global solutions for abstract neutral differential equations

Nonlinear Analysis 72 (2010) 2210–2218

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Nonlinear Analysis

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Global solutions for abstract neutral differential equationsEduardo HernándezDepartamento de Matemática, I.C.M.C. Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos SP, Brazil

a r t i c l e i n f o

Article history:Received 31 August 2009Accepted 19 October 2009

MSC:primary 35R1034K40secondary 34K30

Keywords:Neutral equationsMild solutionsGlobal solutionsSemigroup of linear operators

a b s t r a c t

We study the existence of global solutions for a class of abstract neutral differentialequation defined on the whole real axis. Some concrete applications related to ordinaryand partial differential equations are considered.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In this article, we study the existence of mild solutions for a class of abstract neutral differential equations of the formddt[x(t)+ g(t, xt)] = Ax(t)+ f (t, xt), t ∈ R, (1.1)

where A is the infinitesimal generator of an hyperbolic C0-semigroup of bounded linear operators (T (t))t≥0 on a Banachspace (X, ‖ · ‖), the history xt : (−∞, 0] → X defined by xt(θ) = u(t+ θ), belongs to some abstract phase spaceB definedaxiomatically and g, f : R×B → X are continuous functions.The literature on related ordinary neutral differential equations is very extensive and we refer the reader to the book

by Hale & Lunel [1] for details. Concerning partial neutral functional– differential equations on bounded intervals, wecite Hale [2], Wu [3–5], Adimy [6] and Hernández et al. [7] for finite delay equations, and Hernández et al. [8,9] andHernández [10] for the case of equations with unbounded delay. To the best of our knowledge, the problem of the existenceof solutions for abstract neutral systems defined on the whole real axis is an untreated topic in the literature, and it is themain motivation of this article.Partial neutral differential equations similar to (1.1) arise, for instance, in the theory of heat conduction in fadingmemory

materials. In the classic theory of heat conduction, it is assumed that the internal energy and the heat flux depends linearlyon the temperature u(·) and on its gradient∇u(·). Under these conditions, the classical heat equation describes sufficientlywell the evolution of the temperature in different types ofmaterials. However, this description is not satisfactory inmaterialswith fading memory. In the theory developed in [11,12], the internal energy and the heat flux are described as functionalsof u(·) and ux(·). The following equation, see [13–16], has been frequently used to describe this phenomena,

ddt

[u(t, x)+

∫ t

−∞

k1(t − s)u(s, x)ds]= c1u(t, x)+

∫ t

−∞

k2(t − s)1u(s, x)ds,

u(t, x) = 0, x ∈ ∂Ω.

E-mail address: [email protected].

0362-546X/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2009.10.020

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E. Hernández / Nonlinear Analysis 72 (2010) 2210–2218 2211

In this equation,Ω ⊂ Rn is open, bounded and has smooth boundary, (t, x) ∈ R×Ω , u(t, x) represents the temperature in xat time t , c is a physical constant and ki : R→ R, i = 1, 2, are the internal energy and the heat flux relaxation, respectively.If we assume k2 ≡ 0, then we can describe the above equation in the abstract form (1.1). In the last section, we considertwo additional applications.We now include some definitions, properties and technicalities used in this work. In this article, (X, ‖ · ‖) is a Banach

space and A is the infinitesimal generator of an analytic semigroup of bounded linear operators, (T (t))t≥0, on (X, ‖ · ‖) suchthat σ(A) ∩ iR = ∅. In this case, the sets σ−(A) = λ ∈ σ(A) : Re(λ) < 0 and σ+(A) = λ ∈ σ(A) : Re(λ) > 0 are closed,disjoint and there exists δ > 0 such that

supRe(λ) : λ ∈ σ−(A) < −δ < 0 < δ < infRe(λ) : λ ∈ σ+(A).

LetΩ ⊂ R2 be an open bounded set with smooth boundary ∂Ω , such that σ+(A) ⊂ Ω ⊆ C+ = λ ∈ C : Re(λ) > 0 andP : X → X be the operator defined by

Px =12π i

∫∂Ω

R(µ; A) xdµ, x ∈ X,

with ∂Ω oriented counterclockwise. In the next result, X1 = P(X), X2 = (I − P)(X) and A1 : X1 → X , A2 : D(A2) =x ∈ D(A) : x ∈ X2, Ax ∈ X2 → X2 are the operators defined by A1x = Ax for x ∈ X1 and A2y = Ay for y ∈ D(A2). TheProposition 1.1 below, follows from the results in [17].

Proposition 1.1. The following properties are valid.

(a) The operator P is a projection, P(X) ⊂ D(An) for all n ∈ N, T (t)Px = PT (t)x for all x ∈ X and T (t)Xi ⊂ Xi for i = 1, 2, andevery t ≥ 0.

