Classical solutions of hyperbolic IBVPs with state dependent delays
We consider the initial boundary-value problem for a system of quasilinear partial functional differential equations of the first order. Using the method of bicharacteristics and the fixed-point theorem we prove the local existence, uniqueness and continuous dependence on data of classical solutio...
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irk-123456789-1749702021-01-30T01:26:56Z Classical solutions of hyperbolic IBVPs with state dependent delays Czernous, W. We consider the initial boundary-value problem for a system of quasilinear partial functional differential equations of the first order. Using the method of bicharacteristics and the fixed-point theorem we prove the local existence, uniqueness and continuous dependence on data of classical solutions of the problem. Розглядається гранична задача з початковими даними для системи квазiлiнiйних функцiонально-диференцiальних рiвнянь з частинними похiдними першого порядку. З допомогою методу бiфуркацiй та теореми про нерухому точку доведено локальне iснування класичних розв’язкiв задачi, їх єдинiсть та неперервну залежнiсть вiд даних 2010 Article Classical solutions of hyperbolic IBVPs with state dependent delays / W. Czernous // Нелінійні коливання. — 2010. — Т. 13, № 4. — С. 556-573. — Бібліогр.: 13 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/174970 517.9 en Нелінійні коливання Інститут математики НАН України |
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We consider the initial boundary-value problem for a system of quasilinear partial functional differential equations of the first order.
Using the method of bicharacteristics and the fixed-point theorem we prove the local existence, uniqueness and continuous dependence on data of classical solutions of the problem. |
| format |
Article |
| author |
Czernous, W. |
| spellingShingle |
Czernous, W. Classical solutions of hyperbolic IBVPs with state dependent delays Нелінійні коливання |
| author_facet |
Czernous, W. |
| author_sort |
Czernous, W. |
| title |
Classical solutions of hyperbolic IBVPs with state dependent delays |
| title_short |
Classical solutions of hyperbolic IBVPs with state dependent delays |
| title_full |
Classical solutions of hyperbolic IBVPs with state dependent delays |
| title_fullStr |
Classical solutions of hyperbolic IBVPs with state dependent delays |
| title_full_unstemmed |
Classical solutions of hyperbolic IBVPs with state dependent delays |
| title_sort |
classical solutions of hyperbolic ibvps with state dependent delays |
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Інститут математики НАН України |
| publishDate |
2010 |
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http://dspace.nbuv.gov.ua/handle/123456789/174970 |
| citation_txt |
Classical solutions of hyperbolic IBVPs with state dependent delays / W. Czernous // Нелінійні коливання. — 2010. — Т. 13, № 4. — С. 556-573. — Бібліогр.: 13 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT czernousw classicalsolutionsofhyperbolicibvpswithstatedependentdelays |
| first_indexed |
2025-07-15T12:06:26Z |
| last_indexed |
2025-07-15T12:06:26Z |
| _version_ |
1837714570505879552 |
| fulltext |
UDC 517 . 9
CLASSICAL SOLUTIONS OF HYPERBOLIC IBVPS
WITH STATE DEPENDENT DELAYS
КЛАСИЧНI РОЗВ’ЯЗКИ ГIПЕРБОЛIЧНИХ ГРАНИЧНИХ ЗАДАЧ
IЗ ПОЧАТКОВИМИ ДАНИМИ ТА ЗАПIЗНЕННЯМ,
ЩО ЗАЛЕЖИТЬ ВIД СТАНУ
W. Czernous
Inst. Math.
Univ. Gdańsk
Wit Stwosz Street 57
80-952 Gdańsk, Poland
We consider the initial boundary-value problem for a system of quasilinear partial functional differential
equations of the first order,
∂tzi(t, x) +
n∑
j=1
ρij(t, x, V (z; t, x))∂xjzi(t, x) = Gi(t, x, V (z; t, x)), 1 ≤ i ≤ m,
where V is a nonlinear operator of Volterra type, mapping bounded (w.r.t. seminorm) subsets of the space
of Lipschitz-continuously differentiable functions, into bounded subsets of this space.
Using the method of bicharacteristics and the fixed-point theorem we prove the local existence, uni-
queness and continuous dependence on data of classical solutions of the problem.
This approach covers systems of the form
∂tzi(t, x) +
n∑
j=1
ρij(t, x, zψ(t,x,z(t,x)))∂xj
zi(t, x) = Gi(t, x, zψ(t,x,z(t,x))), 1 ≤ i ≤ m,
where (t, x) 7→ z(t,x) is the Hale operator, and all the components of ψ may depend on (t, x, z(t,x)). More
specifically, problems with deviating arguments and integro-differential systems are included.
Розглядається гранична задача з початковими даними для системи квазiлiнiйних функцiональ-
но-диференцiальних рiвнянь з частинними похiдними першого порядку
∂tzi(t, x) +
n∑
j=1
ρij(t, x, V (z; t, x))∂xj
zi(t, x) = Gi(t, x, V (z; t, x)), 1 ≤ i ≤ m,
де V — нелiнiйний оператор типу Вольтерра, що вiдображає обмеженi вiдносно семiнорми пiд-
множини простору диференцiйовних функцiй, похiдна яких задовольняє умову Лiпшиця, в обме-
женi пiдмножини цього простору.
З допомогою методу бiфуркацiй та теореми про нерухому точку доведено локальне iснуван-
ня класичних розв’язкiв задачi, їх єдинiсть та неперервну залежнiсть вiд даних.
c© W. Czernous, 2010
556 ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 4
CLASSICAL SOLUTIONS OF HYPERBOLIC IBVPS WITH STATE DEPENDENT DELAYS 557
Цей пiдхiд можна застосувати до систем типу
∂tzi(t, x) +
n∑
j=1
ρij(t, x, zψ(t,x,z(t,x)))∂xj
zi(t, x) = Gi(t, x, zψ(t,x,z(t,x))), 1 ≤ i ≤ m,
де (t, x) 7→ z(t,x) — оператор Хейла i всi компоненти ψ можуть залежати вiд (t, x, z(t,x)). Зокре-
ма, вiн може бути застосовним до всiх задач з аргументами, що вiдхиляються, та iнтегро-
диференцiальних задач.
1. Introduction. We formulate the functional differential problem. Let a > 0, h0 ∈ R+, R+ =
= [0,+∞), and b = (b1, . . . , bn) ∈ Rn
+, h = (h1, . . . , hn) ∈ Rn
+ be given. We use the sets
E = [0, a]× [−b, b], D = [−h0, 0]× [−h, h].
Let c̄ = (c1, . . . , cn) = b+ h and
E0 = [−h0, 0]× [−c̄, c̄], ∂0E = [0, a]× ([−c̄, c̄] \ (−b, b)), Ω = E0 ∪ E ∪ ∂0E.
