mathematical and computer modelling

Applied Mathematics and Computation 186 (2007) 294–301
www.elsevier.com/locate/amc

On state-dependent delay partial neutral
functional–differential equations
Eduardo Herna´ndez M.

a,*

, Mark A. McKibben

b

a

b

Departamento de Matema´tica, ICMC, Universidade de Sa˜o Paulo, Caixa Postal 668, 13560-970 Sa˜o Carlos SP, Brazil
Department of Mathematics and Computer Science, Goucher College, 1021 Dulaney Valley Road, Baltimore, MD 21204, USA

Abstract
In this paper, we study the existence of mild solutions for a class of abstract partial neutral functional–differential equations with state-dependent delay.
Ó 2006 Elsevier Inc. All rights reserved.
Keywords: Abstract Cauchy problem; Neutral equations; State-dependent delay; Semigroup of linear operators; Unbounded delay

1. Introduction
The purpose of this article is establish the existence of mild solutions for a class of abstract neutral functional–differential equations with state-dependent delay described by the form
d
Dðut Þ ¼ ADðut Þ þ F ðt; xqðt;xt Þ Þ;
dt
x0 ¼ u 2 B;

t 2 I ¼ ½0; aŠ;

ð1:1Þ
ð1:2Þ

where A is the infinitesimal generator of a compact C0-semigroup of bounded linear operators (T(t))tP0 on a
Banach space X; the function xs : (À1, 0] ! X, xs(h) = x(s + h), belongs to some abstract phase space B described axiomatically; F, G are appropriate functions; and Dw = w (0) À G(t, w), where w is in B.
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 equation has received a significant amount of attention
in the last years, see for instance [1–11] and the references therein. We also cite [12,9,13] for the case of neutral
differential equations with dependent delay. The literature related to partial functional–differential equations
with state-dependent delay is limited, to our knowledge, to the recent works [14,15].
Abstract neutral differential equations arise in many areas of applied mathematics. For this reason, they
have largely been studied during the last few decades. The literature related to ordinary neutral differential
*

Corresponding author.
E-mail addresses: (E. Herna´ndez M.), (M.A. McKibben).

0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2006.07.103


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295

equations is very extensive, thus, we refer the reader to [16] only, which contains a comprehensive description
of such equations. Similarly, for more on partial neutral functional–differential equations and related issues we

refer to Adimy and Ezzinbi [17], Hale [18], Wu and Xia [19] and [20] for finite delay equations, and Herna´ndez
and Henriquez [21,22] and Herna´ndez [23] for unbounded delays.
2. Preliminaries
Throughout this paper, A : D(A) & X ! X is the infinitesimal generator of a compact C0-semigroup of line is a positive constant such that kT ðtÞk 6 M
e for every
ear operators (T(t))tP0 on a Banach space X and M
t 2 I = [0, a]. For background information related to semigroup theory, we refer the reader to Pazy [24].
In this work we will employ an axiomatic definition for the phase space B which is similar to those introduced in [25]. Specifically, B will be a linear space of functions mapping (À1, 0] into X endowed with a seminorm k Á kB , and satisfies the following axioms:
(A) If x : (À1, r + b] ! X, b > 0, is such that xj[r, r+b] 2 C([r, r + b] : X) and xr 2 B, then for every
t 2 [r, r + b] the following conditions hold:
(i) xt is in B,
(ii) kxðtÞk 6 H kxt kB ,
(iii) kxt kB 6 Kðt À rÞ supfkxðsÞk : r 6 s 6 tg þ Mðt À rÞkxr kB ,
where H > 0 is a constant; K, M : [0, 1) ! [1, 1), K is continuous, M is locally bounded, and H, K,
M are independent of x(Æ).
(A1) For the function x(Æ) in (A), the function t ! xt is continuous from [r, r + b] into B.
(B) The space B is complete.
Example 2.1 (The phase spaces Cg, C 0g ). Let g : (À1, 0] ! [1, 1) be a continuous, non-decreasing function
with g(0) = 1, which satisfies conditions (g-1), (g-2) of [25]. Briefly, this means that the function cðtÞ :¼
supÀ1gðhÞ is locally bounded for t P 0 and that g(h) ! 1 as h ! À1.
Let Cg(X) be the vector space consisting of the continuous functions u such that ug is bounded on (À1, 0],
and let C 0g ðX Þ be the subspace of Cg(X) containing precisely those functions u for which uðhÞ
gðhÞ ! 0 as h ! À1.
The spaces Cg and C 0g , endowed with the norm kukg :¼ suph60 kuðhÞk
,
are
both
phases
spaces which satisfy

