b b x b i i. ââ. =â. {. } [ , ]:. , where 1 ⤠i ⤠n, and. Î = Î. Î i i i n. +. â. = (. ) âª. âª. 1 . 2. Bicharacteristics of Nonlinear Equations. We begin with assumptions on f.
Ukrainian Mathematical Journal, Vol. 58, No. 6, 2006
GENERALIZED SOLUTIONS OF MIXED PROBLEMS FOR FIRST-ORDER PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS W. Czernous
UDC 517.9
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.
1. Introduction We formulate the functional differential problem. Let a > 0 , h0 ∈ R+ , R+ = [0, + ∞) , b = (b1, … , bn ) ∈ R , and h = (h1, … , hn ) ∈ R+n be given, where bi > 0 for 1 ≤ i ≤ n. We define the sets n
E = [ 0, a ] × [ – b, b ] ,
D = [ – h0 , 0 ] × [ – h, h ] .
Let c = (c1, … , cn ) = b + h and E0 = [− h0 , 0] × [− c , c ], ∂0 E = [0, a] × ([− c , c ] \ (− b, b)) ,
E∗ = E0 ∪ E ∪ ∂0 E .
Assume that z : E * → R and (t, x ) ∈ E are fixed. We define a 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. Elements of the space C ( D, R ) are denoted by w, w , and so on. We denote by ⋅ 0 the supren
mum norm in the space C ( D, R ) . We set Ω = E × C ( D, R ) × R . Let f : Ω → R, α 0 : [0, a] → R ,
ϕ : E0 ∪ ∂0 E → R ,
α ′ : E → Rn ,
α′ = (α1, … , α n ) ,
Gdansk University of Technology, Gdansk, Poland. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 804 – 828, June, 2006. Original article submitted July 13, 2004. 904
0041–5995/06/5806–0904
© 2006
Springer Science+Business Media, Inc.
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
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be given functions. Denote α(t, x ) = (α 0 (t ), α′(t, x )) , ( t, x ) ∈ E . We require that α ( t, x ) ∈ E for ( t, x ) ∈ E and α0 ( t ) ≤ t for t ∈ [ 0, a ] . We deal with the following mixed problem: ∂t z(t, x ) = f (t, x, zα (t, x ), ∂ x z(t, x )) , z ( t, x ) = ϕ ( t, x )
on
(1)
E0 ∪ ∂0 E ,
(2)
where ∂ x z = (∂ x1 z, … , ∂ xn z) . A function z˜ : [− h0 , ξ] × [− c , c ] → R , where 0 < ξ ≤ a, is a generalized solution of (1), (2) if it is continuous and the following conditions are satisfied: (i) the derivatives (∂ x1 z˜, … , ∂ xn z˜ ) = ∂ x z˜ exist on [0, ξ] × [−b, b] , and the function z˜( ⋅, x ) : [− h0 , ξ] → R is absolutely continuous on [0, ξ] for every x ∈[− b, b] ; (ii) for every x ∈[− b, b], Eq. (1) is satisfied for almost all t ∈[0, ξ] and condition (2) is satisfied on ( E0 ∪ ∂0 E) × [− h0 , ξ] × Rn . Note that our hereditary setting contains well-known delay structures as particular cases. The simplest differential equation with deviated variables is obtained in the following way: Put h0 = 0 and n
h = 0 and assume that F : E × R × R → R is a given function. Consider the operator f defined as follows: f ( t, x, w, q ) = F ( t, x, w (0, 0 ), q ) ,
( t, x, w, q ) ∈ Ω .
(3)
Then f (t, x, zα (t, x ), q) = F (t, x, z(α (t, x )), q) , and Eq. (1) is equivalent to the differential equation with deviated variables ∂t z (t, x ) = F (t, x, z(α (t, x )), ∂ x z(t, x )) .
(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 following way: Suppose that n β 0 : [ 0, a ] → R and β ′ : E → R , β ′ = ( β1 , … , βn ) are given functions and − h0 ≤ β 0 (t ) − α 0 (t ) ≤ 0 ,
− h ≤ β′(t, x ) − α′(t, x ) ≤ h ,
( t, x ) ∈ E .
(5)
( t, x, w, q ) ∈ Ω .
(6)
For the function F given above, we define the operator f as follows: f ( t, x, w, q ) = F ( t, x, w (β0 ( t ) – α 0 ( t ), β ′ ( t, x ) – α ′ ( t, x )), q ) , Then f (t, x, zα (t, x ), q) = F (t, x, z(β (t, x )), q) ,
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W. CZERNOUS
where β ( t, x ) = ( β0 ( t ), β ′ ( t, x )), and Eq. (1) is equivalent to ∂t z (t, x ) = F (t, x, z(β (t, x )), ∂ x z(t, x )) .
(7) n
Now consider differential integral equations. Suppose that γ0 : [ 0, a ] → R and γ ′ : E → R , γ ′ = ( γ1 , … , γn ) are given functions and − h0 ≤ γ 0 (t ) − α 0 (t ) ≤ 0 ,
− h ≤ γ ′(t, x ) − α′(t, x ) ≤ h ,
( t, x ) ∈ E .
(8)
For the functions β and F given above, we define the operator f in the following way: ⎛ f ( t, x, w, q ) = F ⎜ t, x, ⎜ ⎝
γ 0 ( t ) − α 0 ( t ) γ ′( t , x ) − α ′( t , x )
∫
⎞
∫ w(τ, y) dy dτ, q⎟⎟ ,
β 0 ( t ) − α 0 ( t ) β ′( t , x ) − α ′( t , x )
⎠
(9)
where ( t, x, w, q ) ∈ Ω . Then γ (t , x ) ⎞ ⎛ f (t, x, zα (t, x ), q) = F ⎜ t, x, ∫ z(τ, y) dy dτ, q⎟ , ⎟ ⎜ ⎠ ⎝ β (t , x )
and (1) reduces to the differential integral equation γ (t , x ) ⎛ ⎞ ∂t z (t, x ) = F ⎜ t, x, ∫ z(τ, y) dy dτ, ∂ x z (t, x )⎟ . ⎜ ⎟ ⎝ ⎠ β (t , x )
(10)
We will discuss the question of the existence of solutions of problem (1), (2). Different classes of weak solutions of mixed problems for partial functional differential equations are considered in literature. Almost linear systems in two independent variables were investigated in [1, 2]. A continuous function is a solution of the mixed problem considered in these papers if it satisfies an integral functional system obtained from the functional differential system by integrating along bicharacteristics. Note that the papers [1, 2] initiated investigations of first-order partial functional differential equations. The class of Carathéodory solutions consists of all functions that are continuous and have their derivatives almost everywhere in the domain, and the set of all points where the differential equation or the system is not satisfied is of Lebesgue measure zero. Existence and uniqueness results for quasilinear systems with initial boundary condition in the class of almost everywhere solutions are given in [3, 4]. The 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 quasilinear equations was investigated in [5]. Functional differential problems considered in these papers are equivalent to integral functional equations obtained by integration along bicharacteristics. Under natural assumptions, continuous solutions of integral functional equations are Carathéodory solutions of the original problems. Generalized solutions in the Cinquini-Cibrario sense for equations without functional dependence were first considered in [6 – 8]. This class of solutions is placed between 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
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
907
of extended assumptions for given functions. Existence results for mixed problems and nonlinear functional differential equations can be found in [9] and [10] (Chaps. IV and V). They are obtained by a quasilinearization procedure and by the construction of integral functional systems for unknown functions and for their derivatives with respect to space variables. Continuous solutions of integral equations lead to generalized solutions of original problems. Note that the monograph [11] contains an exposition of generalized solutions in the Cinquini-Cibrario sense for nonlinear equations and systems without functional variable. Broad classes of solutions to mixed functional differential equations were investigated in [10, 12 – 14]. Only derivative of unknown functions with respect to t appear in the equations considered in these papers. Viscosity solutions of mixed problems for functional differential equations were first considered in [15, 16]. Uniqueness results were based on the method of differential inequalities. Existence theorems were obtained by using the method of vanishing viscosity. Further bibliographical information concerning hyperbolic functional differential equations can be found in [10]. The present paper is a generalization of the existence results for nonlinear functional differential equations with initial boundary conditions given in [9] and [10] (Chap. V). There are several 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 following property: Let sign ∂q f = (sign ∂q1 f , … , sign ∂qn f ) . The following condition is important in [10]: the function sign ∂q f is constant on Ω . We omit this 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 assumption made in [10] by changing variables in the unknown function in the 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(t, x ) : D → R . The domain of the function z(t, x ) considered in [10] has the form D =
[− h0 , 0] × [0, h′] × [− h0′′, 0] ⊂ R1+ n , where h′ = (h1, … , hk ) and h′′ = (hk +1, … , hn ) . In our case, we set D = [− h0 , 0] × [− h, h] . It follows that the class of differential equations with deviated variables considered in the present paper is more general than the corresponding class of equations that 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, Eq. (1) contains the functional variable zα(t, x ) It is easy to see that the class of differential equations covered by our theory is more general than the corresponding class considered in [10].
