ρ(A)and

|R(λ,A)ⁿ| ≤ ^M0

(λ−ω)^{n forn}∈_Nandλ>ω, (1.2) whereρ(A)is the resolvent set ofAandR(λ,A) = (λI−A)⁻¹ forλ∈ ρ(A).

L is a bounded linear operator from B to X, where B is a normed linear space of functions mapping (−_{∞, 0}]into X satisfying the fundamental axioms introduced by Hale and Kato in [14]. For everyt∈R, the history function xt is defined by

xt(θ) =x(t+θ) forθ ≤0.

BCorresponding author. Email: essebbar@live.fr

(H1) The nonlinearity f :R⁺× B →Xis continuous such that

|f(t,φ)| ≤q(t)kφk_B for (t,φ)∈[_0,∞)× B, (1.3) whereq:[0,∞)→[0,∞)is a continuous function.

(H2) The functionqin (1.3) satisfies Z _t+1

t q(s)ds→_{0 when}t→_{∞. (1.4)}

The conditions (1.3) and (1.4) means that the nonlinearity f(t,xt) in equation (1.1) is “small”

as t → ∞. As a consequence, the solutions of equation (1.1) are expected to have similar asymptotic properties as the solutions of the following unperturbed equation





 d

dtx(t) =Ax(t) +L(_xt) _{for t}≥0, x₀= φ∈ B.

The so-called Perron’s theorem for the asymptotic behavior of solutions of differential equa- tions have been the subject of many studies, see [7,22–25]. For ordinary differential equations, we refer the reader to the books [8,9,11,16]. Let us recall Perron’s theorem for ordinary differential equations, which is presented in a form given by Coppel [9].

Theorem. [9]Consider the following ordinary differential equation





 d

dtx(t) =Bx(t) +g(t,x(t)) for t≥0 x(0) =x0∈_Cⁿ,

(1.5)

where B is an n×n constant complex matrix and g:[_0,∞)×_Cⁿ →_Cⁿis a continuous function such that

|g(t,z)| ≤γ(t)|z| for t≥0and z ∈_Cⁿ, whereγ:[0,∞)→[0,∞)is a continuous function satisfying

Z _t+1

t

γ(s)ds→0 as t→_∞.

If x(·)is a solution of equation(1.5), then either

x(t) =0 for all large t, or

tlim→∞

log|x(t)|

t =_Reλ₀, whereλ₀ is one of the eigenvalues of the matrix B.

Recently in [4,5], the authors showed that the asymptotic exponential behavior of the solutions of the linear nonautonomous differential equation

d

dtx(t) = B(t)x(t) fort ≥0, (1.6)

in Cⁿ, persists under sufficiently small nonlinear perturbations. More precisely, they showed that if all Lyapunov exponents of equation (1.6) are limits, then the same can be said about the solutions of the following perturbed nonlinear system

d

dtx(t) =B(t)x(t) + f(t,x(t)) _fort≥_0.

In [24], the author proved a Perron theorem for equation (1.1), when A = 0, the delay is finite and the space X is finite dimensional. In [21], the authors treated the case when X is infinite dimensional and the delay is infinite. They assumed thatAis densely defined inXand satisfies the Hille–Yosida condition(H0), which is equivalent, by the Hille–Yosida theorem, to Abeing the infinitesimal generator of a strongly continuous semigroup on X. They assumed that this semigroup is compact and they used the variation of constants formula established in [18] for some specific phase space.

In this work, we are interested in studying the asymptotic behavior of solutions of equation (1.1) when Ais not necessarily densely defined and for a general class of phase spacesB. We use the variation of constants formula established in [3]. Since the nonlinear part of equation (1.1) is assumed to be “small” in some sense ((1.3) and (1.4)), we describe the asymptotic be- havior of solutions in terms of the growth bound of the semigroup solution of the unperturbed linear system (Theorem5.1). This result is then refined in the form of a Perron type theorem where the asymptotic behavior of solutions is compared to the essential growth bound of the semigroup solution of the unperturbed linear system. Unlike in [21,24], we do not need to assume any kind of compactness (see Remark 5.4). A condition of compactness is, however, needed in order to describe the asymptotic behavior of solutions in terms of the parameters of the system (Corollary5.8).

This work is organized as follows. In Section2, we state the fundamental axioms onBthat will be used in this work, and we recall some spectral properties of the semigroup solution of equation (1.1) with f =0. In Section3, we recall the variation of constants formula established in [3] and we give the spectral decomposition of the phase space which plays an important role in the whole of this work. Section 4 is devoted to study the asymptotic behavior of so- lutions of equation (1.1) with respect to the invariant subspaces corresponding to the spectral decomposition. In Section5, we describe the asymptotic behavior of solutions in terms of the growth bound and the essential growth bound of the semigroup solution of the unperturbed linear system. An example of a Lotka–Volterra model with diffusion is given to illustrate our studies.