gðhÞ
axioms (A), (A-1), (B), see [25, Theorem 1.3.6] for details. Moreover, in this case K(t) = 1, for every t P 0.
Example 2.2 (The phase space Cr · Lp(g;X)). Assume that g : ðÀ1; ÀrÞ ! R is a Lebesgue integrable function and that there exists a non-negative and locally bounded function c such that g(n + h) 6 c(n) g(h), for
all n 6 0 and h 2 (À1, Àr) nNn, where Nn  (À1, Àr) is a set with Lebesgue measure zero. The space
Cr · Lp(g;X) consists of all classes of functions u : (À1, 0] ! X such that u is continuous on [Àr, 0] and
1
gp kuk 2 Lp ððÀ1; ÀrÞ; X Þ. The seminorm in Cr · Lp(g;X) is defined by
kukB :¼ supfkuðhÞk : Àr 6 h 6 0g þ

Z

Àr
p

gðhÞkuðhÞk dh

1=p
:

À1

If g(Æ) satisfies the conditions (g-5), (g-6) and (g-7) in the nomenclature of [25], then B ¼ C r  Lp ðg; X Þ satisfies
axioms (A), (A1), (B) (see [25, TheoremR 1.3.8] for details). Moreover, when r = 0 and p = 2, we have that
0
1=2
H = 1, M(t) = c(Àt)1/2, and KðtÞ ¼ 1 þ ð Àt gðhÞ dhÞ for t P 0.
Remark 2.1. Let u 2 B and t 6 0. The notation ut represents the function ut : (À1, 0] ! X defined by
ut(h) = u(t + h). Consequently, if the function x(Æ) in axiom (A) is such that x0 = u, then xt = ut. We observe
that ut is well defined for every t < 0 since the domain of u(Æ) is (À1, 0].
We also note that, in general, ut 62 B; consider, for example, the characteristic function X½l;0Š , l < Àr < 0, in

the space Cr · Lp(g; X).


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The terminology and notations are those generally used in functional analysis. In particular, for Banach
spaces (Z, k Æ kZ), (W, k Æ kW), the notation LðZ; W Þ stands for the Banach space of bounded linear operators
from Z into W, and we abbreviate this notation to LðZÞ when Z = W. Moreover, Br(x, Z) denotes the closed
ball with center at x and radius r > 0 in Z and for a bounded function n : I ! Z and t 2 [0, a] we employ the
notation knkZ,t for
knkZ;t ¼ supfknðsÞkZ : s 2 ½0; tŠg:

ð2:3Þ

We will simply write knkt when no confusion arises.
The remainder of the paper is divided into two sections. The existence of mild solutions for the abstract
Cauchy problem (1.1)–(1.2) is studied in Section 3, and Section 4 is devoted to a discussion of some applications. To conclude the current section, we recall the following well-known result, referred to as the Leray
Schauder Alternative, for convenience.
Theorem 2.1 [26, Theorem 6.5.4]. Let D be a convex subset of a Banach space X and assume that 0 2 D. Let
G : D ! D be a completely continuous map. Then, either the set {x 2 D : x = kG(x), 0 < k < 1} is unbounded or the
map G has a fixed point in D.
3. Existence results
In this section we discuss the existence of mild solutions for the abstract system (1.1)–(1.2). We begin by
introducing the following conditions:
(Hu) Let RðqÀ Þ ¼ fqðs; wÞ : ðs; wÞ 2 I Â B; qðs; wÞ 6 0g. The function t ! ut is well defined and continuous
from RðqÀ Þ into B, and there exists a continuous and bounded function J u : RðqÀ Þ ! ð0; 1Þ such that
kut kB 6 J u ðtÞkukB for every t 2 RðqÀ Þ.
(H1) The function F : I Â B ! X satisfies the following properties:

(a) The function F(Æ, w) : I ! X is strongly measurable, for every w 2 B.
(b) The function F ðt; ÁÞ : B ! X is continuous, for each t 2 I.
(c) There exist a continuous non decreasing function W : [0, 1) ! (0, 1) and an integrable function
m : I ! [0, 1) such that
kF ðt; wÞk 6 mðtÞW ðkwkB Þ;

ðt; wÞ 2 I Â B:

(H2) The function G : R Â B ! X is continuous and there exists LG > 0 such that
kGðt; w1 Þ À Gðt; w2 Þk 6 LG kw1 À w2 kB ;

ðt; wi Þ 2 I Â B:

(H3) Let S(a) = {x : (À1, a] ! X : x0 = 0; x 2 C([0, a] : X)} endowed with the norm of uniform convergence on
[0, a] and y : (À1, a] ! X be the function defined by y0 = u on (À1, 0] and y(t) = T(t)u(0) on [0, a].
Then, for every bounded set Q such that Q & S (a), the set of functions {t ! G(t, xt + yt) : x 2 Q} is equicontinuous on [0, a].
Remark 3.2. We remark that condition Hu is frequently satisfied by functions that are continuous and
bounded. In fact, if the space B satisfies axiom C2 in [25], then there exists a constant L > 0 such that
kukB 6 Lsuph60 kuðhÞk for every u 2 B that is continuous and bounded, see [25, Proposition 7.1.1] for details.
Consequently,
kut kB 6 L

suph60 kuðhÞk
kukB
kukB

for every u 2 B n f0g continuous and bounded and every t 6 0. We also observe that Cr · Lp(g; X) satisfies
axiom C2 if g(Æ) is integrable on (À1, r], see [25, p. 10].

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Motivated by general semigroup theory, we adopt the following concept of mild solution.
Definition 3.1. A function x : (À1, a] ! X is a mild solution of the abstract Cauchy problem (1.1)–(1.2) if
x0 = u, xqðs;xs Þ 2 B, for every s 2 I, and
Z t
T ðt À sÞF ðs; xqðs;xs Þ Þ ds; t 2 I:
xðtÞ ¼ T ðtÞðuð0Þ À Gð0; uÞÞ þ Gðt; xt Þ þ
0

The proof of Lemma 3.1 is routine; the details are left to the reader.
Lemma 3.1. Let x : (À1, a] ! X be a function such that x0 = u and xj½0;aŠ 2 Cð½0; aŠ : X Þ. Then
kxs kB 6 ðM a þ J u0 ÞkukB þ K a supfkxðhÞk; h 2 ½0; maxf0; sgŠg;

s 2 RðqÀ Þ [ ½0; aŠ;

where J u0 ¼ supt2RðqÀ Þ J u ðtÞ and the notation in (2.3) has been used.
Now, we can prove our first existence result.
Theorem 3.2. Let u 2 B and assume that conditions H1, H2, Hu hold. If


Z
W ðnÞ a
e
K a LG þ M lim inf
mðsÞ ds < 1;
n!1þ
n

0
then there exists a mild solution of (1.1)–(1.2).
Proof. Consider the space Y = {u 2 C(I : X) : u(0) = u(0)} endowed with the uniform convergence topology,
and define the operator C : Y ! Y by
Z t
CxðtÞ ¼ T ðtÞðuð0Þ À Gð0; uÞÞ þ Gðt; xt Þ þ
T ðt À sÞF ðs; xqðs;xs Þ Þ ds; t 2 I;
0

where x : ðÀ1; aŠ ! X is the extension of x to (À1, a] such that x0 ¼ u. From our assumptions it is easy to see
that Cx 2 Y.
 : ðÀ1; aŠ ! X be the extension of u to (À1, a] such that u
 ðhÞ ¼ uð0Þ on I. We claim that there exists
Let u
r > 0 such that CðBr ð
ujI ; Y ÞÞ & Br ð
ujI ; Y Þ. Indeed, suppose to the contrary that this assertion is false. Then, for
every r > 0 there exist xr 2 Br ð
ujI ; Y Þ and tr 2 I such that r < kCxr(tr) À u(0)k. Then, from Lemma 3.1 we find that
r < kCxr ðtr Þ À uð0Þk
6 kT ðtr Þuð0Þ À uð0Þk þ kT ðtr ÞGð0; uÞ À Gð0; uÞk þ kGðtr ; ðxr Þtr Þ À Gð0; uÞk
Z tr
þ
kT ðtr À sÞkkF ðs; ðxr Þqðs;ðxr Þs Þ Þk ds
0