The paper is organized as follows: In Sec. 2, we prove results on the existence and uniqueness and on the regularity of bicharacteristics for nonlinear mixed problems. The integral functional equations generated by (1), (2) are investigated in Sec. 3. It is shown that, under natural assumptions on given functions, there exists a sequence of successive approximations and it is convergent. The main results on the existence of generalized solutions and on the continuous dependence of solutions on initial boundary conditions are presented in Sec. 4. An application to equations with deviated variables is given. The following function spaces will be needed in our considerations: Let Et∗ = E∗ ∩ ([− h0 , t ] × Rn ) and Et = [0, t ] × [− b, b] , where 0 ≤ t ≤ a . We denote by
⋅
t
the supremum norm in the spaces C( Et∗, R) and
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W. CZERNOUS
C( Et∗, Rn ) . Analogously, ⋅
(t )
denotes the supremum norm in C( Et , R) and C( Et , Rn ). We denote by Mn × n
the class of all n × n matrices with real elements. For x ∈ Rn and X ∈ Mn × n , where x = ( x1, … , xn ) and X = [ xij ]i, j = 1,…, n , we set n
n
x
=
∑
j =1
xi
and
X
= max
1≤ j ≤ n
∑ i =1
xij .
The product of two matrices is denoted by “ ”. If X ∈ Mn × n , then X T is the transpose matrix. By “ ° ” we n denote the scalar product in R . Let ⋅ 0 denote the supremum norm in the space C ( D, R ) . Let C 0,1( D, R) be the set of all w ∈C( D, R) such that the derivatives (∂ x1 w, … , ∂ xn w) = ∂ x w exist on D and ∂ x w ∈C( D, Rn ). For w ∈ C 0,1( D, R), we set
w
1
=
w
0
+ max { ∂ x w(t, x ) : (t, x ) ∈ D} .
We denote by CL0,1( D, R) the class of all w ∈C 0,1( D, R) such that
w
1, L
=
w
1
w
1, L
< + ∞, where
⎧ ∂ w(t, x ) − ∂ x w(t , x ) ⎫ + sup ⎨ x : (t, x ), (t , x ) ∈ D, (t, x ) ≠ (t , x )⎬ . t − t + x − x ⎩ ⎭
We consider the spaces n
Ω = E × C ( D, R ) × R ,
Ω(1) = [− b, b] × C 0,1( D, R) × Rn ,
and Ω(1, L) = [− b, b] × CL0,1( D, R) × Rn . Let Θ be the class of all functions γ ∈C( R+ , R+ ) that are nondecreasing on R+ . We now define several more function spaces. Given s = (s0 , s1, s2 ) ∈ R+3 , we denote by C1, L [s] the set of all functions ϕ ∈C ( E0 ∪ ∂0 E, R) for which the following conditions are satisfied: (i) there exists ∂ x ϕ (t, x ) for (t, x ) ∈ E0 ∪ ∂0 E , (ii) the following estimates are satisfied on E0 ∪ ∂0 E : ϕ (t, x ) ≤ s0 ,
ϕ (t, x ) − ϕ (t , x ) ≤ s1 t − t ,
∂ x ϕ(t, x ) − ∂ x ϕ(t , x )
∂ x ϕ (t, x ) ≤ s1 ,
≤ s2 [ t − t + x − x ] .
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
909
Let ϕ ∈C1, L [s] be given and let 0 < c ≤ a, d = (d0 , d1, d2 ) ∈ R3+ , and di ≥ si for i = 0, 1, 2. We 1, L ∗ consider the space Cϕ, c [d ] of all functions z : Ec → R for which the following conditions are satisfied: (i) z ∈C ( Ec∗, R) and z(t, x ) = ϕ(t, x ) on ( E0 ∪ ∂0 E ) ∩ ([− h0 , c] × Rn ) , (ii) there exists ∂ x z(t, x ) on Ec∗ and the estimates z(t, x ) ≤ d0 ,
z(t, x ) − z(t , x ) ≤ d1 t − t ,
∂ x z(t, x ) − ∂ x z(t , x )
∂ x z (t, x ) ≤ d1 ,
≤ d2 [ t − t + x − x
]
are satisfied on Ec∗ . Let p = ( p0 , p1) ∈ R+2 , p0 ≥ s1 , p1 ≥ s2 . We denote by Cc0, L [ p] the set of all functions v: Ec → Rn such that, for (t, x ) ∈ Ec , the estimates v(t, x ) ≤ p0 and v (t, x ) − v (t , x ) ≤ p1 [ t − t + x − x
]
hold on Ec . We prove that, under suitable assumptions on f, α, and ϕ and for sufficiently small c such that 1, L 0, L 0 < c ≤ a, there exists a solution z of problem (1), (2) such that z ∈ Cϕ, c [d ] and ∂ x z ∈ Cc [ p]. Let Δ+i = {x ∈[− b, b] : xi = bi } ,
Δ−i = {x ∈[− b, b] : xi = − bi } ,
where 1 ≤ i ≤ n, and Δ =
n
∪ (Δ+i i =1
)
∪ Δ−i .
2. Bicharacteristics of Nonlinear Equations We begin with assumptions on f. Assumption H [ ∂q f ] . Assume that a function f : Ω → R of variables ( t, x, w, q ) , q = ( q1 , … , qn) , is satisfies the following conditions: (i)
n
f ( ⋅, x, w, q) : [0, a] → R is measurable for every ( x, w, q ) ∈ [ – b, b ] × C ( D, R ) × R , and the partial derivatives (∂q1 f ( P), … , ∂qn f ( P)) = ∂q f ( P) , exist for ( x, w, q) ∈Ω(1) and almost all t ∈ [ 0, a ] ;
P = ( t, x, w, q ),
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W. CZERNOUS
(ii) ∂q f ( ⋅, x, w, q) : [0, a] → Rn is measurable, there is B ∈ Θ such that ∂ q f (t, x, w, q) ≤ B ( w 1)
for ( x, w, q) ∈Ω(1) and almost all t ∈ [ 0, a ],
and there is C ∈ Θ such that, for ( x, w, q) ∈Ω(1, L) , ( x , q ) ∈[− b, b] × Rn , and w ∈C 0,1( D, R) and almost all t ∈ [ 0, a ], we have
(
∂ q f (t, x , w + w , q ) − ∂ q f (t, x, w, q)
≤ C w
1, L
)[
x−x + w
1+
q − q ];
(iii) there is κ > 0 such that, for 1 ≤ i ≤ n, we have ∂qi f (t, x, w, q) ≥ κ
for
( x, w, q) ∈ Δ+i × C 0,1( D, R) × Rn
and ∂qi f (t, x, w, q) ≤ – κ
for
( x, w, q) ∈ Δ−i × C 0,1( D, R) × Rn
for almost all t ∈ [ 0, a ] . Assumption H [ α ] . Assume that functions α : [ 0, a ] → [ 0, a ] and α ′ : E → [ – b, b ] satisfy the following conditions: (i) α 0 ( t ) ≤ t for t ∈ [ 0, a ] and there is r0 ∈ R+ such that α 0 (t ) − α 0 (t ) ≤ r0 t − t
on
[ 0, a ] ;
1 (ii) α′ is of class C and
∂ xα′(t, x ) ≤ r0
on
E;
(iii) there is r1 ∈ R+ such that ∂ xα′(t, x ) − ∂ xα′(t , x )
≤ r1 x − x
on
E.
Let r = ( r0 , r1 ) . 1, L 0, L Suppose that ϕ ∈C1, L[s], z ∈Cϕ, c [d ] , and u ∈ Cc [ p]. Consider the Cauchy problem η ′ ( τ ) = – ∂q f (τ, η(τ), zα ( τ, η( τ)), u(τ, η(τ))) ,
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 Eq. (1) corresponding to ( z, u ) . Let I(t, x ) be the domain of g [z, u]( ⋅, t, x ) and let δ[z, u](t, x ) be the left end of the maximal interval on which the bicharacteristic g [z, u]( ⋅, t, x ) is defined. Let
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
911
P [z, u](τ, t, x ) = (τ, g [z, u](τ, t, x ), zα ( τ, g[ z, u]( τ, t, x )), u (τ, g [z, u](τ, t, x ))) . We prove a lemma on bicharacteristics. Lemma 1. Suppose that Assumptions H [∂q f ] and H [ α ] are satisfied and ϕ, ϕ ∈ C1, L [s] ,
1, L z ∈ Cϕ, c [d ] ,
1, L z ∈ Cϕ, c [d ] ,
u, u ∈ Cc0, L [ p] ,
are given. Then the solutions g [z, u]( ⋅, t, x ) a n d g [z , u ]( ⋅, t, x ) exist on the intervals
I(t, x ) a n d I(t, x )
such that, for ξ = δ [z, u](t, x ) and ξ = δ [z , u ](t, x ) , we have g [z, u](ξ, t, x ) ∈Δ a n d g [z , u ](ξ, t, x ) ∈Δ. The bicharacteristics are unique on I(t, x ) and I(t, x ) . Moreover, we have the estimates g [z, u](τ, t, x ) − g [z, u](τ, t , x ) ≤ C [ t − t + x − x
]
(12)
for τ ∈I(t, x ) ∩ I(t , x ) , (t, x ), (t , x ) ∈ Ec , and
g [z, u](τ, t, x ) − g [z , u ](τ, t, x )
≤ C
τ
∫[
z−z
ξ
+ ∂x z − ∂x z
(ξ)
+ u−u
t
(ξ)
] dξ
(13)
for τ ∈I(t, x ) ∩ I(t, x ) , (t, x ) ∈ Ec , where I(t, x ) is the domain of the bicharacteristic g [z , u ]( ⋅, t, x ) ,
{
}
C = max 1, B(d˜ ), C ( d ) exp {cC ( d ) [1 + r0 (d1 + d2 ) + p1]} and d˜ = d0 + d1,
d = d0 + d1 + d2 .