2 Phase space, integral solutions and spectral analysis

The choice of the phase space B plays an important role in the qualitative analysis of partial functional differential equations with infinite delay. In fact, the choice of B affects some properties of solutions. In this work, we employ an axiomatic definition of the phase space B which has been introduced at first by Hale and Kato [14]. We assume that (B,k·k_B) is a normed space of functions mapping (−_∞, 0] into a Banach space (X,|·|) and satisfying the following fundamental axioms.

(A) There exist a positive constantNand functionsK,Ke:[0,∞)→[0,∞), withKcontinuous and Ke locally bounded, such that if a function x : (−_∞,a] → X is continuous on [σ,a] with x_σ ∈ B_{, for some}σ <a, then for allt∈[σ,a]:

(i)x_t ∈ B;

(ii)t7→ x_tis continuous with respect tok.k_B on[σ,a]; (iii)N|x(t)| ≤ kx_tk_B ≤K(t−σ) sup

σ≤s≤t

|x(s)|+Ke(t−σ)kx_σk_B. (B) Bis a Banach space.

(C) If(φ_n)_n≥0 is a Cauchy sequence in B which converges compactly to a function φ, then φ∈ B_andkφ_n−φk_B →_{0 as}n→_∞.

As a consequence of Axiom(A), we deduce the following result.

Lemma 2.1([19]). Let C₀₀((−_∞, 0],X)be the space of continuous functions mapping(−_∞, 0]into X with compact supports. Then C₀₀((−_{∞, 0}],X)⊂ B. In addition, for a<0, we have

kφk_B ≤K(−a)sup

θ≤0

|φ(θ)|, for anyφ∈C00((−_{∞, 0}],X)with the support included in[a, 0].

In the sequel, we assume that the operator A satisfies the Hille–Yosida condition (H0).

Consider the following Cauchy problem





 d

dtx(t) = Ax(t) +L(x_t) + f(t) fort≥0 x0= φ∈ B.

(2.1) Definition 2.2([3]). Letφ∈ B. A functionx:R→Xis called an integral solution of equation (2.1) onRif the following conditions hold

(i)xis continuous on[_0,∞)_, (ii) x₀ =φ,

(ii) Z _t

0

x(s)ds∈D(A)_fort≥_0, (iv)x(t) =φ(0) +A

Z _t

0 x(s)ds+

Z _t

0

[L(xs) + f(s)]dsfort≥0.

If x is an integral solution of equation (2.1), then from the continuity of x, we have x(t) ∈ D(A), for all t ≥ 0. In particular,φ(0)∈ D(A). Let us introduce the part A₀of the operator Ain D(A)defined by

(D(A0):={x∈ D(A): Ax∈ D(A)}, A₀x:= Axforx∈ D(A₀).

Lemma 2.3 ([26]). Assume that (H0) holds. Then A₀ generates a strongly continuous semigroup (T₀(t))_t≥0on D(A).

For the existence of integral solutions, one has the following result:

Theorem 2.4([6,26]). Assume that(H0)holds. Then, for allφ∈ Bsuch thatφ(0)∈ D(A), equation (2.1)has a unique integral solution x onR. Moreover, x is given by







x(t) =T0(t)φ(0) + lim

λ→_∞ Z _t

0 T0(t−s)B_λ[L(xs) + f(s)]ds for t ≥0, x0=φ,

where B_λ :=λR(_λ,A)forλ> ω.

In the sequel of this work, for simplicity, integral solutions are called solutions. A solution of equation (2.1) is denoted byx(·,φ,L,f). The phase spaceB_A of equation (2.1) is given by

B_A:= {φ∈ B: φ(0)∈D(A)}.

For eacht ≥0,V(t)is the bounded linear operator defined onB_Aby V(t)φ= xt(·,φ,L, 0),

where x(·,φ,L, 0)is the solution of the homogeneous equation





 d

dtx(t) = Ax(t) +L(x_t) fort ≥0, x0= φ.

We have the following result:

Proposition 2.5([2, Proposition 2]). Assume that(H0)holds. Then(V(t))_t≥0is a strongly contin- uous semigroup onB_A. Moreover,(V(t))_t≥0 satisfies the following translation property

(V(t)φ) (θ) =

(V(t+θ)φ(0) for t+θ ≥0, φ(t+θ) for t+θ ≤0.

LetA_Vdenote the infinitesimal generator of the semigroup (V(t))_t≥0 onB_A.