(b) A1(X1) ⊂ X1, σ(A1) = σ+(A) and R(λ : A1) = R(λ : A)|X1 for all λ ∈ ρ(A1). Moreover, A1 is the generator of a C0-group(TA1(t))t≥0 on X1 and TA1(t) = T (t)|X1 for every t ≥ 0.

(c) σ(A2) = σ−(A), R(λ : A2) = R(λ : A)|X2 for all λ ∈ ρ(A2), A2 is the generator of an uniformly stable C0-semigroup(TA2(t))t≥0 on X2, TA2(t) = T (t)|X2 for every t ≥ 0 and T (t) = TA1(t)+ TA2(t) for each t ≥ 0.

(d) There are constants di,Mi, i ∈ N, such that ‖AiT (t)(I − P)‖ ≤ Mie−γ t t−i and ‖AiT (−t)P‖ ≤ die−γ t for all t ≥ 0 and everyi ∈ N.

From Lunardi [17] we also remark the following result. In this result, [D(A)] denote the domain of A endowed with thegraph norm.

Proposition 1.2. Let f ∈ L∞(R; X). If u ∈ C1(R; X) ∩ C(R; [D(A)]) is a bounded solution of

x′(t) = Ax(t)+ f (t), t ∈ R,

then

u(t) =∫ t

−∞

T (t − τ)(I − P)f (τ )dτ −∫∞

tT (t − τ)Pf (τ )dτ , ∀ t ∈ R.

In thiswork,we employ an axiomatic definition for the phase spaceB, which is similar to the oneused in [18]. Specifically,B is a vector space of functions mapping (−∞, 0] into X endowed with a seminorm ‖ · ‖B , such that the next axioms hold.

(A) If x : (−∞, σ + a) 7→ X , a > 0, σ ∈ R, is continuous on [σ , σ + a) and xσ ∈ B, then for every t ∈ [σ , σ + a) thefollowing hold:(i) xt is inB;(ii) ‖x(t)‖ ≤ H‖xt‖B ;(iii) ‖xt‖B ≤ K(t − σ) sup‖x(s)‖ : σ ≤ s ≤ t +M(t − σ)‖xσ‖B,where H > 0 is a constant; K ,M : [0,∞) 7→ [1,∞), K is continuous, M is locally bounded and H, K ,M areindependent of x(·).

(A1) For the function x(·) in (A), the function t → xt is continuous from [σ , σ + a) intoB.(B) The spaceB is complete.(C2) If (ϕn)n∈N is a uniformly bounded sequence in C((−∞, 0], X) formed by functions with compact support and ϕn → ϕ

in the compact-open topology, then ϕ ∈ B and ‖ϕn − ϕ‖B → 0 as n→∞.

Example 1.1. The phase space Cr × Lp(η, X).

2212 E. Hernández / Nonlinear Analysis 72 (2010) 2210–2218

Let r ≥ 0, 1 ≤ p <∞ and let η : (−∞,−r] 7→ R be a non-negative measurable function, which satisfies the conditions(g-5)–(g-6) in the terminology of [18]. Briefly, this means that η is locally integrable and there exists a non-negative locallybounded function ζ on (−∞, 0] such thatη(ξ+θ) ≤ ζ (ξ)η(θ), for all ξ ≤ 0 and θ ∈ (−∞,−r)\Nξ , whereNξ ⊆ (−∞,−r)is a set with Lebesgue measure zero.The spaceB = Cr × Lp(η, X) consists of all classes of functions ϕ : (−∞, 0] 7→ X such that ϕ is continuous on [−r, 0],

Lebesgue-measurable, and η‖ϕ‖p is Lebesgue integrable on (−∞,−r). The seminorm in Cr × Lp(η, X) is given by

‖ ϕ ‖B := sup‖ϕ(θ)‖ : −r ≤ θ ≤ 0 +(∫

−r

−∞

η(θ)‖ϕ(θ)‖pdθ)1/p

.

The space B = Cr × Lp(η, X) satisfies axioms (A), (A1), and (B). Moreover, when r = 0 and p = 2, we can take H = 1,

K(t) = 1+(∫ 0−t η(θ) dθ

)1/2andM(t) = γ (−t)1/2 for t ≥ 0 (see [18, Theorem 1.3.8] for details).

Remark 1.1. In the rest of this article, we assume that L is a positive constant such that ‖ϕ‖B ≤ L supθ≤0 ‖ϕ(θ)‖ for eachϕ ∈ B bounded continuous. See [18, Proposition 7.1.1] for additional details on this assumption.

