GENERALIZED SOLUTIONS OF MIXED PROBLEMS FOR FIRST-ORDER PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS

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 in- equalities. Classical solutions of integral functional equations lead to generalized solutions of the original problem. Differential equations with deviated variables and differential integral prob- lems 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 …,b_n) ∈ Rⁿ, and h = ( ,h₁ …,h_n)∈R₊ⁿ be given, where b_i > 0 for 1 ≤ i ≤ n. We define the sets

E = [0, a]×[–b, b], D = [–h0, 0]×[–h, h]. Let c = ( ,c₁ …,c_n) = +b h and

E₀ = −[ h₀, ]0 × −[ c c, ],

∂0E = [ , ]0 a × −

(

[ c c, ] \ (−b b, )

)

_{, E}^∗ = E0 ∪ E ∪ ∂0E.

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 supre- mum norm in the space C(D, R). We set Ω = E × C(D, R)× Rⁿ. Let

f : Ω → R, ϕ: E₀∪∂₀E → R,

α0: [ , ]0 a → 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.

be given functions. Denote α( , )t x = (α0( ),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:

∂tz t x( , ) = f t x z( , , _{α( , )}t x,∂xz t x( , )), (1)

z(t, x) = ϕ(t, x) on E₀∪∂₀E, (2) where ∂_xz ∂_xz ∂_x z

= ( ,…, n )

1 .

A function ˜ : [z −h0, ] [ξ × −c c, ]→ R, where 0 < ξ ≤ a, is a generalized solution of (1), (2) if it is con- tinuous and the following conditions are satisfied:

(i) the derivatives (∂xz˜, ,∂x z˜) ∂xz˜

n

1 … = 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∪∂₀E)× −[ 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 h = 0 and assume that F: E × R × Rⁿ → 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 z( , , _{α( , )t x} , )q = F t x z

(

, ,

(

α( , ) ,t x

) )

q , and Eq.(1) is equivalent to the differential equation with deviated variables

∂tz t x( , ) = F t x z( , ,

(

α( , ) ,t x

)

∂_xz 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 β0 : [0, a] → R and β′: E → Rⁿ, β′ = (β1,…, βn) are given functions and

−h₀ ≤ β₀( )t − α₀( )t ≤ 0, − ≤ ′h β( , )t x − ′α( , )t x ≤ h, (t, x)∈E. (5) For the function F given above, 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( , , _{α( , )t x} , )q = F t x z( , , ( ( , )), )β t x q ,

where β(t, x)= (β0(t), β′(t, x)), and Eq.(1) is equivalent to

∂_tz t x( , ) = F t x z( , , ( ( , )),β t x ∂_xz t x( , )). (7)

Now consider differential integral equations. Suppose that γ0 : [0, a] → R and γ′: E → Rⁿ, γ′ = (γ1,…, γn) are given functions and

−h₀ ≤ γ₀( )t − α₀( )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 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

f t x z( , , _{α( , )t x}, )q = F t x z y dy d q

t x t x

, , ( , ) ,

( , ) ( , )

τ τ

β γ

∫

⎛

⎝⎜⎜ ⎞

⎠⎟⎟, and (1) reduces to the differential integral equation

∂tz 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 for partial functional differential equations are con- sidered in literature. Almost linear systems in two independent variables were investigated in [1, 2]. A continu- ous 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 pa- pers [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 equa- tions 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 differ- ential 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 Cara- théodory sense. It is important that both inclusions are strict. This class of solutions is investigated in the case

of extended assumptions for given functions. Existence results for mixed problems and nonlinear functional dif- ferential 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 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 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∂q f, ,sign∂q f)

n

1 … .

The following condition is important in [10]: the function sign∂qf is constant on Ω. We omit this condition in our considerations. It is assumed in [9] that the function sign∂qf 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 func- tional 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 …,h_k) and h′′ = (h_k+1,…,h_n). 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 z_{α( , )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 se- quence of successive approximations and it is convergent. The main results on the existence of generalized solu- tions 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 E_t^∗ = E^∗∩ ([−h t₀, ]×Rⁿ) and E_t = [ , ] [0 t × −b b, ], where 0 ≤ t ≤ a. We denote by ⋅ _t the supremum norm in the spaces C E R( _t^∗, ) and

C E R( _t^∗, ⁿ). Analogously, ⋅ _{( )t} denotes the supremum norm in C E R( , ) and C E R_t ( ,_t ⁿ). We denote by M_{n n×} the class of all n × n matrices with real elements. For x∈Rⁿ and X∈M_{n n×} , where x = ( ,x₁ …,x_n) and

X = [x_{ij i j}]_{, = …1, ,n}, we set

x = x_i

j n

∑

= 1

and X = max

1≤ ≤_{j n}

∑

=1 ^ij i

n

x .