For a bounded subsetBof a Banach spaceY, the Kuratowski measure of noncompactness α(B)is defined by

α(B):=inf{d>0 : there exist finitely many sets of diameter at mostdwhich cover B}. Moreover, for a bounded linear operator KonY, we defineα(K)by

α(K):=inf{k>0 : α(K(B))≤kα(B) for any bounded setBofY}.

Definition 2.6 ([12]). Let C be a densely defined operator on a Banach space Y. Let σ(C) denote the spectrum of the operator C. The essential spectrum ofC denoted byσess(C)is the set of λ∈ σ(C)such that one of the following conditions holds:

(i) Im(λI− C)is not closed,

(ii) the generalized eigenspace M_λ(C):=^S_k≥1Ker(λI− C)^k is of infinite dimension, (iii) λis a limit point ofσ(C)\ {λ}.

The essential radius ofC is defined by

r_ess(C) =sup{|λ|:λ∈σ_ess(C)}.

Let(R(t))_t≥0 be aC₀-semigroup on a Banach spaceY andA_R its infinitesimal generator.

Definition 2.7 ([12]). The growth boundω₀(R)of theC₀-semigroup (R(t))_t≥0 is defined by ω₀(R):=inf

(

ω∈_R: sup

t≥0

e^−ωt|R(t)|<_∞ )

.

Definition 2.8 ([19]). The essential growth bound ω_ess(R) of the C₀-semigroup (R(t))_t≥0 is defined by

ω_ess(R):= lim

t→_∞

logα(R(t))

t =inf

t>0

logα(R(t))

t . (2.2)

The relation betweenr_ess(R(t))andω_ess(R)is given by the following formula

r_ess(R(t)) =e^tωess(R). (2.3) From the spectral mapping inclusione^tσess(AR) ⊂σ_ess(R(t))and the formula (2.3), one can see that

σess(A_R)⊂ {λ∈ σ(A_R): Reλ≤ ωess(R)}.

This means that if λ ∈ σ(A_R) and Reλ > ω_ess(R), then λ does not belong to σ_ess(A_R). Thereforeλis an isolated eigenvalue ofA_R.

The spectral bounds(A_R)of the infinitesimal generatorA_R is defined by s(A_R):=sup{Reλ: λ∈ σ(A_R)}.

Recall the following formula

ω₀(R) =max{ω_ess(R),s(A_R)}.

3 Variation of constants formula and spectral decomposition of the phase space B

We introduce the following sequence of linear operators Θⁿ mapping X intoB_A defined for n>ω andy∈ Xby

(_Θⁿy) (θ) =

((nθ+1)Bny for −_n¹ ≤θ ≤0, 0 forθ< −¹_n,

where Bn is the bounded operator defined for sufficiently large n by Bn := nR(n,A). For y ∈ X, the function Θⁿy belongs toC₀₀((−_{∞, 0}],X)with the support included in[−1, 0]. By Lemma2.1, we deduce thatΘⁿy∈ B_and

k_Θⁿyk_B ≤ NKe (1)|y|, (3.1) whereNe :=sup

n>ω

|B_n|. In addition we have for eachy∈ X

(_Θⁿy) (0) =Bny=nR(n,A)y∈ D(A). It follows thatΘⁿy∈ B_A.

Now we give the variation of constants formula for equation (2.1) established in [3].

Theorem 3.1([3]). Assume that(H0)holds. Then, for allφ∈ B_A, the solution x(·,φ,L,f)of equation (2.1)satisfies the following variation of constants formula

xt(·,φ,L,f) =V(t)φ+ lim

n→_∞ Z _t

0 V(t−s)_Θⁿf(s)ds for t ≥0. (3.2)

The spectral decomposition of the phase space provides a powerful tool to analyze the asymptotic behavior of solutions. We know that each λ ∈ σ(A_V) with Reλ > ω_ess(V)is an isolated eigenvalue of the operator A_V. Let ρ>ω_ess(V)be such that

σ(A_V)∩(_iR+ρ) =_∅.

Consider the set

Σρ:={λ∈σ(A_V): Reλ≥ρ}. (3.3) From [12, Corollary 2.11, Chapter IV and Theorem 3.1, Chapter V], the setΣ_ρ is finite and we have the following decomposition of the phase spaceB_A

B_A=Uρ⊕Sρ, (3.4)

where Uρ andSρ are closed subspaces of B_A which are invariant under(V(t))_t≥0. The sub- space U_ρ is finite-dimensional. For every sufficiently small ε > 0, there exists C_ε > 0 such that

(kV(t)φk_B ≤C_εe^(ρ−ε)tkφk_B fort≥0 and φ∈ S_ρ

kV(t)φk_B ≤Cεe^(ρ+ε)tkφk_B fort≤0 and φ∈Uρ. (3.5) In what follows,V^Uρ(t)andV^Sρ(t)denote the restrictions ofV(t)onU_ρandS_ρ respectively.

