Journal of Applied Mathematics and Stochastic Analysis, 15:2 (2002), 125-140.
ON A CLASS OF NONCLASSICAL HYPERBOLIC
EQUATIONS WITH NONLOCAL CONDITIONS
ABDELFATAH BOUZIANI
Centre Universitaire d'Oum El Baouagui
´
´
Departement
de Mathematiques
´
B.P. 565, Oum El Baouagui 04000, Algerie
(Received January, 1998; Revised September, 2000)
This paper proves the existence, uniqueness and continuous dependence of a
solution of a class of nonclassical hyperbolic equations with nonlocal boundary
and initial conditions. Results are obtained by using a functional analysis method
based on an a priori estimate and on the density of the range of the linear operator
corresponding to the abstract formulation of the considered problem.
Key words: Nonclassical Hyperbolic Equations, Nonlocal Boundary Conditions, A Priori Estimate, Strong Solutions.
AMS subject classifications: 35L15, 35L20, 35L99, 35B45, 35D05.
1. Introduction
In this article, we prove the existence, uniqueness and continuous dependence on a solution
of a class of nonclassical hyperbolic partial differential equations with nonlocal boundary and
initial conditions. Results are obtained by using a functional analysis method based on an a
priori estimate and on the density of the range of the operator corresponding to the abstract
formulation of the considered problem. The precise statement of the problem is of the form:
Let ,  !, X:  ! Ð: œ "ß #Ñ and UÀ œ ÖÐBß >" ß ># Ñ − ‘$ À !  B  , , !  >"  X" , ! 

>#  X# ×. Find a function ÐBß >" ß ># Ñ Ä @ÐBß >" ß ># Ñ, where ÐBß >" ß ># Ñ − U , satisfying the
equation
_@À œ
`#@
`>" `>#

` ˆ
`@ ‰
`B +ÐBß >" ß ># Ñ `B
œ fÐBß >" ß ># Ñß
Ð"Þ"Ñ
the initial conditions
j" @À œ @ÐBß !ß ># Ñ œ FÐBß ># Ñß
ÐBß ># Ñ − Ð!ß ,Ñ ‚ Ð!ß X# Ñß
j# @À œ @ÐBß >" ß !Ñ œ GÐBß >" Ñ,
ÐBß >" Ñ − Ð!ß ,Ñ ‚ Ð!ß X" Ñ,
Ð"Þ#Ñ
the integral condition
' @ÐBß >" ß ># Ñ.B œ IÐ>" ß ># Ñ
,
Ð>" ß ># Ñ − Ð!ß X" Ñ ‚ Ð!ß X#Ñß
Ð"Þ$Ñ
!
and one of the following boundary conditions:
`@Ð!ß>" ß># Ñ
`B
œ .Ð>" ß ># Ñ,
Ð># ß ># Ñ − Ð!ß X" Ñ ‚ Ð!ß X# Ñß
Printed in the U.S.A. ©2002 by North Atlantic Science Publishing Company
Ð"Þ%+Ñ
125
126
ABDELFATAH BOUZIANI
' B@ÐBß >" ß ># Ñ.B œ ;Ð>" ß ># Ñ,
,
Ð>" ß ># Ñ − Ð!ß X" Ñ ‚ Ð!ß X# Ñß
Ð"Þ%,Ñ
!
where +ß 0 ß Fß Gß Iß . and ; are given functions.
Mixed problems with integral conditions for parabolic equations are studied by different
method in Batten [1], Cannon [14], Kamynin [22], Ionkin [19], Cannon and van der Hoek
[17, 18], Yurchuk [25], Benouar and Yurchuk [2], Cahlon, Kulkarni, and Shi [15], Cannon,
Esteva, and van der Hoek [16], Byszewski [11], Lin [23], Shi [24], Bouziani [3-5], BouzianiBenouar [7], Jones, Jumarhon, McKee, and Scott [19], and Jumarhon and McKee [20]. A
mixed problem with integral condition for a second order pluriparabolic equation has been
investigated by Bouziani [6]. Nonlocal nonlinear hyperbolic problems were studied by
Byszewski [9-11], by Byszewski and Lakshmikantham [12], and by Byszewski and
Papageorgiou [13]. A mixed problem with integral conditions for a second order classical
hyperbolic equation has been treated under growth conditions in Bouziani and Benouar [8].
The results of the paper are generalizations of those given in [8-12]. These findings are also
continuations of those obtained by the author in [8].
The paper is organized as follows. In Section 2, we state three assumptions on the functions involved in problem (1.1)-(1.4) and we reduce the posed problem to one with
homogeneous boundary conditions. In addition, we present an abstract formulation of the
considered problem and we define the strong solution of the problem. In Section 3, the
uniqueness and continuous dependence of the solution are established. Finally, in Section 4
we prove the existence of the strong solution and offer remarks on its generalizations.
2. Preliminaries
First, we begin with the following assumptions on function +:
Assumption A1:
-! Ÿ + Ÿ -" ;
`+
`B

`+
Ÿ -# à `>
Ÿ -$ Ð: œ "ß #Ñ in U Þ
:
Assumption A2:
-% Ÿ
`+
`>: ß
-& Ÿ
`#+
`>#:
Ÿ -' ,
`#+
`>: `B
Ÿ -( Ð: œ "ß #Ñ;
$
`$+
, `+
`>" `>## `># `>#"
`#+
`#+
`># `>" ß `>" `>#
Ÿ -) ;

