Generalized solutions of mixed problems for first-order partial functional differential equations
A theorem on the existence of solutions and their continuous dependence upon initial boundary conditions is proved. The method of bicharacteristics is used to transform the mixed problem into a system of integral functional equations of the Volterra type. The existence of solutions of this system is...
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| Дата: | 2006 |
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| Мова: | Англійська |
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Institute of Mathematics, NAS of Ukraine
2006
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509599031885824 |
|---|---|
| author | Czernous, W. Черноус, В. |
| author_facet | Czernous, W. Черноус, В. |
| author_sort | Czernous, W. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:56:00Z |
| description | A theorem on the existence of solutions and their continuous dependence upon initial boundary conditions is proved. The method of bicharacteristics is used to transform the mixed problem into a system of integral functional equations of the Volterra type. The existence of solutions of this system is proved by the method of successive approximations using theorems on integral inequalities. Classical solutions of integral functional equations lead to generalized solutions of the original problem. Differential equations with deviated variables and differential integral problems can be obtained from the general model by specializing given operators. |
| first_indexed | 2026-03-24T02:43:39Z |
| format | Article |
| fulltext |
UDC 517.9
W. Czernous (Gdańsk Univ. Technology, Poland)
GENERALIZED SOLUTIONS OF MIXED PROBLEMS
FOR FIRST-ORDER PARTIAL FUNCTIONAL
DIFFERENTIAL EQUATIONS
UZAHAL|NENI ROZV’QZKY MIÍANYX ZADAÇ
DLQ FUNKCIONAL|NYX DYFERENCIAL|NYX RIVNQN|
Z ÇASTYNNYMY POXIDNYMY PERÍOHO PORQDKU
A theorem on the existence of solutions and their continuous dependence upon initial boundary
conditions is proved. The method of bicaracteristics is used to transform the mixed problem into a
system of integral functional equations of the Volterra type. The existence of solutions of this system is
proved by a method of successive approximations and by using theorems on integral inequalities.
Classical solutions of integral functional equations lead to generalized solutions of the original problem.
Differential equations with deviated variables and differential integral problems can be obtained from a
general model by specializing given operators.
Dovedeno teoremu pro isnuvannq rozv’qzkiv ta ]x neperervnu zaleΩnist\ vid poçatkovyx hranyç-
nyx umov. Dlq peretvorennq mißano] zadaçi u systemu intehral\nyx funkcional\nyx rivnqn\
typu Vol\terra vykorystano metod bixarakterystyk. Isnuvannq rozv’qzkiv ci[] systemy dovede-
no za dopomohog metodu poslidovnyx nablyΩen\ ta teorem pro intehral\ni nerivnosti. Klasyçni
rozv’qzky intehral\nyx funkcional\nyx rivnqn\ pryvodqt\ do uzahal\nenyx rozv’qzkiv poçatko-
vo] zadaçi. Iz zahal\no] modeli za dopomohog konkretyzaci] zadanyx operatoriv moΩna otrymaty
dyferencial\ni rivnqnnq iz zminnymy, wo vidxylqgt\sq, ta dyferencial\ni intehral\ni zadaçi.
1. Introduction. We formulate the functional differential problem. Let a > 0,
h R0 ∈ + , R+ = + ∞[ , )0 , b b b Rn
n= … ∈( , , )1 , and let h h h Rn
n= … ∈ +( , , )1 be given,
where bi > 0 for 1 ≤ i ≤ n. We define the sets
E = [ 0, a ] × [ – b, b ] , D = [ – h0 , 0 ] × [ – h, h ] .
Let c c c b hn= … = +( , , )1 and
E h c c0 0 0= − × −[ , ] [ , ],
∂0 0E a c c b b= × − −[ , ] [ , ] \ ( , )( ) , E E E E∗ = 0 0∪ ∪ ∂ .
Suppose that z : E
* → 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 x( ),+ +τ 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. Elements of the space C ( D, R ) will be denoted by
w, w and so on. We denote by ⋅ 0 the supremum norm in the space C ( D, R ) . Put
Ω = E × C ( D, R ) × R
n
and let
f : Ω → R , ϕ ∂: E E R0 0∪ → ,
α0 0: [ , ]a R→ , ′ →α : E Rn , ′ = …α α α( , , )1 n ,
be given functions. We denote α α α( , ) ( ( ), ( , ))t x t t x= ′0 , ( t, x ) ∈ E . We require that
α ( t, x ) ∈ E for ( t, x ) ∈ E and α0 ( t ) ≤ t for t ∈ [ 0, a ] . We will deal with the
following mixed problem:
∂ ∂αt t x xz t x f t x z z t x( , ) ( , , , ( , ))( , )= , (1)
© W. CZERNOUS, 2006
804 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
GENERALIZED SOLUTIONS OF MIXED PROBLEMS … 805
z ( t, x ) = ϕ ( t, x ) on E E0 0∪ ∂ , (2)
where ∂ ∂ ∂x x xz z z
n
= …( , , )
1
.
A function ˜ : [ , ] [ , ]z h c c R− × − →0 ξ where 0 < ξ ≤ a, is a generalized solution
of (1), (2) if it is continuous and
(i) the derivatives ( ˜, , ˜) ˜∂ ∂ ∂x x xz z z
n1
… = exist on [ , ] [ , ]0 ξ × − b b and the func-
tion ˜( , ) : [ , ]z x h R⋅ − →0 ξ is absolutely continuous on [ , ]0 ξ for each x b b∈ −[ , ];
(ii) for each x b b∈ −[ , ] equation (1) is satisfied for almost all t ∈[ , ]0 ξ and
condition (2) holds on ( ) [ , ]E E h Rn
0 0 0∪ ∂ ξ× − × .
Note that our hereditary setting contains well-known delay structures as particular
cases.
The simplest differential equation with deviated variables is obtained in the follow-
ing way: put h0 = 0, h = 0, and suppose that F : E × R × R
n
→ R is a given functi-
on. Consider the operator f defined by
f ( t, x, w, q ) = F ( t, x, w ( 0, 0 ), q ) , ( t, x, w, q ) ∈ Ω . (3)
Then
f t x z qt x( , , , )( , )α = F t x z t x q( ( ) ), , ( , ) ,α
and equation (1) is equivalent to the differential equation with deviated variables
∂t z t x( , ) = F t x z t x z t xx( , , ( , ) , ( , ))( )α ∂ . (4)
We require that α ( t, x ) ∈ E for ( t, x ) ∈ E and α0 ( t ) ≤ t for t ∈ [ 0, a ] .
A general class of equations with deviated variables can be obtained in the follow-
ing way: suppose that β0 : [ 0, a ] → R , β ′ : E → R
n
, β ′ = ( β1 , … , βn ) are given
functions and
− ≤ − ≤h t t0 0 0 0β α( ) ( ) , − ≤ ′ − ′ ≤h t x t x hβ α( , ) ( , ) , ( t, x ) ∈ E . (5)
For the above given function F, we define the operator f as follows:
f ( t, x, w, q ) = F ( t, x, w ( β0 ( t ) – α0 ( t ), β ′ ( t, x ) – α ′ ( t, x )), q ) , ( t, x, w, q ) ∈ Ω . (6)
Then
f t x z qt x( , , , )( , )α = F t x z t x q( , , ( ( , )), )β ,
where β ( t, x ) = ( β0 ( t ), β ′ ( t, x )) and equation (1) is equivalent to
∂t z t x( , ) = F t x z t x z t xx( , , ( ( , )), ( , ))β ∂ . (7)
Now we consider differential integral equations. Suppose that γ0 : [ 0, a ] → R ,
γ ′ : E → R
n
, γ ′ = ( γ1 , … , γn ) are given functions and
− ≤ − ≤h t t0 0 0 0γ α( ) ( ) , − ≤ ′ − ′ ≤h t x t x hγ α( , ) ( , ) , ( t, x ) ∈ E . (8)
For the above given functions β and F, we define the operator f in the following
way:
f ( t, x, w, q ) = F t x w y dy d q
t t
t t
t x t x
t x t x
, , ( , ) ,
( ) ( )
( ) ( )
( , ) ( , )
( , ) ( , )
β α
γ α
β α
γ α
τ τ
0 0
0 0
−
−
′ − ′
′ − ′
∫ ∫
, (9)
where ( t, x, w, q ) ∈ Ω . Then
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
806 W. CZERNOUS
f t x z qt x( , , , )( , )α = F t x z y dy d q
t x
t x
, , ( , ) ,
( , )
( , )
τ τ
β
γ
∫
and (1) reduces to the differential integral equation
∂t z t x( , ) = F t x z y dy d z t x
t x
t x
x, , ( , ) , ( , )
( , )
( , )
τ τ ∂
β
γ
∫
. (10)
We will discuss the question of the existence of solutions of problem (1), (2).
Different classes of weak solutions of mixed problems to partial functional diffe-
rential problems are considered in literature. Almost linear systems in two-independent
variables were investigated in [1, 2]. A continuous function is a solution of a mixed
problem considered in the papers if it satisfies an integral functional system arising
from functional differential system by integrating along bicharacteristics. Note that pa-
pers [1, 2] initiated investigations of first order partial functional differential equations.
The class of Carathéodory solutions consists of all functions which are continuous,
have their derivatives almost everywhere in a domain and the set of all points where the
differential equation or the system is not fulfilled is of Lebesgue measure zero. The
existence and uniqueness results for quasilinear systems with initial boundary condition
in the class of almost everywhere solutions are given in [3, 4]. Right-hand sides of
equations contain operators of the Volterra type and unknown functions depend on two
variables. A general class of mixed problems and Carathéodory solutions for quasili-
near equations is investigated in [5]. Functional differential problems considered in
these papers are equivalent to integral functional equations which are obtained by
integration along bicharacteristics. Under natural assumptions, continuous solutions of
functional integral equation are Carathéodory solutions of original problems.
Generalized solutions in the Cinquini Cibrario sense for equations without a func-
tional dependence were first treated in [6 – 8]. This class of solutions is placed betwe-
en classical solutions and solutions in the Carathéodory sense. It is important that both
inclusions are strict. This class of solutions is investigated in the case that assumptions
for given functions are extended. Existence results for mixed problems and nonlinear
functional differential equations can be found in [9] and [10] (Chapters IV, V). They
are obtained by a quasilinearization procedure and by construction of functional integ-
ral systems for unknown functions and for their derivatives with respect to spacial vari-
ables. Continuous solutions of integral equations lead to generalized solutions of origi-
nal problems.
Note that the monograph [11] contains an exposition of generalized solutions in the
Cinquini Cibrario sense for nonlinear equations and systems without the functional
variable.
Wide classes of solutions to mixed functional differential equations are investigated
in [10, 12 – 14]. The only derivative with respect to t of unknown functions appear in
equations considered in the above papers.