Π^Uρ and Π^Sρ denote the projections on Uρ and Sρ respectively. V^Uρ(t)_t∈R is a group of operators. Letε_ρ>0 be such that σ(A_V)∩λ∈_C:ρ−ε_ρ≤Reλ≤ρ+ε_ρ =_{∅. Put}

ρ₁ :=ρ−ε_ρ and ρ₂ :=ρ+ε_ρ. (3.6) We deduce from (3.5) that there exists a constantC_ρ >0 such that for eacht ≥0

V^Sρ(t)≤C_ρe^ρ1t and

V^Uρ (−t)≤C_ρe^−ρ2t. We introduce the new norm defined onB_A by

|φ|_B :=sup

t≥0

e^−ρ1t

V^Sρ(t)_Π^Sρφ

_B+sup

t≥0

e^ρ2t

V^Uρ(−t)_Π^Uρφ _B.

Lemma 3.2 ([10,21,24]). The two norms k·k_B and|·|_B are equivalent, namely, for all φ ∈ B_A, we have

kφk_B ≤ |φ|_B ≤C₂kφk_B, (3.7) where C₂ := C_ρ

Π^Sρ

+Π^Uρ

. In addition, for allφ∈ B_A

|φ|_B =_Π^Sρφ

B+_Π^Uρφ

B. (3.8)

The corresponding operator norms

V^Sρ(t)and

V^Uρ(−t)satisfy

V^Sρ(t)

≤e^ρ1t and

V^Uρ(−t)

≤e^−ρ2t for t≥0. (3.9)

4 Asymptotic behavior of the solutions in the invariant subspaces

In this section, we give the lemmas that will be used for the proof of our main results. First, we give sufficient conditions which insure the existence of global solutions for equation (1.1).

Theorem 4.1. Assume that(H0)and(H1)hold. Let φ∈ B_A. If the nonlinearity f :R⁺× B → X is locally Lipschitz with respect to the second argument, then equation (1.1) has a unique solution x which is defined onR.

Proof. Since the nonlinearity f is locally Lipschitz with respect to the second argument, then using the same argument as in [13, Theorem 3.4], one can prove that there exists a maximal in- terval of existence(−_∞,bφ)and a unique solutionx(·,φ)of equation (1.1) defined on(−_∞,bφ) and either b_φ = _∞ or lim sup_t→b⁻

φ

|x(t,φ)| = _∞. Now using the fact that the nonlinearity f satisfies (1.3), we deduce using the same approach as in [13, Corollary 3.5] thatb_φ =_∞.

Theorem 4.2. Assume that (H0)and (H1)hold. Let φ ∈ B_A. If the C₀−semigroup (T₀(t))_t≥0 is compact, then equation(1.1)has at least a solution x which is defined onR.

Proof. As in [1, Theorem 17], the proof uses the Schauder fixed point theorem to prove the existence of at least a solution defined on a maximal interval of existence(−_∞,bφ). It follows again as in Theorem4.1thatb_φ =_∞.

Remark 4.3. Unlike in Theorem4.1, the global solution provided by Theorem4.2 is not nec- essarily unique.

For the rest of this work, we assume the global existence of a solution xof equation (1.1).

The following lemma is essential for the rest of the paper.

Lemma 4.4. Suppose that(H0)–(H2)hold. Let x be a solution of equation(1.1). Then for anyε> 0, there exists a constant C(ε)≥1such that

kx_tk_B ≤C(ε)e^{(ω0(V)+ε)(t−σ)}exp

NKe (1)C(ε)

Z _t

σ

q(s)ds

kxσk_B for0≤ σ≤t. (4.1) In particular, there exists a constant C₁≥0such that for m∈_Nand m≤t≤ m+1, we have

1

C₁kx_m+1k_B ≤ kx_tk_B ≤C₁kx_mk_B. (4.2) Proof. Using the variation of constants formula (3.2), we have for 0≤σ≤t

x_t =V(t−σ)x_σ+ lim

n→∞

Z _t

σ

V(t−s)_Θⁿf(s,x_s)ds. (4.3) Letε>0. Then, there existsC(ε)≥1 such that

kV(t)k ≤C(ε)e^(ω0(V)+ε)t fort≥_{0. (4.4)} It follows from (3.1), (4.3) and (4.4) that

kx_tk_B ≤ kV(t−σ)k kx_σk_B+ _lim

n→_∞ Z _t

σ

kV(t−s)k k_Θⁿf(s,x_s)k_Bds

≤ C(ε)e^{(ω0(V)+ε)(t−σ)}kxσk_B+NKe (1)C(ε)

Z _t

σ

e^{(ω0(V)+ε)(t−s)}|f(s,x_s)|ds

≤ C(ε)e^{(ω0(V)+ε)(t−σ)}kx_σk_B+NKe (1)C(ε)

Z _t

σ

e^{(ω0(V)+ε)(t−s)}q(s)kx_sk_Bds.