e þ 1ÞH kukB þ kT ðtr ÞGð0; uÞ À Gð0; uÞk þ LG ðK a kxr À uð0Þktr þ ðM a þ HK a þ 1ÞkukÞ
6 ðM
Z tr
e

þM
mðsÞW ððM a þ J u0 ÞkukB þ K a ðkxr À uð0Þka þ kuð0ÞkÞÞ ds
0

e þ 1ÞH Þkuk þ kT ðtr ÞGð0; uÞ À Gð0; uÞk þ LG ðK a r þ ðM a þ HK a þ 1ÞkukÞ
6 ðð M
B
Z tr
e
þM
mðsÞW ððM a þ J u0 þ H ÞkukB þ K a rÞ ds
0

and hence



Z a
e lim inf W ðnÞ
1 6 K a LG þ M
mðsÞ ds ;
n!1
n
0
which contradicts our assumption.
ujI ; Y ÞÞ & Br ð
ujI ; Y Þ and consider the decomposition C = C1 + C2 where
Let r > 0 be such that CðBr ð
C1 xðtÞ ¼ T ðtÞðuð0Þ À Gð0; uÞÞ þ Gðt; xt Þ; t 2 I;
Z t

C2 xðtÞ ¼
T ðt À sÞF ðs; xqðs;xs Þ Þ ds; t 2 I:
0


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From the proof of [14, Theorem 2.2], we know that C2 is completely continuous. Moreover, using the phase
space axioms we find that
kC1 uðtÞ À C1 vðtÞk 6 LG K a ku À vka ; t 2 I;
which proves that C1 is a contraction on Br ð
ujI ; Y Þ, so that C is a condensing operator on Br ð
ujI ; Y ÞÞ.
The existence of a mild solution for (1.1)–(1.2) is now a consequence of [27, Theorem 4.3.2]. This completes
the proof. h
Theorem 3.3. Let conditions Hu, H1, H3 be satisfied. Assume that q(t, w) 6 t for every ðt; wÞ 2 I Â B, that G is
completely continuous, and that there exist positive constants c1, c2 such that kGðt; wÞk 6 c1 kwkB þ c2 for every
ðt; wÞ 2 I Â B. If l = 1 À Kac1 > 0 and
Z 1
e Ka Z a
M
ds
;
mðsÞ ds <
W
ðsÞ
l
0

C
where


h
i
e kGð0; uÞk þ c1 ðM a þ M
e HK a kuk þ K a M
e HK a ÞkukB þ c2 ;
C ¼ M a þ J u0 þ M
l
then there exists a mild solution of (1.1)–(1.2).
Proof. On the space BC ¼ fu : ðÀ1; aŠ ! X ; u0 ¼ 0; ujI 2 CðI; X Þg
kuka = sups2[0, a]ku(s)k, we define the operator C : BC ! BC by

0; t 2 ðÀ1; 0Š;
Rt
CxðtÞ ¼
T ðtÞGð0; uÞ À Gðt; xt Þ þ 0 T ðt À sÞF ðs; xqðs;xs Þ Þ ds; t 2 I;

endowed

with

the

norm

where x : ðÀ1; aŠ ! X is defined by the relation x ¼ y þ x on (À1, a]. In preparation for using Theorem 2.1,
we establish a priori estimates for the solutions of the integral equation z = kCz, k 2 (0, 1). Let xk be a solution

of z = kCz, k 2 (0, 1). If ak(s) = suph2[0, s]kxk(h)k, then from Lemma 3.1 and the fact that qðs; ðxk Þs Þ 6 s, we find
that
Z
e
kxk ðtÞk 6 kT ðtÞGð0; uÞk þ c1 kxt kB þ c2 þ M

t

0

e HK a ÞkukB þ K a ak ðsÞÞ ds
mðsÞW ððM a þ J u þ M

e HK a ÞkukB þ K a aðtÞÞ þ c2
e kGð0; uÞk þ c1 ððM a þ M
6M
Z t
e
e HK a ÞkukB þ K a ak ðsÞÞ ds:
þM
mðsÞW ððM a þ J u þ M
0

