Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 1, 131–141

Existence and Uniqueness of Solutions to a Class of

Stochastic Functional Partial Differential Equations via Integral Contractors

J. Turo

Abstract. Existence and uniqueness theorem for first order stochastic functional partial differential equations with the white noise as a coefficient is proved. In the proof the characteristics method and the concept of integral contractors are used.

Keywords: Stochastic functional partial differential equations, integral contractors

AMS subject classification: 60 H 15, 35 R 60, 35 R 10

1. Introduction

Let ℝ^{m} denote the m-dimensional Euclidean space with the norm | · |, and let B = [−r, 0] × [−d, +d] where r ≥ 0 and d = (d_{1}, ..., d_{m}) ∈ ℝ_{+}^{m} (in particular, it may be d_{i} = ∞ for some i, 1 ≤ i ≤ m) and ℝ_{+} = [0, ∞). Put I = [0, T] × ℝ^{m}, I_{0} = [−r, 0] × ℝ^{m} and D = I ∪ I_{0}, where T > 0. Let (Ω, F, P) be a complete probability space. We assume that there is a set of sub-σ-algebras F_{t} (t ∈ [0, T]) in F such that F_{s} ⊂ F_{t} if s ≤ t. Let w(t; ω) be a p-dimensional standard Brownian motion process adapted to F_{t} such that F(w(t + h; ω) − w(t; ω), h > 0) is independent on F_{t} (t ∈ [0, T]), where F(w(ξ), ξ > 0) denotes the σ-algebra generated by the process w(ξ) (ξ > 0). For any function u : D × Ω → ℝ^{n} and a fixed (t, x; ω) ∈ I × Ω we define the Hale-type operator u_{(t,x)}(ω) : B → ℝ^{n} by

u_{(t,x)}(ω)(τ, θ) = u(t + τ, x + θ; ω) ((τ, θ) ∈ B, ω ∈ Ω).

Let L_{2} be the space of all random variables ξ : Ω → ℝ^{n} with finite L_{2}-norm ||ξ||_{2} = \{E|ξ|^{2}\}^{\frac{1}{2}} , where E is an expectation. Denote by C_{B} = C(B, L_{2}) the space of all continuous processes v : B → L_{2}. Let L(ℝ^{p}, ℝ^{m}) be the space of all linear maps from ℝ^{p} into ℝ^{m}.

J. Turo: Tech. Univ. of Gdansk, 80-952 Gdansk, Narutowicza 11/12, Poland e-mail: turo@mif.pg.gda.pl ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

Let us consider the functional partial differential equations of first order with a random coefficient

\begin{matrix}& \frac{∂u}{∂t}(t, x; ω) + \{a(t, x; ω) + b(t, x; ω)ẇ(t; ω)\}\frac{∂u}{∂x}(t, x; ω) \\ & = f(t, x, u_{(t,x)}(ω); ω) + g(t, x, u_{(t,x)}(ω); ω)ẇ(t; ω) ((t, x) ∈ I) \\ u(t, x; ω) & = φ(t, x; ω) ((t, x) ∈ I_{0}, ω ∈ Ω) \end{matrix} \} (1)

where

a : I × Ω → ℝ^{m}
b : I × Ω → L(ℝ^{p}, ℝ^{m})
f : I × C_{B} × Ω → ℝ^{n}
g : I × C_{B} × Ω → L(ℝ^{p}, ℝ^{n})
φ : I_{0} × Ω → ℝ^{n}

and ẇ(t; ω) is the formal derivative of the process w(t; ω), namely the so called white noise. Equation (1) contains as particular cases the equations investigated in [3, 5 - 8].

Now, we consider the stochastic functional integral equation

\begin{matrix}u(t, x) & = φ(0, y(0; t, x)) \\ & + ∫_{0}^{t} f(s, y(s; t, x), u_{(s,y(s;t,x))})ds \\ & + ∫_{0}^{t} g(s, y(s; t, x), u_{(s,y(s;t,x))})dw(s) ((t, x) ∈ I) \\ u(t, x) & = φ(t, x) ((t, x) ∈ I_{0}) \end{matrix} \}  (2)

where y(s; t, x) is a solution of the stochastic integral equation

y(s) = x + ∫_{s}^{t} a(τ, y(τ)) dτ + ∫_{s}^{t} b(τ, y(τ)) dw(τ).  (3)

We can notice the close analogy between our consideration and the common theory of partial differential equations of first order. In this sense we call the stochastic process {y(s; t, x), s ≤ t} the characteristic line of equation (1) through (t, x) and equation (2) can be considered as equation (1) integrated along the characteristic line.