Let (Z, ‖ · ‖Z ) and (W , ‖ · ‖W ) be Banach spaces. In this article, we denote by L(Z,W ) the space of bounded linearoperators from Z into W endowed with norm of operators denoted ‖ · ‖L(Z,W ), and we write L(Z) and ‖ · ‖L(Z), whenZ = W . In addition, Br(x, Z) denotes the closed ball with center at x and radius r in Z , and (Cb(R; Z), ‖ · ‖Cb(Z)) is the spaceformed for all the bounded continuous functions from R into Z endowed with the sup-norm denoted by ‖ · ‖Cb(Z), and wewrite ‖ · ‖Cb when nonconfusion on the space Z arise.This article has three sections. In the next section, we discuss the existence of mild solutions for the Eq. (1.1). In the last

section, some concrete applications to ordinary and partial differential equations are considered.

2. Existence of solutions

In this section, we discuss the existence of solutions for the neutral equation (1.1). By considering Proposition 1.2 andthe development in [8], we introduce the next concept of solution.

Definition 2.1. A function u ∈ Cb(R; X) is called a mild solution of the neutral Eq. (1.1) if

u(t) = −g(t, ut)−∫ t

−∞

AT (t − s)(I − P)g(s, us)ds+∫∞

tAT (t − s)Pg(s, us)ds

+

∫ t

−∞

T (t − s)(I − P)T (t − s)f (s, us)ds−∫∞

tT (t − s)Pf (s, us)ds, t ∈ R.

In this article, we always assume that the following assumption is verified.

(Hg) There are a Banach space (Y , ‖ · ‖Y ) continuously included in X and functions H, H ∈ L1loc([0,∞), (0,∞)) such thatg ∈ C([0, a] ×B; Y ), ‖AT (t)‖L(Y ,X) ≤ H(t) and ‖AT (t)(I − P)‖L(Y ,X) ≤ e−γ tH(t) for all t ≥ 0.

Remark 2.2. The condition (Hg) is linked to the integrability of the function s→ AT (t − s)gu(s), since except trivial cases,the operator function AT (·) is not integrable in the operator topology on [0, b], for b > 0. We refer the reader to [7,10] foradditional details on this type of condition in the theory of neutral equations.

Remark 2.3. In the rest of this work, for functions H ∈ Cb(R × B, X) and u ∈ Cb(R; X) we use the notation Hu for thefunction Hu : R→ X given by Hu(t) = H(t, ut). In addition, ic denote the inclusion map from Y into X .

We establish now our first existence result.

Theorem 2.1. Assume f ∈ C(R×B, X), g ∈ C(R×B, Y ) and there are Lg > 0 and Lf ∈ L1loc(R,R+) such that

‖g(t, ϕ1)− g(t, ϕ2)‖Y ≤ Lg‖ϕ1 − ϕ2‖B,‖f (t, ϕ1)− f (t, ϕ2)‖ ≤ Lf (t)‖ϕ1 − ϕ2‖B,

for all (t, ϕi) ∈ R×B . Suppose, in addition, there areψi ∈ B , i = 1, 2, such that g(·, ψ1) ∈ Cb(R; Y ) and f (·, ψ2) ∈ Cb(R; X).If

Λ = LLg

[‖ic‖L(Y ,X) +

∫∞

0e−γ sH(s)ds+

d1γ

]+L sup

t∈R

(M0

∫ t

−∞

e−γ (t−s)Lf (s)ds+ d0

∫∞

te−γ (t−s)Lf (s)ds

)< 1,

then there exists a unique mild solution of (1.1).


Proof. On the space Cb(R; X), we define the map Γ : Cb(R; X)→ Cb(R; X) by

Γ u(t) = −gu(t)−∫ t

−∞

AT (t − s)(I − P)gu(s)ds+∫∞

tAT (t − s)Pgu(s)ds

+

∫ t

−∞

T (t − s)(I − P)fu(s)ds−∫∞

tT (t − s)Pfu(s)ds.

From the condition Hg and the estimate∫ t

−∞

‖AT (t − s)(I − P)gu(s)‖ds ≤∫ t

−∞

e−γ (t−s)H(t − s)(Lg‖us − ψ1‖B + ‖g(·, ψ1)‖Cb(Y ))ds

≤ (Lg(L‖u‖Cb(X) + ‖ψ1‖B)+ ‖g(·, ψ1)‖Cb(Y ))∫∞

0e−γ sH(s)ds,

we infer that the function t → AT (t − s)(I − P)gu(s) is integrable on (−∞, t] for all t ∈ R and the function t →∫ t−∞AT (t − s)(I − P)gu(s)ds belongs to Cb(R; X). Similarly, from the estimate∫