Ÿ -* in U Þ
In Assumptions A1-A2, we assume that -3 Ð3 œ !ß á ß *Ñ are positive constants. We also,
assume that the functions F and G satisfy the following:
Assumption A3: Functions F and G satisfy the compatibility conditions:
' FÐBß ># Ñ.B œ IÐ!ß ># Ñß
,
!
` FÐ!ß># Ñ
`B
œ .Ð!ß ># Ñß
' GÐBß >" Ñ.B œ IÐ>" ß !Ñß
,
!
` GÐ!ß>" Ñ
`B
œ .Ð>" ß !Ñ
– ' BFÐBß ># Ñ.B œ ;Ð!ß ># Ñß ' BGÐBß >" Ñ.B œ ;Ð>" ß !Ñ, respectively—ß
,
!
,
!
Class of Nonclassical Hyperbolic Equations
127
and
FÐBß !Ñ œ GÐBß !Ñ
where B − Ð!ß ,Ñß >: − Ò!ß X: ÓÐ: œ "ß #Ñ.
Let us reduce problem (1.1)-(1.3) and (1.4+Ñ [respectively (1.4,Ñ] with nonhomogeneous
boundary conditions (1.3), (1.4+Ñ [respectively (1.4,Ñ] to an equivalent problem with
homogeneous conditions by assuming that we are able to find a function ? œ ?ÐBß >" ß ># Ñ
defined as follows:
?ÐBß >" ß ># Ñ œ @ÐBß >" ß ># Ñ  Y ÐBß >" ß ># Ñß
where
Y ÐBß >" ß ># ÑÀ œ
B
#, Ð#,
 $BÑ.Ð>" ß ># Ñ 
Y ÐBß >" ß ># ÑÀ œ ", Ð  "),B#  "#, # B  "ÑIÐ>" ß ># Ñ 
$B#
,$ IÐ>" ß ># Ñ
"#
#
, Ð$B
 #,BÑ;Ð>"ß >#Ñß respectively‘Þ
Consequently, we have to find a function ÐBß >" ß ># Ñ Ä ?ÐBß >" ß ># Ñ, which is a solution of the
problem
_? œ fÐBß >" ß ># Ñ  _Y œ 0 ß
Ð#Þ"Ñ
j" ? œ ?ÐBß !ß ># Ñ œ FÐBß ># Ñ  j" Y œ :ÐBß ># Ñß
Ð#Þ#Ñ
j# ? œ ?ÐBß >" ß !Ñ œ GÐBß >" Ñ  j# Y œ <ÐBß >" Ñß
' ?ÐBß >" ß ># Ñ.B œ !ß
,
Ð#Þ$Ñ
!
`?Ð!ß>" ß># Ñ
`B
œ!
– ' B?ÐBß >" ß ># Ñ.B œ !, respectively—Þ
Ð#Þ%+Ñ
,
Ð#Þ%,Ñ
!
We assume that the functions : and < satisfy conditions of the form (2.3), (2.4+)
[respectively (2.4,Ñ], i.e.:
Assumption A3w À
' :ÐBß !Ñ.B œ !,
` :Ð!ß># Ñ
`B
œ ! – ' B:ÐBß !Ñ.B œ !, respectively—ß
' <ÐBß !Ñ.B œ !,
` <Ð!ß>" Ñ
`B
œ ! – ' B<ÐBß !Ñ.B œ !, respectively—
,
!
,
!
,
!
,
!
and
:ÐBß !Ñ œ <ÐBß !ÑÞ
In order to write down an abstract formulation of the problem we will need suitable
function spaces. Let F#" Ð!ß ,Ñ be a Hilbert space defined by the author in general form in [3].
128
ABDELFATAH BOUZIANI
Here, we reformulate the author's definition. For this purpose, let G! Ð!ß ,Ñ be the vector
space of continuous functions with compact support in Ð!ß ,Ñ. Since such functions are
Lebesgue integrable with respect to .B, we can define on G! Ð!ß ,Ñ the bilinear form ÐÐ † ß † ÑÑ
given by
ÐÐ?ß =ÑÑÀ œ ' gB ? † gB =.Bß
,
Ð#Þ&Ñ
!
where
gB ?À œ ' ?Ð0 ß >Ñ. 0 Þ
B
!
We recall that ÐÐ † ß † ÑÑ is a scalar product on G! Ð!ß ,Ñ for which G! Ð!ß ,Ñ is not complete.
We denote by F#" Ð!ß ,Ñ a completion of G! Ð!ß ,Ñ for a scalar product defined by (2.5). The
norm of function ? in F#" Ð!ß ,Ñ is defined by
² ? ² F#" Ð!ß,Ñ œ ÐÐ?ß ?ÑÑ"Î# œ ² gB ? ² P# Ð!ß,Ñ Þ
Lemma 1: For B − Ð!ß ,Ñ, we have
² ? ² #F " Ð!ß,Ñ Ÿ
#
,#
#
² ? ² #P# Ð!ß,Ñ Þ
Proof: See Corollary 1 for 7 œ " in Bouziani [3].
The spaces P# ÐÐ!ß X: Ñß P# Ð!ß ,ÑÑß P# ÐÐ!ß X: Ñß F#" Ð!ß ,ÑÑ, L " ÐÐ!ß X: Ñß P#Ð!ß ,ÑÑ,
"
L ÐÐ!ß X: Ñ, F#" Ð!ß ,ÑÑ Ð: œ "ß #Ñ, P# ÐÐ!ß X" Ñ ‚ Ð!ß X# Ñß F#" Ð!ß ,ÑÑ are defined as usual.
To problems (2.1)-(2.3), (2.4+Ñ [Ð#Þ%,Ñ, respectively] we assign the operator P œ
Ð_ß j" ß j# Ñ with the domain HÐPÑ consisting of functions ? belonging to P# ÐÐ!ß X" Ñ ‚
#
`? `? ` # ?
Ð!ß X# Ñß F#" Ð!ß ,ÑÑ satisfying the condition `>
ß ,
ß `? ß ` ?# − P# ÐÐ!ß X" Ñ ‚
" `># `>" `># `B `B
Ð!ß X# Ñß F#" Ð!ß ,ÑÑ and conditions (2.3), (2.4+Ñ ÒÐ#Þ%,Ñ, respectively]. The operator P is considered from F to J , where F is the Banach space consisting of functions ? −
P# ÐÐ!ß X" Ñ ‚ Ð!ß X# Ñß F#" Ð!ß ,ÑÑ having finite norms
² ? ² FÀ œ
² ?Ð † ß † ß 7# Ñ ² #L " ÐÐ!ßX" ÑßF " Ð!ß,ÑÑ
sup
#
! Ÿ 7# Ÿ X#
"
#
 sup
² ?Ð † ß 7" ß † Ñ ² #L " ÐÐ!ßX# ÑßF " Ð!ß,ÑÑ
#
! Ÿ 7" Ÿ X"
and satisfying conditions (2.3), (2.4+Ñ [(2.4,), respectively], and J is a Hilbert space of
vector-valued functions Ð0 ß :ß <Ñ with the finite norms
² Ð0 ß :ß <Ñ ² J À œ Š ² 0 ² #P# ÐÐ!ßX" Ñ‚Ð!ßX# ÑßF " Ð!ß,ÑÑ  ² : ² #L " ÐÐ!ßX# ÑßP# Ð!ß,ÑÑ
#
 ² G ² #L " ÐÐ!ßX" ÑßP# Ð!ß,ÑÑ ‹ Þ
"
#
Class of Nonclassical Hyperbolic Equations
129

Finally, we give a definition of a strong solution. Let P be the closure of the operator P

with the domain HÐP Ñ.
Definition: A solution of the operator equation

P ? œ Ð0 ß :ß <Ñ
is called a strong solution of problem (2.1)-(2.3), (2.4+Ñ [(2.4,), respectively].
3. Uniqueness and Continuous Dependence of the Solution
In this section we will prove an a priori estimate. The uniqueness and continuous dependence
of the solution upon the data presented here are direct consequences of the a priori estimate.
Theorem 1: Under Assumption A", the solution of problem Ð#Þ"Ñ-Ð#Þ$Ñ, Ð#Þ%+Ñ Òrespectively, Ð#Þ%,ÑÓ satisfies the following a priori estimate:
² ? ² F Ÿ - ² P? ² J ß
Ð$Þ"Ñ
where -  ! is a constant independent of ?.
Proof: Taking the scalar product, in F#" Ð!ß ,Ñ, of equation (2.1) and Q ?À œ
`?
#Š `>