The concept of contractors by Altman [1] has been used by Constantin [2] to prove the existence and uniqueness solutions of a stochastic integral equation. A particular case of equation (2) has been studied in [4] under the condition that the functions f and g satisfy a Lipschitz condition with respect to the last variable (see also [5]). In this paper, using the characteristics and integral contractors methods, we obtain more general conditions for the existence and uniqueness of solutions to equation (2).

2. The existence of characteristic lines

Denote by C = C([0, T], L_{2}) the space of all processes y : [0, T] → L_{2} which are continuous and adapted to the F_{t} (t ∈ [0, T]). We consider on C the norm ||y|| = sup_{t∈[0,T]} ||y(t)||_{2}.

Define the integral operators J̃_{1} and J̃_{2} on C by

\begin{aligned}(J̃_{1}y)(t) & = ∫_{0}^{t} y(s) ds \\ (J̃_{2}y)(t) & = ∫_{0}^{t} y(s) dw(s). \end{aligned}

It is easy to seen (see [2]) that

\begin{matrix}||J̃_{1}y|| & ≤ T ||y|| \\ ||J̃_{2}y|| & ≤ \sqrt{T} ||y|| \end{matrix} \}  (y ∈ C).  (4)

Assumption (H_{1}). Suppose the following:

(i) The functions a : I × Ω → ℝ^{m} and b : I × Ω → L(ℝ^{p}, ℝ^{m}) are such that a(·, y(·)), b(·, y(·)) ∈ C for all y ∈ C.

(ii) For each t ∈ [0, T] and y ∈ L_{2} there exist bounded linear operators Γ̃_{i}(t, y) (i = 1, 2) on C such that ||Γ̃_{i}(t, y)|| are continuous in (t, y) and there is a constant Q̃ > 0 such that

∥(Γ̃_{i}(t, y(t))v)(t)∥_{2} ≤ Q̃||v(t)||_{2}

for every v ∈ C.

(iii) There exist continuous functions γ̃_{i} : ℝ_{+} → ℝ_{+} with γ̃_{i}(0) = 0 (i = 1, 2) such that for each t ∈ [0, T] and y, v ∈ C we have

\begin{matrix}∥a(t, y(t) + v(t) + (J̃_{1}Γ̃_{1}(t, y(t))v)(t) + (J̃_{2}Γ̃_{2}(t, y(t))v)(t)) \\ −a(t, y(t)) − (Γ̃_{1}(t, y(t))v)(t)∥_{2} & ≤ γ̃_{1}(||v(t)||_{2}) \end{matrix}

and

\begin{matrix}∥b(t, y(t) + v(t) + (J̃_{1}Γ̃_{1}(t, y(t))v)(t) + (J̃_{2}Γ̃_{2}(t, y(t))v)(t)) \\ −b(t, y(t)) − (Γ̃_{2}(t, y(t))v)(t)∥_{2} & ≤ γ̃_{2}(||v(t)||_{2}). \end{matrix}

The vector of functions (a, b) satisfying assumption (H_{1}) is said to have a bounded integral vector contractor (Γ̃_{1}, Γ̃_{2}) with nonlinear majorants (γ̃_{1}, γ̃_{2}) with respect to C.

Remark 1. If γ̃_{i} = α_{i}t (t ∈ ℝ_{+}) where α_{i} > 0 (i = 1, 2) are constants, we have that the vector functions (a, b) has a bounded integral vector contractor (Γ̃_{1}, Γ̃_{2}). These conditions are weaker than the usual Lipschitz condition. Indeed, if Γ̃_{i} = 0 (i = 1, 2), the condition in assumption (H_{1}) reduces to Lipschitz condition on a and b.