When it does not lead to misunderstading, we write Ut = U ∩ ([−∞, t] × Rn) for U ⊂ R1+n
and t ∈ [0, a]. The symbol U◦ denotes the interior of U . For k, l being arbitrary positive
integers, we denote by Mk×l the class of all k × l matrices with real elements, and we choose
the norms in Rk and Mk×l to be ∞-norms: ‖y‖ = ‖y‖∞ = max1≤i≤k |yi| and ‖A‖ = ‖A‖∞ =
= max1≤i≤k
∑l
j=1 |aij |, respectively, where A = [aij ]i=1,...,k,j=1,...,l. The product of two matri-
ces is denoted by “∗”. For U ⊂ R1+n and a normed space Y , equipped with the norm ‖ · ‖Y ,
we define C(U, Y ) to be the set of all continuous functions w : U → Y ; this space is equipped
with the usual supremum norm ‖w‖C(U,Y ) = supP∈U ‖w(P )‖Y . We write it simply C(U) when
no confusion can arise.
Put X = C(D,Rm). Let V : C(Ω,Rm) × E → X , in variables (z; t, x), be a nonlinear
Volterra operator. By the Volterra property we mean that for z, z̄ ∈ C(Ω,Rm) and t ∈ [0, a],
z
∣∣∣
Ωt
≡ z̄
∣∣∣
Ωt
implies V (z; τ, x) ≡ V (z̄; τ, x) for (τ, x) ∈ Et.
Let
ρij , Gi : E ×X → R, 1 ≤ i ≤ m, 1 ≤ j ≤ n, and ϕ : E0 ∪ ∂0E → Rm
be given. We consider the hyperbolic functional differential system
∂tzi(t, x) +
n∑
j=1
ρij(t, x, V (z; t, x)) ∂xjzi(t, x) = Gi(t, x, V (z; t, x)), 1 ≤ i ≤ m, (1)
augmented with the initial boundary condition
z(t, x) = ϕ(t, x) (2)
on E0 ∪ ∂0E. A function z̃ ∈ C1(Ωc,R), where 0 < c ≤ a, is a classical solution of (1), (2) if it
satisfies (1) on Ec and condition (2) holds on E0 ∪ ∂0Ec.
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 4
558 W. CZERNOUS
Note that different models of the functional dependence in partial equations are used in
literature. The first group of results is connected with initial problems for equations
∂tz(t, x) = G(t, x, z, ∂xz(t, x)) (3)
where the variable z represents the functional argument. This model is suitable for differential
functional inequalities generated by initial problems considered on the Haar pyramid. Exi-
stence results for (3) can be characterized as follows: theorems have simple assumptions and
their proofs are very natural (see [1, 2]). Unfortunately, a small class of differential functional
problems is covered by this theory. There are a lot of papers concerning initial value problems
for equations
∂tz(t, x) = H(t, x,W [z](t, x), ∂xz(t, x)) (4)
where W is an operator of Volterra type and H is defined on finite-dimensional Euclidean
space. The main assumptions in existence theorems for (4) concern the operator W . They are
formulated [3, 4] in terms of inequalities for norms in some functional spaces.
A new model of a functional dependence is proposed in [5, 6]. Partial equations have the
form
∂tz(t, x) = F (t, x, z(t,x), ∂xz(t, x)) (5)
where z(t,x) is a functional variable. This model is well-known for ordinary functional differenti-
al equations (see, for example, [7 – 9]). It is also very general since equations with deviating
variables, integral differential equations, and equations of forms (3) and (4) can be obtained
from (5) by specifying the operator F . In the paper we use the model (5). In existence results,
concerning partial differential equations with state dependent delays [10 – 12], Carathéodory
type or semiclassical solutions were considered and the functional variable was
z(ψ0(t),ψ′(t,x,z(t,x))).
We deal in this paper with a slightly wider class of deviating functions, admitting functional
variable of the form
z(ψ0(t,x,z(t,x)),ψ
′(t,x,z(t,x)))
and we consider classical solutions of the respective problem. Cases of more (or less) compli-
cated deviating functions are also covered by our operator formulation.
Delay systems with state dependent delays occur as models for the dynamics of diseases
when the mechanism of infection is such that the infectious dosage received by an individual
has to reach a threshold before the resistance of the individual is broken down and as a result
the individual becomes infectious. A prototype of such model was proposed in [13].
The aim of this paper is to prove a theorem on the existence and continuous dependence
of classical solutions to (1), (2). The paper is organized as follows. In Section 2 we prove a
result on the existence and regularity of bicharacteristics, having assumed our conditions on
the operator V . In the next section, the method of bicharacteristics is used to transform the
Cauchy problem into a system of integral equations. A fixed-point equation is constructed.
The Section 4 contains the main result. Application of our approach to the systems with state
dependent delays is described in the last section.
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 4
CLASSICAL SOLUTIONS OF HYPERBOLIC IBVPS WITH STATE DEPENDENT DELAYS 559
2. Bicharacteristics. Let U ⊂ R1+n and k be a positive integer. For z : U → Rk and
(t, x) ∈ U , denote
∂xz(t, x) = [∂xjzi(t, x)]i=1,...,k,j=1,...,n ∈Mk×n
and
∂z(t, x) = [∂xjzi(t, x)]i=1,...,k,j=0,...,n ∈Mk×(n+1),
where ∂x0 ≡ ∂t.
For a fixed p ∈ R+, we consider the space
C1[p](U,Rk) =
{
w ∈ C(U,R) : w is continuously differentiable on U◦ and ‖∂w‖C(U◦) ≤ p
}
.
Similarly, for p = (p1, p2) ∈ R2
+, we define
C1,L[p](U,Rk) =
{
w ∈ C1[p1](U,Rk) : |∂w|C0,L(U◦) ≤ p2
}
,
where |z|C0,L(U) = supP 6=P̄ ;P,P̄∈U ‖z(P )− z(P̄ )‖ · ‖P − P̄‖−1. We denote
C1,L(U,Rk) =
⋃
p∈R2
+
C1,L[p](U,Rk).
We are now able to define the function space, in which we seek the solutions to (1), (2). The
symbol C1,L
∂ [p] is short for C1,L[p](E0 ∪ ∂0E,Rm). Given p ∈ R2
+, ϕ ∈ C1,L
∂ [p], and d ∈ R2
+
such that dj ≥ pj , j = 1, 2, we set
C1,L
ϕ,c [d] =
{
z ∈ C1,L[d](Ωc,Rm) : z ≡ ϕ on E0 ∪ ∂0Ec
}
.
We prove that under suitable assumptions on ρ, G, V , ϕ, on the parameters p, d, and for suffici-
ently small c ∈ (0, a], there exists a solution z̄ of problem (1), (2) such that z̄ ∈ C1,L
ϕ,c [d].