Proof. The existence and uniqueness of solutions of (11) follows from the classical theorem on Carathéodory solutions of initial-value problems. The function g [z, u]( ⋅, t, x ) satisfies the integral equation g [z, u](τ, t, x ) = x −
τ
∫ ∂q f (P [z, u](ξ, t, x)) dξ .
(14)
t
1, L Since z ∈Cϕ, c [d ] , condition (ii) of Assumption H [ α ] shows that
zα ( τ, y) − zα ( τ, y )
1
≤ r0 (d1 + d2 ) y − y
for all (τ, y), (τ, 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
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W. CZERNOUS
g [z, u](τ, t, x ) − g [z, u](τ, t , x )
{
≤ max 1, B(d˜ )
}[ t − t
+ x − x ] + C( d )[1 + r0 (d1 + d2 ) + p1]
τ
∫
g[z, u](ξ, t, x ) − g[z, u](ξ, t , x ) dξ ,
t
τ ∈I(t, x ) ∩ I(t , x ) . We now obtain (12) by the Gronwall inequality. For (τ, y), (τ, y ) ∈ Ec , we have zα ( τ, y) − zα ( τ, y )
1
≤
z − z
τ
+ ∂x z − ∂x z
+ r0 (d1 + d2 ) y − y .
( τ)
It follows that the function g[z, u]( ⋅, t, x ) − g[z , u ]( ⋅, t, x ) satisfies the integral inequality g [z, u](τ, t, x ) − g [ z , u ](τ, t, x ) ≤ C( d )
τ
∫[
z−z
ξ
+ ∂x z − ∂x z
(ξ)
+ u−u
(ξ)
t
+ C( d )[1 + r0 (d1 + d2 ) + p1]
τ
∫
] dξ
g[z, u](ξ, t, x ) − g[ z , u ](ξ, t, x ) dξ ,
t
τ ∈I(t, x ) ∩ I(t, x ) . We get (13) from the Gronwall inequality. This proves Lemma 1. We now give a lemma on the regularity of the function δ [ z, u ] . Lemma 2. Suppose that Assumptions H [∂q f ] and H [ α ] are satisfied and ϕ, ϕ ∈ C1, L [s] ,
1, L z ∈ Cϕ, c [d ] ,
1, L z ∈ Cϕ, c [d ] ,
u, u ∈ Cc0, L [ p] .
Then the functions δ[z, u] and δ[ z , u ] are continuous on Ec . Moreover, we have the estimates δ [z, u](t, x ) − δ [z, u](t , x ) ≤
2C [ t − t + x − x ], κ
(15)
where (t, x ), (t , x ) ∈ Ec , and 2C δ [z, u](t, x ) − δ [ z , u ](t, x ) ≤ κ where (t, x ) ∈ Ec .
t
∫[ 0
z−z
ξ
+ ∂x z − ∂x z
(ξ)
+ u−u
(ξ)
] dξ ,
(16)
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
913
Proof. The continuity of δ[z, u] and δ[z , u ] on E c follows from classical theorems on continuous dependence on initial conditions for Carathéodory solutions of initial-value problems. We now 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 where solutions of problem (11) are defined on [0, t ] and [0, t ] ). Now assume that 0 ≤ δ[z, u](t, x ) < δ[z, u](t , x ) . Then, for δ[z, u](t , x ) , we have g [z, u](ζ, t , x ) ∈Δ and there exists i, 1 ≤ i ≤ n, such that following two cases are possible:
ζ =
gi[z, u](ζ, t , x ) = bi . The
(i) gi[z, u](ζ, t , x ) = bi ; (ii) gi[z, u](ζ, t , x ) = − bi . Consider case (i). Let x = ( x1 , … , xn) and x˜ = ( x1, … , xi −1, bi , xi +1, … , xn ) . We have the following estimate for (t, x ) ∈ Ec : ∂qi f (t, x, zα (t, x ), u(t, x )) − ∂qi f (t, x˜ , zα (t, x˜ ), u(t, x˜ )) ≤ c˜ (bi − xi ) ,
(17)
where c˜ = C( d )[1 + r0 (d1 + d2 ) + p1]. Thus, ∂qi f (t, x, zα (t, x ), u(t, x )) ≥
κ 2
for (t, x ) ∈ Ec such that bi − xi ≤ κ (2c˜ )−1. It follows from Lemma 1 that bi − gi [z, u]( ζ, t, x ) = gi [z, u]( ζ, t , x ) − gi [z, u]( ζ, t, x ) ≤
κ 2c˜
for (t, x ), (t , x ) ∈ Ec such that t−t + x−x
≤
κ . 2c˜ C
(18)
Then we get
(
)
∂qi f ζ, g [z, u]( ζ, t, x ), zα (ζ, g [ z, u](ζ, t, x )), u ( ζ, g [z, u](ζ, t, x )) ≥
κ > 0, 2
and, consequently, ∂t gi [z, u](δ [z, u](t , x ), t, x ) < 0 for (t, x ), (t , x ) ∈ Ec satisfying (18). It can easily be seen that gi [z, u]( ⋅, t, x ) is decreasing on the interval (δ [z, u](t, x ), δ [z, u](t , x )) . Therefore, bi − gi [z, u]( τ, t, x ) ≤
κ , 2c˜
914
W. CZERNOUS
and the estimate ∂ qi f (τ, g [z, u]( τ, t, x ), zα ( τ, g [ z, u]( τ, t, x )) , u (τ, g [z, u](τ, t, x ))) ≥
κ 2
holds for τ ∈(δ [z, u](t, x ), δ [z, u](t , x )) and (t, x ), (t , x ) ∈ Ec such that (18) is satisfied. Then –
κ [δ [z, u](t , x ) − δ [z, u](t, x )] 2 δ[ z, u](t , x )
≥ –
∂qi f (τ, g [z, u](τ, t, x ), zα ( τ, g [ z, u]( τ, t, x )), u (τ, g [z, u](τ, t, x ))) dτ
∫
δ[ z, u](t, x )
= gi [z, u](δ [z, u](t , x ), t, x ) − gi [z, u](δ [z, u](t, x ), t, x ) ≥ gi [z, u](δ [z, u](t , x ), t, x ) − gi [z, u](δ [z, u](t , x ), t , x ) ≥ – C [ t − t + x − x ]. Thus, the proof of (15) for (t, x ), (t , x ) ∈ Ec such that (18) holds is completed in case (i). In a similar way, we prove (ii). Let (t, x ), (t , x ) ∈ Ec be arbitrary. We set M = x − x + t − t . There exists K ∈ N such that ( K − 1)
κ κ < M ≤ K . ˜ ˜ 2cC 2cC
Let ε ∈ R , ε = 1 / K. For j = 0, … , K, we set x ( j ) = j ε x + (1 − j ε) x ,
t ( j ) = j ε t + (1 − j ε) t .
Note that (t (0), x (0) ) = (t, x ) , (t ( K ), x ( K ) ) = (t , x ) and x ( j ) − x ( j +1) + t ( j ) − t ( j +1) =
M κ ≤ K 2c˜ C
for j = 0, … , K – 1. It is easy to see that
x−x
K −1
=
∑
x ( j ) − x ( j +1)
j =0
and
t−t
K −1
=
∑
t ( j ) − t ( j +1) .
j =0
Then δ [z, u](t, x ) − δ [z, u](t , x ) ≤
K −1
∑
δ [z, u](t ( j ), x ( j ) ) − δ [z, u](t ( j +1), x ( j +1) )
j =0
≤
K −1
∑
[
2C ( j ) t − t ( j +1) + x ( j ) − x ( j +1) κ j =0
]
=
2C [ t − t + x − x ]. κ
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
915
Thus we see that estimate (15) holds for all (t, x ), (t , x ) ∈ Ec . Now consider estimate (16). The inequality is obvious if δ[z, u](t, x ) = δ[z , u ](t, x ) = 0. Now assume that 0 ≤ δ[z, u](t, x ) < δ[z , u ](t, x ) . Then, for ζ = δ [z , u ](t, x ) , we have δ [z , u ](ζ, t, x ) ∈Δ and there is i, 1 ≤ i ≤ n, such that gi [z , u ](ζ, t, x ) = bi . The following two cases are possible: (i) gi [z , u ](ζ, t, x ) = bi ; (ii) gi [z , u ](ζ, t, x ) = − bi . Consider case (i). We have estimate (17) for (t, x ) ∈ Ec . It follows from Lemma 1 that gi [z, u](ζ, t, x ) − gi [z , u ](ζ, t, x ) ≤ cC
[ z−z t +
∂x z − ∂x z
(t )
+ u−u
(t )
].