Definition 1. Let H be the family of all functions γ ∈ C^{1}(ℝ_{+}, ℝ_{+}) satisfying γ(0) = 0 and γ′(t) ∈ [0, 1) (t ∈ ℝ_{+}).

Lemma 1 (see [2]). If γ ∈ H, we have that γ is non-decreasing on ℝ_{+}, γ(t) < t for t > 0 and ∑_{k=0}^{∞} γ^{(k)}(t) < ∞ for t ∈ ℝ_{+}, where γ^{(k)} denotes the k-th iterate of γ.

Remark 2. Examples of functions γ ∈ H are γ(t) = αt (t ∈ ℝ_{+}), where α ∈ (0, 1), γ(t) = \frac{t^{2}}{t+1} (t ∈ ℝ_{+}), γ(t) = t − arctan t (t ∈ ℝ_{+}), and γ(t) = t − ln(1 + t) (t ∈ ℝ_{+}).

Lemma 2 (see [2], but also [9]). Let us suppose the following:

(i) The functions a : I × Ω → ℝ^{m} and b : I × Ω → L(ℝ^{p}, ℝ^{m}) are such that if y_{n} → y in C, then a(·, y_{n}(·)) → a(·, y(·)) and b(·, y_{n}(·)) → b(·, y(·)) in C for n → ∞.

(ii) Assumption (H_{1}) is satisfied with the vector of nonlinear majorants (γ̃_{1}, γ̃_{2}) such that T γ̃_{1} + \sqrt{T} γ̃_{2} = γ̃ ∈ H.

Then equation (3) has a unique solution y(s) = y(s; t, x) in C.

Remark 3. Note that y satisfies the group property

y(s; τ, y(τ; t, x)) = y(s; t, x)  (5)

for τ ∈ [s, t] and (t, x) ∈ I, since y(s; t, x) is the unique solution of equation (3).

3. Assumptions and lemma

Let C_{Y} = C(Y, L_{2}) be the space of all processes v : Y → L_{2} which are continuous bounded and adapted to the F_{t} for each x, where Y ⊂ I or Y ⊂ D (let F_{t} = F_{0} for −r ≤ t ≤ 0). We consider on C_{Y} the norm ||v||_{Y} = sup_{(t,x)∈Y} ||v(t, x)||_{2}. Define the integral operators J_{1} and J_{2} on C_{I} by

\begin{aligned}(J_{1}u)(t, x) & = ∫_{0}^{t} u(s, x) ds \\ (J_{2}u)(t, x) & = ∫_{0}^{t} u(s, x) dw(s). \end{aligned}

For these operators we have analogous estimates as in (4)

\begin{matrix}||J_{1}u||_{I} & ≤ T ||u||_{I} \\ ||J_{2}u||_{I} & ≤ \sqrt{T} ||u||_{I} \end{matrix} \} (u ∈ C_{I}).

Put

\begin{aligned}f[u](s; t, x) & = f(s, y(s; t, x), u_{(s,y(s;t,x))}) \\ g[u](s; t, x) & = g(s, y(s; t, x), u_{(s,y(s;t,x))}). \end{aligned}

Assumption (H_{2}). Suppose the following:

(i) The functions f : I × C_{B} × Ω → ℝ^{n} and g : I × C_{B} × Ω → L(ℝ^{p}, ℝ^{n}) are such that f[u](·) ∈ C_{I} for u ∈ C_{B} and g[u](·) ∈ C_{I} for y ∈ C.

(ii) For each s ∈ [0, T], y ∈ L_{2} and u ∈ C_{D} there exist bounded linear operators Γ_{i}(s, y, u) (i = 1, 2) on C_{D} such that ||Γ_{i}(s, y, u)|| are continuous in (s, y, u) and there is a constant Q > 0 such that

∥(Γ_{i}[y, u]v)(s; t, x)∥_{2} ≤ Q∥v(s, y(s; t, x))∥_{2}

for every v ∈ C_{D}, where

(Γ_{i}[y, u]v)(s; t, x) = (Γ_{i}(s, y(s; t, x), u_{(s,y(s;t,x))})v)(s, y(s; t, x)).