Viscosity solutions of mixed problems for functional differential equations were
first treated in [15, 16]. Uniqueness results were based on a differential inequalities
method. Existence theorems were obtained by using the vanishing viscosity method.
Further bibliographical information concerning hyperbolic functional differential
equations can be found in the monograph [10].
The paper is a generalization of existence results for nonlinear functional differen-
tial equations with initial boundary conditions which are presented in [9] and [10]
(Chapter V). There are the following differences between the above mentioned results
and our theorems.
I. It is assumed in [10] that the function f of the variables ( t, x, w, q ) has the fol-
lowing property. Write
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
GENERALIZED SOLUTIONS OF MIXED PROBLEMS … 807
sign ∂q f = ( , , )sign sign∂ ∂q qf f
n1
… .
The following condition is important in [10]: the function sign ∂q f is constant on Ω .
We have omitted the above condition in our considerations. It is assumed in [9] that
the function sign ∂q f is constant on Ω . This condition can be reduced to the assump-
tion adopted in [10] by changing variables in an unknown function in a differential
functional equation.
II. The functional dependence in partial differential equations is based on the use of
the Hale operator ( , ) ( , )t x z t x→ where z D Rt x( , ) : → . The domain of the function z t x( , )
considered in [10] has the form D h h h R n= − × ′ × − ′′ ⊂ +[ , ] [ , ] [ , ]0 0
10 0 0 , where ′h =
= ( , , )h hk1 … , ′′ = …+h h hk n( , , )1 . In our case we put D h h h= − × −[ , ] [ , ]0 0 . It foll-
ows that the class of differential equations with deviated variables considered in the pa-
per is more general than an adequate class of equations which can be obtained from
[10].
The same conclusion can be drawn for differential integral equations.
III. The right-hand sides of the equations considered in [10] depend on the
functional variable z t x( , ) . In our considerations, equation (1) contains the functional
variable z t xα( , ) It is easy to see that the class of differential equations which is cover-
ed by our theory is more general than a suitable class considered in [10].
The paper is organized as follows. In Section 2 we prove results on the existence
and uniqueness and on the regularity of bicharacteristics for nonlinear mixed problems.
Integral functional equations generated by (1), (2) are investigated in Section 3. It is
shown that under natural assumptions on given functions there exists a sequence of
successive approximations and it is covergent. The main results on the existence of ge-
neralized solutions and on the continuous dependence of solutions on initial boundary
conditions are presented in Section 4. An application to equations with deviated vari-
ables is given.
The following function spaces will be needed in our considerations. Write Et
∗ =
= E h t Rn∗ − ×∩ ([ , ] )0 and E t b bt = × −[ , ] [ , ]0 where 0 ≤ t ≤ a . We will denote by
⋅ t the supremum norm in the spaces C E Rt( , )∗ and C E Rt
n( , )∗ . Analogously, we
will use the symbol ⋅ ( )t to denote the supremum norm in C E Rt( , ) and C E Rt
n( , ).
We will denote by Mn n× the class of all n × n matrices with real elements. For
x Rn∈ , X Mn n∈ × , where x x xn= …( , , )1 and
X = [ ] , , ,xij i j n= …1
we put
x = xi
j
n
=
∑
1
and X = max
1 1≤ ≤ =
∑
j n
ij
i
n
x .
The product of two matrices is denoted by “ � ”. If X Mn n∈ × then X
T is the trans-
pose matrix. We use the symbol “ ° ” to denote the scalar product in R
n.
Let us denote by ⋅ 0 the supremum norm in the space C ( D, R ) . Let C D R0 1, ( , )
be the set of all w C D R∈ ( , ) such that the derivatives ( , , )∂ ∂ ∂x x xw w w
n1
… = exist
on D and ∂x
nw C D R∈ ( , ) . For w C D R∈ 0 1, ( , ) we put
w 1 = w w t x t x Dx0 + ∈{ }max ( , ) : ( , )∂ .
We denote by C D RL
0 1, ( , ) the class of all w C D R∈ 0 1, ( , ) such that w L1, < + ∞
where
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
808 W. CZERNOUS
w L1, = w
w t x w t x
t t x x
t x t x D t x t xx x
1 + −
− + −
∈ ≠
sup
( , ) ( , )
: ( , ), ( , ) , ( , ) ( , )
∂ ∂
.
We will consider the spaces
Ω = E × C ( D, R ) × R
n
, Ω( )1 = [ , ] ( , ),− × ×b b C D R Rn0 1
and
Ω( , )1 L = [ , ] ( , ),− × ×b b C D R RL
n0 1 .
Let Θ be the class of all functions γ ∈ + +C R R( , ) which are nondecreasing on R+ .
Now we define some further function spaces. Given s s s s R= ∈ +( , , )0 1 2
3 , we de-
note by C sL1, [ ] the set of all functions ϕ ∂∈C E E R( , )0 0∪ such that
(i) there exists ∂ ϕx t x( , ) for ( , )t x E E∈ 0 0∪ ∂ ,
(ii) the estimates ϕ ( , )t x s≤ 0,
and
ϕ ϕ( , ) ( , )t x t x− ≤ s t t1 − , ∂ ϕx t x( , ) ≤ s1,
∂ ϕ ∂ ϕx xt x t x( , ) ( , )− ≤ s t t x x2 − + −[ ],
are satisfied on E E0 0∪ ∂ .
Let ϕ ∈C sL1, [ ] be given and let 0 < c ≤ a, d d d d R= ∈ +( , , )0 1 2
3 , d si i≥ for
i = 0, 1, 2, we consider the space C dc
L
ϕ,
, [ ]1 of all functions z E Rc: ∗ → such that
(i) z C E Rc∈ ∗( , ) and z t x t x( , ) ( , )= ϕ on ( ) ([ , ] )E E h c Rn
0 0 0∪ ∩∂ − × ,
(ii) there exists ∂xz t x( , ) on Ec
∗ and the estimates z t x d( , ) ≤ 0 ,
and
z t x z t x( , ) ( , )− ≤ d t t1 − , ∂xz t x( , ) ≤ d1,
∂ ∂x xz t x z t x( , ) ( , )− ≤ d t t x x2 − + −[ ],
are satisfied on Ec
∗.
Let p p p R= ∈ +( , )0 1
2 , p s0 1≥ , p s1 2≥ . We denote by C pc
L0, [ ] the set of all
functions v: E Rc
n→ such that for ( , )t x Ec∈ the estimates v( , )t x p≤ 0 and
v v( , ) ( , )t x t x− ≤ p t t x x1 − + −[ ]
hold on Ec . We will prove that under suitable assumptions on f, α and ϕ, and for
sufficiently small c with 0 < c ≤ a, there exists a solution z of the problem (1),
(2) such that z C dc
L∈ ϕ,
, [ ]1 , ∂x c
Lz C p∈ 0, [ ]. Write
∆i
+ = x b b x bi i∈ − ={ }[ , ] : , ∆i
− = x b b x bi i∈ − = −{ }[ , ] : ,
where 1 ≤ i ≤ n, and
∆ =
∆ ∆i i
i
n
+ −
=
( )∪∪
1
.
2. Bicaracteristics of nonlinear equations. We begin with assumptions on f.
Assumption H [ ∂∂∂∂q f ] . Suppose that the function f : Ω → R of the variables ( t, x,
w, q ) , q = ( q1 , … , qn ) , is such that
1) f x w q a R( , , , ) : [ , ]⋅ →0 is measurable for each ( x, w, q ) ∈ [ – b, b ] × C ( D, R ) ×
× R
n
and the partial derivatives
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
GENERALIZED SOLUTIONS OF MIXED PROBLEMS … 809
( ( ), , ( ))∂ ∂q qf P f P
n1
… = ∂q f P( ) , P = ( t, x, w, q ) ,
exist for ( , , ) ( )x w q ∈Ω 1 and for almost all t ∈ [ 0, a ] ;
2) ∂q
nf x w q a R( , , , ) : [ , ]⋅ →0 is measurable and there is B ∈ Θ such that
∂q f t x w q( , , , ) ≤ B w( )1 for ( , , ) ( )x w q ∈Ω 1 and for almost all t ∈ [ 0, a ]
and there is C ∈ Θ such that for ( , , ) ( , )x w q L∈Ω 1 , ( , ) [ , ]x q b b Rn∈ − × ,
w C D R∈ 0 1, ( , ) and for almost all t ∈ [ 0, a ] we have
∂ ∂q qf t x w w q f t x w q( , , , ) ( , , , )+ − ≤ C w x x w q qL1 1,( ) − + + −[ ];
3) there is κ > 0 such that for 1 ≤ i ≤ n we have
∂qi
f t x w q( , , , ) ≥ κ for ( , , ) ( , ),x w q C D R Ri
n∈ × ×+∆ 0 1
and
∂qi
f t x w q( , , , ) ≤ – κ for ( , , ) ( , ),x w q C D R Ri
n∈ × ×−∆ 0 1
for almost all t ∈ [ 0, a ] .
Assumption H [ αααα ] . Suppose that the functions α : [ 0, a ] → [ 0, a ] and α ′ : E →
→ [ – b, b ] are such that
1) α0 ( t ) ≤ t for t ∈ [ 0, a ] and there is r0 ∈ R+ such that
α α0 0( ) ( )t t− ≤ r t t0 − on [ 0, a ] ;
2) α′ is of class C1 and
∂ αx t x′( , ) ≤ r0 on E ;
3) there is r1 ∈ R+ such that
∂ α ∂ αx xt x t x′ − ′( , ) ( , ) ≤ r x x1 − on E .
Write r = ( r0 , r1 ) .
Suppose that ϕ ∈C sL1, [ ] and z C dc
L∈ ϕ,
, [ ]1 , u C pc
L∈ 0, [ ]. We consider the Cauchy
problem
η ′ ( τ ) = – ∂ τ η τ τ η τα τ η τq f z u( , ( ), , ( , ( )))( , ( )) , η ( t ) = x , (11)
and denote by g z u t x[ , ]( , , )⋅ its solution in the Carathéodory sense. The function
g z u t x[ , ]( , , )⋅ is the bicharacteristic of equation (1) corresponding to ( z, u ) . Let I t x( , )
be the domain of g z u t x[ , ]( , , )⋅ and δ[ , ]( , )z u t x is the left end of the maximal inter-
val on which the bicharacteristic g z u t x[ , ]( , , )⋅ is defined. Write
P z u t x[ , ]( , , )τ = τ τ τ τα τ τ, [ , ]( , , ), , , [ , ]( , , )( , [ , ]( , , )) ( )g z u t x z u g z u t xg z u t x( ) .
We prove a lemma on bicharacteristics.