The product of two matrices is denoted by “”. If X∈M_{n n×} , then X^T is the transpose matrix. By “°” we denote the scalar product in Rⁿ.

Let ⋅ ₀ denote the supremum norm in the space C(D, R). Let C^{0 1,} ( , ) be the set of all wD R ∈C D R( , ) such that the derivatives (∂xw, ,∂x w) ∂xw

n

1 … = exist on D and ∂xw∈C D R( , n). For w ∈ C^{0 1,} ( , ), weD R set

w ₁ = w ₀ + ^max

{

∂_xw t x( , ) : ( , )t x ∈D

}

.

We denote by C_L^{0 1,} ( , ) the class of all wD R ∈C^{0 1,} ( , ) such that wD R _1,L < + ∞, where

w _1,L = w w t x w t x

t t x x t x t x D t x t x

x x

1 + −

− + − ∈ ≠

⎧⎨

⎩

⎫⎬

sup ( , ) ( , ) ⎭

: ( , ), ( , ) , ( , ) ( , )

∂ ∂ .

We consider the spaces

Ω = E × C(D, R)× Rⁿ, Ω^{( )1} = [−b b, ] × C^{0 1,} ( , )D R × Rⁿ,

and

Ω^{( , )1 L} = [−b b, ] × C_L^{0 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 = ( , ,s s s_{0 1 2})∈R₊³, we denote by C^1,L[ ] the set ofs all functions ϕ∈C E( ₀∪∂₀E R, ) for which the following conditions are satisfied:

(i) there exists ∂ ϕx ( , ) for (, )t x t x ∈E0∪∂0E, (ii) the following estimates are satisfied on E₀∪∂₀E:

ϕ( , )t x ≤ s0, ϕ( , )t x −ϕ( , )t x ≤ s t₁ −t , ∂ ϕx ( , ) t x ≤ s₁,

∂ ϕx ( , )t x −∂ ϕx ( , )t x ≤ s₂

[

t− +t x−x

]

^.

Let ϕ ∈C^1,L[ ] be given and let 0 s < c ≤ a, d = ( ,d d d_{0 1}, ₂)∈R₊³, and d_i ≥ s_i for i = 0, 1, 2. We consider the space C_ϕ,^1,L_c[ ]d of all functions z E: _c^∗→R for which the following conditions are satisfied:

(i) z∈C E R( _c^∗, ) and z t x( , ) = ϕ( , )t x on (E₀∪∂₀E)∩ ([−h c₀, ]×Rⁿ),

(ii) there exists ∂_xz t x( , ) on E_c^∗ and the estimates

z t x( , ) ≤ d₀, z t x( , )−z t x( , ) ≤ d t₁ −t , ∂_xz t x( , ) ≤ d₁,

∂_xz t x( , )−∂_xz t x( , ) ≤ d₂

[

t− +t x−x

]

are satisfied on E_c^∗.

Let p = (p₀,p₁)∈R₊², p₀ ≥ s₁, p₁ ≥ s₂. We denote by C_c^0,L[ ] the set of all functions v: Ep _c→Rⁿ such that, for ( , )t x ∈E_c, the estimates v( , )t x ≤ p₀ and

v( , )t x −v( , )t x ≤ p₁

[

t− +t x−x

]

hold on E_c. We prove that, under suitable assumptions on f, α, and ϕ and for sufficiently small c such that 0 < c ≤ a, there exists a solution z of problem (1), (2) such that z ∈C_ϕ,^1,L_c[ ]d and ∂_xz∈C_c^0,L[ ]. Let p

Δ+i = x

{

∈ −^[ b b^{, ] :} x_i = b_i

}

^, Δ−i = x

{

∈ −^[ b b^{, ] :} x_i = −b_i

}

^,

where 1 ≤ i ≤ n, and

Δ = Δ_i Δ_i

i

n + −

=

(

^∪

)

∪

1

.