(iii) There exist continuous functions γ_{i} : ℝ_{+} → ℝ_{+} with γ_{i}(0) = 0 (i = 1, 2) such that for each u, v ∈ C_{D} and y ∈ C we have

∥f[u + v + J_{1}Γ_{1}[y, u]v + J_{2}Γ_{2}[y, u]v](s; t, x) − f[u](s; t, x) − (Γ_{1}[y, u]v)(s; t, x)∥_{2}

≤ γ_{1}(||v_{(s,y(s;t,x))}||_{B})

and

∥g[u + v + J_{1}Γ_{1}[y, u]v + J_{2}Γ_{2}[y, u]v](s; t, x) − g[u](s; t, x) − (Γ_{2}[y, u]v)(s; t, x)∥_{2}
≤ γ_{2}(||v_{(s,y(s;t,x))}||_{B})

for all s ∈ [0, T] and (t, x) ∈ I.

Lemma 3. If assumption (H_{2})/(ii) is satisfied and y is a solution of equation (3) and u(·, y(·; t, x)) ∈ C_{D}, then for every h ∈ C_{I} such that h(0; x) = 0, there is a unique solution v ∈ C_{D} to the stochastic integral equation

\begin{matrix}v(t; x) & = h(t; x) \\ & − ∫_{0}^{t}(Γ_{1}[y, u]v)(s; t, x) ds \\ & − ∫_{0}^{t}(Γ_{2}[y, u]v)(s; t, x) dw(s) ((t, x) ∈ I) \\ v(t, x) & = 0 ((t, x) ∈ I_{0}). \end{matrix} \}  (6)

Proof. Define an operator K on C_{D} as

\begin{matrix}(Kv)(t; x) = \\ \{ & \begin{matrix}h(t; x) − ∫_{0}^{t}(Γ_{1}[y, u]v)(s; t, x) ds − ∫_{0}^{t}(Γ_{2}[y, u]v)(s; t, x) dw(s) & if (t, x) ∈ I \\ 0 & if (t, x) ∈ I_{0}. \end{matrix} \end{matrix}

It is obvious that K maps C_{D} into itself. Let us introduce the norm

||v||_{∗} = sup_{(t,x)∈D} {e^{−λt}||v(t, x)||_{2}}

where λ > Q^{2}(T + 1). Now we prove that K is a contraction. Indeed, since (x + y)^{2} ≤ 2(x^{2} + y^{2}) we have

\begin{matrix}∥(Kv_{1})(t, x) − (Kv_{2})(t, x)∥_{2}^{2} \\ ≤ 2Q^{2}(T + 1) ∫_{0}^{t} ∥v_{1}(s, y(s; t, x)) − v_{2}(s, y(s; t, x))∥_{2}^{2}ds \\ ≤ 2Q^{2}(T + 1)∥v_{1} − v_{2}∥_{∗}^{2} ∫_{0}^{t} e^{2λs}ds \\ ≤ \frac{Q^{2}(T + 1)}{λ}(e^{2λt} − 1)∥v_{1} − v_{2}∥_{∗}^{2} \end{matrix}  (7)

for all (t, x) ∈ I. Multiplying (7) by e^{−2λt} we obtain

||Kv_{1} − Kv_{2}||_{∗}^{2} ≤ \frac{Q^{2}(T + 1)}{λ}||v_{1} − v_{2}||_{∗}^{2}

thus

||Kv_{1} − Kv_{2}||_{∗} ≤ q||v_{1} − v_{2}||_{∗}

where q = [\frac{Q^{2}(T +1)}{λ}]^{\frac{1}{2}} . The assertion of Lemma 3 now follows from the Banach fixed point theorem

4. The main results

We are now in the position to prove the main results.

Theorem. Let us suppose the following:

(i) φ : I_{0} → L_{2} is continuous and F_{0}-adapted for each x, and φ(0, y(0; t, x)) is independent on {w(t), t ∈ [0, T]} for each x.