"
`?
`># ‹,
we obtain
`?
½ `>
½
"
`
`>#
#
F#" Ð!ß,Ñ

`
`>"
`?
½ `>
½
#
F#" Ð!ß,Ñ
 ' +ÐBß >" ß ># Ñ `>`# ?# .B  #'
,
!
 ' +ÐBß >" ß ># Ñ `>`" ?# .B
,
#
,
!
!
`+ÐBß>" ß># Ñ
? gB
`B
`?
œ # Š_?ß `>

"
Š `>`" 
`?
`># ‹.B
Ð$Þ#Ñ
`?
`># ‹F " Ð!ß,Ñ Þ
#
Integrating (3.2) over Ð!ß 7" Ñ ‚ Ð!ß 7# Ñ, where !  7" Ÿ X" and !  7# Ÿ X# , we have
' ½ `?ÐBß>" ß7# Ñ ½
`>"
7"
!
F#" Ð!ß,Ñ
7" ß># Ñ
 ' ½ `?ÐBß
½
`>#
7#
!
.>"  ' ' +ÐBß >" ß 7# ÑÐ?ÐBß >" ß 7# ÑÑ# .B.>"
! !
!
.>#  ' ' +ÐBß 7" ß ># ÑÐ?ÐBß 7" ß ># ÑÑ# .B.>#
,
#
F#" Ð!ß,Ñ
`?
œ #' ' Š0 ß `>

"
7"
7"
,
#
7#
!
7#
! !
`?
`># ‹F " Ð!ß,Ñ .>" >#
#
ÐBß>" Ñ
 ' ½ ` <`>
½
"
7"
!
ÐBß># Ñ
 ' ' +ÐBß >" ß !ÑÐ<ÐBß >" ÑÑ# .B.>"  ' ½ ` :`>
½
#
,
7"
7#
! !
!
 ' ' +ÐBß !ß ># ÑÐ:ÐBß ># ÑÑ# .B.>#
,
! !
7#
#
F#" Ð!ß,Ñ
#
F#" Ð!ß,Ñ
.>"
.>#
Ð$Þ$Ñ
130
ABDELFATAH BOUZIANI
" ß># Ñ
 ' ' ' ’ `+ÐBß>

`>"
, 7"
! !
7#
!
 #' ' '
, 7" 7#
! !
!
`+ÐBß>" ß># Ñ
“?# .B.>" .>#
`>#
`+ÐBß>" ß># Ñ
`?
?gB Š `>
`B
"
`?
`># ‹.B.>" .># Þ

Observe that
`?
#' ' Š0 ß `>

"
7"
!
7#
!
' '
7" 7#
!
!
`?
`># ‹F " Ð!ß,Ñ .>" .>#
#
#
`?
.>" .>#
½ `>
½
" F " Ð!ß,Ñ
#
Ÿ #' '
7" 7 #
!
!
' '
7" 7 #
!
!
² 0 ² #F " Ð!ß,Ñ .>" .>#
#
#
`?
.># .># Þ
½ `>
½
# F " Ð!ß,Ñ
Ð$Þ%Ñ
#
It is easy to see that
 #' ' '
, 7"
! !
7#
!
`+ÐBß>" ß># Ñ
`?
?gB Š `>
`B
"

`?
`># ‹.B.>" .>#
Ð$Þ&Ñ
" ß># Ñ
Ÿ #' ' ' Š `+ÐBß>
‹ ?# .B.>" .>#
`B
, 7" 7#
! !
' '
7" 7#
!
!
#
!
`?
”½ `>" ½
#
F#" Ð!ß,Ñ
`?
 ½ `>
½
#
#
F#" Ð!ß,Ñ
•.>" .># Þ
From (3.3)-(3.5) and from Assumption A1, we get
'
7"
!
'
” ² ?ÐBß >" ß 7# Ñ ² P# Ð!ß,Ñ  ½
#
7#
!
' '
7" 7#
Ÿ -"!
!
!
#
`?ÐBß>" ß7# Ñ
½ " •.>"
`>"
F Ð!ß,Ñ
” ² ?ÐBß 7" ß ># Ñ ² P# Ð!ß,Ñ  ½
#
#
#
`?ÐBß7" ß># Ñ
½ " •.>#
`>#
F Ð!ß,Ñ
`<
² 0 ² #F " Ð!ß,Ñ .>" .>#  ' ” ² < ² #P# Ð!ß,Ñ  ½ `>
½
"
#
7"
!
`:
 ' ” ² : ² #P# Ð!ß,Ñ  ½ `>
½
#
7#
!
' '
7" 7#
 -""
#
!
!
” ² ? ² P# Ð!ß,Ñ  ½ `>" ½
`?
#
F#" Ð!ß,Ñ
#
F#" Ð!ß,Ñ
where
-"! œ
#
maxÐ%ß-" Ñ
minÐ-! ß"Ñ
#
F#" Ð!ß,Ñ
•.>"
Ð$Þ'Ñ
•.>#
`?
 ½ `>
½
#
#
F#" Ð!ß,Ñ
•.>" .># ß
Class of Nonclassical Hyperbolic Equations
131
and
-"" œ
maxÐ#-# ß"Ñ
minÐ-! ß"Ñ Þ
To finish the proof of Theorem 1, we will need the following result:
Lemma 2: If 0" and 0# are nonnegative functions on the rectangle Ð!ß X" Ñ ‚ Ð!ß X# Ñ, 0" is
integrable on Ð!ß X" Ñ ‚ Ð!ß X# Ñ, and 0# is nondecreasing in Ð!ß X" Ñ ‚ Ð!ß X# Ñ with respect to
each of its variables separately, then
0" Ð7" ß 7# Ñ Ÿ expÐ#-Ð7"  7# ÑÑ0# Ð7" ß 7# Ñ
is a consequence of the inequality
' 0" Ð>" ß 7# Ñ.>"  ' 0" Ð7" ß ># Ñ.>#
7"
0 " Ð 7" ß 7# Ñ Ÿ -
7#
!
 0# Ð7" ß 7# ÑÞ
!
The proof of the above lemma is analogous to the proof of Lemma 1 in Bouziani [6].
Continuing the proof of Theorem 1, we apply Lemma 2 to (3.6). For this purpose, we
denote the left-hand side of (3.6) by 0" Ð7" ß 7# Ñ, and the sum of three first integrals on the
right-hand side of (3.6) by 0# Ð7" ß 7# Ñ. This procedure eliminates the last integral of the righthand side of (3.6) and yields:
'
7"
!
” ² ?ÐBß >" ß 7# Ñ ² P# Ð!ß,Ñ  ½
#
#
`?ÐBß>" ß7# Ñ
½ " •.>"
`>"
F Ð!ß,Ñ
#
7" ß># Ñ
 ' ” ² ?ÐBß 7" ß ># Ñ ² #P# Ð!ß,Ñ  ½ `?ÐBß
½
`>#
7#
!
' '
7" 7#
Ÿ -"" expÐ#-"! Ð7"  7# ÑÑ
!
'
7"
!
 '
7#
!
!
`<
” ² : ² P# Ð!ß,Ñ  ½ `># ½
#
`:
F#" Ð!ß,Ñ
•.>#
Ð$Þ(Ñ
² 0 ² #F " Ð!ß,Ñ .>" .>#
#
” ² < ² P# Ð!ß,Ñ  ½ `>" ½
#
#
#
F#" Ð!ß,Ñ
#
F#" Ð!ß,Ñ
•.>"
•.># Þ
According to Lemma 1, we bound below the first and the third terms on the left-hand side
of (3.7) and we bound above the third and the fifth terms on the right-hand side of (3.7).
Consequently, we obtain
#
² ?ÐBß >" ß 7# Ñ ² #L " ÐÐ!ßX" ÑßF " Ð!ß,ÑÑ  ² ?ÐBß 7" ß ># Ñ ² L
" ÐÐ!ßX ÑßF " Ð!ß,ÑÑ
#
#
Ÿ -"# Š ² 0 ²
#
Ð$Þ)Ñ
#
P# ÐÐ!ßX" Ñ‚Ð!ßX# ÑßF#" Ð!ß,ÑÑ
 ²<²
 ² : ² #L " ÐÐ!ßX# ÑßP# Ð!ß,ÑÑ ‹,
#
L " ÐÐ!ßX" ÑßP# Ð!ß,ÑÑ
132
ABDELFATAH BOUZIANI
where
-"" expÐ#-"! ÐX" X# ÑÑmaxŠ"ß ,# ‹
#
-"# œ
minŠ"ß ,## ‹
Þ
Since the right-hand side of inequality (3.8) does not depend on 7" and 7# , then, in the lefthand side of (3.8) we can take the supremum with respect to 7: from ! to X: Ð: œ "ß #Ñ.
"Î#
Therefore, we obtain (3.1) with - œ -"# . The proof of Theorem 1 is complete.
Proposition 1: The operator P from F to J has a closure.
The proof of the above proposition is similar to the proof of Proposition 1 in Bouziani [4].
Theorem 1 can be extended to cover strong solutions by passing to the limit.
Corollary 1: Under the assumptions of Theorem ", there exists a positive constant - such
that