Lemma 1. Suppose that Assumptions H [ ]∂q f , H [ α ] are satisfied and let
ϕ ϕ, [ ],∈ C sL1 , z C dc
L∈ ϕ,
, [ ]1 , z C dc
L∈ ϕ,
, [ ]1 , u u C pc
L, [ ],∈ 0 ,
be given. Then the solutions g z u t x[ , ]( , , )⋅ and g z u t x[ , ]( , , )⋅ exist on the inter-
vals I t x( , ) and I t x( , ) such that for ξ δ= [ , ]( , )z u t x , ξ δ= [ , ]( , )z u t x we have
g z u t x[ , ]( , , )ξ ∈∆ and g z u t x[ , ]( , , )ξ ∈∆ . The bicharacteristics are unique on
I t x( , ) and I t x( , ) . Moreover we have the estimates
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
810 W. CZERNOUS
g z u t x g z u t x[ , ]( , , ) [ , ]( , , )τ τ− ≤ C t t x x− + −[ ] (12)
for τ ∈I It x t x( , ) ( , )∩ , ( , ), ( , )t x t x Ec∈ , and
g z u t x g z u t x[ , ]( , , ) [ , ]( , , )τ τ− ≤ C z z z z u u d
t
x x
τ
ξ ξ ξ∂ ∂ ξ∫ − + − + −[ ]( ) ( )
(13)
for τ ∈I It x t x( , ) ( , )∩ , ( , )t x Ec∈ , where I t x( , ) is the domain of the bicharacteristic
g z u t x[ , ]( , , )⋅ and
C = max , ( ˜), exp ( )1 1 0 1 2 1B d C d cC d r d d p( ){ } ( ) + + +[ ]{ }
and
d̃ = d d0 1+ , d = d d d0 1 2+ + .
Proof. The existence and uniqueness of solutions of (11) follows from the classi-
cal theorem on Carathéodory solutions of initial problems. The function g z u t x[ , ]( , , )⋅
satisfies the integral equation
g z u t x[ , ]( , , )τ = x f P z u t x d
t
q− ( )∫
τ
∂ ξ ξ[ , ]( , , ) . (14)
Since z C dc
L∈ ϕ,
, [ ]1 condition 2 of Assumption H [ α ] shows that
z zy yα τ α τ( , ) ( , )−
1
≤ r d d y y0 1 2( )+ −
for all ( , ), ( , )τ τy y Ec∈ . It follows from Assumption H [ ]∂q f that the function
g z u t x g z u t x[ , ]( , , ) [ , ]( , , )⋅ − ⋅ satisfies the integral inequality
g z u t x g z u t x[ , ]( , , ) [ , ]( , , )τ τ− ≤ max , ( ˜)1 B d t t x x{ } − + −[ ] +
+ C d r d d p g z u t x g z u t x d
t
( ) + + + −∫[ ( ) ] [ , ]( , , ) [ , ]( , , )1 0 1 2 1
τ
ξ ξ ξ ,
τ ∈I It x t x( , ) ( , )∩ .
Now we obtain (12) by the Gronwall inequality. For ( , ), ( , )τ τy y Ec∈ we have the
estimate
z zy yα τ α τ( , ) ( , )−
1
≤ z z z z r d d y yx x− + − + + −τ τ∂ ∂ ( ) ( )0 1 2 .
It follows that the function g z u t x g z u t x[ , ]( , , ) [ , ]( , , )⋅ − ⋅ satisfies the integral inequa-
lity
g z u t x g z u t x[ , ]( , , ) [ , ]( , , )τ τ− ≤
≤ C d z z z z u u d
t
x x( ) − + − + −[ ]∫
τ
ξ ξ ξ∂ ∂ ξ( ) ( ) +
+ C d r d d p g z u t x g z u t x d
t
( ) + + + −∫[ ( ) ] [ , ]( , , ) [ , ]( , , )1 0 1 2 1
τ
ξ ξ ξ ,
τ ∈I It x t x( , ) ( , )∩ .
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GENERALIZED SOLUTIONS OF MIXED PROBLEMS … 811
We get (13) from the Gronwall inequality. This proves Lemma 1.
Now we give a lemma on a regularity of the function δ [ z, u ] .
Lemma 2. Suppose that Assumptions H [ ]∂q f , H [ α ] are satisfied and
ϕ ϕ, [ ],∈ C sL1 , z C dc
L∈ ϕ,
, [ ]1 , z C dc
L∈ ϕ,
, [ ]1 , u u C pc
L, [ ],∈ 0 .
Then the functions δ[ , ]z u a n d δ[ , ]z u are continuous on Ec . Moreover we
have the estimates
δ δ[ , ]( , ) [ , ]( , )z u t x z u t x− ≤ 2C
t t x x
κ
− + −[ ] (15)
where ( , ), ( , )t x t x Ec∈ and
δ δ[ , ]( , ) [ , ]( , )z u t x z u t x− ≤ 2
0
C
z z z z u u d
t
x xκ
∂ ∂ ξξ ξ ξ∫ − + − + −[ ]( ) ( ) (16)
where ( , )t x Ec∈ .
Proof. The continuity of δ[ , ]z u and δ[ , ]z u on Ec follows from classical theo-
rems on continuous dependence on initial conditions for Carathéodory solutions of ini-
tial problems. Now we prove (15). This estimate is obvious in the case δ[ , ]( , )z u t x =
= δ[ , ]( , )z u t x = 0 (i.e., in the case when solutions of problem (11) are defined on
[ , ]0 t and [ , ]0 t ). Suppose now that 0 ≤ δ[ , ]( , )z u t x < δ[ , ]( , )z u t x . Then for
ζ δ= [ , ]( , )z u t x we have g z u t x[ , ]( , , )ζ ∈∆ and there exists i, 1 ≤ i ≤ n, such
that g z u t x bi i[ , ]( , , )ζ = . Two possibilities can occur, either (i) g z u t x bi i[ , ]( , , )ζ =
or (ii) g z u t x bi i[ , ]( , , )ζ = − . Consider the case (i). Let x = ( x1 , … , xn ) , x̃ =
= ( , , , , , , )x x b x xi i i n1 1 1… …− + . We have the estimate
∂ ∂α αq t x q t xi i
f t x z u t x f t x z u t x( , , , ( , )) ( , ˜, , ( , ˜))( , ) ( , ˜)− ≤ ˜ ( )c b xi i− , (17)
for ( , )t x Ec∈ , where ˜ [ ( ) ]c C d r d d p= ( ) + + +1 0 1 2 1 . Thus
∂ αq t xi
f t x z u t x( , , , ( , ))( , ) ≥ κ
2
for ( , )t x Ec∈ such that b x ci i− ≤ −κ ( ˜)2 1. It follows from Lemma 1, that
b g z u t xi i− [ , ] , ,( )ζ = g z u t x g z u t xi i[ , ] , , [ , ] , ,( ) ( )ζ ζ− ≤ κ
2c̃
,
for ( , ), ( , )t x t x Ec∈ such that
t t x x− + − ≤ κ
2c̃C
. (18)
Then we get
∂ ζ ζ ζ ζα ζ ζq g z u t xi
f g z u t x z u g z u t x, [ , ] , , , , ( , [ , ]( , , ))( )
( , [ , ]( , , ))( ) ≥ κ
2
> 0,
and consequently,
∂ δt ig z u z u t x t x[ , ]( [ , ]( , ), , ) < 0
for ( , ), ( , )t x t x Ec∈ satisfying (18). It can be easily seen that g z u t xi[ , ]( , , )⋅ is de-
creasing on the interval ( [ , ]( , ), [ , ]( , ))δ δz u t x z u t x . Therefore
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812 W. CZERNOUS
b g z u t xi i− [ , ]( , , )τ ≤ κ
2c̃
and the estimate
∂ τ τ τ τα τ τq g z u t xi
f g z u t x z u g z u t x, [ , ] , , , , ( , [ , ]( , , ))( ) ( , [ , ]( , , ))( ) ≥ κ
2
holds for τ δ δ∈( [ , ]( , ), [ , ]( , ))z u t x z u t x and ( , ), ( , )t x t x Ec∈ such that (18) is satis-
fied. Then
–
κ δ δ
2
[ ][ , ]( , ) [ , ]( , )z u t x z u t x− ≥
≥ –
δ
δ
α τ τ∂ τ τ τ τ τ
[ , ]( , )
[ , ]( , )
( , [ , ]( , , )), [ , ] , , , , ( , [ , ]( , , ))( )
z u t x
z u t x
q g z u t xi
f g z u t x z u g z u t x d∫ ( ) =
= g z u z u t x t x g z u z u t x t xi i[ , ]( [ , ]( , ), , ) [ , ]( [ , ]( , ), , )δ δ− ≥
≥ g z u z u t x t x g z u z u t x t xi i[ , ]( [ , ]( , ), , ) [ , ]( [ , ]( , ), , )δ δ− ≥
≥ – C t t x x− + −[ ].
Thus the proof of (15) for ( , ), ( , )t x t x Ec∈ , such that (18) holds, is completed in the
case (i). In a similar way we prove (ii). Let ( , ), ( , )t x t x Ec∈ be arbitrary. We put
M x x t t= − + − . There exists K ∈ N such that
( )
˜
K
cC
− 1
2
κ < M ≤ K
cC
κ
2˜
.
Let ε ∈ R , ε = 1 / K. For j = 0, … , K we put
x j x j xj( ) ( )= + −ε ε1 , t j t j tj( ) ( )= + −ε ε1 .
Note that ( )( ) ( ), ( , )t x t x0 0 = , ( )( ) ( ), ( , )t x t xK K = and
x x t tj j j j( ) ( ) ( ) ( )− + −+ +1 1 = M
K
≤ κ
2c̃C
for j = 0, … , K – 1. It is easy to see that
x x− = x xj j
j
K
( ) ( )− +
=
−
∑ 1
0
1
and t t− = t tj j
j
K
( ) ( )− +
=
−
∑ 1
0
1
.
Then we have
δ δ[ , ]( , ) [ , ]( , )z u t x z u t x− ≤
≤ δ δ[ , ] , [ , ] ,( ) ( )( ) ( ) ( ) ( )z u t x z u t xj j j j
j
K
− + +
=
−
∑ 1 1
0
1
≤
≤ 2 1 1
0
1 C
t t x xj j j j
j
K
κ
( ) ( ) ( ) ( )− + −[ ]+ +
=
−
∑ = 2C
t t x x
κ
− + −[ ].
Thus we see that (15) holds true for all ( , ), ( , )t x t x Ec∈ . Now we consider esti-
mate (16). The inequality is obvious if δ[ , ]( , )z u t x = δ[ , ]( , )z u t x = 0. Suppose now
that 0 ≤ δ[ , ]( , )z u t x < δ[ , ]( , )z u t x . Then for ζ δ= [ , ]( , )z u t x we have
δ ζ[ , ]( , , )z u t x ∈∆ and there is i, 1 ≤ i ≤ n, such that g z u t x bi i[ , ]( , , )ζ = . Two
possibilities can happen, either (i) g z u t x bi i[ , ]( , , )ζ = or (ii) g z u t x bi i[ , ]( , , )ζ = − .