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) f( , , , ) : [ , ]⋅ x w q 0 a → R is measurable for every (x, w, q) ∈[–b, b] × C(D, R) × Rⁿ, and the par- tial derivatives

(∂q f P( ), ,∂q f P( ))

n

1 … = ∂qf P( ), P = (t, x, w, q), exist for ( , , )x w q ∈Ω^{( )1} and almost all t ∈[0, a];

(ii) ∂qf( , , , ) : [ , ]⋅ x w q 0 a → Rn is measurable, there is B ∈Θ such that

∂qf 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^{( , )1L}, ( , ) [x q ∈ −b b, ]×Rⁿ, and w∈C^{0 1,} ( , ) andD R almost all t ∈[0, a], we have

∂qf t x w( , , +w q, )−∂qf 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

∂q_if t x w q( , , , ) ≥ κ for ( , , )x w q ∈Δ⁺_i × C^{0 1,} ( , )D R × Rⁿ

and

∂_qif 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[αααα]. Assume that functions α: [0, a] → [0, a] and α′: E →[–b, b] satisfy the follow- ing conditions:

(i) α0(t)≤ t for t ∈[0, a] and there is r0∈R+ such that α0( )t −α0( )t ≤ r t₀ −t on [0, a];

(ii) α′ is of class C¹ and

∂ α_x ′( , ) t x ≤ r0 on E;

(iii) there is r₁∈R₊ such that

∂ αx ′( , )t x − ∂ αx ′( , ) t x ≤ r x₁ − x on E.

Let r = (r0, r1).

Suppose that ϕ ∈C^1,L[ ], zs ∈C_ϕ,^1,L_c[ ]d , and u∈C_c^0,L[ ]. Consider the Cauchy problem p

η′(τ) = –∂_qf( , ( ),τ η τ z_{α 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 bichar- acteristic 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

P z u[ , ]( , , )τ t x =

(

τ, [ , ]( , , ),g z u τ t x z_{α τ}( , [ , ]( , , ))_{g z u τt x} ,u

(

τ, [ , ]( , , )g z u τ t x

) )

^.

We prove a lemma on bicharacteristics.

Lemma 1. Suppose that Assumptions H [∂qf and H] [α] are satisfied and ϕ ϕ, ∈C^1,L[ ]s , z ∈C_ϕ,^1,L_c[ ]d , z ∈C_ϕ,^1,L_c[ ]d , u u, ∈C_c^0,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 ∈E_c, and

g z u[ , ]( , , )τ t x − g z u[ , ]( , , )τ t x ≤ C z z z z u u d

t

x x

τ

ξ ∂ ∂ ξ ξ ξ

∫ [

^{− + − ( ) + − ( )}

]

⁽¹³⁾

for τ ∈I_{( , )t x} ∩I_{( , )t x} , ( , )t x ∈E_c, 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⁺r d₀( ₁⁺d₂)⁺p₁

^{] }}

and

d = d˜ ₀ + d₁, d = d₀ + d₁ + d₂.

Proof. The existence and uniqueness of solutions of (11) follows from the classical theorem on Cara- théodory solutions of initial-value problems. The function g z u[ , ]( , , )⋅ t x satisfies the integral equation

g z u[ , ]( , , )τ t x = x f P z u t x d

t

−

∫

^τ∂q

(

[ , ]( , , )^ξ

)

^ξ. (14)

Since z∈C_ϕ,^1,L_c[ ]d , condition (ii) of Assumption H[α] shows that

z_{α τ( , )y} − z_{α τ( , )y 1} ≤ r d₀( ₁+d₂) y−y

for all ( , ), ( , )τ y τ y ∈E_c. It follows from Assumption H [∂qf 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

≤ 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_{( , )t x} ∩ I_{( , )t x} .

We now obtain (12) by the Gronwall inequality. For ( , ), ( , )τ y τ y ∈E_c, we have z_{α τ( , )y} − z_{α τ( , )y 1} ≤ z − z _τ + ∂_xz − ∂_xz _{( )τ} + r d₀( ₁+d₂) y−y .