Thus, we have bi − gi [z, u](ζ, t, x ) = gi [z , u ](ζ, t, x ) − gi [z, u](ζ, t, x ) ≤
κ , 2c˜
for (t, x ) ∈ Ec and z, z , u, and u such that z−z
t
+ ∂x z − ∂x z
(t )
+ u−u
(t )
≤
κ . ˜ 2ccC
(19)
Then we get
(
)
∂ qi f ζ, g [z, u](ζ, t, x ), zα (ζ, g [ z, u](ζ, t, x )) , u (ζ, g [z, u](ζ, t, x )) ≥
κ > 0, 2
and, consequently, ∂t gi [z, u](δ [z , u ](t, x ), t, x ) < 0 for (t, x ) ∈ Ec and for z, z , u, and u satisfying (19). It can easily be seen that gi [z, u]( ⋅, t, x ) is decreasing on (δ [z, u](t, x ), δ [z , u ](t, x )) . Therefore, bi − gi [z, u]( τ, t, x ) ≤
κ 2c˜
and the estimate ∂qi f (τ, g [z, u](τ, t, x ), zα ( τ, g [ z, u]( τ, t, x )), u (τ, g [z, u](τ, t, x ))) ≥
κ 2
holds for τ ∈(δ [z, u](t, x ), δ [z , u ](t, x )) , (t, x ) ∈ Ec , and z, z , u, and u such that (19) is satisfied. Then
916
W. CZERNOUS
–
κ [δ [z , u ](t, x ) − δ [z, u](t, x )] 2 δ[ z , u ](t, x )
∂qi f (τ, g [z, u](τ, t, x ), zα ( τ, g [ z, u]( τ, t, x )), u (τ, g [z, u](τ, t, x ))) dτ
∫
≥ –
δ[ z, u](t, x )
≥ gi [z, u](δ [z , u ](t, x ), t, x ) − gi [z , u ](δ [z , u ](t, x ), t, x ) δ[ z , u ](t, x )
[ z−z ξ +
∫
≥ –C
t
t
≥ –C∫ 0
[ z−z ξ +
∂x z − ∂x z
∂x z − ∂x z
(ξ)
(ξ)
+ u−u
+ u−u
(ξ)
(ξ)
] dξ
] dξ .
Thus, the proof of (16) for (t, x ) ∈ Ec and for z, z , u, and u such that (19) holds is completed in case (i). In a similar way, we prove (ii). 1, L 1, L 0, L Let z ∈Cϕ, [ p] be arbitrary. We set M = z − z t + ∂ x z − ∂ x z (t ) + c [d ] , z ∈ Cϕ, c [d ] , and u, u ∈ C u−u
(t ) .
There exists K ∈ N such that ( K − 1)
κ κ < M ≤ K . ˜ ˜ 2ccC 2ccC
Let ε ∈ R , ε = 1 / K . For j = 0, … , K, we set z j = j ε z + (1 − j ε) z
u j = j εu + (1 − j ε) u
and
on Ec .
Note that z0 = z , u0 = u , zK = z , uK = u , and z j − z j +1
t
+ ∂ x z j − ∂ x z j +1
(t )
+ u j − u j +1
(t )
=
κ M ≤ ˜ 2ccC K
for j = 0, … , K – 1. It is easy to see that z−z
K −1 t
=
∑
j =0
∂x z − ∂x z
z j − z j +1 t ,
K −1 (t )
=
∑
j =0
and u−u
Then
K −1 (t )
=
∑
j =0
u j − u j +1
(t )
.
∂ x z j − ∂ x z j +1
(t )
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
δ [z, u](t, x ) − δ [z , u ](t, x ) ≤
K −1
∑
j =0
≤
K −1
∑
j =0
2C = κ
917
δ [z j , u j ](t, x ) − δ [z j +1, u j +1](t, x )
2C κ
∫[ t
z j − z j +1
ξ
+ ∂ x z j − ∂ x z j +1
(ξ)
+ u j − u j +1
0
t
∫[
z−z
ξ
+ ∂x z − ∂x z
(ξ)
+ u−u
0
(ξ)
(ξ)
] dξ
] dξ .
1, L 1, L 0, L Therefore, estimate (16) holds for all z ∈Cϕ, c [d ] , z ∈ Cϕ, c [d ] , and u, u ∈ Cc [ p]. This completes the proof of the Lemma 2.
3. Integral Functional Equations Denote by CL( D, R) the set of all continuous and real functions defined on C( D, R) and by norm in CL( D, Rn ). We now formulate next assumptions on f.
⋅
Assumption H [ f ] . Suppose that the Assumption H [∂q f ] and the following conditions are satisfied: (i) there is B0 ∈ Θ such that f (t, x, w, q) ≤ B0 ( w
0
)
on
Ω;
(ii) the partial derivatives (∂ x1 f ( P), … , ∂ xn f ( P)) = ∂ x f ( P),
P = ( t, x, w, q ),
and the Fréchet derivative ∂w f ( P) exist for ( x, w, q) ∈Ω and almost all t ∈ [ 0, a ] ; (iii) the estimates ∂ x f (t, x, w, q)
≤ B( w
1
),
∂w f (t, x, w, q)
are satisfied for ( x, w, q) ∈Ω(1) and almost all t ∈ [ 0, a ] ; (iv) the terms ∂ x f (t, x , w + w, q ) − ∂ x f (t, x, w, q) , ∂w f (t, x , w + w, q ) − ∂w f (t, x, w, q)
≤ B( w
1
),
the
918
W. CZERNOUS
are bounded from above by
(
C w
1, L
)[
x−x + w
1
+ q−q
]
for ( x, w, q) ∈Ω(1, L) , ( x , q ) ∈[− b, b] × Rn , w ∈C 0,1( D, R), and almost all t ∈ [ 0, a ] . Remark 1. We give a theorem on the existence of solutions of problem (1), (2). For the simplicity of the formulation of the result, we have assumed the same estimates for the derivatives ∂ x f , ∂w f , and ∂q f . We have also assumed the Lipschitz condition for these derivatives with the same coefficient. n 0, L ∗ Suppose that ϕ ∈C1, L [s]. Let Cϕ, such that u(t, x ) = c [ p] be the class of all functions u : Ec → R ∂ x ϕ(t, x ) on ( E0 ∪ ∂0 E ) ∩ ([− h0 , c] × Rn ) and u equations generated by (1), (2). We write
∈Cc0, L [ p]. We now formulate the system of integral
Ec
Q [z, u](t, x ) = (δ [z, u](t, x ), g [z, u](δ [z, u](t, x ), t, x )) , Φ[z, u](t, x ) = ϕ (Q [z, u](t, x )) , ψ[z, u](t, x ) = ∂ x ϕ (Q [z, u](t, x )) , W [z, u](τ, t, x ) = uα ( τ, g [ z, u]( τ, t, x ))
ψ = ( ψ1 , … , ψn) , ∂ x α (τ, g [z, u](τ, t, x )) .
0, L 1, L Given ϕ ∈C1, L[s], z ∈Cϕ, c [d ] , and u ∈ Cϕ, c [ p], where 0 < c ≤ a, we define t
F [z, u](t, x ) = Φ[z, u](t, x ) +
∫ [ f (P [z, u](τ, t, x)) − ∂q f (P [z, u](τ, t, x)) u(τ, g [z, u](τ, t, x)) ] dτ ,
δ[ z, u](t, x )
and t
G [z, u](t, x ) = ψ [z, u](t, x ) +
∫ [ ∂ x f (P [z, u](τ, t, x))
+ ∂w f ( P [z, u](τ, t, x )) W [z, u](τ, t, x ) ] dτ .
δ[ z, u](t, x )
Consider the system of integral functional equations z ( t, x ) = F [ z, u ] ( t, x ),
u ( t, x ) = G [ z, u ] ( t, x ),
(20)
t
g [ z, u ] ( τ, t, x ) = x +
∫ ∂q f (P[z, u](ξ, t, x)) dξ
(21)
τ
with initial-boundary conditions z = ϕ,
u = ∂x ϕ on
( E0 ∪ ∂0 E ) ∩ ([− h0 , c] × Rn ) .
(22)
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
919
Remark 2. The integral functional system (20) – (22) is obtained in the following way: First, we introduce an additional unknown function u, where u = ∂ x z . Then we consider the linearization of (1) with respect to u, i.e., ∂t z (t, x ) = f (U ) + ∂q f (U ) (∂ x z (t, x ) − u (t, x )) ,
(23)
where U = (t, x, zα (t, x ), u(t, x )) . By virtue of Eq. (1), we get the following differential system for the unknown function u : ∂t u (t, x ) = ∂ x f (U ) + ∂q f (U ) [∂ x u (t, x )]T + ∂w f (U )(∂ x z)α (t, x ) ∂ x α (t, x ) .
(24)
Finally, we set ∂ x z = u in (24). System (23), (24) has the following property: The differential equations of bicharacteristics for (23) and (24) are the same and have the form (11). If we consider (23) and (24) along the bicharacteristics g [z, u]( ⋅, t, x ) , then we obtain d z (τ, g [z, u](τ, t, x )) = f ( P[z, u](τ, t, x )) − ∂q f ( P[z, u](τ, t, x )) u (τ, g [z, u](τ, t, x )) , dτ and d u (τ, g [z, u](τ, t, x )) = ∂ x f ( P[z, u](τ, t, x )) + ∂w f ( P[z, u](τ, t, x )) W [z, u](τ, t, x ) . dτ By integrating the above equations on [δ [z, u](t, x ), t ] with respect to τ, we get (20) and (21). The existence results for (20) – (22) are based on the following method of successive approximations. Assume that ϕ ∈C1, L [s] and Assumption H [ f ] is satisfied. We define a sequence
{z(m), u(m)},
z(m) : Ec∗ → R ,
u(m) : Ec∗ → Rn ,
in the following way: Let z(0) : Ec∗ → R and u(0) : Ec∗ → Rn be arbitrary functions such that z(0) ∈Cϕ1,,Lc [d ], (0) (0) ( m) ( m) 0, L u(0) ∈ Cϕ, , u ) ∈Cϕ1,,Lc [d ] × Cϕ0,, cL [ p] are known c [ p] , and ∂ x z (t, x ) = u (t, x ) on Ec . In this case, if ( z
functions, then u(m +1) is a solution of the equation
u = G(m)[u]
(25)
with the initial-boundary condition u ( t, x ) = ∂ x ϕ (t, x )
on
( E0 ∪ ∂0 E ) ∩ ([− h0 , c] × Rn )
and z(m +1) (t, x ) = F [ z(m), u(m +1) ](t, x ) z
( m +1)
on
Ec , (26)
(t, x ) = ϕ ( t, x ) on
( E0 ∪ ∂0 E ) ∩ ([− h0 , c] × R ) , n
920
W. CZERNOUS
where G(m) = (G1(m), … , Gn(m) ) is defined by t
∫
G(m)[u](t, x ) = ψ [ z(m), u](t, x ) +
[ ∂x f (P[z(m), u](τ, t, x))
δ[ z ( m ) , u](t, x )
]
+ ∂w f ( P [ z(m), u](τ, t, x )) W (m)[ z(m), u](τ, t, x ) dτ , and W (m)[ z, u](τ, t, x ) = (u(m) )α ( τ, g [ z, u]( τ, t, x ))
∂ x α (τ, g [z, u](τ, t, x )) .