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GENERALIZED SOLUTIONS OF MIXED PROBLEMS … 813
Consider the case (i). We have the estimate (17) for ( , )t x Ec∈ .
It follows from Lemma 1, that
g z u t x g z u t xi i[ , ]( , , ) [ , ]( , , )ζ ζ− ≤ cC z z z z u ut x x t t− + − + −[ ]∂ ∂ ( ) ( ) .
Thus we have
b g z u t xi i− [ , ]( , , )ζ = g z u t x g z u t xi i[ , ]( , , ) [ , ]( , , )ζ ζ− ≤ κ
2c̃
,
for ( , )t x Ec∈ and z, z , u, u such that
z z z z u ut x x t t− + − + −∂ ∂ ( ) ( ) ≤ κ
2ccC˜
. (19)
Then we get
∂ ζ ζ ζ ζα ζ ζq g z u t xi
f g z u t x z u g z u t x, [ , ] , , , , ( , [ , ]( , , ))( ) ( , [ , ]( , , ))( ) ≥ κ
2
> 0,
and consequently,
∂ δt ig z u z u t x t x[ , ]( [ , ]( , ), , ) < 0
for ( , )t x Ec∈ and for z , z , u , u satisfying (19). It can be easily seen that
g z u t xi[ , ]( , , )⋅ is decreasing on ( [ , ]( , ), [ , ]( , ))δ δz u t x z u t x . Therefore
b g z u t xi i− [ , ]( , , )τ ≤ κ
2c̃
and the estimate
∂ τ τ τ τα τ τq g z u t xi
f g z u t x z u g z u t x, [ , ] , , , , ( , [ , ]( , , ))( ) ( , [ , ]( , , ))( ) ≥ κ
2
holds for τ δ δ∈( [ , ]( , ), [ , ]( , ))z u t x z u t x , ( , )t x Ec∈ and z, z , u, u such that (19) is
satisfied. Then
–
κ δ δ
2
[ ][ , ]( , ) [ , ]( , )z u t x z u t x− ≥
≥ –
δ
δ
α τ τ∂ τ τ τ τ τ
[ , ]( , )
[ , ]( , )
( , [ , ]( , , )), [ , ] , , , , ( , [ , ]( , , ))( )
z u t x
z u t x
q g z u t xi
f g z u t x z u g z u t x d∫ ( ) ≥
≥ g z u z u t x t x g z u z u t x t xi i[ , ]( [ , ]( , ), , ) [ , ]( [ , ]( , ), , )δ δ− ≥
≥ – C z z z z u u d
t
z u t x
x x
δ
ξ ξ ξ∂ ∂ ξ
[ , ]( , )
( ) ( )∫ − + − + −[ ] ≥
≥ – C z z z z u u d
t
x x
0
∫ − + − + −[ ]ξ ξ ξ∂ ∂ ξ( ) ( ) .
Thus the proof of (16) for ( , )t x Ec∈ and for z, z , u , u such that (19) holds, is
completed in the case (i). In a similar way we prove (ii).
Let z C dc
L∈ ϕ,
, [ ]1 , z C dc
L∈ ϕ,
, [ ]1 , u, u C pL∈ 0, [ ] be arbitrary. We put M = z z t− +
+ ∂ ∂x x tz z− ( ) + u u t− ( ) . There exists K ∈ N such that
( )
˜
K
ccC
− 1
2
κ < M ≤ K
ccC
κ
2 ˜
.
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814 W. CZERNOUS
Let ε ∈ R , ε = 1 / K . For j = 0, … , K we put
zj = j z j zε ε+ −( )1 and uj = j u j uε ε+ −( )1 on Ec .
Note that z z0 = , u u0 = , z zK = , u uK = and
z z z z u uj j t x j x j t j j t
− + − + −+ + +1 1 1∂ ∂
( ) ( )
= M
K
≤ κ
2ccC˜
for j = 0, … , K – 1. It is easy to see that
z z t− = z zj j t
j
K
− +
=
−
∑ 1
0
1
, ∂ ∂x x tz z− ( ) = ∂ ∂x j x j t
j
K
z z− +
=
−
∑ 1
0
1
( )
and
u u t− ( ) = u uj j t
j
K
− +
=
−
∑ 1
0
1
( )
.
Then we have
δ δ[ , ]( , ) [ , ]( , )z u t x z u t x− ≤
≤ δ δ[ , ]( , ) [ , ]( , )z u t x z u t xj j j j
j
K
− + +
=
−
∑ 1 1
0
1
≤
≤
j
K
j j x j x j j j
t
C
z z z z u u d
=
−
+ + +∑ ∫ − + − + −[ ]
0
1
1 1 1
0
2
κ
∂ ∂ ξ
ξ ξ ξ( ) ( )
=
= 2
0
C
z z z z u u d
t
x xκ
∂ ∂ ξξ ξ ξ∫ − + − + −[ ]( ) ( ) .
Therefore (16) holds true for all z C dc
L∈ ϕ,
, [ ]1 , z C dc
L∈ ϕ,
, [ ]1 , u , u C pc
L∈ 0, [ ]. This
completes the proof of the Lemma 2.
3. Integral functional equations. We denote by CL D R( , ) the set of all
continuous and real functions defined on C D R( , ) and by ⋅ � the norm in
CL D Rn( , ) . We formulate now next assumptions on f.
Assumption H [ f ] . Suppose that the Assumption H [ ]∂q f is satisfied and
1) there is B0 ∈ Θ such that
f t x w q( , , , ) ≤ B w0 0( ) on Ω ;
2) the partial derivatives
( ( ), , ( ))∂ ∂x xf P f P
n1
… = ∂x f P( ), P = ( t, x, w, q ) ,
and the Fréchet derivative ∂w f P( ) exist for ( , , )x w q ∈Ω and for almost all t ∈
∈ [ 0, a ] ;
3) the estimates
∂x f t x w q( , , , ) ≤ B w 1( ) , ∂w f t x w q( , , , )
�
≤ B w 1( ) ,
are satisfied for ( , , ) ( )x w q ∈Ω 1 and for almost all t ∈ [ 0, a ] ;
4) the terms
∂ ∂x xf t x w w q f t x w q( , , , ) ( , , , )+ − ,
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GENERALIZED SOLUTIONS OF MIXED PROBLEMS … 815
∂ ∂w wf t x w w q f t x w q( , , , ) ( , , , )+ −
�
are bounded from above by
C w x x w q qL1 1,( ) − + + −[ ]
for ( , , ) ( , )x w q L∈Ω 1 , ( , ) [ , ]x q b b Rn∈ − × , w C D R∈ 0 1, ( , ) and for almost all t ∈
∈ [ 0, a ] .
Remark 1. We give a theorem on the existence of solutions of the problem (1),
(2). For simplicity of formulation of the result we have assumed the same estimates for
the derivatives ∂x f , ∂w f , ∂q f . We also have assumed the Lipschitz condition for
these derivatives with the same coefficient.
Suppose that ϕ ∈C sL1, [ ]. Let C pc
L
ϕ,
, [ ]0 be the class of all functions u E Rc
n: ∗ →
such that u t x t xx( , ) ( , )= ∂ ϕ on ( ) ([ , ] )E E h c Rn
0 0 0∪ ∩∂ − × and u C pE c
L
c
∈ 0, [ ].
Now we formulate a system of inegral equations which are generated by (1), (2). Write
Q z u t x[ , ]( , ) = ( )[ , ]( , ), [ , ]( [ , ]( , ), , )δ δz u t x g z u z u t x t x ,
Φ[ , ]( , )z u t x = ϕ( )[ , ]( , )Q z u t x ,
ψ[ , ]( , )z u t x = ∂ ϕx Q z u t x( )[ , ]( , ) , ψ = ( ψ1 , … , ψn ) ,
W z u t x[ , ]( , , )τ =
u g z u t xg z u t x xα τ τ ∂ α τ τ( , [ , ]( , , )) ( , [ , ]( , , ))� .
Given ϕ ∈C sL1, [ ], z C dc
L∈ ϕ,
, [ ]1 , and u C pc
L∈ ϕ,
, [ ]0 , where 0 < c ≤ a. We define
F z u t x[ , ]( , ) = Φ[ , ]( , )z u t x +
+
δ
τ ∂ τ τ τ τ
[ , ]( , )
( ) ( )[ , ]( , , ) [ , ]( , , ) ( , [ , ]( , , ))
z u t x
t
qf P z u t x f P z u t x u g z u t x d∫ −[ ]� ,
and
G z u t x[ , ]( , ) = ψ ∂ τ
δ
[ , ]( , ) [ , ]( , , )
[ , ]( , )
( )z u t x f P z u t x
z u t x
t
x+ [∫ +
+ ∂ τ τ τw f P z u t x z u t x dW( )[ , ]( , , ) [ , ]( , , )] .
We will consider the following system of functional integral equations:
z ( t, x ) = F [ z, u ] ( t, x ) , u ( t, x ) = G [ z, u ] ( t, x ) , (20)
g [ z, u ] ( τ, t, x ) = x f P z u t x dq
t
+ ∫ ∂ ξ ξ
τ
( [ , ]( , , )) (21)
with initial-boundary conditions
z = ϕ, u = ∂x ϕ on ( ) ([ , ] )E E h c Rn
0 0 0∪ ∩∂ − × . (22)
Remark 2. Integral functional system (20) – (22) is obtained in the following way.
We introduce first an additional unknown function u, where u zx= ∂ . Then we con-
sider the linearization of (1) with respect to u, i.e.,
∂t z t x( , ) = f U f U z t x u t xq x( ) ( ) ( ( , ) ( , ))+ −∂ ∂� , (23)
where U = ( , , , ( , ))( , )t x z u t xt xα . By virtue of equation (1) we get the differential
system for the unknown function u :
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816 W. CZERNOUS
∂t u t x( , ) =
∂ ∂ ∂x q x
Tf U f U u t x( ) ( ) [ ( , )]+ � +
+ ∂ ∂ ∂ ααw x t x xf U z t x( )( ) ( , )( , ) � . (24)
Finally we put ∂x z u= in (24). System (23), (24) has the following property: the
differential equations of bicharacteristics for (23) and for (24) are the same and they
have the form (11).
If we consider (23), (24) along the bicharacteristics g z u t x[ , ]( , , )⋅ , we obtain
d
d
z g z u t x
τ
τ τ( , [ , ]( , , )) = f P z u t x f P z u t x u g z u t xq( [ , ]( , , )) ( [ , ]( , , )) ( , [ , ]( , , ))τ ∂ τ τ τ− � ,
and
d
d
u g z u t x
τ
τ τ( , [ , ]( , , )) = ∂ τ ∂ τ τx wf P z u t x f P z u t x W z u t x( [ , ]( , , )) ( [ , ]( , , )) [ , ]( , , )+ .