It follows that

e^{−(ω0(V)+ε)t}kxtk_B ≤C(ε)e^{−(ω0(V)+ε)σ}kxσk_B+NKe (1)C(ε)

Z _t

σ

e^{−(ω0(V)+ε)s}kxsk_Bq(s)ds.

Gronwall’s lemma implies that for 0≤σ≤ t

e^{−(ω0(V)+ε)t}kx_tk_B ≤C(ε)e^{−(ω0(V)+ε)σ}kx_σk_Bexp

NKe (1)C(ε)

Z _t

σ

q(s)ds

.

Therefore we get the inequality (4.1). Now letm∈_Nandm≤t≤ m+1. By takingε=1 and σ=min (4.1), we get

kx_tk_B ≤C(1)e^{(ω0(V)+1)(t−m)}kx_mk_Bexp

NKe (1)C(1)

Z _t

mq(s)ds

≤C₁kxmk_B. where C₁ := C(1)max

1,e^(ω0(V)+1) e^{NKe (1)C(1)Q} andQ:=sup_m≥0Rm+1

m q(s)dswhich is finite by (1.4). Similarly, we get

kx_m+1k_B ≤C₁kx_tk_B.

Remark 4.5. By (4.2) and (3.7), we can see that form∈_Nandm≤ t≤m+₁ 1

C₃|x_m+1|_B ≤ |x_t|_B ≤ C₃|x_m|_B, (4.5) whereC₃ :=C₁C₂.

For the rest of this section, we suppose that(H0)–(H2) hold and we fix a real number ρ such thatρ> ωess(V)andσ(A_V)∩(_iR+ρ) =∅. Recall thatUρ andSρ are the subspaces in the decomposition of the phase spaceB_A given by (3.4). Let xbe a solution of equation (1.1).

Define for m∈_N

x^U(m):= Π^Uρx_m

B, x^S(m):= Π^Sρx_m

B (4.6)

and

qe(m):= NKe (1)C₁C²₂max{1,e^ρ1,e^ρ2}

Z _m+1

m q(s)ds, (4.7)

whereρ₁andρ₂ are the real numbers defined by (3.6).

Lemma 4.6. The following estimations hold

x^S(m+1)≤ e^ρ1x^S(m) +q_e(m)x^S(m) +x^U(m), (4.8) and

x^U(m+1)≥ e^ρ2x^U(m)−q_e(m)x^S(m) +x^U(m). (4.9) Proof. Using the variation of constants formula (3.2), we obtain for eachm∈_N

x_m+1=V(1)xm+ lim

n→_∞ Z _m+1

m V(m+1−s)_Θⁿf(s,xs)ds. (4.10)

By projecting the formula (4.10) onto the subspace S_ρ and using (3.9), (3.7), (3.1), (1.3) and (4.2), we have

Π^Sρxm+₁

_B ≤V^Sρ(1)_Π^Sρxm

_B+ lim

n→_∞ Z _m+1

m

V^Sρ(m+1−s)_Π^Sρ_Θⁿf(s,xs) _Bds

≤e^ρ1 Π^Sρxm

_B+ lim

n→_∞ Z _m+1

m e^ρ1(m+1−s)

Π^SρΘⁿf(s,xs) _Bds

≤e^ρ1 Π^Sρx_m

B+ lim

n→_∞ Z _m+1

m e^ρ1(m+1−s)C₂ Π^Sρ

k_Θⁿf(s,x_s)k_Bds

≤e^ρ1 Π^Sρx_m

B+NKe (1)C²₂max{1,e^ρ1}

Z _m+1 m

|f(s,x_s)|ds

≤e^ρ1 Π^Sρx_m

B+NKe (1)C²₂max{1,e^ρ1}

Z _m+₁

m q(s)kx_sk_Bds

≤e^ρ1

Π^Sρx_m

B+NKe (1)C²₂max{1,e^ρ1}C₁kx_mk_B

Z _m+1

m q(s)ds

≤e^ρ1

Π^Sρx_m

B+NKe (1)C₁C₂²max{1,e^ρ1}

Z _m+1

m

q(s)ds|x_m|_B. Using (3.8) and the above inequality, we conclude that (4.8) holds.