∞

t‖AT (t − s)Pgu(s)‖ds ≤ d1

∫ t

−∞

e−γ (t−s)(‖ic‖L(Y ,X)Lg‖us − ψ1‖B + ‖g(·, ψ1)‖Cb(X))ds

≤d1γ(Lg(‖ic‖L(Y ,X)L‖u‖Cb(X) + ‖ψ1‖B)+ ‖g(·, ψ1)‖Cb(X)),

we deduce that t →∫∞

t AT (t − s)Pgu(s)ds ∈ Cb(R; X). Arguing as above, we can complete the proof that Γ u ∈ Cb(R, X).On the other hand, for u, v ∈ Cb(R; X)we see that

‖Γ u(t)− Γ v(t)‖ ≤ Lg‖ic‖L(Y ,X)L‖u− v‖Cb(X)

+ Lg

(∫ t

−∞

e−γ (t−s)H(t − s)ds+ d1

∫∞

teγ (t−s)ds

)L‖u− v‖Cb(X)

+

(M0

∫ t

−∞

e−γ (t−s)Lf (s)ds+ d0

∫∞

teγ (t−s)Lf (s)ds

)L‖u− v‖Cb(X)

≤ Λ‖u− v‖Cb(X),

which prove that Γ is a contraction and there exists a unique mild solution of (1.1). The proof is complete.

Next, we establish the existence of mild solutions through fixed points criterions for condensing operators. To this end,we consider two additional Lemmas. In Lemma 2.1 below, C0(R; X) = u ∈ Cb(R; X) : limt→±∞ u(t) = 0.

Lemma 2.1. A set B ⊂ C0(R; X) is relatively compact in C0(R; X) if and only if B(t) = x(t) : x ∈ B is relatively compact in Xfor every t ∈ R, B is equicontinuous and limt→±∞ ‖ x(t) ‖= 0 uniformly for x ∈ B.

Lemma 2.2. If the semigroup (T (t))t≥0 is compact, then ic : Y → X is compact.

Proof. We next prove that Br(0, Y ) is relatively compact in X for all r > 0. Let r > 0, y ∈ Br(0, Y ) and 0 < ε < t . Usingthat A

∫ t0 T (s)ds = T (t)− I , the condition Hg and that A is closed, we get

y = T (t)y− AT (ε)∫ t−ε

0T (t − ε − s)yds− A

∫ t

t−εT (t − s)yds,

= T (t)y− T (ε)∫ t−ε

0AT (t − ε − s)yds−

∫ t

t−εAT (t − s)yds. (2.1)

Since ‖∫ µ0 AT (s)yds‖ ≤

∫ µ0 H(s)‖y‖Yds, from (2.1) we obtain

Br(0, Y ) ⊂ T (t)Br1(0, X)+ T (ε)Br2,ε (0, X)+ Br3,ε (0, X), (2.2)

where r1 = r‖ic‖L(Y ,X), r2,ε = r∫ t−ε0 H(s)ds and r3,ε = r

∫ tt−ε H(s)ds. Since the operators T (t) and T (ε) are compact and

r3,ε → 0 as ε→ 0, we can conclude from (2.2) that Br(0, Y ) is relatively compact in X . This completes the proof.

Theorem 2.2. Assume that the semigroup (T (t))t≥0 is compact, the function g(·) verify the assumptions in Theorem 2.1 andthere are mf ∈ C(R,R+) and a non-decreasing function W : [0,∞)→ [0,∞) such that ‖f (t, ψ)‖ ≤ mf (t)W (‖ψ‖B) for all


(t, ψ) ∈ R×B . Suppose, in addition,

Λ = supt∈R

[M0

∫ t

−∞

e−γ (t−τ)mf (τ )dτ + d0

∫∞

teγ (τ−t)mf (τ )dτ

]<∞, (2.3)

limt→−∞

∫ t

−∞

e−γ (t−τ)mf (τ )dτ + limN→∞

supt≥N

∫ t

Ne−γ (t−τ)mf (τ )dτ = 0, (2.4)

limt→∞

∫∞

teγ (t−τ)mf (τ )dτ + lim

N→−∞supt≤N

∫ N

teγ (t−τ)mf (τ )dτ = 0, (2.5)

LgL[‖ic‖L(Y ,X)

(1+

d1γ

)+

∫∞

0e−γ sH(s)ds

]+Λ lim sup

r→∞

W (Lr)r

< 1. (2.6)

Then, there exists a unique mild solution of (1.1).Proof. Let Γ (·) be the map introduced in the proof of Theorem 2.1. To begin, we prove that there exists r > 0 such thatΓ Br(0, Cb(R; X)) ⊂ Br(0, Cb(R; X)). Let C1, C2 be the constants given by

C1 = (1+ ‖ic‖L(Y ,X))(Lg‖ψ1‖B + ‖g(·, ψ1)‖Cb(Y ))

C2 =[1+

∫∞

0e−γ sH(s)ds+

d1γ

].