By integrating the above equations on [ [ , ]( , ), ]δ z u t x t with respect to τ, we get (20),
(21).
The existence results for (20) – (22) is based on the following method of the suc-
cessive approximations. Suppose that ϕ ∈C sL1, [ ] and that Assumption H [ f ] are
satisfied. We define the sequence
{ }( ) ( ),z um m , z E Rm
c
( ) : ∗ → , u E Rm
c
n( ) : ∗ → ,
in the following way. Let z E Rc
( ) :0 ∗ → , u E Rc
n( ) :0 ∗ → be arbitrary functions such
that z C dc
L( )
,
, [ ]0 1∈ ϕ , u C pc
L( )
,
, [ ]0 0∈ ϕ and ∂x z t x( )( , )0 = u t x( )( , )0 on E c . Now, if
( )( ) ( )
,
,
,
,, [ ] [ ]z u C d C pm m
c
L
c
L∈ ×ϕ ϕ
1 0 are known functions, then u m( )+1 is a solution of the
equation
u = G um( )[ ] (25)
with the initial-boundary condition
u ( t, x ) = ∂ ϕx t x( , ) on ( ) ([ , ] )E E h c Rn
0 0 0∪ ∩∂ − ×
and
z t xm( )( , )+1 = F z u t xm m[ ]( ) ( ), ( , )+1 on Ec ,
(26)
z t xm( )( , )+1 = ϕ ( t, x ) on ( ) ([ , ] )E E h c Rn
0 0 0∪ ∩∂ − × ,
where G G Gm m
n
m( ) ( ) ( )( ), ,= …1 is defined by
G u t xm( )[ ]( , ) = ψ ∂ τ
δ
[ ] ( [ ] )( )
[ , ]( , )
( ), ( , ) , ( , , )
( )
z u t x f P z u t xm
z u t x
t
x
m
m
+ [∫ +
+ ∂ τ τ τw
m m mf P z u t x W z u t x d( [ ] ) [ ]( ) ( ) ( ), ( , , ) , ( , , )] ,
and
W z u t xm( )[ , ]( , , )τ =
( )( )
( , [ , ]( , , )) ( , [ , ]( , , ))u g z u t xm
g z u t x xα τ τ ∂ α τ τ� .
We wish to emphasize that the main difficulty in carrying out this construction is the
problem of the existence of the sequence { }( ) ( ),z um m .
Put
r̃ = 1 0 1 2 1+ + +r d d p( ) ,
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GENERALIZED SOLUTIONS OF MIXED PROBLEMS … 817
A0 = C p r
B d
rcC d( ) ( ˜) ˜1 2
0 0+ + ( )
κ
+ CcB d p r p r( ˜)( )1 0
2
0 1+ ,
A = A s C B d0 2 1 2 1+ + +
κ
( ( ˜)) ,
B = C c r B d p C d p B d˜ ( ˜) ( ˜)+ ( ) +( )[ ]0 1 +
+ ( )( ) ( ˜) ( ( ˜))B d p B d
C
s C B d0 0 0 11 2 1 2 1+ +
+ + +
κ κ
.
Assumption H [ c, d, p ] . Suppose that the constants c, d = ( d0, d1, d2 ) , p = ( p0,
p1 ) satisfy the conditions
d1 = p0 ≥ s cB d p r1 0 01+ +( ˜)( ),
d2 = p1 ≥ max ( ˜)( ), ,{ }B d p r A B1 0 0+ ,
d0 ≥ s c B d B d p0 0 0 0+ +[ ]( ) ( ˜) .
It follows from the continuity of B B C0, , ∈Θ that there are d, p and sufficiently
small c ∈ ( 0, a ] satisfying Assumption H [ c, d, p ] .
Lemma 3. If ϕ ∈C sL1, [ ] and Assumptions H [ α ] , H [ f ], H [ c, d, p ] are sa-
tisfied then G m( ) : C pc
L
ϕ,
, [ ]0 → C pc
L
ϕ,
, [ ]0 .
Proof. Suppose that u C pc
L∈ ϕ,
, [ ]0 . Then
G u t xm( )[ ]( , ) ≤ s B d c p r1 0 01+ +( ˜) ( ) on Ec . (27)
Now we prove that the function G um( )[ ] satisfy the Lipschitz condition with constant
p1. If ( t, x ) , ( , )t x Ec∈ then
G u t x G u t xm m( ) ( )[ ]( , ) [ ]( , )− ≤ A Afϕ + ,
where
Aϕ = ∂ ϕ ∂ ϕx
m
x
mQ z u t x Q z u t x( [ , ]( , )) ( [ , ]( , ))( ) ( )− ,
Af =
=
δ
∂ τ ∂ τ τ τ
[ , ]( , )
( ) ( ) ( ) ( )
( )
( [ ] ) ( [ ] ) [ ] ), ( , , ) , ( , , ) , ( , ,
z u t x
t
x
m
w
m m m
m
f P z u t x f P z u t x W z u t x d∫ +[ ] –
–
δ
∂ τ ∂ τ τ τ
[ , ]( , )
( ) ( ) ( ) ( )
( )
( [ ] ) ( [ ] ) [ ] ), ( , , ) , ( , , ) , ( , ,
z u t x
t
x
m
w
m m m
m
f P z u t x f P z u t x W z u t x d∫ +[ ] .
It follows from Lemmas 1, 2 that
Aϕ ≤ s C B d t t x x2 1 2 1+ +
− + −[ ]
κ
( ( ˜)) ,
and
Af ≤ B d p r t t A t t x x( ˜)( )1 0 0 0+ − + − + −[ ].
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818 W. CZERNOUS
Then using Assumption H [ c, d, p ] we get
G u t x G u t xm m( ) ( )[ ]( , ) [ ]( , )− ≤ p t t x x1 − + −[ ]
on Ec . This inequality together with (27) imply Lemma 3.
Lemma 4. Suppose that ϕ ∈C sL1, [ ] and Assumptions H [ α ] , H [ f ], H [ c, d,
p ] are satisfied. Then there exists exactly one function u C pc
L∈ ϕ,
, [ ]0 satisfying the
equation u G um= ( )[ ].
Proof. Lemma 3 shows that G C p C pm
c
L
c
L( )
,
,
,
,: [ ] [ ]ϕ ϕ
0 0→ . It follows that there is
A > 0 such that for u, ˜ [ ],
,u C pc
L∈ ϕ
0 we have
G u t x G u t xm m( ) ( )[ ]( , ) [ ˜]( , )− ≤ A u u d
t
0
∫ − ˜
( )τ τ . (28)
Now we define the norm in the space C pc
L
ϕ,
, [ ]0 as follows
u λ = max ( , ) : ( , )u t x e t x Et
c
− ∈{ }λ ,
where λ > A . It is easy to see that ( ),
, [ ],C pc
L
ϕ λ
0 ⋅ is a Banach space. Now we prove
that there exists q ∈[ , )0 1 such that
G u G um m( ) ( )[ ] [ ˜]−
λ
≤ q u u− ˜ λ for u, ˜ [ ],
,u C pc
L∈ ϕ
0 . (29)
According to (28), we have
G u t x G u t xm m( ) ( )[ ]( , ) [ ˜]( , )− ≤
≤ A u u d
t
0
∫ − ˜
( )ξ ξ = A u u e e d
t
0
∫ − −˜
( )ξ
λξ λξ ξ ≤
≤ A u u e d
t
− ∫˜ λ
λξ ξ
0
= A u u e t
λ λ
λ− −˜ ( )1 ≤ A u u e t
λ λ
λ− ˜
for ( , )t x Ec∈ . Then we have
G u t x G u t x em m t( ) ( )[ ]( , ) [ ˜]( , )− −λ ≤ A u u
λ λ− ˜ , ( , )t x Ec∈ .
It follows that the estimate (29) holds with q A= −λ 1. By the Banach fixed point
theorem there exists exactly one u C pc
L∈ ϕ,
, [ ]0 satisfying the equation u G um= ( )[ ].
This completes the proof of Lemma 4.
Suppose that ϕ ∈C sL1, [ ]. Let us denote by Tϕ the set of all functions ω: E R∗ →
such that
(i) ω is continuous,
(ii)
ω ∂E E0 0∪ = ϕ.
Write T = { : }i di > 0 . The following compatibility condition for the problem (1), (2)
will be needed in our considerations.
Assumption Hc [ f, ϕϕϕϕ ] . Suppose that
1) if ω, ω̃ ϕ∈T then
f t x qt x( ), , ,( , )ωα = f t x qt x( ), , ˜ ,( , )ωα
for x ∈∆ , q Rn∈ and for almost all t ∈ [ 0, a ] ;
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GENERALIZED SOLUTIONS OF MIXED PROBLEMS … 819
2) there is a function ψ ∂: 0 E Rn→ , ψ = ( ψ1 , … , ψn ) , such that
∂ ϕt t x( , ) = f t x t xt x( ), , ˜ , ( , )( , )ϕ ψα for x ∈∆ and for almost all t ∈ [ 0, a ] , (30)
where ϕ̃ ϕ∈T and ψ ∂ ϕi xt x t x
i
( , ) ( , )= for ( , ) [ , ]t x a i i∈ × + −0 ∆ ∆∪ , i ∈T .
Remark 3. Relation (30) can be considered as an assumption on ϕ for i ∈T and
t ∈ [ 0, c ] , x i i∈ + −∆ ∆∪ and (30) is the equation for ψi t x( , ) if i n∈ …{ , , } \1 T ,
t ∈ [ 0, c ] , x i i∈ + −∆ ∆∪ .
Remark 4. Suppose that h0 0= , h = 0 and F E R R Rn: × × → is a given
function and f is defined by (3). Then (1) is equivalent to (4). If
αi t x( , ) = bi for x i∈ +∆ and αi t x( , ) = – bi for x i∈ −∆ ,
where 1 ≤ i ≤ n then condition 1 of the Assumption Hc [ f, ϕ ] is satisfied. For the
above given F, consider the operator f given by (6) and the differential integral
equation with deviated variables (7). If
βi t x( , ) ≥ bi for x i∈ +∆ and βi t x( , ) ≤ – bi for x i∈ −∆ ,
where 1 ≤ i ≤ n then condition 1 of the Assumption Hc [ f, ϕ ] is satisfied. If f is
defined by (9) then we have differential integral equation (10). If
βi t x( , ), γ i t x( , ) ≥ bi for x i∈ +∆ and βi t x( , ), γ i t x( , ) ≤ – bi for x i∈ −∆ ,
where 1 ≤ i ≤ n then condition 1 of the Assumption Hc [ f, ϕ ] is satisfied.