² ? ² F Ÿ - ² P ? ² Jß
where - does not depend on ?.
Corollary 1 implies the following:
Corollary 2: A strong solution of Ð#Þ"Ñ-Ð#Þ$Ñ, Ð#Þ%+Ñ Òrespectively, Ð#Þ%,ÑÓ is unique, if it
exists, and it depends continuously on Ð0 ß :ß <Ñ.


Corollary 3: The range VÐP Ñ of solutions for the closure P of P is closed in J ,

 "
VÐP Ñ œ VÐPÑ and P œ P" , where P" is the unique extension of P" Ðby continuity
from VÐPÑ to VÐPÑÑ.
4. Existence of the Solution
In this section we concentrate on the existence of the strong solution of problem (2.1)-(2.3),
Ð#Þ%+Ñ [respectively, Ð#Þ%,ÑÓ. The main idea is to demonstrate that the range VÐPÑ is dense in
J.
Theorem 2: Suppose that Assumptions A" and A# are satisfied. Then, for arbitrary
0 − P# ÐÐ!ß X" Ñ ‚ Ð!ß X# Ñß F#" Ð!ß ,ÑÑß : − L " ÐÐ!ß X# Ñß P# Ð!ß ,ÑÑ and < − L "ÐÐ!ß X"Ñ,
P# Ð!ß ,ÑÑ, problem Ð#Þ"Ñ-Ð#Þ$Ñ, Ð#Þ%+Ñ Òrespectively, Ð#Þ%,ÑÓ admits a strong solution
 "
? œ P Ð0 ß :ß <Ñ œ P" Ð0 ß :ß <Ñ.
Proof: To prove the existence of a strong solution of problem (2.1)-(2.3), (2.4+Ñ
[respectively Ð#Þ%,Ñ] for all Ð0 ß :ß <Ñ − J , it remains to prove that VÐPÑ œ J . To this end,
we establish the density in the special case, where ? belongs to H! ÐPÑ, which is equal to the
set of all ? − HÐPÑ vanishing in a neighborhood of >" œ ! and ># œ !. Then we proceed to
the general case.
Proposition 2: Let the assumptions of Theorem # be satisfied. If for 1 −
P# ÐÐ!ß X" Ñ ‚ Ð!ß X# Ñà F#" Ð!ß ,ÑÑ and for all ? − H! ÐPÑ, we have
' ' ÐÐ_?ß 1ÑÑ.>" .># œ ! a? − H! ÐPÑÞ
X"
!
X#
Ð%Þ"Ñ
!
Then 1 vanishes almost everywhere in U.
Proof of Proposition 2: By the fact that relation (4.1) holds for any function ? − H! ÐPÑ,
we express it in a special form. Let
Class of Nonclassical Hyperbolic Equations
Ú
!ß
Ý
>" >#
? œ Û ' ' `#?
Ý
` 7" ` 7# . 7" . 7# ß
Ü =" =#
`#?
`>" `>#
for : œ "ß # and let
133
! Ÿ > : Ÿ =: ß
Ð%Þ#Ñ
=: Ÿ >: Ÿ X:
be a solution of
?
+Ð5ß >" ß ># Ñ `>`" `>
œ g>:‡ 1: œ ' 1: . 7: ß
#
X:
#
Ð%Þ$Ñ
>:
where 5 is a fixed number belonging to Ò!ß X Ó. Formulas (4.2) and (4.3) imply that ? belongs
to H=" ß=# ÐPÑ, which denotes the set of all ? − HÐPÑ vanishing in the neighborhood of >: œ =:
and >:  =: Ð: œ "ß #Ñ. Putting =: œ ! Ð: œ "ß #Ñ, we have that ? − H! ÐPÑ. It follows from
(4.2) and (4.3) that
#
1œ
:œ"
1: œ  Š `>`" 
`
`># ‹
?
Š+Ð5ß >" ß ># Ñ `>`" `>
‹Þ
#
#
Ð%Þ%Ñ
To prove that the function 1, defined by (4.4), belongs to P# ÐÐ=" ß X" Ñ ‚
Ð=# ß X# Ñß F#" Ð!ß ,ÑÑ, we apply the following result:
Lemma 3: If the assumptions of Proposition # are satisfied, then the function ?,
$
$
?
?
defined by Ð%Þ#Ñ, has derivatives of the form `>`" `>
and `>`# `>
belonging to
#
#
P# ÐÐ=" ß X" Ñ ‚ Ð=# ß X# Ñß F#" Ð!ß ,ÑÑ.
Proof: See Lemma 2 in Bouziani [3].
Relation (4.4) implies that (4.1) can be written in the form
#
"
#
#
?
?
 ' ' ŠŠ `>`" `>
ß ` Š+Ð5ß >" ß ># Ñ `>`" `>
‹‹‹.>" .>#
# `>"
#
X"
!
X#
!
#
#
?
?
 ' ' ŠŠ `>`" `>
ß ` Š+Ð5ß >" ß ># Ñ `>`" `>
‹‹‹.>" .>#
# `>#
#
X"
!
X#
Ð%Þ&Ñ
!
#
` ˆ
`? ‰ `
?
 ' ' ŠŠ `B
+ÐBß >" ß ># Ñ `B
ß `>" Š+Ð5ß >" ß ># Ñ `>`" `>
‹‹‹.>" .>#
#
X"
!
X#
!
#
` ˆ
`? ‰ `
?
 ' ' ŠŠ `B
+ÐBß >" ß ># Ñ `B
ß `># Š+Ð5ß >" ß ># Ñ `>`" `>
‹‹‹.>" .># œ !Þ
#
X" X#
!
!
Integrating by parts, each term of (4.5), we obtain
' ' +Ð5ß =" ß ># Ñ ŠgB `
X#
,
! =#
œ' ' '
,
! ="
X"
=#
X#
#
#
?ÐBß=" ß># Ñ
`>" `># ‹ .B.>#
#
#
`+Ð5ß>" ß># Ñ
?
ŠgB `>`" `>
‹ .B.>" .>#
`>"
#
#
#
?
?
 #' ' ŠŠ `>`" `>
ß ` Š+Ð5ß >" ß ># Ñ `>`" `>
‹‹‹.>" .># ß
# `>"
#
X" X#
!
!
Ð%Þ'Ñ
134
ABDELFATAH BOUZIANI
' ' +Ð5ß >" ß =# ÑŠgB `
X"
,
! ="
œ' ' '
, X"
! ="
X#
=#
`+Ð5ß>" ß># Ñ
`>#
#
#
?ÐBß>" ß=# Ñ
`>" `># ‹ .B.>"
?
ŠgB `>`" `>
‹ .B.>" .>#
#
#
#
Ð%Þ(Ñ
#
#
?
?
 #' ' ŠŠ `>`" `>
ß ` Š+Ð5ß >" ß ># Ñ `>`" `>
‹‹‹.>" .># ß
# `>#
#
X" X#
!
' ' ’`
, X"
#
! ="
!
+ÐBß>" ßX# Ñ
+Ð5ß >" ß X# Ñ
`>#"