(ii) The functions f : I × C_{B} × Ω → ℝ^{n} and g : I × C_{B} × Ω → L(ℝ^{p}, ℝ^{n}) are such that if u^{n} → u in C_{B}, then f[u^{n}](·) → f[u](·) and g[u^{n}](·) → g[u](·) for n → ∞.

(iii) Assumptions of Lemma 2 are satisfied.

(iv) Assumption (H_{2}) is satisfied with the vector of nonlinear majorants (γ_{1}, γ_{2}) such that T γ_{1} + \sqrt{T} γ_{2} = γ ∈ H.

Then equation (2) has a unique solution in C_{D}.

Proof. Consider the sequence {u^{n}} defined by

\begin{matrix}u^{n+1}(t, x) & = u^{n}(t, x) − v^{n}(t, x) \\ & − ∫_{0}^{t}(Γ_{1}[y, u^{n}]v^{n})(s; t, x) ds \\ & − ∫_{0}^{t}(Γ_{2}[y, u^{n}]v^{n})(s; t, x) dw(s) ((t, x) ∈ I) \\ u^{n+1}(t, x) & = φ(t, x) ((t, x) ∈ I_{0}) \end{matrix} \}  (8)

where

\begin{matrix}v^{n}(t, x) & = u^{n}(t, x) − φ(0, y(0; t, x)) \\ & − ∫_{0}^{t} f[u^{n}](s; t, x) ds \\ & − ∫_{0}^{t} g[u^{n}](s; t, x) dw(s) ((t, x) ∈ I) \\ v^{n}(t, x) & = 0 ((t, x) ∈ I_{0}) \end{matrix} \}  (9)

and u^{0} ∈ C_{D}. We will now demonstrate that the auxiliary sequence {v^{n}} is such that ||v^{n}||_{D} → 0 as n → ∞. By (8) - (9) applying (5) we deduce that

v^{n+1}(t, x) = ∫_{0}^{t} f[u^{n}](s; t, x) ds + ∫_{0}^{t} g[u^{n}](s; t, x) dw(s)
− ∫_{0}^{t}(Γ_{1}[y, u^{n}]v^{n})(s; t, x) ds − ∫_{0}^{t}(Γ_{2}[y, u^{n}]v^{n})(s; t, x) dw(s)
− ∫_{0}^{t} f[u^{n} − v^{n} − J_{1}Γ_{1}[y, u^{n}]v^{n} − J_{2}Γ_{2}[y, u^{n}]v^{n}](s; t, x) ds
− ∫_{0}^{t} g[u^{n} − v^{n} − J_{1}Γ_{1}[y, u^{n}]v^{n} − J_{2}Γ_{2}[y, u^{n}]v^{n}](s; t, x) dw(s)
((t, x) ∈ I)
v^{n+1}(t, x) = 0 ((t, x) ∈ I_{0}).

Using assumption (H_{2})/(iii) on f and g we obtain

||v^{n+1}||_{D} ≤ T ∥f[u^{n} − v^{n} − J_{1}Γ_{1}[y, u^{n}]v^{n} − J_{2}Γ_{2}[y, u^{n}]v^{n}](s; t, x)
− f[u^{n}](s; t, x) − Γ_{1}[y, u^{n}](−v^{n})(s; t, x)∥_{D}
+ \sqrt{T} ∥g[u^{n} − v^{n} − J_{1}Γ_{1}[y, u^{n}]v^{n} − J_{2}Γ_{2}[y, u^{n}]v^{n}](s; t, x)
− g[u^{n}](s; t, x) − Γ_{2}[y, u^{n}](−v^{n})(s; t, x)∥_{D}
≤ γ(||v_{(s,y(s;t,x))}^{n}||_{B})
≤ γ(||v^{n}||_{D})

since ||v_{(t,x)}^{n}||_{B} ≤ ||v^{n}||_{D}. Thus ||v^{n+1}||_{D} ≤ γ^{(n+1)}(||v^{0}||_{D}). Since lim_{n→∞}γ^{(n)}(t) = 0 for all t ∈ ℝ_{+} (see Lemma 1) we get lim_{n→∞}||v^{n}||_{D} = 0.