From (2.6), there are r > 0 and θ ∈ (0, 1) such that

C1C2 + LgLs[‖ic‖L(Y ,X)

(1+

d1γ

)+

∫∞

0e−γ τH(τ )dτ

]+W (Ls)Λ < θs,

for all s ≥ r . Under these conditions, for u ∈ Br(0, Cb(R; X)) and t ∈ Rwe get

‖Γ u(t)‖ ≤ Lg‖ic‖L(Y ,X)‖ut − ψ1‖B + ‖ic‖L(Y ,X)‖g(t, ψ1)‖Y

+

∫ t

−∞

e−γ (t−τ)H(t − τ)(Lg‖uτ − ψ1‖B + ‖g(τ , ψ1)‖Y )dτ

+ d1

∫∞

teγ (t−τ)(Lg‖ic‖L(Y ,X)‖uτ − ψ1‖B + ‖ic‖L(Y ,X)‖g(τ , ψ1)‖Y )dτ

+W (Lr)(M0

∫ t

−∞

e−γ (t−τ)mf (τ )dτ + d0

∫∞

teγ (t−τ)mf (τ )dτ

)≤ Lg‖ic‖L(Y ,X)Lr + C1 + LgLr

∫∞

0e−γ sH(s)ds+ C1

∫∞

0e−γ sH(s)ds

+d1γLg‖ic‖L(Y ,X)Lr +

d1C1γ+W (Lr)Λ

≤ C1C2 + LgLr[‖ic‖L(Y ,X) +

∫∞

0e−γ sH(s)ds+

d1γ‖ic‖L(Y ,X)

]+W (Lr)Λ

≤ θr,

which shows that Γ u ∈ Br(0, Cb(R; X)) and Γ Br(0, Cb(R; X)) ⊂ Br(0, Cb(R; X)).To prove that Γ (·) is a condensing operator on Br(0, Cb(R; X)), we introduce the decomposition Γ (·) =

∑3i=1 Γi(·),

where

Γ1u(t) = −gu(t)−∫ t

−∞


tAT (t − s)Pgu(s)ds,

Γ2u(t) =∫ t

−∞

T (t − s)(I − P)fu(s)ds,

Γ3u(t) = −∫∞


It is easy to see that the operators Γi, i = 1, 2, 3, are continuous and

‖Γ1u− Γ1v‖Cb(X) ≤ LgL[‖ic‖L(Y ,X)

(1+

d1γ

)+

∫∞

0e−γ sH(s)ds

]‖u− v‖Cb(X),

which shows that Γ1 is a contraction on Br(0, Cb(R; X)).


We next prove that Γ2 and Γ3 are completely continuous on Br(0, Cb(R; X)). We divide the rest of the proof in severalsteps.Step 1. For all t ∈ R, the set Γ2Br(0, Cb(R; X))(t) = Γ2u(t) : u ∈ Br(0, Cb(R; X)) is relatively compact in X .For t ∈ R, u ∈ Br(0, Cb(R; X)) and ε > 0, we see that

Γ2u(t) = T (ε)∫ t−ε

−∞

T (t − ε − τ)(I − P)fu(τ )dτ +∫ t

t−εT (t − τ)(I − P)fu(τ )dτ .

By noting now that∥∥∥∥∫ t−ε

−∞

T (t − ε − τ)(I − P)fu(τ )dτ∥∥∥∥ ≤ W (Lr)Λ,∥∥∥∥∫ t

t−εT (t − τ)(I − P)fu(τ )dτ

∥∥∥∥ ≤ M0W (Lr) ∫ t

t−εmf (τ )dτ ,

we obtain

Γ1Br(0, Cb(R; X))(t) ⊂ T (ε)Br∗(0, X)+ Cε, (2.7)

where r∗ = W (Lr)Λ and diam(Cε) ≤ M0W (Lr)∫ tt−ε mf (τ )dτ . Since T (ε) is compact, diam(Cε)→ 0 as ε → 0 and u(·) is

arbitrary, from (2.7) we infer that Γ2Br(0, Cb(R; X))(t) is relatively compact in X .Step 2. The set Γ2Br(0, Cb(R; X)) = Γ2u : u ∈ Br(0, Cb(R; X)) is equicontinuous.Let ε > 0 and t ∈ R. Since U = Γ2Br(0, Cb(R; X))(t) is relatively compact in X , there exists δ > 0 such that