Let Y stand forC(D,Mm×(n+1)). Write ρi = (ρi1, . . . , ρin), 1 ≤ i ≤ m. For the convenience
of calculations, we consider m Fréchet derivatives ∂wρi(t, x, w) ∈ L(X,Rn), 1 ≤ i ≤ m, rather
than mn Fréchet derivatives ∂wρij(t, x, w), 1 ≤ i ≤ m, 1 ≤ j ≤ n. We are interested in
estimating it in the norm ‖ · ‖L(Y,Mn×(n+1)), since we use the notation
∂wρi(t, x, w)δ =
(
∂wρi(t, x, w)δ0, . . . , ∂wρi(t, x, w)δn
)
∈ Mn×(n+1) (6)
for δ ∈ Y , δ = (δ0, . . . , δn), δj ∈ X , 0 ≤ j ≤ n.
Assumption H [ρ]. Suppose that ρ : E×X → Mm×n, in the variables (t, x, w), is continuous
and
1) the derivatives: ∂xρi(t, x, w) and the Fréchet derivative ∂wρi(t, x, w) exist for (t, x, w) ∈
∈ E × C1,L(D,Rm), 1 ≤ i ≤ m,
2) for 1 ≤ i ≤ m, ∂xρi and ∂wρi are continuous in t on E × C1,L(D,Rm),
3) there is a non-negative constant A such that, for 1 ≤ i ≤ m,
‖ρi(t, x, w)‖, ‖∂xρi(t, x, w)‖, ‖∂wρi(t, x, w)‖L(Y,Mn×(n+1)) ≤ A on E × C1,L(D,Rm)
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 4
560 W. CZERNOUS
and
‖∂xρi(t, x, w)− ∂xρi(t, x̄, w̄)‖,
‖∂wρi(t, x, w)− ∂wρi(t, x̄, w̄)‖L(Y,Mn×(n+1)) ≤ A
(
‖x− x̄‖+ ‖w − w̄‖X
)
for (t, x, w), (t, x̄, w̄) ∈ E × C1,L(D,Rm),
4) there is κ > 0 such that, for 1 ≤ i ≤ m, 1 ≤ j ≤ n,
ρij(t, x, w) < −κ on [0, a]× {bj} × C1,L(D,Rm),
ρij(t, x, w) > κ on [0, a]× {−bj} × C1,L(D,Rm).
Assumption H [V ]. The operator V : C(Ω,Rm) × E → X is such that for every d ∈ R2
+
there are d̄ ∈ R2
+, L ∈ R+ such that:
1) for z ∈ C1[d1](Ω,Rm), (t, x) ∈ E,
‖∂V (z; t, x)‖Y ≤ d̄1,
2) for z ∈ C1,L[d](Ω,Rm) and (t, x), (t, x̄) in E,
‖∂V (z; t, x)‖Y ≤ d̄1, ‖∂V (z; t, x)− ∂V (z; t, x̄)‖Y ≤ d̄2‖x− x̄‖,
3) for every z, z̄ ∈ C1[d1](Ω,Rm) and (t, x) ∈ E,
‖V (z; t, x)− V (z̄; t, x)‖X ≤ L‖z − z̄‖C(Et)
.
Suppose that ϕ ∈ C1,L
∂ [p] and z ∈ C1,L
ϕ,c [d]. For 1 ≤ i ≤ m, and a point (t, x) ∈ Ec, we
consider the Cauchy problem
η′(τ) = ρi(τ, η(τ), V (z; τ, η(τ))), η(t) = x, (7)
and denote by gi[z](·, t, x) = (gi1[z](·, t, x), . . . , gin[z](·, t, x)) its classical solution. This function
is the bicharacteristic of the i-th equation of (1), corresponding to z. Let δi[z](t, x) be the left
end of the maximal interval on which the bicharacteristic gi[z](·, t, x) is defined. Write
Qi[z](τ, t, x) = (τ, gi[z](τ, t, x), V (z; τ, gi[z](τ, t, x))).
We prove a lemma on bicharacteristics.
Lemma 2.1. Suppose that Assumptions H[ρ], H[V ] are satisfied and let ϕ, ϕ̄ ∈ C1,L
∂ [p]
and z ∈ C1,L
ϕ,c [d], z̄ ∈ C1,L
ϕ̄.c [d], be given. Then, for 1 ≤ i ≤ m, the solutions gi[z](·, t, x)
and gi[z̄](·, t, x) exist on intervals [δi[z](t, x), c] and [δi[z̄](t, x), c], respectively, and are unique.
If ξ = δi[z](t, x) > 0 then gi[z](ξ, t, x) ∈ ∂0E ∩ E. Moreover, the estimates
‖∂gi[z](τ, t, x)‖ ≤ C, ‖∂gi[z](τ, t, x)− ∂gi[z](τ, t̄, x̄)‖ ≤ Qmax{|t− t̄|, ‖x− x̄‖} (8)
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 4
CLASSICAL SOLUTIONS OF HYPERBOLIC IBVPS WITH STATE DEPENDENT DELAYS 561
and
‖gi[z](τ, t, x)− gi[z̄](τ, t, x)‖ ≤ Ā
∣∣∣∣∣
τ∫
t
‖z − z̄‖C(Es) ds
∣∣∣∣∣, (9)
|δi[z](t, x)− δi[z̄](t, x)|, ≤ 2Āκ−1
∣∣∣∣∣
t∫
0
‖z − z̄‖C(Es) ds
∣∣∣∣∣ (10)
hold with constants depending only on data and on c, d, p :
C = (A+ 1)ecB, Q = [(1 + C)B + C̃]ecB, Ā = ALecB, (11)
where
B = A(1 + d̄1), C̃ = C2cA[(1 + d̄1)2 + d̄2] (12)
and d̄ = (d̄1, d̄2) ∈ R2
+ is the parameter from Assumption H[V ], corresponding to d.