Now from (3.9), we have forφ∈Uρ

V^Uρ(1)φ

_B ≥e^ρ2|φ|_B.

By projecting the formula (4.10) onto the subspace Uρ using (3.9), (3.7), (3.1), (1.3), (4.2) and (3.8), we deduce that

Π^Uρx_m+1

B =

V^Uρ(1)

Π^Uρx_m+ lim

n→_∞ Z _m+1

m V^Uρ(m−s)_Π^Uρ_Θⁿf(s,x_s)ds

B

≥e^ρ2

Π^Uρxm+ lim

n→_∞ Z _m+1

m V^Uρ(m−s)_Π^Uρ_Θⁿf(s,xs)ds B

≥e^ρ2 Π^Uρx_m

B−e^ρ2 lim

n→_∞ Z _m+1

m

V^Uρ(m−s)_Π^Uρ_Θⁿf(s,x_s) Bds

≥e^ρ2x^U(m)−e^ρ2 lim

n→_∞ Z _m+1

m e^ρ2(m−s)

Π^UρΘⁿf(s,x_s) Bds

≥e^ρ2x^U(m)−e^ρ2 lim

n→_∞ Z _m+1

m e^ρ2(m−s)C₂ Π^Uρ

k_Θⁿf(s,x_s)k_Bds

≥e^ρ2x^U(m)−NKe (1)e^ρ2max 1,e^−ρ2

Z _m+1

m C₂²|f(s,x_s)|ds

≥e^ρ2x^U(m)−NKe (1)e^ρ2C²₂max 1,e^−ρ2

Z _m+1

m

q(s)C₁kx_mk_Bds

≥e^ρ2x^U(m)−NKe (1)C₁C₂²max{e^ρ2, 1}

Z _m+1

m q(s)ds

x^U(m) +x^S(m). Therefore, we get the estimation (4.9).

In what follows, we assume that the solution x is nontrivial, that is, kxtk_B > 0 for t ≥ 0.

We have the following lemma.

Lemma 4.7. Either

mlim→_∞

x^U(m)

x^S(m) =0 (4.11)

or

mlim→_∞

x^S(m)

x^U(m) =0. (4.12)

Proof. The proof follows the same approach as in [21,24]. From (3.7), one can see that|x_t|_B >0 fort ≥0. Suppose that (4.11) fails, then there existsε0>0 such that

x^U(m) x^S(m) ≥ ε₀,

for infinitely many m. Next we will show that (4.12) must hold. From (1.4) and (4.7) we can see that

mlim→_∞qe(m) =0. (4.13)

By (4.13), there existsm₁ ≥0 such that form≥m₁ e^ρ2−¹+ε₀

ε₀ eq(m)>₀

and e^ρ1 + (1+ε₀)_eq(m)

ε₀e^ρ2 −(1+ε₀)_eq(m) < ¹

ε₀. (4.14)

Since (4.11) fails then there existsm₂≥ m₁such that x^U(m₂)≥ε₀x^S(m₂). Next we will show that for allm≥m₂

x^U(m)≥ε₀x^S(m). (4.15)

Suppose by induction that this inequality holds for some m≥ m2. Then it follows from (4.8) that

x^S(m+1)≤ e^ρ1x^U(m)

ε₀ +_eq(m)^x

U(m)

ε₀ +q_e(m)x^U(m)

= e^ρ1

ε₀ +^qe(m)

ε₀ +q_e(m)

x^U(m). Now from (4.9) we have

x^U(m+1)≥e^ρ2x^U(m)−q_e(m)^x

U(m)

ε₀ −_eq(m)x^U(m)

=

e^ρ2− ^qe(m) ε0

−q_e(m)

x^U(m)_{. (4.16)}

It follows that

x^S(m+1)≤ e^ρ1

ε₀ + ^qe(m)

ε₀ +_eq(m)

x^U(m)

≤ e^ρ1

ε₀ + ^qe(m)

ε₀ +_eq(m)

1

e^ρ2−_eq(m)−^qe(m)

ε0

x^U(m+1)

= ^e

ρ1+q_e(m) +ε0qe(m) ε₀e^ρ2−ε₀qe(m)−q_e(m)^x

U(m+1).

Now from (4.14), we deduce that

x^U(m+₁)≥ε₀x^S(m+₁).