Now we prove the main lemma in this section.
Lemma 5. If Assumptions H [ α ] , H [ f ], H [ c, d, p ], Hc [ f, ϕ ] are satisfied,
then for any m ≥ 0 we have
∂x
mz t x( )( , ) = u t xm( )( , ) on Ec (31)
and z C dm
c
L( )
,
, [ ]∈ ϕ
1 .
Proof. We prove (31) by induction. It follows from the definition of the sequence
{ }( ) ( ),z um m that (31) is satisfied for m = 0. Supposed now that condition (31) hold
for a given m ≥ 0, we will prove that the function z m( )+1 given by (26) satisfies (31).
Put
∆( )( , , )m t x x = z t x z t x u t x x xm m m( ) ( ) ( )( , ) ( , ) ( , ) ( )+ + +− − −1 1 1 � , (32)
where ( , ), ( , )t x t x Ec∈ . We prove that there exists C̃ R∈ + such that
∆ ( , , )t x x ≤ C̃ x x− 2 on Ec . (33)
It follows from (25), (26) that
∆( )( , , )m t x x =
= F z u t x F z u t x G u t x x xm m m m m m[ ] [ ] [ ]( ) ( ) ( ) ( ) ( ) ( ), ( , ) , ( , ) ( , ) ( )+ + +− − −1 1 1 � .
(34)
Write
g t xm( )( , , )τ = g z u t xm m[ ] )( ) ( ), ( , ,+1 τ , δ( )( , )m t x = δ[ ] )( ) ( ), ( ,z u t xm m+1 ,
S t xm( )( , ) = δ δ( ) ( ) ( )( , ), ( ( , ), , )m m mt x g t x t x( )
and
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820 W. CZERNOUS
Q s t x xm( )( , , , , )τ = sP z u t x s P z u t xm m m m[ ] [ ]( ) ( ) ( ) ( ), ( , , ) ( ) , ( , , )+ ++ −1 11τ τ .
Suppose that δ δ( ) ( )( , ) ( , )m mt x t x≤ . Similar arguments apply to the case δ( )( , )m t x >
> δ( )( , )m t x . For simplicity of formulation of properties of the function ∆( )m we
define
A t x xm( )( , , ) = ϕ ϕ( ) ( )( ) ( )( , ) ( , )S t x S t xm m− –
– ∂ ϕ δ δx
m m m m mS t x g t x t x g t x t x( )( ) ( ) ( ) ( ) ( )( , ) ( ( , ), , ) ( ( , ), , )� −[ ] –
– ∂ ϕ δ δt
m m mS t x t x t x( )( ) ( ) ( )( , ) ( , ) ( , )−[ ]
and
B t x xm( )( , , ) = ∂ ϕ δ δt
m m mS t x t x t x( )( ) ( ) ( )( , ) ( , ) ( , )−[ ] +
+ ∂ ϕ δ δx
m m m m mS t x g t x t x g t x t x x x( )( ) ( ) ( ) ( ) ( )( , ) ( ( , ), , ) ( ( , ), , ) ( )� − − −[ ],
C t x xm( )( , , , )τ =
τ
∂ ξ ∂ ξ ξ
t
q
m m
q
m mf P z u t x f P z u t x d∫ + +−[ ]( [ ] ) ( [ ] )( ) ( ) ( ) ( ), ( , , ) , ( , , )1 1 ,
and
C m( ) = C Cm
n
m
1
( ) ( ), ,…( ).
Moreover we put
Λ( )( , , )m t x x =
δ
δ
τ τ
( )
( )
( , )
( , )
( ) ( )( [ ] ), ( , , )
m
m
t x
t x
m mf P z u t x d∫ +1 –
–
δ
δ
∂ τ τ τ τ
( )
( )
( , )
( , )
( ) ( ) ( ) ( )( [ ] ) ), ( , , ) ( , ( , , )
m
m
t x
t x
q
m m m mf P z u t x u g t x d∫ + +1 1� .
It follows from (34) that
∆( )( , , )m t x x = ϕ ϕ( ) ( )( ) ( )( , ) ( , )S t x S t xm m− +
+
δ
τ τ τ
( )( , )
( ) ( ) ( ) ( )( [ ] ) ( [ ] ), ( , , ) , ( , , )
m t x
t
m m m mf P z u t x f P z u t x d∫ + +−[ ]1 1 –
–
δ
∂ τ τ τ τ
( )( , )
( ) ( ) ( ) ( )( [ ] ) ), ( , , ) ( , ( , , )
m t x
t
q
m m m mf P z u t x u g t x d∫ + +1 1� +
+
δ
∂ τ τ τ τ
( )( , )
( ) ( ) ( ) ( )( [ ] ) ), ( , , ) ( , ( , , )
m t x
t
q
m m m mf P z u t x u g t x d∫ + +1 1� +
+ Λ
( ) ( ) ( )( , , ) ( , ) ( )[ ]m m mt x x G u t x x x− −+1 � .
Having disposed this preliminary step, we apply the Hadamard mean value theorem to
the difference
f P z u t x f P z u t xm m m m( [ ] ) ( [ ] )( ) ( ) ( ) ( ), ( , , ) , ( , , )+ +−1 1τ τ .
We thus get
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GENERALIZED SOLUTIONS OF MIXED PROBLEMS … 821
∆( )( , , )m t x x = ∆ ∆( ) ( )( , , ) ˜ ( , , )m mt x x t x x+ ,
where
∆ ( , , )t x x = A t x xm( )( , , ) +
+
δ
∂ τ
( )( , )
( )( )( , , , , )
m t x
t
x
mf Q s t x x∫ ∫ [
0
1
–
– ∂ τ τ τ τx
m m m mf P z u t x ds g t x g t x d( [ ] )( ) ( ) ( ) ( ), ( , , ) ( , , ) ( , , )+ ] −[ ]1 � +
+
δ
∂ τ
( )( , )
( )( )( , , , , )
m t x
t
w
mf Q s t x x∫ ∫ [
0
1
–
– ∂ τ τ
α τ τ α τ τw
m m
g t x
m
g t x
mf P z u t x ds z z dm m( [ ] )( ) ( )
( , ( , , ))
( )
( , ( , , ))
( ), ( , , ) ( ) ( )
+ ] −[ ]1 +
+
δ
∂ τ
( )( , )
( )( )( , , , , )
m t x
t
q
mf Q s t x x∫ ∫ [
0
1
–
– ∂ τ τ τ τ τ τq
m m m m m mf P z u t x ds u g t x u g t x d( [ ] ) ) )( ) ( ) ( ) ( ) ( ) ( ), ( , , ) ( , ( , , ) ( , ( , , )+ + +] −[ ]1 1 1� +
+
δ
∂ τ
( )( , )
( ) ( )( [ ] ), ( , , )
m t x
t
w
m mf P z u t x∫ +1 [ −z z
g t x
m
g t x
m
m mα τ τ α τ τ( , ( , , ))
( )
( , ( , , ))
( )
( ) ( ) –
– W z u t x g t x g t x dm m m m m( ) ( ) ( ) ( ) ( )[ ] ( ), ( , , ) ( , , ) ( , , )+ − ]1 τ τ τ τ�
and
˜ ( , , )( )∆ m t x x = B t x xm( )( , , ) +
+
δ
∂ τ τ τ τ
( )( , )
( ) ( ) ( ) ( )( [ ] ) [ ], ( , , ) ( , , ) ( , , ) ( )
m t x
t
x
m m m mf P z u t x g t x g t x x x d∫ + − − −1 � +
+
δ
∂ τ
( )( , )
( ) ( )( [ ] ), ( , , )
m t x
t
w
m mf P z u t x∫ +1 ×
× [ − − − ]+W z u t x g t x g t x x x dm m m m m( ) ( ) ( ) ( ) ( )[ ] [ ], ( , , ) ( , , ) ( , , ) ( )1 τ τ τ τ� –
–
δ
∂ τ
( )( , )
( ) ( )( [ ] ), ( , , )
m t x
t
q
m mf P z u t x∫ [ +1 –
–
∂ τ τ τ τq
m m m m mf P z u t x u g t x d t x x( [ ] ) )( ) ( ) ( ) ( ) ( ), ( , , ) ( , ( , , ) ( , , )+ +] +1 1� Λ .
We will write an estimate for ˜ ( , , )( )∆ m t x x . Since g t xm( )( , , )⋅ satisfies (14), we have
g t x g t x x xm m( ) ( )( , , ) ( , , ) ( )τ τ− − − = C t x xm( )( , , , )τ .
Substituting the above relation into ˜ ( , , )( )∆ m t x x and changing the order of integtals
where necessary we obtain
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822 W. CZERNOUS
˜ ( , , )( )∆ m t x x = D t x xm( )( , , ) +
+
δ
∂ τ ∂ τ τ τ
( )( , )
( ) ( ) ( ) ( ) ( )( [ ] ) ( [ ] ), ( , , ) , ( , , ) ( , , )
m t x
t
q
m m
q
m m mf P z u t x f P z u t x E t x d∫ + +−[ ]1 1 � ,
where
D t x xm( )( , , ) =
δ
δ
τ ∂ ϕ τ
( )
( )
( , )
( , )
( ) ( ) ( )( [ ] ) ( ), ( , , ) ( , )
m
m
t x
t x
m m
t
mf P z u t x S t x d∫ + −[ ]1 +
+
δ
δ
∂ ϕ τ τ ∂ τ τ
( )
( )
( , )
( , )
( ) ( ) ( ) ( ) ( )( ) ) ( [ ] )( , ) ( , ( , , ) , ( , , )
m
m
t x
t x
x
m m m
q
m mS t x u g t x f P z u t x d∫ −[ ]+ +1 1�
and
E t xm( )( , , )τ = – u g t xm m( ) ( )( ), ( , , )+1 τ τ +
+ ∂ ϕ ∂ ξ
δ
τ
x
m
t x
x
m mS t x f P z u t x
m
( ) ( [ ] )( )
( , )
( ) ( )( , ) , ( , , )
( )
+ [∫ +1 +
+ ∂ ξ ξ ξw
m m m m mf P z u t x W z u t x d( [ ] ) [ ]( ) ( ) ( ) ( ) ( ), ( , , ) , ( , , )+ + ]1 1 .