Proof. Let z ∈ C1,L
ϕ,c [d]. The existence and uniqueness of solutions of (7) follow from the
theorem on classical solutions of initial problems. From another classical theorem on differenti-
ation of solutions with respect to the initial data it follows that the derivative ∂gi[z] exists and
fulfils the integral equations
∂gi[z](τ, t, x) =
[
− ρi(t, x, V (z; t, x))
∣∣ I ]
+
+
τ∫
t
[
∂xρi(Qi[z](s, t, x)) + ∂wρi(Qi[z](s, t, x))∂V (z; s, gi[z](s, t, x))
]
∗
∗ ∂gi[z](s, t, x)ds (13)
where
[
− ρi(t, x, V (z; t, x))
∣∣ I ]
denotes concatenation of the matrix −ρi(t, x, V (z; t, x)) with
the identity matrix. It follows from (13), from Assumptions H[ρ], H[V ] that ∂gi[z](·, t, x) satisfy
the integral inequality
‖∂gi[z](τ, t, x)‖ ≤ A+ 1 +B
∣∣∣∣∣
τ∫
t
‖∂gi[z](s, t, x)‖ds
∣∣∣∣∣,
and from the Gronwall lemma we get the first estimate in (8). Hence we derive the inequality
‖∂gi[z](τ, t, x)− ∂gi[z](τ, t̄, x̄)‖ ≤ (B + C̃) max{|t− t̄|, ‖x− x̄‖}+ CB|t− t̄|+
+B
∣∣∣∣∣∣
τ∫
t
‖∂gi[z](s, t, x)− ∂gi[z](s, t̄, x̄)‖ds
∣∣∣∣∣∣
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 4
562 W. CZERNOUS
which, by the Gronwall lemma, implies that
‖∂gi[z](τ, t, x)− ∂gi[z](τ, t̄, x̄)‖ ≤ Q1 max{|t− t̄|, ‖x− x̄‖}+Q0|t− t̄| ≤
≤ (Q0 +Q1) max{|t− t̄|, ‖x− x̄‖},
with Q0 = CB exp(cB) and Q1 = (C̃ +B) exp(cB), yielding the second estimate in (8).
We now prove (9). The function gi[z](τ, t, x) satisfies the following relation:
gi[z](τ, t, x) = x+
τ∫
t
ρi(s, gi[z](s, t, x), V (z; s, gi[z](s, t, x)))ds.
This leads to
‖gi[z](τ, t, x)− gi[z̄](τ, t, x)‖ ≤ B
∣∣∣∣∣
τ∫
t
‖gi[z](s, t, x)− gi[z̄](s, t, x)‖ds
∣∣∣∣∣+
+AL
∣∣∣∣∣
τ∫
t
‖z − z̄‖C(Es) ds
∣∣∣∣∣.
Again from the Gronwall inequality we obtain
‖gi[z](τ, t, x)− gi[z̄](τ, t, x)‖ ≤ AL exp(cB)
∣∣∣∣∣
τ∫
t
‖z − z̄‖C(Es) ds
∣∣∣∣∣,
and hence (9).
Now we proceed to the proof of (10), fixing (t, x) ∈ Ec and beginning by a local version of
the estimate, that is, under the condition that
‖z − z̄‖C(Et)
≤ κ
2(cBĀ+AL)
. (14)
Since (10) is obvious if both values of δ are zero, we may assume that 0 ≤ δi[z](t, x) < δi[z̄](t, x).
Denoting ζ = δi[z̄](t, x), we then have |gij [z̄](ζ, t, x)| = bj for some coordinate j. Let us focus
on the case gij [z̄](ζ, t, x) = bj ; the opposite case gij [z̄](ζ, t, x) = −bj is then treated analogously.
By virtue of (9) and (14),
|ρij(Qi[z](ζ, t, x))− ρij(Qi[z̄](ζ, t, x))| ≤ B‖gi[z](ζ, t, x)− gi[z̄](ζ, t, x)‖+AL‖z − z̄‖C(Et)
≤
≤ BĀ
t∫
0
‖z − z̄‖C(Es) ds+AL‖z − z̄‖C(Et)
≤ κ
2
.
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CLASSICAL SOLUTIONS OF HYPERBOLIC IBVPS WITH STATE DEPENDENT DELAYS 563
This, together with the condition 4 of Assumption H [ρ] applied to ρij(Qi[z̄](ζ, t, x)), gives
ρij(Qi[z](ζ, t, x)) ≤ ρij(Qi[z̄](ζ, t, x))+
+ |ρij(Qi[z](ζ, t, x))− ρij(Qi[z̄](ζ, t, x))| < −κ+
κ
2
= −κ
2
.
Consequently,
∂tgij [z](ζ, t, x) < −κ
2
< 0,
and hence gij [z](·, t, x) is decreasing on the interval (ξ, ζ) for some ξ ∈ [δi[z](t, x), ζ). This fact,
and the estimate
bj − gij [z](ζ, t, x) = gij [z̄](ζ, t, x)− gij [z](ζ, t, x) ≤
≤ cĀ‖z − z̄‖C(Et)
≤ κcĀ
2(cBĀ+AL)
≤ κ
2B
≤ κ
2A
,
imply
bj − gij [z](s, t, x) ≤
κ
2A
for s ∈ (ξ, ζ].
Let us now define β : Rn → Rn by β(x) = (x1, . . . , xj−1, bj , xj+1, . . . , xn) and notice that the
condition 4 of Assumption H [ρ] may be applied to give
ρij(s, β(gi[z](s, t, x)), V (z; s, gi[z](s, t, x))) < −κ, s ∈ (ξ, ζ].
Then, for s ∈ (ξ, ζ],
ρij(Qi[z](s, t, x)) = ρij(s, gi[z](s, t, x), V (z; s, gi[z](s, t, x))) ≤
≤ ρij(s, β(gi[z](s, t, x)), V (z; s, gi[z](s, t, x)))+
+A
(
bj − gij [z](s, t, x)
)
≤
≤ −κ+A
κ
2A
= −κ
2
.
Note that the last inequality implies ξ = δi[z](t, x), otherwise it would be 0 = ∂tgij [z](ξ, t, x) =
= ρij(Qi[z](ξ, t, x)) for some ξ ∈ (δi[z](t, x), ζ). Hence this inequality holds for s ∈ [δi[z](t, x),
δi[z̄](t, x)], yielding
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−κ
2
(
δi[z̄](t, x)− δi[z](t, x)
)
≥
≥
δi[z̄](t,x)∫
δi[z](t,x)
ρij(Qi[z](s, t, x)) ds =
= gij [z](δi[z̄](t, x), t, x)− gij [z](δi[z](t, x), t, x) ≥
≥ gij [z](δi[z̄](t, x), t, x)− gij [z̄](δi[z̄](t, x), t, x) ≥
≥ −Ā
∣∣∣∣∣
δi[z̄](t,x)∫
t
‖z − z̄‖C(Es) ds
∣∣∣∣∣ ≥
≥ −Ā
t∫
0
‖z − z̄‖C(Es) ds.
Since this proves (10) for z, z̄ satisfying (14), the argument of convexity of C1,L
∂ [p] and
C1,L[d′](Ωc,Rm) completes the proof.
Lemma 2.2. Suppose that Assumptions H[ρ], H[V ] are satisfied and let ϕ ∈ C1,L
∂ [p], z ∈
∈ C1,L
ϕ,c [d] be given. Then, for 1 ≤ i ≤ m,
δi[z] ∈ C(Ec) ∩ C1[B̃](E(i,∂)
c [z],R),
where
E(i,∂)
c [z] = {(t, x) ∈ Ec : δi[z](t, x) > 0}◦ and B̃ = Cκ−1.