Consequently,
e kGð0; uÞk þ c1 ððM a þ M
e HK a Þkuk þ K a aðtÞÞ þ c2
kak ðtÞk 6 M
B
Z t
e

e HK a ÞkukB þ K a ak ðsÞÞ ds
þM
mðsÞW ððM a þ J u þ M
0

and so
1 e
e HK a ÞkukB þ c2 Š
½ M kGð0; uÞk þ c1 ðM a þ M
l
e Z t
M
e HK a ÞkukB þ K a ak ðsÞÞds:
mðsÞW ððM a þ J u þ M
þ
l 0
e HK a Þkuk þ K a aðtÞ, we obtain after a rearrangement of terms that
By denoting nk ðtÞ ¼ ðM a þ J u0 þ M
B
kak ðtÞk 6

e kGð0; uÞk þ c1 ðM a þ M
e HK a Þkuk þ K a ½ M
e HK a ÞkukB þ c2 Š
nk ðtÞ 6 ðM a þ J u0 þ M
l
e Ka Z t
M
mðsÞW ðnk ðsÞÞ ds:
þ

l
0

ð3:4Þ


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299

Denoting by bk(t) the right-hand side of (3.4), it follows that
b0k ðtÞ 6
and hence
Z bk ðtÞ
bk ð0Þ¼C

e Ka
M
mðtÞW ðbk ðtÞÞ
l
Z 1
e Ka Z a
M
ds
ds
6
;
mðsÞ ds <
W ðsÞ
W

ðsÞ
l
0
C

which implies that the set of functions {bk(Æ) : k 2 (0, 1)} is bounded in CðI : RÞ. Thus, the set {xk(Æ) : k 2 (0, 1)}
is bounded on I.
To prove that C is completely continuous, we consider the decomposition C = C1 + C2 introduced in the
proof of Theorem 3.2. From the proof of [14, Theorem 2.2] we know that C2 is completely continuous and
from the assumptions on G we infer that C1 is a compact map. It remains to show that C1 is continuous. Let
ðun Þn2N be a sequence in BC and u 2 BC such that un ! u. From the phase space axioms we infer that
ðun Þs ! us uniformly on [0, a] as n ! 1 and that U ¼ ½0; aŠ  fðun Þs ; us : s 2 ½0; aŠ; n 2 Ng is relatively compact
in ½0; aŠ  B. Thus, G is uniformly continuous on U, so that Gðs; ðun Þs Þ ! Gðs; us Þ uniformly on [0, a] as
n ! 1, which shows that C1 is continuous.
These remarks, in conjunction with Theorem 2.1, enable us to conclude that there exists a mild solution for
(1.1) and (1.2). The proof is complete. h
4. Examples
We conclude this work with two applications of our previous abstract results. In the sequel, X = L2([0, p])
and A : D(A) & X ! X is the operator Af = f00 with domain D(A) :¼ {f 2 X : f00 2 X, f(0) = f(p) = 0}. It is well
known that A is the infinitesimal generator of a compact C0-semigroup of bounded linear operators
(T(t))tP0 on X. Moreover, A has discrete spectrum, the eigenvalues are Àn2, n 2 N, with corresponding nor1=2
malized P
eigenvectors zn ðnÞ :¼ ðp2 Þ sinðnnÞ, the set fzn : n 2 Ng is an orthonormal basis of X, and
1
Àn2 t
T ðtÞx ¼ n¼1 e
< x, zn > zn for x 2 X. Consider the differential system


Z t

d
uðt; nÞ þ
a1 ðs À tÞuðs; nÞ ds
dt
À1

Z t
2 
o
¼ 2 uðt; nÞ þ
a1 ðs À tÞuðs; nÞ ds
on
À1
Z t
þ
a2 ðs À tÞuðs À q1 ðtÞq2 ðkuðtÞkÞ; nÞ ds; t 2 I ¼ ½0; aŠ; n 2 ½0; pŠ;
ð4:5Þ
À1

together with the initial conditions
uðt; 0Þ ¼ uðt; pÞ ¼ 0; t P 0;
uðs; nÞ ¼ uðs; nÞ; s 6 0; 0 6 n 6 p:

ð4:6Þ
ð4:7Þ

In the sequel, B ¼ C 0 Â L2 ðg; X Þ is the space introduced in Example 2.2; u 2 B with the identification
u(s)(s) = u(s, s); the functions ai : R ! R, qi : [0, 1) ! [0, 1), i = 1, 2, are continuous; and
!1=2
Z 0

2
ðai ðsÞÞ
ds
Li ¼
< 1; i ¼ 1; 2:
gðsÞ
À1
By defining the operators D; G; F : I Â B ! X and q : I Â B ! R by
DðwÞ ¼ wð0; nÞ À GðwÞðnÞ
Z 0
GðwÞðnÞ ¼ À
a1 ðsÞwðs; nÞ ds;
À1


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300

F ðt; wÞðnÞ ¼

Z

0

a2 ðsÞwðs; nÞ ds;
À1

qðs; wÞ ¼ s À q1 ðsÞq2 ðkwð0ÞkÞ;
we can transform system (4.5)–(4.7) into the abstract system (1.1)–(1.2). Moreover, G, F are bounded linear

operators, kGkLðB;X Þ 6 L1 and kF kLðB;X Þ 6 L2 .
The next result is an immediate consequence of Theorem 3.2.
Theorem 4.4. Let u 2 B be such that condition Hu holds and assume that
Z 0
1=2 !
1þ
gðhÞ dh
ðL1 þ aL2 Þ < 1:

ð4:8Þ

Àa

Then there exists a mild solution of (4.5)–(4.7).
The proof of Corollary 4.1 follows directly from Theorem 4.4 and Remark 3.2.
Corollary 4.1. Let u 2 B continuous and bounded, and assume that (4.8) holds. Then, there exists a mild solution
of (4.5)–(4.7) on I.
To conclude this section, we briefly consider the differential system
d
o2
½uðt; nÞ þ uðt À r; nފ ¼ 2 ½uðt; nÞ þ uðt À r; nފ þ a1 ðtÞb1 ðuðt À rðkuðtÞkÞ; nÞÞ;
dt
on
t 2 I ¼ ½0; aŠ; n 2 ½0; pŠ;

ð4:9Þ

uðt; 0Þ ¼ uðt; pÞ ¼ 0;
uðs; nÞ ¼ uðs; nÞ; s 6 0; n 2 ½0; pŠ:

ð4:10Þ
ð4:11Þ

For this system, we take u 2 B ¼ C 0g ðX Þ and assume that the functions a1 : I ! R, b1 : R Â J ! R,
r : R ! Rþ are continuous and that there exist positive constants d1, d2 such that jb1(t)j 6 d1jtj + d2 for every
t 2 R.
Let D, G : B ! X , F : ½0; aŠ  B ! X and q : ½0; aŠ  B ! R be the operators defined by D(w)(n) =
w(0, n) À G(w)(n), G(w)(n) = Àw(Àr, n), F(t, w)(n) = a1(t)b1(w(0, n)) and q(t, w) = t À r(kw(0)k). Using these
definitions, we can represent the system (4.8)–(4.10) in the abstract form (1.1) and (1.2). Moreover, G is a
bounded linear operator on B with kGðwÞkLðB;X Þ 6 gðÀrÞ; F is continuous and kF ðt; wÞk 6
a1 ðtÞ½d 1 kwkB þ d 2 pŠ for all ðt; wÞ 2 I  B. As such, the following results follow from Theorem 3.2 and Remark
3.2.
Ra
Theorem 4.5. If u 2 B satisfies condition Hu and gðÀrÞ þ d 1 0 a1 ðsÞ ds < 1, then there exists a mild solution of
(4.9)–(4.11).
Corollary 4.2. If u is continuous and bounded on (À1, 0] and gðÀrÞ þ d 1
solution of (4.9)–(4.11).

Ra
0

a1 ðsÞ ds < 1, then there exists a mild

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