We wish to emphasize that the main difficulty in carrying out this construction is the problem of the existence of the sequence {z(m), u(m)}. We set r˜ = 1 + r0 (d1 + d2 ) + p1, ⎡2 B(d˜ ) ⎤ A0 = C (1 + p0 r0 ) ⎢ + r˜ cC ( d )⎥ + C cB( d˜ ) ( p1 r02 + p0 r1) , ⎣ κ ⎦ 2 A = A0 + s2 C ⎡1 + (1 + B(d˜ ))⎤ , ⎢⎣ ⎥⎦ κ
[(
)] + (B0 (d0 ) + p0 B (d˜ )) ⎛⎝1 + 2κC ⎞⎠ + s1C ⎡⎢⎣1 + κ2 (1 + B(d˜ ))⎤⎥⎦ .
B = C c r˜ B(d˜ ) + p0 C ( d ) + p1 B (d˜ )
Assumption H [ c, d, p ] . Assume that the constants c, d = ( d0, d 1 , d 2 ) , and p = ( p0, p 1 ) satisfy the conditions d1 = p0 ≥ s1 + cB (d˜ )(1 + p0 r0 ), d2 = p1 ≥ max {B (d˜ )(1 + p0 r0 ), A , B}, d0 ≥ s0 + c [ B0 (d0 ) + B (d˜ ) p0 ] . It follows from the continuity of B0 , B, C ∈Θ that there are d, p, and sufficiently small c ∈ ( 0, a ] that satisfy Assumption H [ c, d, p ] . Lemma 3. If ϕ ∈C1, L [s] and Assumptions H [ α ] , H [ f ], and H [ c, d, p ] are satisfied, then G(m) : 0, L 0, L Cϕ, c [ p] → Cϕ, c [ p] . 0, L Proof. Suppose that u ∈Cϕ, c [ p]. Then
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
921
G(m)[u](t, x ) ≤ s1 + B (d˜ ) c (1 + p0 r0 )
(27)
on
Ec .
We now prove that the function G(m)[u] satisfies the Lipschitz condition with constant p1 . If ( t, x ) , (t , x ) ∈ Ec , then G(m)[u](t, x ) − G(m)[u](t , x )
≤ Aϕ + A f ,
where Aϕ = t
Af =
∫
∂ x ϕ (Q [z(m), u](t, x )) − ∂ x ϕ (Q [z(m), u](t , x )) ,
[∂x f (P[z(m), u](τ, t, x)) + ∂w f (P[z(m), u](τ, t, x)) W (m)[z(m), u](τ, t, x)] dτ
δ[ z ( m ) , u](t, x )
t
– δ[ z
(m)
∫
[∂ x f (P[z(m), u](τ, t , x )) + ∂w f (P[z(m), u](τ, t , x )) W (m)[z(m), u](τ, t , x )] dτ
.
, u](t , x )
It follows from Lemmas 1 and 2 that 2 Aϕ ≤ s2 C ⎡1 + (1 + B(d˜ ))⎤ [ t − t + x − x ] , ⎢⎣ ⎥⎦ κ and A f ≤ B(d˜ )(1 + p0 r0 ) t − t + A0 [ t − t + x − x ]. Then, using Assumption H [ c, d, p ], we get G(m)[u](t, x ) − G(m)[u](t , x )
≤ p1 [ t − t + x − x
]
on Ec . This inequality, together with (27), yields Lemma 3. Lemma 4. Suppose that ϕ ∈C1, L[s] and Assumptions H [ α ] , H [ f ], a n d H [ c, d, p ] are satisfied. 0, L ( m) Then there exists exactly one function u ∈Cϕ, c [ p] satisfying the equation u = G [u]. Proof. Lemma 3 shows that G(m) : Cϕ0,, cL[ p] → Cϕ0,, cL[ p] . It follows that there is A > 0 such that, for u, u˜ ∈Cϕ0,, cL[ p], we have t
G(m)[u](t, x ) − G(m)[u˜ ](t, x ) ≤ A ∫ u − u˜ 0
( τ ) dτ .
(28)
922
W. CZERNOUS
0, L We now define the norm in the space Cϕ, c [ p] as follows:
u
λ
= max
{ u(t, x) e− λt : (t, x) ∈ Ec } ,
where λ > A . It is easy to see that (Cϕ0,, cL[ p], ⋅ λ ) is a Banach space. Let us prove that there exists q ∈[0, 1) such that G(m)[u] − G(m)[u˜ ]
λ
≤ q u − u˜
for
λ
u, u˜ ∈Cϕ0,, cL[ p].
(29)
According to (28), we have t
G
( m)
[u](t, x ) − G
( m)
[u˜ ](t, x ) ≤ A ∫ u − u˜
t
(ξ) dξ
0
λ
(ξ) e
A u − u˜ λ
λ (e
e dξ
0
t
≤ A u − u˜
− λξ λξ
= A ∫ u − u˜
∫e
λξ
dξ =
0
λt
− 1) ≤
A u − u˜ λ
λe
λt
for (t, x ) ∈ Ec . Then A u − u˜ G(m)[u](t, x ) − G(m)[u˜ ](t, x ) e−λt ≤ λ
λ,
(t, x ) ∈ Ec .
It follows that estimate (29) holds with q = Aλ−1 . By virtue of the Banach fixed-point theorem, there exists ( m) 0, L exactly one u ∈Cϕ, c [ p] satisfying the equation u = G [u]. This completes the proof of Lemma 4. Assume that ϕ ∈C1, L[s]. Let T ϕ denote the set of all functions ω: E∗ → R satisfying the following conditions: (i) ω is continuous; (ii) ω
E0 ∪ ∂ 0 E
= ϕ.
Let T = {i : di > 0}. The following compatibility condition for problem (1), (2) is necessary for our considerations: Assumption Hc [ f, ϕ ] . Assume that the following conditions are satisfied: ˜ ∈Tϕ , then (i) if ω, ω ˜ α (t , x ) , q) f (t, x, ω α (t, x ), q) = f (t, x, ω for x ∈Δ, q ∈ Rn , and almost all t ∈ [ 0, a ] ;
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
923
(ii) there is a function ψ : ∂0 E → Rn , ψ = ( ψ1 , … , ψn) , such that ∂t ϕ (t, x ) = f (t, x, ϕ˜ α (t, x ), ψ (t, x ))
for x ∈Δ and almost all t ∈ [ 0, a ] ,
(30)
where ϕ˜ ∈Tϕ and ψ i (t, x ) = ∂ xi ϕ(t, x ) for (t, x ) ∈[0, a] × Δ+i ∪ Δ−i , i ∈T . Remark 3. Relation (30) can be considered as an assumption on ϕ for i ∈ T , t ∈ [ 0, c ] , and x ∈ Δ i+ ∪ Δ i− , and (30) is an equation for ψ i (t, x ) if i ∈{1, … , n} \ T , t ∈ [ 0, c ] , and x ∈ Δ+i ∪ Δ−i . Remark 4. Suppose that h0 = 0 , h = 0 , F : E × R × Rn → R is a given function, and f is defined by (3). Then (1) is equivalent to (4). If αi (t, x ) = bi
for x ∈ Δ+i
and
αi (t, x ) = – bi
for x ∈ Δ−i ,
where 1 ≤ i ≤ n, then condition (i) of 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 ∈ Δ+i
and
βi (t, x ) ≤ – bi
for x ∈ Δ−i ,
where 1 ≤ i ≤ n, then condition (i) of Assumption Hc [ f, ϕ ] is satisfied. If f is defined by (9), then we have the differential integral equation (10). If βi (t, x ), γ i (t, x ) ≥ bi
for x ∈ Δ+i
and
βi (t, x ), γ i (t, x ) ≤ – bi
for x ∈ Δ−i ,
where 1 ≤ i ≤ n, then condition (i) of the Assumption Hc [ f, ϕ ] is satisfied. We now prove the main lemma in this section. Lemma 5. If Assumptions H [ α ] , H [ f ], H [ c, d, p ], and Hc [ f, ϕ ] are satisfied, then, for any m ≥ 0, we have ∂ x z(m) (t, x ) = u(m) (t, x )
on
Ec
(31)
and z(m) ∈Cϕ1,,Lc [d ]. Proof. We prove (31) by induction. It follows from the definition of the sequence {z(m), u(m)} that (31) is satisfied for m = 0. Assuming that condition (31) is satisfied for a given m ≥ 0, we prove that the function z(m +1) given by (26) satisfies (31). We set Δ(m) (t, x, x ) = z(m +1) (t, x ) − z(m +1) (t, x ) − u(m +1) (t, x ) ( x − x ) ,
(32)
924
W. CZERNOUS
where (t, x ), (t, x ) ∈ Ec . We prove that there exists C˜ ∈ R + such that Δ (t, x, x ) ≤ C˜ x − x
2
on
Ec .