Proof. Fix z ∈ C1,L
ϕ,c [d] and i, 1 ≤ i ≤ m. Once it is done, we may introduce the notation:
f = δi[z], U for the set E(i,∂)
c [z], and W for the set
E(i,0)
c [z] = {(t, x) ∈ Ec : δi[z](t, x) = 0} .
We first prove that f ∈ C1[B̃](U,R). Let us temporarily fix (t, x) ∈ U and set ξ = f(t, x). Since
by the Lemma 2.1, gi[z] is of class C1 in all variables, and by Assumption H[ρ], ∂τgi[z](τ, t, x) 6=
6= 0 at (ξ, t, x), the existence and continuity of the gradient ∂f at the point (t, x) follow from the
implicit function theorem applied to ±bj − gij [z](ξ, t, x) = 0, 1 ≤ j ≤ n. By the same token,
fixing j, we may calculate the gradient with the help of the formula
∂xk
f(t, x) = − ∂xk
gij [z](ξ, t, x)
ρij(Qi[z](ξ, t, x))
, 0 ≤ k ≤ n,
and estimate it by
‖∂f‖C(U) ≤
C
κ
,
independently of i, j.
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CLASSICAL SOLUTIONS OF HYPERBOLIC IBVPS WITH STATE DEPENDENT DELAYS 565
Remark that
Ec = U ∪W ∪ {c} × (−b, b) ∪
n⋃
k=1
∆(k),
where ∆(k) = {(t, x) ∈ Ec : |xk| = bk}, 1 ≤ k ≤ n. The continuity of f on W is obvious, and
on U follows from differentiability. Extending ρ, V and z in a natural way, we could replace Ec
with [−ε, c]× [−b, b] in the formulation of (7) and in the consequent results on bicharacteristics
— including differentiability of the new function f̃ = δ̃i[z] on the set analogous to E(i,∂)
c [z] and
containing f̃−1({0}). By the uniqueness of bicharacteristics, f̃ is identical with f on U. Hence,
for (t, x) ∈ f̃−1({0}) = U ∩W, the one-sided limit vanishes:
lim
(t̄,x̄)→(t,x)
(t̄,x̄)∈U
f(t̄, x̄) = 0.
The continuity of f on U ∪W follows now from the fact that f vanishes on the other side, i.e.,
on W = W . Then, by extending the data in the opposite direction, we get the continuity on
U ∪W ∪ {c} × (−b, b). Finally, from the condition 4 of the Assumption H[ρ] follows easily that
f satisfies the local Lipschitz condition, with the uniform constant max{1, Cκ−1}, at each point
of ∆(k), 1 ≤ k ≤ n. This shows that f ∈ C(Ec) and completes the proof.
3. Functional integral system. Let W stand for L(C(D,Mm×n),M1×n). The expression
∂wGi(t, x, w)δ, for δ ∈ C(D,Mm×n), is to be interpreted in a way analogous to (6); for the
sake of simplicity of calculations, we use ‖ ·‖W (rather than ‖ ·‖L(X,R)) for measuring the values
of ∂wGi.
Assumption H [ρ,G]. The Assumption H[ρ] is fulfilled, G : E ×X → Rm, in the variables
(t, x, w), is continuous and, for 1 ≤ i ≤ m,
1) the derivative ∂xG(t, x, w) and the Fréchet derivative ∂wG(t, x, w) exist for (t, x, w) ∈
∈ E × C1,L(D,Rm),
2) for (t, x, w), (t, x̄, w̄) ∈ E × C1,L(D,Rm),
‖G(t, x, w)‖, ‖∂xG(t, x, w)‖, ‖∂wGi(t, x, w)‖W ≤ A,
‖Gi(t, x, w)−Gi(t̄, x, w)‖ ≤ A|t− t̄|,
‖∂xGi(t, x, w)− ∂xGi(t, x̄, w̄)‖, ‖∂wGi(t, x, w)− ∂wGi(t, x̄, w̄)‖W ≤ A
(
‖x− x̄‖+ ‖w − w̄‖X
)
with the same constant A as in the Assumption H[ρ].
Write
Si[z](t, x) = ( δi[z](t, x), gi[z](δi[z](t, x), t, x) ).
We define the operator F = (F1, . . . , Fm) on C1,L
ϕ,c [d] by the formula
Fiz(t, x) = ϕi(Si[z](t, x)) +
t∫
δi[z](t,x)
Gi(Qi[z](s, t, x))ds on Ec, 1 ≤ i ≤ m, (15)
Fz ≡ ϕ on E0 ∪ ∂0Ec.
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Remark 3.1. The right-hand side of (15) is obtained in the following way. We consider each
equation of (1) along its bicharacteristic:
∂tzi(τ, gi[z](τ, t, x)) + ∂xzi(τ, gi[z](τ, t, x)) ∗ ρi(τ, gi[z](τ, t, x), V (z; τ, gi[z](τ, t, x))) =
= Gi(τ, gi[z](τ, t, x), V (z; τ, gi[z](τ, t, x)))
from which, using (7), we get
d
dτ
zi(τ, gi[z](τ, t, x)) = Gi(τ, gi[z](τ, t, x), V (z; τ, gi[z](τ, t, x))).
By integrating the latter equation with respect to τ , and adding the initial value, we get the
right-hand side of (15).
Assumption H[c, d, V ]. The Assumption H[V ] is fulfilled, and the constant c ∈ (0, a] is
small enough so to satisfy, together with d and p,
d1 ≥ p1C +A+ cCB, (16)
d2 ≥ p1(Q+BĈ) + p2(C2 + ÂĈ) +B + cBQ+ C̃ + 2BĈ, (17)
with constantsB,C, C̃,Q defined in (11), (12), and with Ĉ = Cmax{1, Cκ−1}, Â = max{1, A}.
Write I = {0 ≤ i ≤ n : hi > 0}. The following compatibility condition for the problem
(1), (2) will be needed in our considerations.
Assumption Hc[G,ϕ]. The equivalence
G(t, x, V (z; t, x)) = G(t, x, V (z̄; t, x)) on ∂0E ∩ E
holds for any z, z̄ ∈ C1,L
ϕ,a [d]. Moreover, there is ψ ∈ C(∂0E,Mm×n) such that, for each k,
1 ≤ k ≤ n, the system of equations
∂tϕi(t, x) +
n∑
j=1
ρij(t, x, V (z; t, x))ψij(t, x) = Gi(t, x, V (z; t, x)), 1 ≤ i ≤ m, (18)
holds on ∆(k) = {(t, x) ∈ E : |xk| = bk} with
ψij ≡ ∂xjϕi on ∆(k) whenever j ∈ I.