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 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_{( , )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 [∂qf and H] [α] are satisfied and ϕ ϕ, ∈C^1,L[ ]s , z ∈C_ϕ,^1,L_c[ ]d , z ∈C_ϕ,^1,L_c[ ]d , u u, ∈C_c^0,L[ ]p .

Then the functions δ[ , ]z u and δ[ , ]z u are continuous on E_c. Moreover, we have the estimates δ[ , ]( , )z u t x −δ[ , ]( , )z u t x ≤ 2 C

t t x x

κ

[

− + −

]

^{, (15)}

where ( , ), ( , )t x t x ∈E_c, and

δ[ , ]( , )z u t x −δ[ , ]( , )z u t x ≤ 2

0

C z z z z u u d

t

x x

κ

∫ [

⁻ _ξ ⁺ ∂ ⁻∂ ^{( )}_ξ ^{+ − ( )}_ξ

]

ξ^{, (16)} where ( , )t x ∈E_c.

Proof. The continuity of δ[ , ]z u and δ[ , ]z u on E_c follows from classical theorems on continuous de- pendence 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 ζ = δ[ , ]( , )z u t x , we have g z u[ , ]( , , )ζ t x ∈Δ and there exists i, 1 ≤ i ≤ n, such that g z u_i[ , ]( , , )ζ t x = b_i. The following two cases are possible:

(i) g z u_i[ , ]( , , )ζ t x = b_i; (ii) g z u_i[ , ]( , , )ζ t x = −b_i.

Consider case (i). Let x = (x₁,…, x_n) and ˜x = ( ,x₁ …,x_i−1, ,b x_{i i+1},…,x_n). We have the following estimate for ( , )t x ∈E_c:

∂qif t x z( , , _{α( , )}t x, ( , ))u t x −∂qif t x z( , ˜, _{α( , ˜)}t x, ( , ˜))u t x ≤ ˜ (c b_i − x_i), (17)

where ˜c = C d( )[1+r d₀( ₁+d₂)+p₁]. Thus,

∂q_if t x z( , , _{α( , )}t x , ( , ))u t x ≥ κ 2 for ( , )t x ∈E_c such that b_i − x_i ≤ κ( ˜)2c ⁻¹. It follows from Lemma 1 that

b_i − g z u_i[ , ]

(

ζ, ,t x

)

= g z u_i[ , ]

(

ζ, ,t x

)

− g z u_i[ , ]

(

ζ, ,t x

)

≤ κ 2 ˜c for ( , ), ( , )t x t x ∈E_c such that

t−t + x− x ≤ κ

2 ˜cC. (18)

Then we get

∂qif

(

ζ, [ , ]g z u

(

ζ, ,t x

)

,z_{α ζ}( , [ , ]( , , ))g z u _ζt x , ( , [ , ]( , , ))u ζ g z u ζ t x

)

^{≥ κ}₂ ^{> 0,}

and, consequently,

∂t ig z u[ , ]( [ , ]( , ), , ) < 0δ z u t x t x

for ( , ), ( , )t x t x ∈E_c satisfying (18). It can easily be seen that g z u_i[ , ]( , , )⋅ t x is decreasing on the interval ( [ , ]( , ), [ , ]( , ))δ z u t x δ z u t x . Therefore,

b_i − g z u_i[ , ]( , , )τ t x ≤ κ 2 ˜c,

and the estimate

∂q_if

(

τ, [ , ] , ,g z u

(

τ t x

)

,z_{α τ}( , [ , ]( , , ))g z u _τt 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 ∈E_c such that (18) 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_if g z u t x z g z u t x u g z u t x d

∫ ( )

= g z u_i[ , ]( [ , ]( , ), , )δ z u t x t x − g z u_i[ , ]( [ , ]( , ), , )δ z u t x t x

≥ g z u_i[ , ]( [ , ]( , ), , )δ z u t x t x − g z u_i[ , ]( [ , ]( , ), , )δ z u t x t x ≥ –C t

[

− +t x−x

]

^.