From (8) we see that

||u^{n+1} − u^{n}||_{D} ≤ ||v^{n}||_{D} + (T + \sqrt{T})Q||v^{n}||_{D}
≤
⋮
≤ (1 + T Q + \sqrt{T} Q)γ^{(n)}(||v^{0}||_{D}).

Since γ^{(n)}(||v^{0}||_{D}) → 0 as n → ∞, we get that {u^{n}} is a Cauchy sequence and thus there exists u ∈ C_{D} such that lim_{n→∞}u^{n} = u.

From (9) and assumption (ii) it follows that u is a solution of equation (2). Let us now prove the uniqueness of solutions to equation (2). Let u, ū be two solutions in C_{D} of equation (2) with u(t, x) = ū(t, x) = φ(t, x) for (t, x) ∈ I_{0}. Then

\begin{matrix}u(t, x) − ū(t, x) & = ∫_{0}^{t} \{f[u](s; t, x) − f[ū](s; t, x)\}ds \\ & + ∫_{0}^{t} \{g[u](s; t, x) − g[ū](s; t, x)\}dw(s) ((t, x) ∈ I) \\ u(t, x) − ū(t, x) & = 0 ((t, x) ∈ I_{0}). \end{matrix} \}  (10)

We denote h(t, x) = u(t, x) − ū(t, x) ((t, x) ∈ I), and let v ∈ C_{D} be a solution to the stochastic integral equation (see Lemma 3)

\begin{matrix}v(t, x) & = h(t, x) \\ & − ∫_{0}^{t}(Γ_{1}[y, ū]v)(s; t, x) ds \\ & − ∫_{0}^{t}(Γ_{2}[y, ū]v)(s; t, x) dw(s) ((t, x) ∈ I) \\ v(t, x) & = 0 ((t, x) ∈ I_{0}). \end{matrix} \}  (11)

By (10), (11) and (5) we get

||v||_{D} ≤ ∥ ∫_{0}^{t} {f[ū + v + J_{1}Γ_{1}[y, ū]v + J_{2}Γ_{2}[y, ū]v)](s; t, x)
− f[ū](s; t, x) − (Γ_{1}[y, ū]v)(s; t, x)}ds∥_{I}
+ ∥ ∫_{0}^{t} {g[ū + v + J_{1}Γ_{1}[y, ū]v + J_{2}Γ_{2}[y, ū]v)](s; t, x)
− g[ū](s; t, x) − (Γ_{2}[ū]v)(s; t, x)}ds∥_{I}
≤ γ(||v||_{D}).

Hence (see Lemma 1) v(t, x) = 0 a.s., (t, x) ∈ D, and so by (11) we obtain h(t, x) = 0, a.s., (t, x) ∈ D, i.e. u = ū in C_{D} and the uniqueness is proved. This completes the proof of the Theorem

Corollary. Let assumptions (i), (iii) and (iv) of Theorem be satisfied. If 3(T^{2} + T)Q^{2} < 1, then equation (2) has a unique solution in C_{D}.

Proof. Let us prove that if u^{n} → u in C_{B}, then f[u^{n}] → f[u] and g[u^{n}] → g[u] in C_{I} for n → ∞. Let v^{n} ∈ C_{D} be a solution to the stochastic integral equation

\begin{matrix}v^{n}(t, x) & = u^{n}(t, x) − u(t, x) \\ & − ∫_{0}^{t}(Γ_{1}[y, u]v^{n})(s; t, x) ds \\ & − ∫_{0}^{t}(Γ_{2}[y, u]v^{n})(s; t, x) dw(s) ((t, x) ∈ I) \\ v^{n}(t, x) & = 0 ((t, x) ∈ I_{0}). \end{matrix} \}

The existence of a such solution follows from Lemma 3. Since (x+y+z)^{2} ≤ 3(x^{2}+y^{2}+z^{2}) we have

\begin{aligned}||v^{n}(t, x)||_{2}^{2} & ≤ 3∥u^{n}(t, x) − u(t, x)∥_{2}^{2} \\ & + 3T Q^{2} ∫_{0}^{t} ∥v^{n}(s, y(s; t, x))∥_{2}^{2}ds + 3Q^{2} ∫_{0}^{t} ∥v^{n}(s, y(s; t, x))∥_{2}^{2}ds \end{aligned}

therefore

||v^{n}||_{D}^{2} ≤ 3||u^{n} − u||_{D}^{2} + 3(T^{2} + T)Q^{2}||v^{n}||_{D}^{2}.