Additionally, if 0 ∈ I, then on {0} × (−b, b) holds
∂tϕi(t, x) + ∂xϕi(t, x) ∗ ρi(t, x, V (z; t, x)) = Gi(t, x, V (z; t, x)), 1 ≤ i ≤ m. (19)
Remark 3.2. Relation (18) may be considered as an assumption on ϕ on
⋃
k∈I ∆(k) and
(18) defines the number ∂xjϕi(t, x) at the points where there is not enough space to define the
partial derivative.
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CLASSICAL SOLUTIONS OF HYPERBOLIC IBVPS WITH STATE DEPENDENT DELAYS 567
Remark 3.3. Let us explain our application of the chain rule to the term ϕi(Si[z](t, x)), made
in the next proof. We write
E(ij,∂)
c [z] =
{
(t, x) ∈ E(i,∂)
c [z] : Si[z](t, x) ∈ (∆(j))◦
}
, 1 ≤ j ≤ n.
Note that for (t, x) ∈ E
(ij,∂)
c [z] we shall not use the partial derivative ∂xjϕi in the expansion of
the differential
d
dxk
ϕi(Si[z](t, x))
(
k is fixed, 0 ≤ k ≤ n
)
.
Fortunately, for those (t, x) the differential
d
dxk
gij [z](δi[z](t, x), t, x) = 0, 0 ≤ k ≤ n,
and the number ∂xjϕi(Si[z](t, x)) is defined even for j /∈ I (by compatibility condition), thus it
is justified to write
d
dxk
ϕi(Si[z](t, x)) = ∂tϕi(Si[z](t, x))∂xk
δi[z](t, x)+
+
n∑
j=1
∂xjϕi(Si[z](t, x))
d
dxk
gij [z](δi[z](t, x), t, x).
Lemma 3.1. Suppose that Assumptions H[ρ,G], H[c, d, V ], Hc[G,ϕ] are satisfied. Then the
operator F maps C1,L
ϕ,c [d] into itself.
Proof. Let z ∈ C1,L
ϕ,c [d]. Write
Φi[z](s, t, x) = ∂xGi(Qi[z](s, t, x)) + ∂wGi(Qi[z](s, t, x)) ∂xV (z; s, gi[z](s, t, x)),
where ∂wGi(Qi)∂xV (z; τ, y) is to be interpreted column-wise. Fix (t, x) ∈ Ec. Once it is done,
we may introduce the notation gi = gi[z](·, t, x) and δi = δi[z](t, x). From (15) and by the
Remark 3.3, for (t, x) ∈ E
(i,∂)
c [z],
∂Fiz(t, x) =
=
[
∂tϕi(δi, gi(δi)) + ∂xϕi(δi, gi(δi)) ∗ ρi(Qi[z](δi, t, x))−Gi(Qi[z](δi, t, x))
]
∂δi[z](t, x)+
+ ∂xϕi(δi, gi(δi)) ∗ ∂gi[z](δi, t, x) +
[
Gi(t, x, V (z; t, x))
∣∣ 0
]
+
+
t∫
δi[z](t,x)
Φi[z](s, t, x) ∗ ∂gi[z](s, t, x) ds, (20)
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568 W. CZERNOUS
where
[
Gi(t, x, V (z; t, x))
∣∣ 0
]
=
(
Gi(t, x, V (z; t, x)), 0, . . . , 0
)
∈ R1+n. Moreover, on
E
(i,0)
c [z] ∩ E◦c ,
∂Fiz(t, x) = ∂xϕi(0, gi(0)) ∗ ∂gi[z](0, t, x) +
[
Gi(t, x, V (z; t, x))
∣∣ 0
]
+
+
t∫
0
Φi[z](s, t, x) ∗ ∂gi[z](s, t, x) ds. (21)
Due to the compatibility condition, and by the continuity of δi[z],
∂Fiz(t, x) = ∂xϕi(δi, gi(δi)) ∗ ∂gi[z](δi, t, x) +
[
Gi(t, x, V (z; t, x))
∣∣ 0
]
+
+
t∫
δi[z](t,x)
Φi[z](s, t, x) ∗ ∂gi[z](s, t, x) ds, (22)
on E◦c .
It follows that ‖∂Fz(t, x)‖ ≤ p1C + CBc+ A on E◦c , which, by the Assumption H[c, d, V ],
implies ‖∂Fz‖C(E◦
c )
≤ d1. Furthermore, for 1 ≤ i ≤ m and for (t, x), (t̄, x̄) ∈ E◦c ,
‖∂Fiz(t, x)− ∂Fiz(t̄, x̄)‖ ≤
≤
∥∥∥∥∂xϕi(Si[z](t, x)) ∗ ∂gi[z](δi[z](t, x), t, x)− ∂xϕi(Si[z](t̄, x̄)) ∗ ∂gi[z](δi[z](t̄, x̄), t̄, x̄)
∥∥∥∥+
+
∣∣∣∣Gi(t, x, V (z; t, x))−Gi(t̄, x̄, V (z; t̄, x̄))
∣∣∣∣+
+
t∫
δi[z](t,x)
∥∥∥∥Φi[z](s, t, x) ∗ ∂gi[z](s, t, x)− Φi[z](s, t̄, x̄) ∗ ∂gi[z](s, t̄, x̄)
∥∥∥∥ds+
+
∣∣∣∣∣
t̄∫
t
∥∥∥∥Φi[z](s, t̄, x̄) ∗ ∂gi[z](s, t̄, x̄)
∥∥∥∥ds
∣∣∣∣∣ +
∣∣∣∣∣
δi[z](t̄,x̄)∫
δi[z](t,x)
∥∥∥∥Φi[z](s, t̄, x̄) ∗ ∂gi[z](s, t̄, x̄)
∥∥∥∥ds
∣∣∣∣∣.
Note that the Lemma 2.2 gives
|δi[z](t, x)− δi[z](t̄, x̄)| ≤ Cκ−1 max{|t− t̄|, ‖x− x̄‖}.
From the above inequalities, Assumption H[ρ,G] and Lemma 2.1 it follows that
‖∂Fiz(t, x)− ∂Fiz(t̄, x̄)‖ ≤
≤
(
p1(Q+BĈ) + p2(C2 + ÂĈ) +B + cBQ+ C̃ +BĈ
)
max{|t− t̄|, ‖x− x̄‖}+BC|t− t̄|,
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CLASSICAL SOLUTIONS OF HYPERBOLIC IBVPS WITH STATE DEPENDENT DELAYS 569
for (t, x), (t̄, x̄) ∈ E◦c , which, in view of the second inequality from the Assumption H[c, d, V ],
gives |∂Fz|C0,L(E◦
c )
≤ d2.