The next claim is
E t xm( )( , , )τ = 0, ( , )t x Ec∈ , τ ∈I t x( , ) . (35)
We have
g s t xm mg( ) ( )( ), , ( , , )τ τ = g s t xm( )( , , ) and δ τ τ( ) ( )( ), ( , , )m mg t x = δ( )( , )m t x
for ( t, x ) ∈ Ec , τ, ( , )s I t x∈ , therefore we get by (25)
u g t xm m( ) ( )( , ( , , ))+1 τ τ = ∂ ϕ δ δx
m m mt x g t x t x( )( ) ( ) ( )( , ), ( ( , ), , ) +
+
δ
τ
∂
( )( , )
( ) ( )( [ ] ), ( , , )
m t x
x
m mf P z u s t x ds∫ +1 +
+
δ
τ
∂
( )( , )
( ) ( ) ( ) ( ) ( )( [ ] ) [ ] ), ( , , ) , ( , , )
m t x
w
m m m m mf P z u s t x W z u s t x ds∫ + +1 1
for ( t, x ) ∈ Ec , τ ∈I t x( , ) and we have E t xm( )( , , )τ = 0. Then we have proved that
˜ ( , , )( )∆ m t x x = D t x xm( )( , , ). We conclude from Assumption Hc [ f, ϕ ] and from
Lemma 2, that there is C1 ∈ R+ such that
D t x xm( )( , , ) ≤ C x x1
2− on Ec
and consequently
˜ ( , , )( )∆ m t x x ≤ C x x1
2− on Ec . (36)
Now we will write an estimate for ∆( )( , , )m t x x . It follows from Assumptions
H [ ]∂q f , H [ f ] that the terms
∂ τ ∂ τx
m
x
m mf Q s t x x f P z u t x( ) ( [ ] )( ) ( ) ( )( , , , , ) , ( , , )− +1 ,
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GENERALIZED SOLUTIONS OF MIXED PROBLEMS … 823
∂ τ ∂ τq
m
q
m mf Q s t x x f P z u t x( ) ( [ ] )( ) ( ) ( )( , , , , ) , ( , , )− +1 ,
∂ τ ∂ τw
m
w
m mf Q s t x x f P z u t x( ) ( [ ] )( ) ( ) ( )( , , , , ) , ( , , )− +1
�
,
are bounded from above by
C d r g t x g t xm m( ) −˜ ( , , ) ( , , )( ) ( )τ τ .
An easy computation shows that
A t x xm( )( , , ) ≤ s g t x t x g t x t xm m m m
2
2( ) ( ) ( ) ( )( ( , ), , ) ( ( , ), , )δ δ− ,
and
z z
g t x
m
g t x
m
m mα τ τ α τ τ( , ( , , ))
( )
( , ( , , ))
( )
( ) ( )−
0
≤ r d g t x g t xm m
0 1
( ) ( )( , , ) ( , , )τ τ− ,
u g t x u g t xm m m m( ) ( ) ( ) ( )( , ( , , )) ( , ( , , ))+ +−1 1τ τ τ τ ≤ p g t x g t xm m
1
( ) ( )( , , ) ( , , )τ τ− .
Since
∂x
mz t x( )( , ) = u t xm( )( , ) on Ec ,
we have
z z W z u t x g t x g t x
g t x
m
g t x
m m m m m m
m mα τ τ α τ τ
τ τ τ
( , ( , , ))
( )
( , ( , , ))
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) [ ] ), ( , , [ ( , , ) ( , , )]− − −+1
0
� ≤
≤ [ ] ( ) ( )( , , ) ( , , )p r p r g t x g t xm m
0 1 1 0
2 2
+ −τ τ .
The above estimates and the definition of ∆( )m imply
∆( )( , , )m t x x ≤ s g t x t x g t x t xm m m m
2
2( ) ( ) ( ) ( )( ( , ), , ) ( ( , ), , )δ δ− +
+ A g t x g t x d
m t x
t
m m∗ ∫ −
δ
τ τ τ
( )( , )
( ) ( )( , , ) ( , , )
2
,
where A∗ = B d p r p r C d r( )[ ]˜ ˜0 1 1 0
2 2+ + ( ) . It follows from Lemmas 1, 2 and from
(14) that
g t x t x g t x t xm m m m( ) ( ) ( ) ( )( ( , ), , ) ( ( , ), , )δ δ− ≤ B d
C
C x x( )˜ 2
κ
+
− .
Then there exists C R2 ∈ + such that
∆( )( , , )m t x x ≤ C x x2
2− . (37)
Adding inequalities (36) and (37), we get (33) and consequently
∂x
mz t x( )( , )+1 = u t xm( )( , )+1 on Ec .
This completes the proof of (31).
Now we prove that z C dm
c
L( )
,
, [ ]+ ∈1 1
ϕ . It is clear that z m( )+1 is continuous on Ec
and z m( )+1 = ϕ on ( ) ([ , ] )E E h c Rn
0 0 0∪ ∩∂ − × . Moreover from (31) it follows
that
∂x
mz t x( )( , )+1 ≤ d1
and
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824 W. CZERNOUS
∂ ∂x
m
x
mz t x z t x( ) ( )( , ) ( , )+ +−1 1 ≤ d t t x x2 − + −[ ]
on Ec
∗. Our assumptions imply the estimates
z t xm( )( , )+1 ≤ d0, z t x z t xm m( ) ( )( , ) ( , )+ +−1 1 ≤ d t t2 −
where ( , ), ( , )t x t x Ec∈ ∗. This completes the proof of Lemma 5.
Now we prove that the sequences { }( )z m and { }( )u m are uniformly convergent on
Ec if the constant c is sufficiently small.
Put
à = A C d p r+ ( ) +( )1 0 0 , B̃ = ˜ ˜( )A r B d+ 0 , D̃ = B B d BeAc+ +( ˜ ˜( ))
˜
1 .
Assumption H [ c ] . Suppose that the Assumption H [ c, d, p ] is satisfied and c is
such small constant that cD̃ < 1.
Lemma 6. If Assumptions H [ α ] , H [ f ] , H [ c ] are satisfied, then the sequen-
ces { }( )z m and { }( )u m are uniformly convergent on Ec .
Proof. For t c∈[ , ]0 and m ≥ 1 we put
Z tm( )( ) = z zm m
t
( ) ( )− −1 , U tm( )( ) = u um m
t
( ) ( )
( )
− −1 .
We prove that for t c∈[ , ]0 we have
U tm( )( )+1 ≤ ˜ ( ) ( )
˜ ( ) ( )Be Z U dAc
t
m m
0
∫ +[ ]τ τ τ . (38)
It follows from Lemmas 1, 2 that we have the estimates
g t x g t xm m( ) ( )( , , ) ( , , )τ τ− −1 ≤ C Z U U d
t
m m m
τ
τ τ τ τ∫ + +[ ]+( ) ( ) ( )( ) ( ) ( )1 (39)
and
δ δ( ) ( )( , ) ( , )m mt x t x− −1 ≤ 2
0
1C
Z U U d
t
m m m
κ
τ τ τ τ∫ + +[ ]+( ) ( ) ( )( ) ( ) ( ) , (40)
where ( , )t x Ec∈ , τ ∈[ , ]0 c . Then we obtain the integral inequality
U tm( )( )+1 ≤ ˜ ( ) ˜ ( ) ( )( ) ( ) ( )A U d B Z U d
t
m
t
m m
0
1
0
∫ ∫+ + +[ ]τ τ τ τ τ ,
where t c∈[ , ]0 . The above estimate and Gronwall inequality imply inequality (38).
An easy computation shows that for t c∈[ , ]0 we have
Z tm( )( )+1 ≤ B Z U U d B d U d
t
m m m
t
m
0
1
0
1∫ ∫+ +[ ] ++ +( ) ( ) ( ) ( )( ) ( ) ( ) ( ˜) ( )τ τ τ τ τ τ
and consequently
Z tm( )( )+1 ≤ B B d Be Z U dAc
t
m m+( ) +[ ]∫( ˜) ˜ ( ) ( )
˜ ( ) ( )
0
τ τ τ , t c∈[ , ]0 .
The above inequality and (38) imply
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GENERALIZED SOLUTIONS OF MIXED PROBLEMS … 825
Z t U tm m( ) ( )( ) ( )+ ++1 1 ≤ ˜ ( ) ( )( ) ( )D Z U d
t
m m
0
∫ +[ ]τ τ τ , t c∈[ , ]0 , (41)
where m ≥ 1.
It follows from Lemma 3 that
Z U( ) ( )1 1+ ≤ 2 0 0( )p d+ , t c∈[ , ]0 .
Then the uniform convergence of the sequences { }( )z m and { }( )u m follows from (41)
and from the condition cD̃ < 1.
This completes the proof of Lemma 6.
4. The existence of solutions of nonlinear mixed problems. We are able to state
the main result on the existence of generalized solutions to problem (1), (2). For a
function ϕ ∈C sL1, [ ] and for a point τ ∈[ , ]0 a we put
ϕ τ = max ( , ) : ( , ) ( ) ([ , ] )ϕ ∂ τt x t x E E h Rn∈ − ×{ }0 0 0∪ ∩ ,
∂ ϕ τx = max ( , ) : ( , ) ( ) ([ , ] )∂ ϕ ∂ τx
nt x t x E E h R∈ − ×{ }0 0 0∪ ∩ .
Theorem 1. If Assumptions H [ α ] , H [ f ] , Hc [ f, ϕ ] and H [ c ] are satisfied
then for every ϕ ∈C sL1, [ ] there exists a solution v : E Rc
∗ → to problem (1), (2).
Moreover v ∈C dc
L
ϕ,
, [ ]1 and ∂ ϕx c
LC pv ∈ ,
, [ ]0 .
If ϕ ∈C sL1, [ ] and v ∈C dc
L
ϕ,
, [ ]1 is a solution of equation (1) with initial boun-
dary condition z t x t x( , ) ( , )= ϕ on E E0 0∪ ∂ then there is Λc R∈ + such that
v v v v− + −t x x t∂ ∂ ( ) ≤ Λc t x x tϕ ϕ ∂ ϕ ∂ ϕ− + −[ ] , 0 ≤ t ≤ c. (42)
Proof. It follows from Lemmas 5 and 6 that there is v ∈C dc
L
ϕ,
, [ ]1 such that
v ( t, x ) = lim ( , )( )
m
mz t x
→∞
, ∂x v ( t, x ) = lim ( , )( )
m
mu t x
→∞
uniformly on Ec . Thus we get
v ( t, x ) = F t xx[ , ]( , )v v∂ ,
∂x v ( t, x ) = G t xx[ , ]( , )v v∂ ,
and
g t xx[ , ]( , , )v v∂ τ =
x f P t x dq x
t
+ ( )∫ ∂ ∂ ξ ξ
τ
[ , ] , ,( )v v .
Moreover, the initial boundary conditions
v = ϕ, ∂x v = ∂x ϕ on ( ) ([ , ] )E E h c Rn
0 0 0∪ ∂ × − ×
are satisfied. It follows from the above relations that v is a generalized solution of
problem (1), (2) on Ec
∗. The proof of the above property of v is similar to the proof
of an edequate properties for initial or initial boundary-value problems considered in
[9] and [10] (Chapter IV). Details are omitted.