Since 3(T^{2} + T)Q^{2} < 1 and lim_{n→∞} ||u^{n} − u||_{D} = 0 we obtain lim_{n→∞} ||v^{n}||_{D} = 0. Writing relation (iii) from Assumption (H_{2}) with u and v^{n}, we get

∥f[u + v^{n} + J_{1}Γ_{1}[y, u]v^{n} + J_{2}Γ_{2}[y, u]v^{n}](s; t, x) − f[u](s; t, x) − (Γ_{1}[y, u]v^{n})(s; t, x)∥_{2}

≤ γ_{1}(||v_{(s,y(s;t,x))}^{n}||_{B})

and

∥g[u + v^{n} + J_{1}Γ_{1}[y, u]v^{n} + J_{2}Γ_{2}[y, u]v^{n}](s; t, x) − g[u](s; t, x) − (Γ_{2}[y, u]v^{n})(s; t, x)∥_{2}

≤ γ_{2}(||v_{(s,y(s;t,x))}^{n}||_{B}).

Thus

\begin{aligned}∥f[u^{n}](s; t, x) − f[u](s; t, x)∥_{2} & ≤ γ_{1}(∥v_{(s,y(s;t,x))}^{n}∥_{B}) + Q∥v^{n}(s, y(s; t, x))∥_{2} \\ ∥g[u^{n}](s; t, x) − g[u](s; t, x)∥_{2} & ≤ γ_{2}(∥v_{(s,y(s;t,x))}^{n}∥_{2}) + Q∥v^{n}(s, y(s; t, x))∥_{2}. \end{aligned}

Hence we get

∥f[u^{n}] − f[u]||_{I} + ||g[u^{n}] − g[u]∥_{I} ≤ sup_{τ∈[0,||v^{n}||_{D}]}{γ_{1}(τ) + γ_{2}(τ)} + 2Q||v^{n}||_{D}

and since v^{n} → 0 as n → ∞ in C_{D} and γ_{i} (i = 1, 2) are continuous with γ_{i}(0) = 0 we obtain the desired property of f and g. The result now follows from Theorem

5. Particular forms of functional dependence

We give now a few examples that show how the Hale-type operator defined in Introduction acts in particular forms of functional dependence such as delays, integrals and other Volterra functionals.

Example 1. Let α, β : I → ℝ^{m+1}, α = (α_{0}, α_{1}, ..., α_{m}) and β = (β_{0}, β_{1}, ..., β_{m}), and let f̃: I × ℝ^{n} × Ω → ℝ^{n} and g̃ : I × ℝ^{n} × Ω → L(ℝ^{p}, ℝ^{n}) be given functions such that −r ≤ α_{0}(t, x) and β_{0}(t, x) ≤ t for (t, x) ∈ I. If we define

\begin{aligned}f(t, x, v) & = f̃(t, x, v(α(t, x) − (t, x))) \\ g(t, x, v) & = g̃(t, x, v(β(t, x) − (t, x))) \end{aligned}

then equation (1) reduces to the differential equation with retarded argument

\begin{matrix}\frac{∂u}{∂t}(t, x) + \{a(t, x) + b(t, x)ẇ(t)\}\frac{∂u}{∂x}(t, x) \\ = f(t, x, u(α(t, x))) + g(t, x, u(β(t, x)))ẇ(t) \end{matrix} ((t, x) ∈ I).