The fact that Fiz are continuous extensions of ϕi, is a simple consequence of the definiti-
on (15); it remains to prove that this extension is of class C1. From (13), (22), and from the
compatibility condition (18) we obtain for (t, x) ∈ ∆(k), 1 ≤ k ≤ n,
lim
(t̄,x̄)→(t,x)
(t̄,x̄)∈E◦
c
∂Fiz(t̄, x̄) = ∂xϕi(t, x) ∗ ∂gi[z](t, t, x) +
[
Gi(t, x, V (z; t, x))
∣∣ 0
]
=
= ∂xϕi(t, x) ∗
[
− ρi(t, x, V (z; t, x))
∣∣ I ]
+
[
Gi(t, x, V (z; t, x))
∣∣ 0
]
=
= ∂ϕi(t, x), 1 ≤ i ≤ m.
If 0 ∈ I, then similar arguments, incurring (19), apply to the case (t, x) ∈ {0} × (−b, b).
Lemma 3.1 is proved.
4. Existence of solutions.
Theorem 4.1. Suppose that ϕ ∈ C1,L
∂ [p], and Assumptions H[ρ,G], H[c, d, V ], Hc[G,ϕ] are
satisfied. Then there exists exactly one solution z̄ ∈ C1,L
ϕ,c [d] of problem (1), (2). Moreover, there
is Λc ∈ R+ such that
‖z̄ − v‖C(Et)
≤ Λc‖ϕ− ψ‖C(E0∪∂0Et), 0 ≤ t ≤ c, (23)
for v ∈ C1,L
ϕ̄,c [d] being a solution of (1) with the initial boundary condition (2) with ϕ replaced by
ψ ∈ C1,L
∂ [p].
Proof. We prove that there exists exactly one z̄ ∈ C1,L
ϕ,c [d] satisfying the equation z = F [z].
Lemma 3.1 shows that F : C1,L
ϕ,c [d] → C1,L
ϕ,c [d]. From the definition (15) of F , from the Lipschitz
continuity of ϕi, and from the Lipschitz continuity (see (9), (10)) of gi and δi with respect to z,
follows easily the existence of an L∗ > 0 such that
‖Fiz(t, x)− Fiz̃(t, x)‖ ≤ L∗
t∫
0
‖z − z̃‖C(Es) ds (24)
for z, z̃ ∈ C1,L
ϕ,c [d], (t, x) ∈ Ec, 1 ≤ i ≤ m. Let λ > L∗. We define a metric in C1,L
ϕ,c [d] by
dλ(z, z̃) = sup
{
‖(z − z̃)(t, x)‖e−λt : (t, x) ∈ Ec
}
.
We now prove that there exists q ∈ [0, 1) such that
dλ(Fz, F z̃) ≤ qdλ(z, z̃). (25)
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According to (24),
‖Fz(t, x)− F z̃(t, x)‖ ≤ L∗
t∫
0
‖z − z̃‖C(Es) ds = L∗
t∫
0
‖z − z̃‖C(Es) e
−λseλsds ≤
≤ L∗dλ(z, z̃)
t∫
0
eλsds =
L∗
λ
dλ(z, z̃)(eλt − 1) ≤ L∗
λ
dλ(z, z̃)eλt
for (t, x) ∈ Ec. Then
‖Fz(t, x)− F z̃(t, x)‖e−λt ≤ L∗
λ
dλ(z, z̃) for all (t, x) ∈ Ec,
which gives (25) with q = L∗λ−1. By the Banach fixed point theorem, there exists a unique
fixed point of F . Denoting this fixed point by z̄, we have for (t, x) ∈ Ec
z̄i(t, x) = ϕi(δi[z̄](t, x), gi[z̄](δi[z̄](t, x), t, x))+
+
t∫
δi[z̄](t,x)
Gi(s, gi[z̄](s, t, x), V (z̄; s, gi[z̄](s, t, x)))ds, 1 ≤ i ≤ m.
Now put ζ = δi[z̄](t, x). For a given x ∈ [−b, b], let us denote y = gi[z̄](ζ, t, x). It follows from
Lemma 2.1 that gi[z̄](s, t, x) = gi[z̄](s, ζ, y) for s, t ∈ [ζ, c] and x = gi[z̄](t, ζ, y). Then we get
z̄i(t, gi[z̄](t, ζ, y)) = ϕi(ζ, y) +
t∫
ζ
Gi(s, gi[z̄](s, ζ, y), V (z̄; s, gi[z̄](s, ζ, y)))ds, 1 ≤ i ≤ m.
(26)
Relations y = gi[z̄](ζ, t, x) and x = gi[z̄](t, ζ, y) are equivalent for x, y ∈ [−b, b]. By differenti-
ating (26) with respect to t and putting again x = gi[z̄](t, ζ, y) we conclude that z̄ satisfies (1).
Since z̄ satisfies initial boundary condition (2), it is a solution of our problem.
We now prove the relation (23). The function v satisfies the integral functional system
z(t, x) = Fz(t, x)
and initial boundary condition (2) with ψ instead of ϕ. It follows easily that there is Λ ∈ R+
such that the integral inequality
‖z̄ − v‖C(Et)
≤ ‖ϕ− ψ‖C(E0∪∂0Et) + Λ
t∫
0
‖z̄ − v‖C(Es) ds, 0 ≤ t ≤ c,
is satisfied. Using the Gronwall inequality, we obtain (23) with Λc = exp(Λc).
Theorem 4.1 is proved.
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CLASSICAL SOLUTIONS OF HYPERBOLIC IBVPS WITH STATE DEPENDENT DELAYS 571
Remark 4.1. Inequalities (16), (17), given in the Assumption H[c, d, V ], have the following
impact on the conditions on the operator V .
We indicate, how to solve those inequalities. Put, for example, d1 = A + 2(1 + A)(1 + p1);
the condition 1 of Assumption H[V ] produces then a corresponding constant d̄1 ≥ 0. Having
performed easy calculations, one can see that condition c ≤ A−1(1 + d̄1)−1 log 2, on c, assures
the fulfilment of (16).
After the construction of d̄1, an example of a suitable value of d2, in terms of d̄1 and of
given constants, may be found using (17) (we shall assume that cd̄2 is appropriately bounded).
Since at this stage d1 and d2 are fixed, the condition 2 of Assumption H[V ] gives d̄2. This leads
to one more constraint on c, of which we assume the stronger one.
The above explained dependence of choice of d2 on d̄1 shows that the condition 2 of the
considered Assumption does not suffice for solvability of inequalities from Assumption H [c,
d, V ], but that the condition 1 has to be added.
5. Systems with state dependent delays. Suppose that z : Ω → R and (t, x) ∈ E are fixed.
We define the function z(t,x) : D → R as follows:
z(t,x)(τ, ξ) = z(t+ τ, x+ ξ), (τ, ξ) ∈ D.