`+ÐBß>" ßX# Ñ `+Ð5ß>" ßX# Ñ
“Ð?ÐBß >" ß X# ÑÑ# .B.>"
`>"
`>"
" ßX# Ñ
 ' ' +ÐBß >" ß X# Ñ+Ð5ß >" ß X# ÑŠ `?ÐBß>
‹ .B.>"
`>"
X"
,
#
Ð%Þ)Ñ
! ="
œ ' ' ' ’`
, X"
! ="
X#
$
=#
+ÐBß>" ß># Ñ
+Ð5ß >" ß ># Ñ
`># `>#"
`+ÐBß>" ß># Ñ `+Ð5ß>" ß># Ñ
`># `>"
`>"

 #' '
,
X#

` # +ÐBß>" ß># Ñ `+Ð5ß>" ß># Ñ
`>#
`>#"
`+ÐBß>" ß># Ñ ` # +Ð5ß>" ß># Ñ
#
`>"
`># `>" “? .B.>" .>#

`+ÐBß>" ß># Ñ
" ß># Ñ
+Ð5ß X" ß ># Ñ?ÐBß X" ß ># Ñ `?ÐBßX
.B.>#
`>"
`>#
! =#
`?
" ß># Ñ
 ' ' ' ’ `+ÐBß>
+Ð5ß >" ß ># Ñ  +ÐBß >" ß ># Ñ `+Ð5`>ß>#" ß># Ñ “Š `>
‹ .B.>" .>#
`>#
"
, X" X#
! ="
#
=#
 #' ' '
,
X"
! ="
 #' ' ' ’ `
,
X" X#
! ="
=#
#
X#
=#
`+ÐBßX" ß># Ñ
`? `?
+Ð5ß >" ß ># Ñ `>
.B.>" .>#
`>"
" `>#
+ÐBß>" ß># Ñ
`>" `B ?

`+ÐBß>" ß># Ñ `?
`#?
`B
`>" “+Ð5 ß >" ß ># ÑgB `>" `># .B.>" .>#
#
` ˆ
`? ‰ `
?
 #' ' ŠŠ `B
+ÐBß >" ß ># Ñ `B
ß `>" Š+Ð5ß >" ß ># Ñ `>`" `>
‹‹‹.>" .># ß
#
X" X#
!
!
' ' ’ `+ÐBßX#" ß># Ñ +Ð5ß X" ß ># Ñ 
`>
,
X#
#
! =#
`+ÐBßX" ß># Ñ `+Ð5ßX" ß>#
“Ð?ÐBß X" ß ># ÑÑ# .B.>#
`>#
`>#
" ß># Ñ
 ' ' +ÐBß X" ß ># Ñ+Ð5ß X" ß ># ÑŠ `?ÐBßX
‹ .B.>#
`>#
, X#
#
! =#
œ ' ' ' ’`
,
! ="

X" X#
=#
$
+ÐBß>" ß># Ñ
+Ð5ß >" ß ># Ñ
`>" `>##
` # +ÐBß>" ß># Ñ `+Ð5ß>" ß># Ñ
`>" `>#
`>#


` # +ÐBß>" ß># Ñ `+Ð5ß>" ß># Ñ
`>"
`>##
`+ÐBß>" ß># Ñ ` # +Ð5ß>" ß># Ñ
#
`>#
`>" `># “? .B.>" .>#
Ð%Þ*Ñ
Class of Nonclassical Hyperbolic Equations
135
`?
" ß># Ñ
 ' ' ' ’ `+ÐBß>
+Ð5ß >" ß ># Ñ  +ÐBß >" ß ># Ñ `+Ð5`>ß>"" ß># Ñ “Š `>
‹ .B.>" .>#
`>"
#
, X" X#
! ="
#
=#
 #' '
,
X"
`+ÐBß>" ßX# Ñ
" ßX# Ñ
+Ð5ß >" ß X# Ñ?ÐBß >" ß X# Ñ `?ÐBß>
.B.>"
`>#
`>"
! ="
 #' ' '
,
X" X#
! ="
=#
 #' ' ' ’ `
, X"
! ="
X#
=#
#
`+ÐBß>" ß># Ñ
`? `?
+Ð5ß >" ß ># Ñ `>
.B.>" .>#
`>#
" `>#
+ÐBß>" ß># Ñ
`># `B ?