(33)
It follows from (25) and (26) that Δ(m) (t, x, x ) = F [z (m) , u(m +1) ](t, x ) − F [z (m) , u(m +1) ](t, x ) − G(m)[u(m +1) ](t, x ) ( x − x ) .
(34)
We write g(m) (τ, t, x ) = g [z(m), u(m +1) ] (τ, t, x ) ,
δ(m) (t, x ) = δ [z(m), u(m+1) ] (t, x ) ,
(
)
S(m) (t, x ) = δ(m) (t, x ), g(m) (δ(m) (t, x ), t, x ) , and Q(m) (s, τ, t, x, x ) = sP [z(m), u(m +1) ](τ, t, x ) + (1 − s) P [z(m), u(m +1) ](τ, t, x ) . Assume that δ(m) (t, x ) ≤ δ(m) (t, x ) . Similar arguments apply to the case δ(m) (t, x ) > δ(m) (t, x ) . For the simplicity of the formulation of properties of the function Δ(m) , we define A(m) (t, x, x ) = ϕ (S(m) (t, x )) − ϕ (S(m) (t, x ))
[
]
– ∂ x ϕ (S(m) (t, x )) g(m) (δ(m) (t, x ), t, x ) − g(m) (δ(m) (t, x ), t, x )
[
]
– ∂t ϕ (S(m) (t, x )) δ(m) (t, x ) − δ(m) (t, x ) ,
[
]
B(m) (t, x, x ) = ∂t ϕ (S(m) (t, x )) δ(m) (t, x ) − δ(m) (t, x )
[
]
+ ∂ x ϕ (S(m) (t, x )) g(m) (δ(m) (t, x ), t, x ) − g(m) (δ(m) (t, x ), t, x ) − ( x − x ) , t
C(m) (τ, t, x, x ) =
∫ [∂q f (P[z
( m)
]
, u(m +1) ](ξ, t, x )) − ∂ q f ( P[z (m) , u(m +1) ](ξ, t, x )) dξ ,
τ
and
(
)
C(m) = C1(m), … , Cn(m) . Moreover, we set
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS δ ( m ) (t , x )
Λ
( m)
(t, x, x ) =
∫
f ( P[z(m), u(m +1) ](τ, t, x )) dτ
δ ( m ) (t , x ) δ ( m ) (t , x )
∂q f ( P[z(m), u(m +1) ](τ, t, x )) u(m +1) (τ, g(m) (τ, t, x )) dτ .
∫
–
δ ( m ) (t , x )
It follows from (34) that Δ(m) (t, x, x ) = ϕ (S(m) (t, x )) − ϕ (S(m) (t, x )) t
+
( m) ( m +1) ∫ [ f (P[z , u ](τ, t, x )) −
δ ( m ) (t , x ) t
–
∫
]
f ( P[z(m), u(m +1) ](τ, t, x )) dτ
∂q f ( P[z(m), u(m +1) ](τ, t, x )) u(m +1) (τ, g(m) (τ, t, x )) dτ
δ ( m ) (t , x ) t
+
∫
∂q f ( P[z(m), u(m +1) ](τ, t, x )) u(m +1) (τ, g(m) (τ, t, x )) dτ
δ ( m ) (t , x )
+ Λ(m) (t, x, x ) − G(m)[u(m +1) ](t, x ) ( x − x ) . Having disposed this preliminary step, we apply the Hadamard mean-value theorem to the difference f ( P[z(m), u(m +1) ](τ, t, x )) − f ( P[z(m), u(m +1) ](τ, t, x )) . We thus get Δ(m) (t, x, x ) = Δ(m) (t, x, x ) + Δ˜ (m) (t, x, x ) , where t
Δ (t, x, x ) = A(m) (t, x, x ) +
1
∫ ∫ [∂ x f (Q
( m)
δ
(m)
(s, τ, t, x, x ))
(t , x ) 0
] [
]
– ∂ x f ( P[z(m), u(m +1) ](τ, t, x )) ds g(m) (τ, t, x ) − g(m) (τ, t, x ) dτ t
1
∫ ∫ [∂w f (Q
( m)
+ δ
(m)
(t , x ) 0
(s, τ, t, x, x ))
925
926
W. CZERNOUS
[
]
– ∂w f ( P[z(m), u(m +1) ](τ, t, x )) ds z(m)
∫ ∫ [∂q f (Q
( m)
δ
(m)
− z( m )
α ( τ, g( m ) ( τ, t, x ))
]dτ
1
t
+
α ( τ, g( m ) ( τ, t, x ))
(s, τ, t, x, x ))
(t , x ) 0
] [
]
– ∂q f ( P[z(m), u(m +1) ](τ, t, x )) ds u(m +1) (τ, g(m) (τ, t, x )) − u(m +1) (τ, g(m) (τ, t, x )) dτ t
[
∂w f ( P[z(m), u(m +1) ](τ, t, x )) z(m)
∫
+
δ ( m ) (t , x )
α( τ, g( m ) ( τ, t, x ))
− z( m )
α( τ, g( m ) ( τ, t, x ))
]
– W (m)[z(m), u(m +1) ](τ, t, x ) (g(m) (τ, t, x ) − g(m) (τ, t, x )) dτ and Δ˜ (m) (t, x, x ) = B(m) (t, x, x ) t
∂ x f ( P[z(m), u(m +1) ](τ, t, x )) [g(m) (τ, t, x ) − g(m) (τ, t, x ) − ( x − x )] dτ
∫
+
δ ( m ) (t , x ) t
∂w f ( P[z(m), u(m +1) ](τ, t, x ))
∫
+
δ ( m ) (t , x )
[
]
× W (m)[z(m), u(m +1) ](τ, t, x ) [g(m) (τ, t, x ) − g(m) (τ, t, x ) − ( x − x )] dτ t
–
∫
[ ∂ f (P[z q
( m)
, u(m +1) ](τ, t, x ))
δ ( m ) (t , x )
]
– ∂q f ( P[z(m), u(m +1) ](τ, t, x )) u(m +1) (τ, g(m) (τ, t, x )) dτ + Λ(m) (t, x, x ) . Let us estimate Δ˜ (m) (t, x, x ) . Since g(m) ( ⋅, t, x ) satisfies (14), we have
g(m) (τ, t, x ) − g(m) (τ, t, x ) − ( x − x ) = C (m) (τ, t, x, x ) . Substituting the above relation into Δ˜ (m) (t, x, x ) and changing the order of integrals where necessary, we obtain
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
927
Δ˜ (m) (t, x, x ) = D(m) (t, x, x ) t
( m) ( m +1) ( m) ( m +1) ( m) ∫ [∂q f (P[z , u ](τ, t, x )) − ∂q f (P[z , u ](τ, t, x))] E (τ, t, x) dτ ,
+
δ ( m ) (t , x )
where δ ( m ) (t , x ) ( m)
D
(t, x, x ) =
( m) ( m +1) ( m) ∫ [ f (P[z , u ](τ, t, x )) − ∂t ϕ (S (t, x))] dτ
δ ( m ) (t , x )
δ ( m ) (t , x )
+
( m) ( m +1) ( m) ( m) ( m +1) ∫ [∂ x ϕ (S (t, x)) − u (τ, g (τ, t, x ))] ∂q f (P[z , u ](τ, t, x )) dτ
δ ( m ) (t , x )
and E(m) (τ, t, x ) = – u(m +1) (τ, g(m) (τ, t, x )) + ∂ x ϕ (S(m) (t, x )) +
τ
( m) ( m +1) ∫ [∂ x f (P[z , u ](ξ, t, x))
δ ( m ) (t , x )
]
+ ∂w f ( P[z(m), u(m +1) ](ξ, t, x )) W (m)[z(m), u (m +1) ](ξ, t, x ) dξ . The next claim is E(m) (τ, t, x ) = 0,
(t, x ) ∈ Ec ,
τ ∈I(t, x ) .
We have g(m) (s, τ, g(m) (τ, t, x )) = g(m) (s, t, x )
and
δ(m) (τ, g(m) (τ, t, x )) = δ(m) (t, x )
for ( t, x ) ∈ Ec and τ, s ∈ I(t, x ) . Therefore, by virtue of (25), we get u(m +1) (τ, g(m) (τ, t, x )) = ∂ x ϕ (δ(m) (t, x ), g(m) (δ(m) (t, x ), t, x )) τ
+
∫
∂ x f ( P[z(m), u(m +1) ] (s, t, x )) ds
δ ( m ) (t , x ) τ
+
∫
∂w f ( P[z(m), u(m +1) ] (s, t, x )) W (m)[z(m), u (m +1) ] (s, t, x )) ds
δ ( m ) (t , x )
(35)
928
W. CZERNOUS
for ( t, x ) ∈ E c and τ ∈I(t, x ) , whence E(m) (τ, t, x ) = 0. Thus, we have proved that Δ˜ (m) (t, x, x ) = D(m) (t, x, x ). We conclude from Assumption Hc [ f, ϕ ] and Lemma 2 that there is C1 ∈ R+ such that D(m) (t, x, x ) ≤ C1 x − x
2
on
Ec
on
Ec .
and, consequently, Δ˜ (m) (t, x, x ) ≤ C1 x − x
2
(36)
We now estimate Δ(m) (t, x, x ) . It follows from Assumptions H [∂q f ] and H [ f ] that the terms ∂ x f (Q(m) (s, τ, t, x, x )) − ∂ x f ( P[z(m), u(m +1) ] (τ, t, x )) , ∂q f (Q(m) (s, τ, t, x, x )) − ∂q f ( P[z(m), u(m +1) ] (τ, t, x )) , ∂w f (Q(m) (s, τ, t, x, x )) − ∂w f ( P[z(m), u(m +1) ] (τ, t, x ))
are bounded from above by C ( d ) r˜ g(m) (τ, t, x ) − g(m) (τ, t, x ) . An easy computation shows that 2
A(m) (t, x, x ) ≤ s2 g(m) (δ(m) (t, x ), t, x ) − g(m) (δ(m) (t, x ), t, x ) , z( m )
α ( τ, g( m ) ( τ, t, x ))
− z( m )
α ( τ, g( m ) ( τ, t, x )) 0
≤ r0 d1 g(m) (τ, t, x ) − g(m) (τ, t, x ) ,
and u(m +1) (τ, g(m) (τ, t, x )) − u(m +1) (τ, g(m) (τ, t, x )) ≤ p1 g(m) (τ, t, x ) − g(m) (τ, t, x ) . Since ∂ x z(m) (t, x ) = u(m) (t, x )
on
Ec ,
we have z( m )
α ( τ, g( m ) ( τ, t, x ))
− z( m )
α ( τ, g( m ) ( τ, t, x ))
− W (m)[z(m), u(m +1) ](τ, t, x ) [g(m) (τ, t, x ) − g(m) (τ, t, x )]
0 2
≤ [ p0 r1 + p1 r02 ] g(m) (τ, t, x ) − g(m) (τ, t, x ) .