Now we prove relation (42). The functions ( , )v v∂x satisfies integral functional
equations (20), (21) and initial boundary conditions (22) with ϕ instead of ϕ . It
follows easily that there are Λ 0 , Λ1 ∈ +R such that the integral inequality
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
826 W. CZERNOUS
v v v v− + −t x x t∂ ∂ ( ) ≤
≤
Λ Λ0 1
0
ϕ ϕ ∂ ϕ ∂ ϕ ∂ ∂ ττ τ− + −[ ] + − + −[ ]∫t x x t x x
t
dv v v v ( ) , 0 ≤ t ≤ c,
is satisfied. Using the Gronwall inequality we obtain (42) with Λ Λ Λc c= 0 1exp( ).
This proves the theorem.
Suppose that β0 : [ 0, a ] → R , β ′ : E → R
n
are given functions and that conditi-
ons (8) are satisfied. We consider the operator f defined by (6). In this case (1) is
equivalent to the differential equation with deviated variables (7). Now we formulate
our existence result for problem (7), (2).
Assumption H [ F ] . Suppose that the function F : E × R × R
n
→ R of the vari-
ables ( t, x, p, q ) satisfies the conditions
1) F ( ⋅ , x, p, q ) : [ 0, a ] → R is measurable for each ( x, p, q ) ∈ [ – b, b ] × R × R
n
and there is B0 ∈ Θ such that
F t x p q( , , , ) ≤ B p0( ) on E × R × R
n
;
2) the partial derivatives
( )( ), , ( )∂ ∂x xF Q F Q
n1
… = ∂x F Q( ), Q = ( t, x, p, q ) ,
( )( ), , ( )∂ ∂p pF Q F Q
n1
… = ∂p F Q( ) ,
( )( ), , ( )∂ ∂q qF Q F Q
n1
… = ∂q F Q( )
exist for ( x, p, q ) ∈ [ – b, b ] × R × R
n
and for almost all t ∈ [ 0, a ] ;
3) the functions
∂x
nF x p q a R( , , , ) : [ , ]⋅ →0 , ∂p F x p q a R( , , , ) : [ , ]⋅ →0 ,
∂q
nF x p q a R( , , , ) : [ , ]⋅ →0
are measurable and there is B R∈ + such that
∂x F t x p q( , , , ) ≤ B , ∂p F t x p q( , , , ) ≤ B , ∂q F t x p q( , , , ) ≤ B
for ( x, p, q ) ∈ [ – b, b ] × R × R
n
and for almost all t ∈ [ 0, a ] ;
4) there is C R∈ + such that the functions ∂x F , ∂p F , ∂q F satisfies the Lip-
schitz condition with respect to ( x, p, q ) ∈ [ – b, b ] × R × R
n
for almost all t ∈ [ 0, a ] .
Assumption H [ ββββ ] . Suppose that the functions β0 : [ 0, a ] → R , β ′ : E → R
n
are
such that
1) 0 ≤ β0 ( t ) ≤ t for t ∈ [ 0, a ] and ′ ∈ −β ( , ) [ , ]t x c c for ( t, x ) ∈ E and for each
i, 1 ≤ i ≤ n, and t ∈ [ 0, a ] we have
βi ( t, x ) ≥ bi for x i∈ +∆ and βi ( t, x ) ≤ bi for x i∈ −∆ ;
2) there is r R0 ∈ + such that
β β0 0( ) ( )t t− ≤ r t t0 − on [ 0, a ] ;
3) β ′ is of class C
1 and ∂ βx t x r′ ≤( , ) 0 on E ;
4) there is r R1 ∈ + such that
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
GENERALIZED SOLUTIONS OF MIXED PROBLEMS … 827
∂ β ∂ βx xt x t x′ − ′( , ) ( , ) ≤ r t t x x1 − + −[ ] on E .
Theorem 2. Suppose that Assumptions H [ α ] , H [ F ] , H [ β ] are satisfied and
ϕ ∈C sL1, [ ] and there is ψ ∂: 0 E Rn→ , ψ = ( ψ1 , … , ψn ) , such that
∂ ϕt t x( , ) = F t x t x t x( , , ( ( , )), ( , ))ϕ β ψ (43)
for x ∈ ∆ and for almost all t ∈ [ 0, a ] .
Then there is c ∈ ( 0, a ] and v : E Rc
∗ → such that v is a solution of (7), (2)
and v ∈C dc
L
ϕ,
, [ ]1 , ∂ ϕx c
LC pv ∈ ,
, [ ]0 .
If ϕ ∈C sL1, [ ] and condition (43) is satisfied with ϕ instead of ϕ and
v ∈C dc
L
ϕ,
, [ ]1 is a solution of equation (7) with initial boundary condition z ( t, x ) =
= ϕ ( , )t x o n E E0 0∪ ∂ then there is Λc R∈ + such that estimate (42) is satis-
fied.
Proof. Write
˜ ( )β0 t = β α0 0( ) ( )t t− , ˜ ( , )′β t x = ′ − ′β α( , ) ( , )t x t x , ( t, x ) ∈ E,
and ˜ ( , )β t x = ( )˜ ( , ), ˜ ( , )β β0 t x t x′ . Then the operator f is defined by
f ( t, x, w, q ) = F t x w t x q( ( ) ), , ˜ ( , ) ,β , ( t, x, w, q ) ∈ Ω ,
and
∂x f t x w q( , , , ) = ∂ βx F t x w t x q( ( ) ), , ˜ ( , ) , +
+ ∂ β ∂ β ∂ βp x xF t x w t x q w t x t x( ( ) ) ( ), , ˜ ( , ) , ˜ ( , ) ˜ ( , )′ ,
∂q f t x w q( , , , ) = ∂ βq F t x w t x q( ( ) ), , ˜ ( , ) , ,
∂w f t x w q w( , , , ) = F t x w t x q w t x( ( ) ) ( ), , ˜ ( , ) , ˜ ( , )β β ,
where ( t, x, q ) ∈ [ 0, a ] × [ – b, b ] × R
n
and w C D R∈ 0 1, ( , ) , w C D R∈ ( , ). It is clear
that
∂x f t x w q( , , , ) ≤ B r r w1 0 0 1+ +[ ]( ) ,
∂q f t x w q( , , , ) ≤ B , ∂w f t x w q( , , , )
�
≤ B .
We conclude from Assumptions H [ f ] and H [ α ] , H [ β ] that
∂ ∂x xf t x w w q f t x w q( , , , ) ( , , , )+ − ≤
≤ C r r w B r r r r w x xL L1 0 0 1
2
1 1 0 0
2
1+ +[ ] + + + +[ ][ ] −( ) ( ), , +
+ C r r w q q c r r B C w wL L1 0 0 1 0 0 1 1+ +[ ] − + + + +( )[ ]( ) ( ), ,
and
∂ ∂q qf t x w w q f t x w q( , , , ) ( , , , )+ − ≤ C r r w x x q q wL1 0 0 1 1+ +( ) − + − +[ ]( ) , ,
∂ ∂q w Lf t x w w q f t x w q C r r w x x q q w( , , , ) ( , , , ) ( ) ,+ − ≤ + +( ) − + − +[ ]�
1 0 0 1 1 .
It follows that all the assumptions of Theorem 1 are satisfied and the assertion follows.
Remark 5. It is important in our considerations that we have assumed the local
with respect to the functional variable Lipschitz condition for the derivatives ∂x f ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
828 W. CZERNOUS
∂w f , ∂q f on some special function spaces. Let us consider the simplest assumption.
Suppose that there is L R∈ + such that
∂ ∂x xf t x w q f t x w q( , , , ) ( , , , )− ≤ L x x w w q q− + − + −[ ]0 (44)
and that suitable estimates for the derivatives ∂w f and ∂q f are satisfied. It is easy
to see that our results are true under the above assumptions. Now we show that our
formulation of the Lipschitz condition is important.
It is easy to see that the operator f defined by (6) does not satisfy condition of the
form (44). Then there is a class of nonlinear equations satisfying Assumption H [ f ]
and do not satisfying conditions of type (44).
Remark 6. In the case when (1) is reduced to the integral functional equation (10),
one can formulate adequate assumptions and prove the particular version of the
existence theorem, as easily as it was done above for the problem (7), (2).
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Received 13.07.2004
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 6
|
| id | umjimathkievua-article-3497 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:43:39Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a9/dbfc99b0db12499168bdb1ebdb09cfa9.pdf |
| spelling | umjimathkievua-article-34972020-03-18T19:56:00Z Generalized solutions of mixed problems for first-order partial functional differential equations Уузагальнені розв'язки мішаних задач для функціональних диференціальних рівнянь з частинними похідними першого порядку Czernous, W. Черноус, В. A theorem on the existence of solutions and their continuous dependence upon initial boundary conditions is proved. The method of bicharacteristics is used to transform the mixed problem into a system of integral functional equations of the Volterra type. The existence of solutions of this system is proved by the method of successive approximations using theorems on integral inequalities. Classical solutions of integral functional equations lead to generalized solutions of the original problem. Differential equations with deviated variables and differential integral problems can be obtained from the general model by specializing given operators. Доведено теорему про існування розв'язків та їх неперервну залежність від початкових граничних умов. Для перетворення мішаної задачі у систему інтегральних функціональних рівнянь типу Вольтерра використано метод біхарактеристик. Існування розв'язків цієї системи доведено за допомогою методу послідовних наближень та теорем про інтегральні нерівності. Класичні розв'язки інтегральних функціональних рівнянь приводять до узагальнених розв'язків початкової задачі. Із загальної моделі за допомогою конкретизації заданих операторів можна отримати диференціальні рівняння із змінними, що відхиляються, та диференціальні інтегральні задачі. Institute of Mathematics, NAS of Ukraine 2006-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3497 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 6 (2006); 804–828 Український математичний журнал; Том 58 № 6 (2006); 804–828 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3497/3731 https://umj.imath.kiev.ua/index.php/umj/article/view/3497/3732 Copyright (c) 2006 Czernous W. |
| spellingShingle | Czernous, W. Черноус, В. Generalized solutions of mixed problems for first-order partial functional differential equations |
| title | Generalized solutions of mixed problems for first-order partial functional differential equations |
| title_alt | Уузагальнені розв'язки мішаних задач для функціональних диференціальних рівнянь з частинними похідними першого порядку |
| title_full | Generalized solutions of mixed problems for first-order partial functional differential equations |
| title_fullStr | Generalized solutions of mixed problems for first-order partial functional differential equations |
| title_full_unstemmed | Generalized solutions of mixed problems for first-order partial functional differential equations |
| title_short | Generalized solutions of mixed problems for first-order partial functional differential equations |
| title_sort | generalized solutions of mixed problems for first-order partial functional differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3497 |
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