The function z(t,x) is the restriction of z to the set [t− h0, t]× [x− h, x+ h] and this restriction
is shifted to the set D. For z : Ω → Rm, z = (z1, . . . , zm), write z(t,x) = ((z1)(t,x), . . . , (zm)(t,x)).
Let ψij : E × C1,L(D,Rm) → R, 1 ≤ i ≤ m, 0 ≤ j ≤ n, be given. Consider the function(
(z1)ψ1(t,x,w), . . . , (zm)ψm(t,x,w)
)
∈ X,
where ψi = (ψi0, . . . , ψin), 1 ≤ i ≤ m, and z : Ω → Rm. We write it zψ(t,x,w) for brevity. We
show that the operator V , defined by
V (z; t, x) = zψ(t,x,z(t,x)) (27)
satisfies Assumption H[V ], provided that certain regularity conditions on ψ are met.
Assumption H [ψ]. Deviating function ψ : E × X → Mm×(n+1) is continuous and, for
1 ≤ i ≤ m,
1) the relations ψi(t, x, w) ∈ Et hold on E ×X ,
2) derivatives: ∂ψi and the Fréchet derivative ∂wψi exist on E × C1,L(D,Rm),
3) there is a non-negative constantA1 independent of i and such that, for (t, x, w), (t, x̄, w̄) ∈
∈ E × C1,L(D,Rm),
‖∂ψi(t, x, w)‖, ‖∂wψi(t, x, w)‖L(Y,M(n+1)×(n+1)) ≤ A1
and
‖∂ψi(t, x, w)− ∂ψi(t, x̄, w̄)‖, ‖∂wψi(t, x, w)− ∂wψi(t, x̄, w̄)‖L(Y,M(n+1)×(n+1))
are bounded from above by A1
(
‖x− x̄‖+ ‖w − w̄‖X
)
.
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572 W. CZERNOUS
In view of the above Assumption, differentiation of (27) gives
∂Vi(z; t, x) ≡ (∂zi)ψi(t,x,z(t,x)) ∗
[
∂ψi(t, x, z(t,x)) + ∂wψi(t, x, z(t,x))(∂z)(t,x)
]
on D,
and, consequently, for z ∈ C1,L
ϕ,c [d] and (t, x), (t, x̄) ∈ E, ‖∂Vi(z; t, x)‖C(D,M1×(n+1)) ≤ d1A1(1+
+d1) and
‖∂Vi(z; t, x)− ∂Vi(z; t, x̄)‖C(D,M1×(n+1)) ≤ A1[d1d2 + (1 + d1)2(d1 + d2A1)]‖x− x̄‖.
Taking maximum (w.r.t. i) on the left-hand sides of these estimates, we obtain the conditions
1, 2 of Assumption H[V ] with d̄1 = d1A1(1 + d1) and d̄2 = A1[d1d2 + (1 + d1)2(d1 + d2A1)].
Fulfilment of the condition 3 of that Assumption follows from the estimates
‖Vi(z; t, x)− Vi(z̄; t, x)‖C(D) ≤ ‖(zi)ψi(t,x,z(t,x)) − (zi)ψi(t,x,z̄(t,x))‖C(D)+
+ ‖(zi − z̄i)ψi(t,x,z̄(t,x))‖C(D) ≤
≤ d1A1‖z − z̄‖C(Et)
+ ‖zi − z̄i‖C(Et)
≤
≤ (d1A1 + 1)‖z − z̄‖C(Et)
, 1 ≤ i ≤ m.
Thus we have proved the following theorem.
Theorem 5.1. Suppose thatϕ ∈ C1,L
∂ [p] and AssumptionsH[ρ,G], H[ψ] are satisfied. Further-
more, assume that the inequalities (16), (17) hold, as well as the compatibility conditions (18),
(19). Then there exists exactly one solution z̄ ∈ C1,L
ϕ,c [d] of the system
∂tzi(t, x) +
n∑
j=1
ρij(t, x, zψ(t,x,z(t,x)))∂xjzi(t, x) = Gi(t, x, zψ(t,x,z(t,x))), 1 ≤ i ≤ m, (28)
augmented with the generalized Cauchy condition (2). Moreover, there is Λc ∈ R+ such that the
Lipschitz condition (23), with respect to data, holds for ψ ∈ C1,L
∂ [p] and for v ∈ C1,L
ϕ̄,c [d] being a
solution of (28) with the initial boundary condition z ≡ ψ on E0 ∪ ∂0Ec.
Assumption H [ρ̄, Ḡ]. Functions ρ̄ : E × Rm → Mm×n, Ḡ : E × Rm → Rm, in variables
(t, x, y), are continuous, uniformly bounded, and
1) Ḡ is Lipschitz continuous in t,
2) the derivatives ∂xρ̄, ∂yρ̄, ∂xḠ, ∂yḠ exist on E × Rm, are continuous in t, and uniformly
bounded,
3) these derivatives are Lipschitz continuous in x and y.
Example 5.1. Suppose that Assumption H[ρ̄, Ḡ] is satisfied and set
ρ(t, x, w) = ρ̄ (t, x, w(0, 0)) , G(t, x, w) = Ḡ (t, x, w(0, 0)) .
Then the Assumption H[ρ,G] is fulfilled and the system (28) takes the form
∂tzi(t, x) +
n∑
j=1
ρ̄ij
(
t, x, z(ψ(t, x, z(t,x)))
)
∂xjzi(t, x) = Ḡi
(
t, x, z(ψ(t, x, z(t,x)))
)
, 1 ≤ i ≤ m,
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CLASSICAL SOLUTIONS OF HYPERBOLIC IBVPS WITH STATE DEPENDENT DELAYS 573
that is, it becomes a system of equations with deviating argument, where the deviation is state
dependent.
Example 5.2. Suppose that Assumption H[ρ̄, Ḡ] is satisfied and set
ρ(t, x, w) = ρ̄
t, x,∫
D
w(τ, ξ) dτ dξ
, G(t, x, w) = Ḡ
t, x,∫
D
w(τ, ξ) dτ dξ
.
Then the Assumption H[ρ,G] is fulfilled and the system (28) takes the form
∂tzi(t, x) +
n∑
j=1
ρ̄ij
t, x,∫
D
zψ(t,x,z(t,x))(τ, ξ) dτ dξ
∂xjzi(t, x) =
= Ḡi
t, x,∫
D
zψ(t,x,z(t,x))(τ, ξ) dτ dξ
, 1 ≤ i ≤ m,
that is, it becomes a system of integro-differential equations, where the domain of integration
is state dependent.
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Received 30.04.09,
after revision — 24.06.09
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