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
929
The above estimates and the definition of Δ(m) yield Δ(m) (t, x, x ) ≤ s2 g(m) (δ(m) (t, x ), t, x ) − g(m) (δ(m) (t, x ), t, x ) ∗
+ A
2
t
∫
2
g(m) (τ, t, x ) − g(m) (τ, t, x ) dτ ,
δ ( m ) (t , x )
where A∗ = B( d˜ )[ p0 r1 + p1 r02 ] + C ( d ) r˜ 2 . It follows from Lemmas 1 and 2 and from (14) that 2C ⎡ ⎤ + C⎥ x − x . g(m) (δ(m) (t, x ), t, x ) − g(m) (δ(m) (t, x ), t, x ) ≤ ⎢ B( d˜ ) κ ⎣ ⎦ Then there exists C2 ∈ R+ such that Δ(m) (t, x, x ) ≤ C2 x − x 2 .
(37)
Adding inequalities (36) and (37), we get (33) and, consequently, ∂ x z(m +1) (t, x ) = u(m +1) (t, x )
on
Ec .
This completes the proof of (31). We now prove that z(m +1) ∈Cϕ1,,Lc [d ]. It is clear that z(m +1) is continuous on E c and z(m +1) = ϕ on ( E0 ∪ ∂ 0 E ) ∩ ([− h0 , c] × Rn ) . Moreover, it follows from (31) that ∂ x z(m +1) (t, x ) ≤ d1 and ∂ x z(m +1) (t, x ) − ∂ x z(m +1) (t , x ) ≤ d2 [ t − t + x − x
]
on Ec∗ . Our assumptions yield the estimates z(m +1) (t, x ) ≤ d0 ,
z(m +1) (t, x ) − z(m +1) (t , x ) ≤ d2 t − t
where (t, x ), (t , x ) ∈ Ec∗ . This completes the proof of Lemma 5. We now prove that the sequences {z(m)} and {u(m)} are uniformly convergent on Ec if the constant c is sufficiently small.
930
W. CZERNOUS
We set A˜ = A + C ( d ) (1 + p0 r0 ) ,
˜ ˜ = B + (1 + B( d˜ )) Be ˜ Ac D .
B˜ = A˜ + r0 B( d˜ ) ,
Assumption H [ c ] . Suppose that Assumption H [ c, d, p ] is satisfied and the constant c is so small that ˜ cD < 1. Lemma 6. If Assumptions H [ α ] , H [ f ] , and H [ c ] are satisfied, then the sequences {z(m)} a n d {u(m)} are uniformly convergent on Ec . Proof. For t ∈[0, c] and m ≥ 1, we set z(m) − z(m −1) ,
Z (m) (t ) =
U (m) (t ) =
t
u(m) − u(m −1)
(t )
.
Let us prove that, for t ∈[0, c] , we have t
[
]
˜ ( m) ( m) ˜ Ac U (m +1) (t ) ≤ Be ∫ Z (τ) + U (τ) dτ .
(38)
0
It follows from Lemmas 1 and 2 that g(m) (τ, t, x ) − g(m −1) (τ, t, x )
≤ C
τ
∫ [Z
( m)
]
(τ) + U (m) (τ) + U (m +1) (τ) dτ
(39)
t
and δ
( m)
(t, x ) − δ
( m −1)
2C (t, x ) ≤ κ
t
∫ [Z
( m)
]
(τ) + U (m) (τ) + U (m +1) (τ) dτ ,
(40)
0
where (t, x ) ∈ Ec and τ ∈[0, c] . Then we obtain the integral inequality U
( m +1)
t
t
0
0
[
]
(t ) ≤ A˜ ∫ U (m +1) (τ) dτ + B˜ ∫ Z (m) (τ) + U (m) (τ) dτ ,
where t ∈[0, c] . The above estimate and the Gronwall inequality yield inequality (38). An easy computation shows that, for t ∈[0, c] , we have
Z
( m +1)
t
0
and, consequently,
[
(t ) ≤ B ∫ Z
( m)
( τ) + U
( m)
( τ) + U
( m +1)
]
t
(τ) dτ + B(d˜ ) ∫ U (m +1) (τ) dτ 0
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
Z (m +1) (t ) ≤
(
˜ ˜ Ac B + B(d˜ ) Be
) ∫ [Z t
( m)
]
(τ) + U (m) (τ) dτ ,
931
t ∈[0, c] .
0
The above inequality and (38) yield t
[
]
˜ Z (m) (τ) + U (m) (τ) dτ , Z (m +1) (t ) + U (m +1) (t ) ≤ D ∫
t ∈[0, c] ,
(41)
0
where m ≥ 1. It follows from Lemma 3 that Z (1) + U (1) ≤ 2 ( p0 + d0 ) ,
t ∈[0, c] .
Then the uniform convergence of the sequences {z(m)} and {u(m)} follows from relation (41) and the condition ˜ < 1. cD This completes the proof of Lemma 6. 4. Existence of Solutions of Nonlinear Mixed Problems We can state the main result on the existence of generalized solutions to problem (1), (2). For a function ϕ ∈C1, L[s] and a point τ ∈[0, a], we set
{
}
ϕ
τ
= max ϕ (t, x ) : (t, x ) ∈ ( E0 ∪ ∂0 E) ∩ ([− h0 , τ] × Rn ) ,
∂x ϕ
τ
= max ∂ x ϕ (t, x ) : (t, x ) ∈ ( E0 ∪ ∂0 E) ∩ ([− h0 , τ] × Rn ) .
{
}
Theorem 1. If Assumptions H [ α ] , H [ f ] , Hc [ f, ϕ ], and H [ c ] are satisfied, then, for every ϕ ∈ 1, L
C
1, L 0, L [s], there exists a solution v : Ec∗ → R of problem (1), (2). Moreover, v ∈Cϕ, c [d ] and ∂ x v ∈ Cϕ, c [ p].
1, L If ϕ ∈C1, L[s] and v ∈Cϕ, c [d ] is a solution of Eq. (1) with initial boundary condition z (t, x ) = ϕ(t, x ) on E0 ∪ ∂0 E , then there is Λ c ∈ R+ such that
v− v
t
+ ∂x v − ∂x v
(t )
≤ Λc [ ϕ − ϕ
t
+ ∂x ϕ − ∂x ϕ
t
],
0 ≤ t ≤ c.
1, L Proof. It follows from Lemmas 5 and 6 that there is v ∈Cϕ, c [d ] such that
v ( t, x ) = lim z(m) (t, x ) , m→∞
∂x v ( t, x ) = lim u(m) (t, x )
uniformly on Ec . Thus, we get v ( t, x ) = F [ v, ∂ x v](t, x ) ,
m→∞
(42)
932
W. CZERNOUS
∂x v ( t, x ) = G [ v, ∂ x v](t, x ) , and t
g [ v, ∂ x v](τ, t, x ) = x +
∫ ∂q f (P[v, ∂ x v](ξ , t, x)) dξ . τ
Moreover, the initial boundary conditions v = ϕ,
∂x v = ∂x ϕ
on
( E0 ∪ ∂0 E) × ([− h0 , c] × Rn )
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 the corresponding properties for initial-value or initial boundary-value problems considered in [9] and [10] (Chap. IV). Details are omitted. We now prove relation (42). The functions ( v, ∂ x v) satisfy the integral functional equations (20) and (21) and the initial boundary conditions (22) with ϕ instead of ϕ. It easily follows that there are Λ 0 , Λ1 ∈ R+ such that the following integral inequality is satisfied: v− v
t
+ ∂x v − ∂x v
(t )
≤ Λ0 [ ϕ − ϕ
t
t
+ ∂x ϕ − ∂x ϕ
t ] + Λ1 ∫ [
v− v
τ
+ ∂x v − ∂x v
( τ)
] dτ ,
0 ≤ t ≤ c.