Thus, the proof of (15) for ( , ), ( , )t x t x ∈E_c such that (18) holds is completed in case (i). In a similar way, we prove (ii). Let ( , ), ( , )t x t x ∈E_c be arbitrary. We set M = x−x + −t t . There exists K ∈N such that

( )

K ˜

−1 cC 2

κ < M ≤ K cC

κ 2 ˜ . Let ε∈R, ε = 1

/

^{K. For j = 0,…}, K, we set

x^{( )j} = j xε + −(1 jε)x, t^{( )j} = j tε + −(1 jε)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

K ≤ κ 2 ˜cC for j = 0,…, K – 1. It is easy to see that

x−x = x ^j x ^j

j K

( ) − ( ⁺)

=

∑

− ¹

0 1

and t−t = t ^j t ^j

j K

( ) − ( ⁺)

=

∑

− ¹

0 1

.

Then

δ[ , ]( , )z u t x − δ[ , ]( , )z u t x ≤ δ[ , ]z u t

(

^{( )j} ,x^{( )j}

)

δ[ , ]z u t

(

^{(j )},x^{(j )}

)

j

K − ^{+ +}

=

∑

− ^{1 1}

0 1

≤ 2 1 1

0

1 C

t ^j t ^j x ^j x ^j

j K

κ

( ) − ( ) + ( ) − ( )

[

^{+ +}

]

=

∑

− ^{= 2C}_κ

^[

^{t− +t x−x}

^]

^.

Thus we see that estimate (15) holds for all ( , ), ( , )t x t x ∈E_c. 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 g z u_i[ , ]( , , )ζ t x = b_i. The following two cases are possible:

(i) g z u_i[ , ]( , , )ζ t x = b_i; (ii) g z u_i[ , ]( , , )ζ t x = −b_i.

Consider case (i). We have estimate (17) for ( , )t x ∈E_c. It follows from Lemma 1 that

g z u_i[ , ]( , , )ζ t x − g z u_i[ , ]( , , )ζ t x ≤ cC

[

z−z _t + ^∂_xz−^∂_xz ^{( )}_t + u−u ^{( )}_t

]

^.

Thus, we have

b_i − g z u_i[ , ]( , , )ζ t x = g z u_i[ , ]( , , )ζ t x − g z u_i[ , ]( , , )ζ t x ≤ κ 2 ˜c, for ( , )t x ∈E_c and z, z , u, and u such that

z−z _t + ∂_xz−∂_xz _{( )t} + u−u _{( )t} ≤ κ

2ccC˜ . (19)

Then we get

∂q_if

(

ζ, [ , ] , ,g z u

(

ζ t x

)

,z_{α ζ}( , [ , ]( , , ))g z u _ζt x , ( , [ , ]( , , ))u ζ g z u ζ t x

)

^{≥ κ}₂ ^{> 0,}

and, consequently,

∂t ig z u[ , ]( [ , ]( , ), , ) < 0δ z u t x t x

for ( , )t x ∈E_c and for z, z , u, and u satisfying (19). It can easily be seen that g z u_i[ , ]( , , )⋅ t x is decreasing on ( [ , ]( , ), [ , ]( , ))δ z u t x δ z u t x . Therefore,

b_i − g z u_i[ , ]( , , )τ t x ≤ κ 2 ˜c and the estimate

∂q_if

(

τ, [ , ] , ,g z u

(

τ t x

)

, z_{α τ}( , [ , ]( , , ))g z u _τt 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 ∈E_c, and z, z , u, and 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_if g z u t x z g z u t x u g z u t x d

∫ ( )

≥ g z u_i[ , ]( [ , ]( , ), , )δ z u t x t x − g z u_i[ , ]( [ , ]( , ), , )δ z u t x t x

≥ –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 ∈E_c and for z, z , u, and u such that (19) holds is completed in case (i). In a similar way, we prove (ii).