Example 2. Suppose that f̃ : I × ℝ^{n} × Ω → ℝ^{n}, g̃ : I × ℝ^{n} × Ω → L(R^{p}, R^{n}), α, β : I → ℝ^{m+1} and k_{i} : I → L(ℝ^{n}, ℝ^{n}) (i = 1, 2) are the given functions. Let

\begin{aligned}f(t, x, v) & = f̃(t, x, ∫_{α(t,x)}^{β(t,x)} k_{1}(s, ξ)v(s − t, ξ − x) dsdξ) \\ g(t, x, v) & = g̃(t, x, ∫_{α(t,x)}^{β(t,x)} k_{2}(s, ξ)v(s − t, ξ − x) dsdξ). \end{aligned}

Then equation (1) reduces to the differential-integral equation

\begin{matrix}\frac{∂u}{∂t}(t, x) + \{a(t, x) + b(t, x)ẇ(t)\}\frac{∂u}{∂x}(t, x) \\ = f(t, x, ∫_{α(t,x)}^{β(t,x)} k_{1}(s, ξ)u(s, ξ) dsdξ) \\ + g(t, x, ∫_{α(t,x)}^{β(t,x)} k_{2}(s, ξ)u(s, ξ)dsdξ)ẇ(t) \end{matrix} ((t, x) ∈ I).

Example 3. Take f̃: I×C(D, ℝ^{n})×Ω → ℝ^{n} and g̃ : I×C(D, ℝ^{n})×Ω → L(ℝ^{p}, ℝ^{n}). Consider the equation

\frac{∂u}{∂t}(t, x) + \{a(t, x) + b(t, x)ẇ(t)\}\frac{∂u}{∂x}(t, x) = f(t, x, u) + g(t, x, u)ẇ(t) ((t, x) ∈ I).

The dependence on the past is expressed by means of so called Volterra condition which reads as follows: if u, ū ∈ C_{D} and u(s, x) = ū(s, x) for (s, x) ∈ [−r, t] × ℝ^{m}, then f̃(t, x, u) = f̃(t, x, ū). The definition of the Volterra condition for g̃ is analogous. There are various possibilities of extending this notation. For instance, if we want to describe

Existence and Uniqueness of Solutions 141

the dependence of f̃ locally on the past and locally on the space, then we can formulate the Volterra-type condition as follows: if u, ū ∈ C_{D} and u(s, ξ) = ū(s, ξ) for (s, ξ) ∈ B + (t, x), then f̃(t, x, u) = f̃(t, x, ū), where B + (t, x) = {(s + t, ξ + x) : (s, ξ) ∈ B} is the translation of the set B. In this case we can define

\begin{aligned}f(t, x, v) & = f̃(t, x, I_{t,x}v(· − t, · − x)) \\ g(t, x, v) & = g̃(t, x, I_{t,x}v(· − t, · − x)) \end{aligned}

where I_{t,x} : C_{B} → C_{D+(−t,−x)} is defined by (I_{t,x}v)(s, ξ) = v(s − t, ξ − x).

Acknowledgement. The author wishes to express his gratitude to the anonymous referees for a number of valuable comments and suggestions.

References

[1] Altman, M.: Inverse differentiability contractors and equations in Banach spaces. Studia Math. 46 (1973), 1 – 15.

[2] Constantin, A.: Study of a stochastic integral equation using integral contractors. Demonstratio Math. 30 (1997), 71 – 83.

[3] Funaki, T.: Construction of a solution of random transport equation with boundary condition. J. Math. Soc. Japan 31 (1979), 719 – 744.

[4] Gikhman, I. I.: The Cauchy problem for second-order nonlinear parabolic stochastic systems (in Russian). In: Statistics and Control of Random Processes (Preila, 1987). Moscow: Nauka 1989, pp. 31 – 36.

[5] Gikhman, I. I. and T. M. Miestechkina: Cauchy problem for first order stochastic partial differential equations (in Russian). Teor. Sluchainykh Protsessov 11 (1983), 25 – 28.

[6] Ogawa, S.: A partial differential equation with the white noise as a coefficient. Z. Wahr. verw. Geb. 28 (1973), 53 – 71.

[7] Turo, J.: First order stochastic partial differential equations. Serdica Bulg. math. publ. 20 (1994), 56 – 67.

[8] Turo, J.: Stochastic functional partial differential equations of first order. Stochastic Anal. Appl. 14 (1996), 245 – 256.

[9] Turo, J.: Solution of a stochastic functional differential equation using integral contractors (to appear).

Received 23.02.1998; in revised form 02.12.1998