0
Using the Gronwall inequality, we obtain (42) with Λ c = Λ 0 exp ( Λ1 c). This proves the theorem. n Suppose that β 0 : [ 0, a ] → R and β ′ : E → R are given functions and conditions (8) are satisfied. Consider the operator f defined by (6). In this case, (1) is equivalent to the differential equation with deviated variables (7). We now formulate our existence result for problem (7), (2). n
Assumption H [ F ] . Assume that a function F : E × R × R → R of the variables ( t, x, p, q ) satisfies the following conditions: (i) F ( ⋅ , x, p, q ) : [ 0, a ] → R is measurable for every ( x, p, q ) ∈ [ – b, b ] × R × R B0 ∈ Θ such that F (t, x, p, q) ≤ B0 ( p )
on
n
E×R×R ;
(ii) the partial derivatives
(∂ x1 F(Q), … , ∂ xn F(Q)) = ∂ x F(Q),
Q = ( t, x, p, q ),
n
and there is
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
933
(∂ p1 F(Q), … , ∂ pn F(Q)) = ∂ p F(Q), (∂q1 F(Q), … , ∂qn F(Q)) = ∂q F(Q) n
exist for ( x, p, q ) ∈ [ – b, b ] × R × R and almost all t ∈ [ 0, a ] ; (iii) the functions ∂ x F ( ⋅, x, p, q) : [0, a] → Rn ,
∂ p F ( ⋅, x, p, q) : [0, a] → R ,
∂q F ( ⋅, x, p, q) : [0, a] → Rn are measurable and there is B ∈ R+ such that ∂ p F (t, x, p, q)
∂ x F (t, x, p, q) ≤ B ,
≤ B,
∂q F (t, x, p, q) ≤ B
n
for ( x, p, q ) ∈ [ – b, b ] × R × R and almost all t ∈ [ 0, a ] ; (iv) there is C ∈ R+ such that the functions ∂ x F , ∂ p F , and ∂q F satisfy the Lipschitz condition n
with respect to ( x, p, q ) ∈ [ – b, b ] × R × R for almost all t ∈ [ 0, a ] . Assumption H [ β ] . Assume that functions β0 : [ 0, a ] → R and β ′ : E → R satisfy the following conditions: n
(i) 0 ≤ β0 ( t ) ≤ t for t ∈ [ 0, a ], β′(t, x ) ∈[− c , c ] for ( t, x ) ∈ E, and, for every i, 1 ≤ i ≤ n, and t ∈ [ 0, a ], we have for x ∈ Δ+i
β i ( t, x ) ≥ bi
and
βi ( t, x ) ≤ bi
for x ∈ Δ−i ;
(ii) there is r0 ∈ R+ such that β0 (t ) − β0 (t ) ≤ r0 t − t 1 (iii) β ′ is of class C and
on
[ 0, a ] ;
∂ x β′(t, x ) ≤ r0 on E ;
(iv) there is r1 ∈ R+ such that ∂ x β′(t, x ) − ∂ x β′(t , x )
≤ r1 [ t − t + x − x
]
on
E.
Theorem 2. Suppose that Assumptions H [ α ] , H [ F ] , and H [ β ] are satisfied, ϕ ∈C1, L [s], a n d there is ψ : ∂0 E → Rn , ψ = ( ψ1 , … , ψn) , such that
934
W. CZERNOUS
∂t ϕ (t, x ) = F (t, x, ϕ (β (t, x )), ψ (t, x ))
(43)
for x ∈ Δ and almost all t ∈ [ 0, a ] . Then there are c ∈ ( 0, a ] and v : Ec∗ → R such that v is a solution of (7), (2),
1, L v ∈Cϕ, c [ d ], a n d
∂ x v ∈Cϕ0,, cL [ p]. 1, L If ϕ ∈C1, L[s], condition (43) is satisfied with ϕ instead of ϕ , a n d v ∈Cϕ, c [d ] is a solution of
Eq. (7) with initial boundary condition z ( t, x ) = ϕ (t, x ) o n E0 ∪ ∂0 E , then there is Λ c ∈ R+ such that estimate (42) is satisfied. Proof. We write β˜ 0 (t ) = β0 (t ) − α 0 (t ) ,
β˜ ′(t, x ) = β′(t, x ) − α′(t, x ) ,
( t, x ) ∈ E,
and β˜ (t, x ) = (β˜ 0 (t, x ), β˜ ′(t, x )) . Then the operator f is defined by f ( t, x, w, q ) = F (t, x, w(β˜ (t, x )), q) ,
( t, x, w, q ) ∈ Ω ,
and ∂ x f (t, x, w, q) = ∂ x F (t, x, w(β˜ (t, x )), q) + ∂ p F (t, x, w(β˜ (t, x )), q) ∂ x w (β˜ (t, x )) ∂ x β˜ ′ (t, x ) , ∂q f (t, x, w, q) = ∂q F (t, x, w(β˜ (t, x )), q) , ∂w f (t, x, w, q) w = F (t, x, w(β˜ (t, x )), q) w (β˜ (t, x )) , n
where ( t, x, q ) ∈ [ 0, a ] × [ – b, b ] × R , w ∈C 0,1( D, R) , and w ∈C( D, R). It is clear that ∂ x f (t, x, w, q)
≤ B [1 + (r0 + r0 ) w
∂q f (t, x, w, q) ≤ B ,
∂w f (t, x, w, q)
1
], ≤ B.
We conclude from Assumptions H [ f ], H [ α ] , and H [ β ] that ∂ x f (t, x , w + w , q ) − ∂ x f (t, x, w, q)
≤
[ C [1 + (r + r ) w 0
[
0
]
2
1, L
+ C 1 + (r0 + r0 ) w
[
]
+ B r1 + r1 + (r0 + r0 )2 w
1, L
]
[
1, L
] x−x
q − q + c + (r0 + r0 ) ( B + C w
1, L
)] w 1,
GENERALIZED SOLUTIONS OF MIXED P ROBLEMS FOR F IRST -ORDER P ARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS
[
1, L
[
1, L
∂q f (t, x , w + w , q ) − ∂q f (t, x, w, q) ≤ C (1 + (r0 + r0 ) w ∂q f (t, x , w + w , q ) − ∂w f (t, x, w, q)
≤ C (1 + (r0 + r0 ) w
935
) x − x + q − q + w 1] , ) x − x + q − q + w 1 ].
Thus, all conditions 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 , ∂w f , and ∂q f on some special function spaces. Let us consider the simplest assumption. Suppose that there is L ∈ R+ such that ∂ x f (t, x, w, q ) − ∂ x f (t, x , w , q ) ≤ L [ x − x + w − w
0
+ q−q
]
(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. We now 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 a condition of the form (44). Then there is a class of nonlinear equations that satisfy Assumption H [ f ] and do not satisfy conditions of the type (44). Remark 6. In the case where (1) is reduced to the integral functional equation (10), one can formulate the corresponding assumptions and prove a particular version of the existence theorem as easily as it has been done above for problem (7), (2). REFERENCES 1. V. E. Abolina and A. D. Myshkis, “Mixed problems for semilinear hyperbolic systems on a plane,” Mat. Sb. (N.S.), 50, 423–442 (1960). 2. A. D. Myshkis and A. S. Slopak, “A mixed problem for systems of differential functional equations with partial derivatives and with operators of the Volterra type,” Mat. Sb. (N.S.), 41, 239–256 (1957). 3. J. Turo, “Global solvability of the mixed problem for first order functional partial differential equations,” Ann. Pol. Math., 52, No. 3, 231–238 (1991). 4. J. Turo, “Mixed problems for quasilinear hyperbolic systems,” in: Proceedings of the Second World Congress of Nonlinear Analysis, Pt. 4, 30, No. 4 (Athens, 1996), Nonlinear Anal. (1997), pp. 2329–2340. 5. T. Człapiński, “On the mixed problem for quasilinear partial differential-functional equations of the first order,” Z. Anal. Anwendungen, 16, No. 2, 463–478 (1997). 6. S. Cinquini, “On hyperbolic systems of (nonlinear) partial differential equations in several independent variables,” Ann. Mat. Pura Appl., 120, No. 4, 201–214 (1979). 7. M. Cinquini-Cibrario, “New research on systems of nonlinear partial differential equations in several independent variables,” Rend. Semin. Mat. Fis. Milano, 52, 531–550 (1982). 8. M. Cinquini-Cibrario, “A class of systems of nonlinear partial differential equations in several independent variables,” Ann. Mat. Pura Appl., 140, No. 4, 223–253 (1985). 9. T. Człapiński, “On the mixed problem for hyperbolic partial differential-functional equations of the first order,” Czech. Math. J., 49 (124), No. 4, 791–809 (1999). 10. Z. Kamont, Hyperbolic Functional Differential Inequalities and Applications, Kluwer, Dordrecht (1999). 11. T. D. Van, M. Tsuji, and N. T. S. Duy, The Characteristic Method and Its Generalization for First-Order Nonlinear Partial Differential Equations, Chapman and Hall/CRC Boca Raton, London (2000). 12. G. A. Kamenskij and A. D. Myshkis, “Mixed functional-differential equations,” Nonlin. Anal., 34, No. 2, 283–297 (1998).
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W. CZERNOUS
13. A. D. Myshkis, “On the solvability of the initial boundary problem for mixed functional differential equation of retarded type,” Funct. Different. Equat., 7, Nos. 3–4, 311–324 (2000). 14. A. D. Myshkis, “Mixed functional differential equations with continuous and piecewise continuous solutions,” Funct. Different. Equat., 9, Nos. 1–2, 221–226 (2002). 15. K. Topolski, “On the uniqueness of viscosity solutions for first order partial differential functional equations,” Ann. Pol. Math., 59, No. 1, 65–75 (1994). 16. K. Topolski, “Classical methods for viscosity solutions of differential-functional inequalities,” Non-Linear World, 4, No. 1, 1–17 (1997).