Let z∈C_ϕ,^1,L_c[ ]d , z∈C_ϕ,^1,L_c[ ]d , and u, u∈C^0,L[ ] be arbitrary. We set M p = z−z _t + ∂xz−∂xz _{( )}t + u−u _{( )t} . There exists K ∈N such that

( )

K ˜

−1 ccC 2

κ < M ≤ K ccC

κ 2 ˜ . Let ε∈R, ε = 1

/

^{K. For j = 0,…}, K, we set

z_j = j zε + −(1 jε)z and u_j = j uε + −(1 jε)u on E_c. Note that z₀ = z, u₀ = u, z_K = z, u_K = u, and

z_j z_j z z u u

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_j z_j

t j

K − ₊

=

∑

− ¹

0 1

, ∂xz−∂xz _{( )}t = ∂x j ∂x j t j

K

z − z ₊

=

∑

− ¹

0 1

( )

and

u−u _{( )t} = u_j u_j

j t

K − ₊

=

∑

− ¹

0 1

( ). Then

δ[ , ]( , )z u t x − δ[ , ]( , )z u t x ≤ δ[ ,z u_{j j}]( , )t x δ[z_j ,u_j ]( , )t x

j

K − _{+ +}

=

∑

− ^{1 1}

0 1

≤

j K

j j x j x j j j

C t

z z z z u u d

=

−

+ + +

∑ ∫

₀¹

[

^{− 1 + − 1 + − 1}

]

0

2

κ _ξ ∂ ∂ _{( )ξ ( )ξ} ξ

= 2

0

C z z z z u u d

t

x x

κ

∫ [

⁻ _ξ ⁺ ∂ ⁻∂ ^{( )}_ξ ^{+ − ( )}_ξ

]

ξ^.

Therefore, estimate (16) holds for all z∈C_ϕ,^1,L_c[ ]d , z∈C_ϕ,^1,L_c[ ]d , and u, u∈C_c^0,L[ ]. This completes the proof ofp 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 ⋅ _ the norm in CL D R( , ⁿ) . We now formulate next assumptions on f.

Assumption H[f]. Suppose that the Assumption H [∂qf 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

(∂x f P( ), ,∂x f P( ))

n

1 … = ∂xf P( ), P = (t, x, w, q),

and the Fréchet derivative ∂wf P( ) exist for ( , , )x w q ∈Ω and almost all t ∈[0, a];

(iii) the estimates

∂xf t x w q( , , , ) ≤ B w

(

₁

)

_, ∂wf t x w q( , , , ) _ ≤ B w

(

₁

)

^,

are satisfied for ( , , )x w q ∈Ω^{( )1} and almost all t ∈[0, a]; (iv) the terms

∂xf t x w( , , +w q, ) − ∂xf t x w q( , , , ) ,

∂wf t x w( , , +w q, ) − ∂wf t x w q( , , , ) _

are bounded from above by

C

(

w _1,L

) ^[

x⁻x ⁺ w ₁ ⁺ q⁻q

^]

for ( , , )x w q ∈Ω^{( , )1L} , ( , ) [x q ∈ −b b, ]×Rⁿ, w∈C^{0 1,} ( , ) , and almost all t D R ∈[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 ∂_xf , ∂_wf , and ∂_qf . We have also assumed the Lipschitz condition for these derivatives with the same coefficient.

Suppose that ϕ ∈C^1,L[ ]. Let Cs _ϕ⁰_,^,_c^L[ ]p be the class of all functions u E: _c^∗ → Rⁿ such that u t x( , ) =

∂ ϕx ( , ) on (t x E₀∪∂₀E) ∩ ([−h c₀, ]×Rⁿ) and u _E C_c ^L p

c ∈ ^0, [ ]. We now formulate the system of integral equations generated by (1), (2). We write

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 = ∂ ϕ_x

(

Q z u t x[ , ]( , ) , ψ = (

)

ψ₁,…, ψ_n),

W z u[ , ]( , , )τ t x = u_{α τ}( , [ , ]( , , ))_{g z u τt x}  ∂ α τ_x ( , [ , ]( , , ))g z u τ t x . Given ϕ ∈C^1,L[ ], zs ∈C_ϕ,^1,L_c[ ]d , and u∈C_ϕ,^0,_c^L[ ]p , 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

f P z u t x qf 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 ^{+ ∂wf P z u}

⁽

[ , ]( , , )^{τ t x}

^{) W}

[ , ]( , , )^{z u τ t x d}

]

^τ.

Consider the system of integral functional 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 _qf P z u t x d

t

+

∫

^{∂ ξ ξ}

τ

( [ , ]( , , )) (21)

with initial-boundary conditions

z = ϕ, u = ∂xϕ on (E₀∪∂₀E) ∩ ([−h c₀, ]×Rⁿ). (22)