‖(T (h) − I)z‖ ≤ ε and∫ t+ht mf (τ )dτ ≤ ε for all z ∈ U and every 0 < h < δ. Then, for u ∈ Br(0, Cb(R; X)) and 0 < h < δ

we get

‖Γ2u(t + h)− Γ2u(t)‖ ≤∥∥∥∥(T (h)− I) ∫ t

−∞

T (t − τ)(I − P)fu(τ )dτ∥∥∥∥+M0W (Lr) ∫ t+h

tmf (τ )dτ

≤ supz∈U‖(T (h)− I)z‖ +M0W (Lr)ε

≤ ε(1+M0W (Lr)),

which shows that the set Γ1Br(0, Cb(R; X)) is right equicontinuous at t . A similar procedure permit us to prove thatΓ1Br(0, Cb(R; X)) is left equicontinuous at t . This completes the proof of the assertion.Step 3. limt→±∞ Γ2u(t) = 0 uniformly for u ∈ Br(0, Cb(R; X)).For ε > 0 be given, we select Nε ∈ N such that

M0

∫ t

−∞

e−γ (t−τ)mf (τ )dτ < ε, t ≤ −Nε,

M0e−γNεr +M0W (Lr) sup

s≥Nε

∫ s

Nεe−γ (s−τ)mf (τ )dτ < ε, t ≥ Nε.

Under these conditions, for u ∈ Br(0, Cb(R; X)) and t ≤ −Nε we get

‖Γ2u(t)‖ ≤ M0

∫ t

−∞

e−γ (t−τ)mf (τ )W (Lr)dτ ≤ εW (Lr),

which proves that limt→−∞ Γ1u(t) = 0 uniformly for u ∈ Br(0,PBb(R; X)). On the other hand, for t ≥ 2Nε andu ∈ Br(0,PBb(R; X))we see that

‖Γ2u(t)‖ ≤ ‖T (t − Nε)∫ Nε

−∞

T (Nε − τ)(I − P)fu(τ )‖dτ +M0

∫ t

Nεe−γ (t−τ)mf (τ )W (Lr)dτ

≤ M0e−γ (t−Nε)‖Γ2u(Nε)‖ +M0W (Lr)

∫ t

Nεe−γ (t−τ)mf (τ )dτ

≤ M0e−γNεr +M0W (Lr) sup

s≥Nε

∫ s

Nεe−γ (s−τ)mf (τ )dτ

≤ ε,

which shows limt→∞ Γ2u(t) = 0 uniformly for u ∈ Br(0, Cb(R; X)).From the above steps and Lemma 2.1, it follows thatΓ2 is completely continuous on Br(0, Cb(R; X)). Moreover, arguing as

abovewe can prove thatΓ3 is also completely continuous on Br(0, Cb(R; X)). This completes the proof thatΓ is a condensingoperator on Br(0, Cb(R; X)).Finally, from [19, Theorem 4.3.2] there exists a fixed point u ∈ Br(0, Cb(R; X)) of Γ . The proof is now complete.


Remark 2.4. The conditions (2.3)–(2.5) are verified in several cases. Consider, for example, mf ∈ Lp(R) for some p ≥ 1 ormf (·) ∈ C0(R; X).

The proof of Theorem 2.3 below, is similar to the proof of Theorem 2.2.We include a sketch of the proof by completeness.

Theorem 2.3. Assume (T (t))t≥0 is compact and f (·) satisfies the conditions in Theorem 2.2. Suppose, in addition, there aremg ∈ C0(R,R+) and a non-decreasing function Wg : [0,∞) → [0,∞) such that ‖g(t, ψ)‖Y ≤ mg(t)Wg(‖ψ‖B) for all(t, ψ) ∈ R×B , the set of functions gu : u ∈ Br(0, Cb(R; X)) is an equicontinuous subset of Cb(R; X) for all r > 0 and

supt∈R

[∫ t

−∞

e−γ (t−τ)H(t − τ)mg(τ )dτ + d1‖ic‖L(Y ,X)

∫∞

teγ (t−τ)mg(τ )dτ

]<∞.

If lim supr→∞1r

[‖ic‖L(Y ,X)Wg(Lr)‖mg‖Cb(R) +Wg(Lr)Θ +W (Lr)Λ

]< 1, then there exists a mild solution of (1.1).