Thus by induction, the inequality (4.15) holds for allm≥m₂. From (4.8) and (4.16), we deduce that form≥m₂

x^S(m+1) x^U(m+1) ≤ ^e

ρ1x^S(m) +_eq(m) x^S(m) +x^U(m)

e^ρ2−q_e(m)− ^eq(_ε^m)

0

x^U(m)

= ^e

ρ1+_eq(m)

e^ρ2−q_e(m)− ^eq(_ε^m)

0

x^S(m)

x^U(m)+ ^eq(m)

e^ρ2−_eq(m)−^qe(_ε^m)

0

. It follows by (4.13) that

lim sup

m→_∞

x^S(m) x^U(m) ≤ ^e

ρ₁

e^ρ2 lim sup

m→_∞

x^S(m) x^U(m)^. That is

1−e^ρ1−ρ2

lim sup

m→_∞

x^S(m) x^U(m) ≤_0.

But sinceρ₁ <ρ₂and lim sup_{m→∞ x}^xU^S((^mm⁾) ≥0, we deduce that lim sup_{m→∞ x}^xU^S((^mm⁾) =0. Therefore

mlim→_∞

x^S(m) x^U(m) =0.

This ends the proof of Lemma4.7.

The proof of Theorem5.3 is based on the following principal lemma.

Lemma 4.8. Either

lim sup

t→_∞

logkx_tk_B

t <ρ (4.17)

or

lim inf

t→_∞

logkx_tk_B

t >ρ. (4.18)

Proof. By Lemma4.7, we have to discuss two cases:

Case 1. Assume that (4.11) holds. Then we havex^U(m) < x^S(m)for all large integersm, where x^U(m)and x^S(m) are given by (4.6). Letε be a positive real number. Then by (4.13), there exists a large positive integerm_ε such that form≥m_ε,

qe(m)<ε and x^U(m)<_x^S(m). (4.19) Using (4.8) and (4.19) we have x^S(m+1)≤(e^ρ1+2ε)x^S(m)form≥mε. It follows that

x^S(m)≤ (e^ρ1 +2ε)^m−mε x^S(m_ε) =K_ε(e^ρ1+2ε)^m,

where K_ε := (e^ρ1+_2ε)^−mεx^S(m_ε) > _{0. For} t ≥ m_ε, we have [t] ≥ m_ε, where [·] is the floor function. Since[t]≤t ≤[t] +1, it follows from (3.7), (3.8), (4.5) and (4.19) that

kx_tk_B ≤ |x_t|_B ≤C₃ x_[t]

B ≤2C₃x^S([t])≤2C₃K_ε(e^ρ1+_2ε)^[t].

Hence,

logkx_tk_B

t ≤ ^log(2C3Kε) t + [t]

t log(e^ρ1+2ε). By takingt →_∞we get

lim sup

t→_∞

logkx_tk_B

t ≤log(e^ρ1+_2ε). Now by takingε→0, we obtain that

lim sup

t→_∞

logkx_tk_B

t ≤log(e^ρ1) =ρ₁< ρ, that is, (4.17) holds.

Case 2.Suppose that (4.12) holds. Note that x^S(m)<x^U(m)for all large integersm. Letε such that 0<ε< ^eρ₂². By (4.13), there exists a large positive integer mε such that form≥mε,

eq(m)<ε and x^S(m)< x^U(m). (4.20) Using (4.9) and (4.20) we havex^U(m+1)≥(e^ρ2 −2ε)x^U(m)form≥mε, which implies that

x^U(m)≥(e^ρ2−2ε)^m−mεx^U(mε) =Kε(e^ρ2 −2ε)^m,

whereK_ε := (e^ρ2−2ε)^−mεx^U(m_ε)>0. Fort≥ m_ε, we have[t] +1≥m_ε. Since[t]≤t ≤[t] +1, it follows from (3.7), (4.5) that

kx_tk_B ≥ |x_t|_B C2

≥ x_[t]+1

_B C2C3

≥ ^x

U([t] +1) C2C3

≥ ^Kε(e^ρ2 −2ε)^[t]+1 C2C3

. Hence,

logkx_tk_B

t ≥ ^log

Kε

C2C3

t +[t] +1

t log(e^ρ2−_2ε). By takingt →_∞we get

lim inf

t→_∞

logkx_tk_B

t ≥log(e^ρ2 −2ε). Now by takingε→0, we obtain that

lim inf

t→_∞

logkxtk_B

t ≥log(e^ρ2) =ρ₂>ρ, that is, (4.18) holds. This completes the proof.

5 Perron’s theorem for equation (1.1)

Let xbe a solution of equation (1.1). If there existst0 ≥0 such thatkxt₀k_B =0, then it follows from Lemma 4.4 that x_t = 0 for all t ≥ t₀. Thus, in what follows, we will only consider the case wherekx_tk_B >0 for all t≥0.