`+ÐBß>" ß># Ñ `?
`#?
`B
`># “+Ð5 ß >" ß ># ÑgB `>" `># .B.>" .>#
#
` ˆ
`? ‰ `
?
 #' ' ŠŠ `B
+ÐBß >" ß ># Ñ `B
ß `># Š+Ð5ß >" ß ># Ñ `>`" `>
‹‹‹.>" .># Þ
#
X" X#
!
!
According to Assumptions A1 and A2, we get
-! '
X#
=#
½
#
` # ?ÐBß=" ß># Ñ
`>" `># ½F " Ð!ß,Ñ .>#
#
#
?
Ÿ -$ ' ' ½ `>`" `>
½
#
X"
="
X#
=#
#
F#" Ð!ß,Ñ
.>" .>#
Ð%Þ"!Ñ
#
#
?
?
 #' ' ŠŠ `>`" `>
ß ` Š+Ð5ß >" ß ># Ñ `>`" `>
‹‹‹.>" .># ß
# `>"
#
X" X#
!
-! ' ½ `
X"
="
!
#
#
?ÐBß>" ß=# Ñ
`>" `># ½F " Ð!ß,Ñ .>"
#
#
?
Ÿ -$ ' ' ½ `>`" `>
½
#
X" X#
="
=#
#
F#" Ð!ß,Ñ
.>" .>#
Ð%Þ""Ñ
?
?
 #' ' ŠŠ `>`" `>
ß ` Š+Ð5ß >" ß ># Ñ `>`" `>
‹‹‹.>" .># ß
# `>#
#
X" X#
!
Ð-& -!  -%# Ñ'
#
!
X"
="
#
" ßX# Ñ
² ?ÐBß >" ß X# Ñ ² P# # Ð!ß,Ñ .>"  -!# ' ½ `?ÐBß>
½
`>"
X"
="
Ÿ Ð-* -"  -' -$  #-) -$  #-(# Ñ' '
X" X#
="
=#
="