Proof. Let Γ be the map introduced in the proof of Theorem 2.1 and consider the decomposition Γ =∑3i=1 Γi, where

Γ1u(t) = −gu(t) and

Γ2u(t) = −∫ t

−∞


tAT (t − s)Pgu(s)ds,

Γ3u(t) =∫ t

−∞

T (t − s)(I − P)fu(s)ds−∫∞


Proceeding as in the proof of Theorem 2.1, we can show there exists r > 0 such that Γ Br(0, Cb(R; X)) ⊂ Br(0, Cb(R; X))and the functions Γ2 and Γ3 are completely continuous.On the other hand, from the assumption on g(·) we know that the set of functions s→ gu(s) : u ∈ Br(0, Cb(R; X)) is

equicontinuous on R and from Lemma 2.2 that Γ1Br(0, Cb(R; X))(t) = Γ1u(t) : u ∈ Br(0, Cb(R; X)) is relatively compactin X for all t ∈ R. By noting now thatmg ∈ C0(R,R+), we infer from Lemma 2.1 that Γ1Br(0, Cb(R; X)) is relatively compactin Cb(R; X) and Γ1 is completely continuous on Br(0, Cb(R; X)).From the above remarks, Γ is completely continuous on Br(0, Cb(R; X)) and from the Schauder’s fixed point theoremwe

assert the existence of a mild solution for (1.1). The proof is complete.

2.1. Neutral equations on finite dimensional spaces

We finish this section considering the particular case in which X is finite dimensional. In this case, our results are easilyapplicable since A is a bounded linear operator and the assumptionHg is automatically verifiedwith Y = X . Next, we assumethat σ(A)∩ iR = ∅ and γ , di andMi are as above. The next theorems are reformulations of our precedent results. The proofswill be omitted.

Theorem 2.4. Assume f , g ∈ C(R×B, X) and there are Lg > 0 and Lf ∈ L1loc(R,R+) such that

‖g(t, ϕ1)− g(t, ϕ2)‖ ≤ Lg‖ϕ1 − ϕ2‖,‖f (t, ϕ1)− f (t, ϕ2)‖ ≤ Lf (t)‖ϕ1 − ϕ2‖,

for all (t, ϕi) ∈ R × B . Suppose, in addition, there are ψi ∈ B , i = 1, 2, such that g(·, ψ1) and f (·, ψ2) are functions in∈ Cb(R; X). If LLg [1+

‖A‖γ(M0 + d0)] + LΛ < 1, whereΛ is defined in (2.3), then there exists a unique mild solution of (1.1).

Theorem 2.5. Assume g(·) verify the assumptions in Theorem 2.4 and there are mf ∈ C(R,R+) and a non-decreasing functionW : [0,∞) → [0,∞) such that ‖f (t, ψ)‖ ≤ mf (t)W (‖ψ‖B) for all (t, ψ) ∈ R × B . Suppose, in addition, the conditions(2.3)–(2.5) are verified and

LgL[1+‖A‖γ(M0 + d0)

]+Λ lim sup

r→∞

W (Lr)r

< 1.

Then, there exists a unique mild solution of (1.1).

Theorem 2.6. Assume f (·) satisfies the conditions in Theorem 2.4, there are mg ∈ C0(R,R+) and a non-decreasing functionWg : [0,∞)→ [0,∞) such that ‖g(t, ψ)‖ ≤ mg(t)Wg(‖ψ‖B) for all (t, ψ) ∈ R×B and the set gu : u ∈ Br(0, Cb(R; X))is a equicontinuous subset of Cb(R; X) for all r > 0. If

Θ = ‖A‖ supt∈R

[M0

∫ t

−∞

e−γ (t−τ)mg(τ )dτ + d0

∫∞

teγ (t−τ)mg(τ )dτ

]<∞,

and lim supr→∞ r−1[Wg(Lr)(‖mg‖Cb(R) +Θ)+ΛW (Lr)

]< 1, then there exists a mild solution of (1.1).


3. Applications

In this section, we discuss some applications of our abstract results. At first, we consider the ordinary neutral differentialequation described on Rn

ddt

[u(t)− λ

∫ t

−∞

C(t − s)u(s)ds]= Au(t)+ λ

∫ t

−∞

B(t − s)u(s)ds− p(t)+ q(t), (3.1)

which arises in the study of the dynamics of income, employment, value of capital stock, and cumulative balance of payment,see [20] for details. In this equation, λ is a real number, the state u(t) ∈ Rn, C(·), B(·) are n× nmatrix continuous functions,A is a constant n× nmatrix, p(·) represents the government intervention and q(·) is the private initiative.To treat this system, we takeB = C0× Lp(η, X)with X = Rn (see Example 1.1) andwe assume that σ(A)∩ iR = ∅. Next,

γ , d0,L andM0 are as in Section 1 and f , g : R×B → X are the functions defined by g(t, ψ) = −λ∫ 0−∞C(−s)ψ(s)ds and

f (t, ψ) =∫ 0−∞B(−s)ψ(s)ds− p(t)+ q(t).