The following result describes the asymptotic behavior of solutions of equation (1.1) in terms of the growth bound of the semigroup solution (V(t))_t≥0 of the unperturbed linear equation, which is natural given the smallness of the nonlinear term.

Theorem 5.1. Suppose that(H0)–(H2)hold. Let x be a solution of equation(1.1)such thatkx_tk_B >0 for t≥0. Then we have

lim sup

t→_∞

logkxtk_B

t ≤ω₀(V). Proof. Letε>0, from Lemma4.4, we deduce that fort ≥0

logkxtk_B

t ≤ ^log(C₀(ε)kx₀k_B)

t +ω₀(V) +ε+NKe (1)C₀(ε) Rt

0 q(s)ds

t . (5.1)

Using (1.4) we can see that

R_t

0q(s)ds

t →_{0 as}t →∞. By takingt→_∞in (5.1), we obtain lim sup

t→_∞

logkx_tk_B

t ≤ω₀(V) +ε. (5.2)

Now by lettingε→0 in (5.2) we obtain the desired estimation.

Corollary 5.2. Ifω₀(V)<0, then the equilibrium point0is globally asymptotically stable.

Now we give the Perron’s theorem for equation (1.1) which constitutes a refinement of Theorem5.1.

Theorem 5.3. Suppose that(H0)–(H2)hold. Let x be a solution of equation(1.1)such thatkx_tk_B >₀ for t≥0. Then either

lim sup

t→_∞

logkx_tk_B

t ≤ω_ess(V), (5.3)

or

tlim→_∞

logkx_tk_B

t =Reλ₀> ω_ess(V), (5.4)

whereλ₀ is an eigenvalue of the operatorA_V.

Proof. We will show that if (5.3) fails, then (5.4) must hold. Suppose that lim sup

t→_∞

logkx_tk_B

t >ω_ess(V). It follows from Theorem5.1that

ω₀(V)> ωess(V). Therefore

ω0(V) =max{s(A_V),ωess(V)}=s(A_V).

It follows that Λ := {λ∈σ(A_V): Reλ> ωess(V)} 6= ∅. We claim that there exist λ0 ∈ _Λ such that

lim sup

t→_∞

logkx_tk_B

t =Reλ₀.

In fact, if lim sup_t→∞^logk_t^xtkB =ρ ∈ {/ Reλ:λ∈_Λ}, withρ>ω_ess(V), then the condition (4.17) in Lemma4.8 fails. Hence we must have

lim inf

t→_∞

logkx_tk_B t >ρ.

However this implies that

ρ=lim sup

t→_∞

logkx_tk_B

t ≥lim inf

t→_∞

logkx_tk_B t >ρ which is a contradiction. Therefore, there existλ₀∈_Λsuch that

lim sup

t→_∞

logkx_tk_B

t =Reλ₀.

Since Reλ₀ > ω_ess(V), then there existsρ₀ ∈ {/ Reλ:λ∈_Λ}such that Reλ₀ >ρ₀ >ω_ess(V). That is

lim sup

t→_∞

logkxtk_B

t =Reλ₀> ρ₀. (5.5)

By applying Lemma4.8toρ₀using (5.5), we obtain lim inf

t→_∞

logkxtk_B

t > ρ₀> ω_ess(V). We claim that

lim sup

t→_∞

logkx_tk_B

t =lim inf

t→_∞

logkx_tk_B

t .

In fact if lim sup_t→∞ ^logk_t^xtkB >lim inf_t→_∞ ^logkxtk_B

t , then there exists ρ₁ ∈ {/ Reλ:λ∈ _Λ}with ρ₁> ω_ess(V)such that

lim sup

t→_∞

logkxtk_B

t > ρ₁ (5.6)

and

lim inf

t→_∞

logkx_tk_B

t < ρ₁. (5.7)

By applying Lemma4.8toρ₁using (5.6) , we obtain lim inf

t→_∞

logkxtk_B t >ρ₁, which contradicts (5.7). Therefore, we have

tlim→_∞

logkx_tk_B

t =_{lim sup}

t→_∞

logkx_tk_B

t =_{lim inf}

t→_∞

logkx_tk_B

t =_Reλ₀. That is (5.4) holds.

Remark 5.4. Unlike in [24], where the author assumed that dim(X) < _∞ and in [21] where the authors assumed that the semigroup generated by A is compact, we do not assume any kind of compactness in Theorem5.3. Thus, the asymptotic behavior of solutions is described in terms of the essential growth bound of the semigroup solution of the unperturbed linear system. That is why the approach we adopted in the proof of Theorem 5.3 differs from the one in [21,24].