#-$# -"#
'
-!#
=#
X#
=#
² ?ÐBß X" ß ># Ñ ² #P# Ð!ß,Ñ .># 
-!#
'
#
=#
P# Ð!ß,Ñ
.>"
Ð%Þ"#Ñ
² ? ² #P# Ð!ß,Ñ .>" .>#
`?
 Ð#-" -$  -$#  #-## Ñ' ' ½ `>
½
"
X" X#
#
#
P# Ð!ß,Ñ
X#
½
.>" .>#
#
`?ÐBßX" ß># Ñ
½ # .>#
`>#
P Ð!ß,Ñ
136
ABDELFATAH BOUZIANI
#
?
 -"# ' ' ½ `>`" `>
½
#
X" X#
="
=#
`?
.>" .>#  -"# ' ' ½ `>
½
#
X" X#
#
F#" Ð!ß,Ñ
="
=#
#
P# Ð!ß,Ñ
.>" .>#
#
` ˆ
`? ‰ `
?
 #' ' ŠŠ `B
+ÐBß @ß >" ß ># Ñ `B
ß `>" Š+Ð5ß >" ß ># Ñ `>`" `>
‹‹‹.>" .># ß
#
X" X#
!
!
Ð-& -!  -%# Ñ'
X#
=#
Ÿ
-#-#
# "- # $ '
!
="
X"
" ß># Ñ
² ?ÐBß X" ß ># Ñ ² #P# Ð!ß,Ñ .>#  -!# ' ½ `?ÐBßX
½
`>#
X"
=#
² ?ÐBß >" ß X# Ñ ² #P# Ð!ß,Ñ .>" 
 Ð-* -"  -' -$  #-) -$  #-(# Ñ' '
-!#
'
#
="
X" X#
="
=#
X" X#
`?
 -"# ' ' ½ `>
½
"
X" X#
="
=#
`?ÐBß> ßX Ñ
½ `>"" # ½
=#
#
P# Ð!ß,Ñ
P# Ð!ß,Ñ
#
?
.>" .>#  -"# ' ' ½ `>`" `>
½
#
="
.>#
Ð%Þ"$Ñ
#
.B.>"
.>" .>#
X" X#
#
P# Ð!ß,Ñ
² ? ² #P# Ð!ß,Ñ .>" .>#
`?
 Ð#-" -$  -$#  #-## Ñ' ' ½ `>
½
#
="
X"
#
=#
#
F#" Ð!ß,Ñ
.>" .>#
#
` ˆ
`? ‰ `
?
 #' ' ŠŠ `B
+ÐBß >" ß ># Ñ `B
ß `># Š+Ð5ß >" ß ># Ñ `>`" `>
‹‹‹.>" .># Þ
#
X!
X"
!
!
Observe that
#-"# -$#
'
-!#
="
Ÿ
² ?ÐBß >" ß X# Ñ ² #P# Ð!ß,Ñ .>"
' ' Œ ² ? ² #P# Ð!ß,Ñ  ½ `? ½
`>#
X"
#-"# -$#
-!#
="
X#
=#
#-"# -$#
'
-!#
=#
Ÿ
X"
#-"# -$#
-!#
X#
Ð%Þ"%Ñ
#
P# Ð!ß,Ñ
² ?ÐBß X" ß ># Ñ ² #P# Ð!ß,Ñ .>#
' ' Œ ² ? ² #P# Ð!ß,Ñ  ½ `? ½
`>"
="
X"
=#
X#
.>" .># ß
Ð%Þ"&Ñ
#
P# Ð!ß,Ñ
Adding inequalities (4.10)-(4.15) and applying (4.5), we get
.>" .># Þ
Class of Nonclassical Hyperbolic Equations
'
X#
=#
”½
#
` # ?ÐBß=" ß># Ñ
`>" `># ½F " Ð!ß,Ñ
" ß># Ñ
 ² ?ÐBß X" ß ># Ñ ² #P# Ð!ß,Ñ  ½ `?ÐBßX
½
`>#
#
 ' ”½ `
X"
#
="
#
?ÐBß>" ß=# Ñ
`>" `># ½F " Ð!ß,Ñ
#
#
P# Ð!ß,Ñ
" ßX# Ñ
 ² ?ÐBß >" ß X# Ñ ² #P# Ð!ß,Ñ  ½ `?ÐBß>
½
`>"
?
Ÿ -"$ ' ' ”½ `>`" `>
½
#
X"
="
137
X#
#
=#
`?
 ½ `>
½
"
#
P# Ð!ß,Ñ
•.>#
#
P# Ð!ß,Ñ
•.>"
Ð%Þ"'Ñ
#
F#" Ð!ß,Ñ
`?
 ½ `>
½
#
 ² ? ² #P# Ð!ß,Ñ
#
P# Ð!ß,Ñ
•.>" .># ß
where
-"$ À œ
maxŒ-$ ß-" -* -$ -' -# -) #-(# 
#-"# -$#
#-# -#
ß#-" -$ -$# #-## -"#  "# $
-!#
-!
.
minŒ-! , #! ß #" Ð-! -& -%# Ñ
-#
Inequality (4.16) is basic in the proof of Proposition 2. In order to apply inequality (4.16),
we introduce a new function ) by the formula
)ÐBß >" ß ># ÑÀ œ ' ?7" . 7"  ' ?7# . 7# Þ
X"
>"
X#
>#
Consequently,
?ÐBß X" ß ># Ñ œ )ÐBß =" ß ># Ñß
?ÐBß >" ß X# Ñ œ )ÐBß >" ß =# Ñ,
`?ÐBßX" ß># Ñ
`>#
`?ÐBß>" ßX# Ñ
`>"
œ
` )ÐBß=" ß># Ñ
ß
`>#
œ
` )ÐBß>" ß=# Ñ
Þ
`>"
Therefore (4.16) yields
'
=#
X#
”½
#
` # ?ÐBß=" ß># Ñ
`>" `># ½F " Ð!ß,Ñ
#
Ð%Þ"(Ñ
" ß># Ñ
 ˆ"  $% -"$ ÐX#  =# щŒ ² )ÐBß =" ß ># Ñ ² #P# Ð!ß,Ñ  ½ ` )ÐBß=
½
`>#
 ' ”½ `
X"
="
#
#
?ÐBß>" ß=# Ñ
`>" `># ½F " Ð!ß,Ñ
#
X" X#
#
$-"$ ' '
`#?
”½ `>" `># ½ "
%
F# Ð!ß,Ñ
=" =#
P# Ð!ß,Ñ
•.>#
 ˆ"  %$ -"$ ÐX"  =" щ
" ß=# Ñ
‚ Œ ² )ÐBß >" ß =# Ñ ² #P# Ð!ß,Ñ  ½ ` )ÐBß>
½
`>"
Ÿ
#
`)
 ² ) ² #P# Ð!ß,Ñ  ½ `>
½
#
#
P# Ð!ß,Ñ
#
P# Ð!ß,Ñ
•.>"
`)
 ½ `>
½
"
#
P# Ð!ß,Ñ
•.>" .># Þ
138
ABDELFATAH BOUZIANI
Hence, if =:!  ! satisfies "  $% -"$ ÐX:  =:! Ñ œ "Î# Ð: œ "ß #Ñ, then (4.17) becomes
' ”½ `
X#
=#
#
#
?ÐBß=" ß># Ñ
`>" `># ½F " Ð!ß,Ñ
 ' ”½ `
X"
="
#
#
" ß># Ñ
 ² )ÐBß =" ß ># Ñ ² #P# Ð!ß,Ñ  ½ ` )ÐBß=
½
`>#
#
?ÐBß>" ß=# Ñ
`>" `># ½F " Ð!ß,Ñ
#
Ÿ
P# Ð!ß,Ñ
" ß=# Ñ
 ² )ÐBß >" ß =# Ñ ² #P# Ð!ß,Ñ  ½ ` )ÐBß>
½
`>"
$-"$ ' '
#
=" =#
X"
#
X#
` ?
”½ `>" `># ½
`)
 ½ `>
½
#
#
#
P# Ð!ß,Ñ
•.>#
#
P# Ð!ß,Ñ
•.>"
Ð%Þ")Ñ
#
F#" Ð!ß,Ñ
 ² ) ² #P# Ð!ß,Ñ
`)
 ½ `>
½
"
#
P# Ð!ß,Ñ
•.>" .># Þ
Applying a variant of Lemma 2 for the case of the reversed intervals to (4.18), to eliminate
the double integrals on the right-hand side in (4.18), we conclude that 1 œ ! almost
everywhere in Ð!ß ,Ñ ‚ Ð=" ß X" Ñ ‚ Ð=# ß X# Ñ. Proceeding in this way, step by step, we prove
that 1 œ ! almost everywhere in U. The proof of Proposition 2 is complete.
To finish the proof of Theorem 2, consider the general case. Let the element
[ œ Ð0 ß :ß <Ñ − J be orthogonal to VÐPÑ, so that
Ð_?ß 0 ÑP# ÐÐ!ßX" Ñ‚Ð!ßX# ÑßF#" Ð!ß,ÑÑ  Ðj" ?ß :ÑL " ÐÐ!ßX# ÑßP# Ð!ß,ÑÑ
Ð%Þ"*Ñ
 Ðj# ?ß <ÑL " ÐÐ!ßX" ÑßP# Ð!ß,ÑÑ œ !, a? − HÐPÑÞ
Putting ? in (4.19) equal to any element of H! ÐPÑ, we get
Ð_?ß 0 ÑP# ÐÐ!ßX" Ñ‚Ð!ßX# ÑßF#" Ð!ß,ÑÑ œ !, a? − H! ÐPÑÞ
Proposition 2 implies that 0 ´ !. Hence, (4.19) implies that
Ðj" ?ß :ÑL " ÐÐ!ßX# ÑßP# Ð!ß,ÑÑ  Ðj# ?ß <ÑL " ÐÐ!ßX" ÑßP#Ð!ß,ÑÑ œ !, a? − HÐPÑÞ
Since j" ? and j# ? are independent and the ranges of values j" and j# are everywhere dense in
L " ÐÐ!ß X# Ñß P# Ð!ß ,ÑÑ, L " ÐÐ!ß X" Ñß P# Ð!ß ,ÑÑ, the above equality implies that : ´ !, and
< ´ ! (recall that : and < satisfy Assumption A3w Ñ. Hence VÐPÑ œ J . Therefore, the proof
of Theorem 2 is complete.
Remark 1: If equation (1.1) is replaced by
_@À œ
`#@
`>" `>#