In the next result, which is a consequence of Theorem 2.4, Lg and Lf are the constants Lg = (∫ 0−∞| C(−s) |2 | η(s) | ds)

12

and Lf = (∫ 0−∞| B(−s) |2 | η(s) | ds)

12 . In this result, we said that a function u ∈ Cb(R; X) is a mild solution of (3.1) if u(·)

is a mild solution of the associated abstract Eq. (1.1).

Proposition 3.3. If LgL[1+ ‖A‖γ (M0 + d0)] + LfL[M0+d0γ] < 1, then there exists a unique mild solution of (1.1).

To complete this section, we consider the partial neutral differential system

∂

∂t

[u(t, ξ)+

∫ 0

−∞

∫ π

0b(s, η, ξ)u(t + s, η)dηds

]=

∂2

∂ξ 2u(t, ξ)+ a0(ξ)u(t, ξ)+

∫ 0

−∞

a(s)u(t + s, ξ)ds, (3.2)

u(t, 0) = u(t, π) = 0, (3.3)

for (t, ξ) ∈ R×[0, π], which arise in control systems described by abstract retarded functional–differential equations withfeedback control governed by proportional integro-differential law, see [8, Examples 4.2] for details.To study this system, we take X = L2([0, π]) and let A be the operator given by Ax = x′′ with domain D(A) :=

x ∈ X : x′′ ∈ X, x(0) = x(π) = 0. It is well known that A is the infinitesimal generator of an analytic semigroup(T (t))t≥0 on X . Furthermore, A has discrete spectrum with eigenvalues of the form −n2, n ∈ N, and correspondingnormalized eigenfunctions given by zn(ξ) := ( 2

π)12 sin(nξ). In addition, zn : n ∈ N is an orthonormal basis for X ,

T (t)x =∑∞

n=1 e−n2t〈x, zn〉zn for all x ∈ X and every t > 0, and Ax = −

∑∞

n=1 n2〈x, zn〉zn for every x ∈ D(A). From the

above, we see that ‖T (t)‖ ≤ e−t for all t ≥ 0.It is also possible to define the fractional powers of A, see [17] for additional details. For x ∈ X and α ∈ (0, 1),

(−A)−αx =∑∞

n=11n2α〈x, zn〉zn, the operator (−A)α : D((−A)α) ⊆ X → X is given by (−A)αx =

∑∞

n=1 n2α〈x, zn〉zn

for all x ∈ D((−A)α) = x ∈ X :∑∞

n=1 n2α〈x, zn〉zn ∈ X. Moreover, for α = 1

2 we have that ‖(−A)−1/2‖ = 1 and

‖ (−A)12 T (t) ‖≤ 1

√2e−t2 t−

12 for all t > 0.

Next, we take B = C0 × Lp(η, X) as phase space, L is the constant in Remark 1.1 and we suppose that the followingconditions hold.

(i) The functions ∂ i

∂ζ ib(τ , η, ζ ), i = 0, 1 are Lebesgue- measurable, b(τ , η, π) = 0, b(τ , η, 0) = 0 for every (τ , η) and

Lg := max

∫ π

0

∫ 0

−∞

∫ π

0η(τ)

(∂ i

∂ζ ib(τ , η, ζ )

)2dηdτdζ : i = 0, 1

<∞.

(ii) The functions a(·), a0(·) are continuous and

Lf = ‖a0‖ +(∫ 0

−∞

| a(s) |12 η(s)ds

) 12

<∞.

Under these conditions, we define the functions f , g : R×B → X by

g(t, ψ)(ξ) =∫ 0

−∞

∫ π

0b(s, η, ξ)ψ(s, η)dηds,

f (t, ψ)(ξ) = a0(ξ)u(t, ξ)+∫ 0

−∞

a(s)ψ(s, ξ)ds,


which permit to represent the systems (3.2)–(3.3) in the abstract form (1.1). Moreover, from (i) and (ii), it follows thatg ∈ C(R × B; X 1

2), f ∈ C(R × B; X), g(t, ·) and f (t, ·) are bounded linear operator, ‖rallelg(t, ·)‖rallelL(B,X 1

2) ≤ Lg and

‖f (t, ·)‖L(B,X) ≤ Lf for each t ∈ R.Proposition 3.4 below, is a consequence of Theorem 2.1. In this result, we said that a function u ∈ Cb(R; X) is a mild

solution of (3.2)–(3.3) if u(·) is a mild solution of the associated abstract system (1.1).

Proposition 3.4. AssumeΛ = LLg[1+ 1

√2

]+ LfL < 1. Then, there exits a unique mild solution u ∈ Cb(R, X) of (3.2)–(3.3).

References

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global solutions for abstract neutral differential equations

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