` ˆ
`@ ‰
`B +ÐBß >" ß ># Ñ `B
œ 0 ÐBß >" ß ># ß @Ñ
then the existence, uniqueness and continuous dependence upon the data of a solution can be
proved by the same method, provided that 0 satisfies a growth condition of the form
µ
± 0 ÐBß >" ß ># ß @Ñ ± Ÿ 0 ÐBß >" ß ># Ñ  G ± @ ± Þ
Class of Nonclassical Hyperbolic Equations
Ú
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Û
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Ý
Ü
139
Remark 2: Our results can be extended to the following nonlocal hyperbolic problem:
`#@
`>" `>#
_@À œ
 signÒÐ"  ± -" ± # ÑÐ"  ± -# ± # ÑÓ
`
`@
`B Ð+ÐBß >" ß ># Ñ `B Ñ
j" @À œ @ ± >" œ!  -" @ ± >" œX" œ FÐBß ># Ñß
j# @À œ @ ± ># œ!  -# @ ± ># œX# œ GÐBß >" Ñß
`@Ð!ß>" ß># Ñ
œ ." Ð>" ß ># Ñß
`B
' @ÐBß >" ß ># Ñ.B œ IÐ>" ß ># Ñß
œ 0 , ÐBß >" ß ># Ñ − Uß -" ß -# − Gß
ÐBß ># Ñ − Ð!ß ,Ñ ‚ Ð!ß X# Ñß
ÐBß >" Ñ − Ð!ß ,Ñ ‚ Ð!ß X"Ñß
Ð>" ß ># Ñ − Ð!ß X" Ñ ‚ Ð!ß X# Ñß
,
Ð>" ß ># Ñ − Ð!ß X" Ñ ‚ Ð!ß X#Ñ
!
Ò' B@ÐBß >" ß ># Ñ.B œ ;Ð>" ß ># Ñß
,
Ð>" ß ># Ñ − Ð!ß X" Ñ − Ð!ß X#Ñ, respectivelyÓÞ
!
Remark 3: Our results can also be extended to the following problem:
#
Ú
` #7
_@À œ `>`" `>@ #  `B
Ý
#7 Ð+ÐBß >" ß ># Ñ@Ñ œ 0 ß
Ý
Ý
Ý
Ý
j
@À
œ
@ÐBß
!ß
># Ñ œ FÐBß ># Ñß
"
Ý
Ý
j# @À œ @ÐBß >" ß !Ñ œ GÐBß >" Ñß
Û
` : @Ð!ß>" ß># Ñ
œ !ß !  : Ÿ 7ß
Ý
`B:
Ý
Ý
Ý ,
Ý
Ý
Ý ' B; @ÐBß >Ñ.B œ !ß : Ÿ ; Ÿ #7  "ß
Ü!
ÐBß >" ß ># Ñ − Uß
ÐBß ># Ñ − Ð!ß ,Ñ ‚ Ð!ß X# Ñß
ÐBß >" Ñ − Ð!ß ,Ñ ‚ Ð!ß X" Ñß
Ð>" ß ># Ñ − Ð!ß X" Ñ ‚ Ð!ß X# Ñß
Ð>" ß ># Ñ − Ð!ß X" Ñ ‚ Ð!ß X# Ñ,
where 7ß :ß ; are integers.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
Batten, Jr., G.W., Second order correct boundary condition for the numerical solution
of the mixed boundary problem for parabolic equations, Math. Comput. 17 (1963), 405413.
Benouar, N.E. and Yurchuk, N.I., Mixed problem with an integral condition for parabolic equations with the Bessel operator, Differ. Urav. 27:12 (1991), 2094-2098.
Bouziani, A., Mixed problem with integral conditions for a certain parabolic equation,
J. of Appl. Math. and Stoch. Anal. 9:3 (1996), 323-330.
Bouziani, A., On a third order parabolic equation with a nonlocal boundary condition,
J. of Appl. Math. and Stoch. Anal., (in press).
Bouziani, A., Solution forte d'un probléme mixte avec condition non locale pour une
´
classe d'equations
paraboliques, Maghreb Math. Rev. 6:1 (1997), 1-17.
Bouziani, A., Mixed problem with nonlocal condition for certain pluriparabolic equations, Hiroshima Math. J. 27:3 (1997), 373-390.
´
Bouziani, A. and Benouar, N.E., Probléme mixte avec conditions integrales
pour une
´
´
classe d'equations
paraboliques, Comptes Rendus de l'Academie
des Sciences Paris 321
(1995), 1177-1182.
Bouziani, A. and Benouar, N.E., Probléme mixte avec conditions intégrales pour une
´
classe d'equations
hyperboliques, Bull. Belgian Math. Soc. Simon Stevan 31 (1996),
125-133.
Byszewski, L., Existence and uniqueness of solutions of nonlocal problems for
hyperbolic equation ?B> œ J ÐBß >ß ?ß ?B Ñ, J. of Appl. Math. and Stoch. Anal. 3:3 (1990),
163-168.
140
ABDELFATAH BOUZIANI
[10] Byszewski, L., Theorem about existence and uniqueness of continuous solutions of
nonlocal problem for nonlinear hyperbolic equation, Appl. Anal. 40 (1991), 173-180.
[11] Byszewski, L., Differential and Functional-Differential Problems with Nonlocal
Conditions, Cracow Univ. of Tech. Monograph 184, Cracow 1995.
[12] Byszewski, L. and Lakshmikantham, V., Monotone iterative technique for nonlocal
hyperbolic differential problem, J. of Math. and Phys. Sci. 26:4 (1992), 345-359.
[13] Byszewski, L. and Papageorgiou, N.S., An application of noncompactness technique to
an investigation of the existence of solutions to a nonlocal multivalued Darboux
problem, J. of Appl. Math. and Stoch. Anal. (1998), (in press).
[14] Cannon, J.R., The solution of the heat equation subject to the specification of energy,
Quart. Appl. Math. 21:2 (1963), 155-160.
[15] Cannon, J.R., The one-dimensional heat equation, In: Encyclopedia of Mathematics
and its Applications 23, Addison-Wesley, Reading, MA, Menlo Park, CA (1984).
[16] Cannon, J.R., Esteva, S.P. and Van der Hoek, J., A Galerkin procedure for the diffusion
equation subject to the specification of mass, SIAM J. Numer. Anal. 24 (1987), 499515.
[17] Cannon, J.R. and Van der Hoek, J., The existence and the continuous dependence for
the solution of the heat equation subject to the specification of energy, Boll. Uni. Math.
Ital. Suppl. 1 (1981), 253-282.
[18] Cannon, J.R. and Van der Hoek, J., An implicit finite difference scheme for the
diffusion of mass in a portion of the domain, In: Numerical Solutions of Partial
Differential Equations (ed. by J. Noye), North-Holland, Amsterdam (1982), 527-539.
[19] Ionkin, N.I., Solution of boundary value problem in heat conduction theorem with
nonlocal boundary conditions, Diff. Urav. 13:2 (1977), 294-304.
[20] Jones, S., Jumarhorn, B., McKee, S. and Scott, J.A., A mathematical model of
biosensor, J. Eng. Math. 30 (1996), 312-337.
[21] Jumarhorn, B. and McKee, S., On the heat equation with nonlinear and nonlocal
boundary conditions, J. Math. Anal. Appl. 190 (1995), 806-820.
[22] Kamynin, N.I., A boundary value problem in the theory of the heat conduction with
nonclassical boundary condition, Th. Vychisl. Mat. Mat. Fiz. 43:6 (1964), 1006-1024.
[23] Lin, Y., Parabolic Partial Differential Equations Subject to Nonlocal Boundary
Conditions, Ph.D. Thesis, Washington State University, Pullman, WA 1988.
[24] Shi, P., Weak solution to an evolution problem with a nonlocal constraint, SIAM J.
Math. Anal. 24:1 (1993), 46-58.
[25] Yurchuk, N.I., Mixed problem with an integral condition for certain parabolic equations, Diff. Urav. 22:19 (1986), 2117-2126.