Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 1, 97–109

Hyperbolic Functional-Differential Equations with Unbounded Delay

Z. Kamont

Abstract. The phase space for quasilinear equations with unbounded delay is constructed. Carathéodory solutions of initial problems are investigated. A theorem on the existence, uniqueness and continuous dependence upon initial data is given. The method of bicharacteristics and integral inequalities are used.

Keywords: Unbounded delay, local existence, Carathéodory solutions AMS subject classification: 35 L 50, 35 D 05

1. Introduction

For any metric spaces U and V we denote by C(U, V) the class of all continuous functions defined on U and taking values in V. We will use vectorial inequalities with the understanding that the same inequalities hold between their corresponding components.

Let

E = [−r_{0}, 0] × [−r, +r] ⊂ ℝ^{1+n}

where r_{0} ∈ ℝ_{+} := [0, +∞) and r = (r_{1}, …, r_{n}) ∈ ℝ_{+}^{n}. Assume that a > 0, (t, x) = (t, x_{1}, …, x_{n}) ∈ [0, a] × ℝ^{n} and z : [−r_{0}, a] × ℝ^{n} → ℝ. We define a function z_{(t,x)} : E → ℝ^{n} by

z_{(t,x)}(τ, s) = z(t + τ, x + s) ((τ, s) ∈ E).

For each (t, x) ∈ [0, a]×ℝ^{n} the function z_{(t,x)} is the restriction of z to the set [t−r_{0}, t]× [x − r, x + r] and this restriction is shifted to the set E. Suppose that

F : [0, a] × ℝ^{n} × C(E, ℝ) × ℝ^{n} → ℝ

is a given function. In this time numerous papers were published concerning various problems for the equation

D_{t}z(t, x) = F (t, x, z_{(t,x)}, D_{x}z(t, x))

where D_{x}z = ( D_{x_{1}}z, …, D_{x_{n}}z) and for adequate weakly coupled hyperbolic systems. The following questions were considered: functional-differential inequalities, uniqueness of initial or initial-boundary value problems, difference-functional inequalities, approximate solutions of initial or initial-boundary value problems, existence of classical or generalized solutions (see [1 - 3, 5 - 8, 10 - 15]). All these problems have the property that the set E is bounded.

Z. Kamont: Univ. of Gdańsk, Inst. Math., ul. Wita Stwosza 57, 80 - 952 Gdańsk, Poland ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

In the paper we start the investigation of first order partial functional-differential equations with unbounded delay. We give sufficient conditions for the existence and uniqueness of Carathéodory solutions of initial problems for quasilinear equations with unbounded delay. We consider functional-differential equations in a Banach space. The theory of ordinary functional-differential equations with unbounded delay is given in monographs [4, 9].

We formulate the problem. Let B be a Banach space with norm ∥ · ∥ and D = (−∞, 0] × [−r, +r] ⊂ ℝ^{1+n} (r ∈ ℝ_{+}^{n}). The norm in ℝ^{n} will also be denoted by ∥ · ∥. For a function z : (−∞, b] × ℝ^{n} → B (b ≥ 0) and for a point (t, x) ∈ [0, b] × ℝ^{n} we define a function z_{(t,x)} : D → B by

z_{(t,x)}(τ, s) = z(t + τ, x + s) ((τ, s) ∈ D).

The phase space X for equations with unbounded delay is a linear space, with norm ∥ · ∥_{X} and consisting of functions mapping the set D into B. Let a > 0 be fixed and suppose that

\begin{aligned}ϱ = (ϱ_{1}, …, ϱ_{n}) & : [0, a] × ℝ^{n} × X → ℝ^{n} \\ f & : [0, a] × ℝ^{n} × X → B \\ φ & : (−∞, 0] × ℝ^{n} → B \end{aligned}

are given functions. We consider the quasilinear equation

D_{t}z(t, x) + ∑_{i=1}^{n} ϱ_{i}(t, x, z_{(t,x)}) D_{x_{i}}z(t, x) = f(t, x, z_{(t,x)})  (1)

with the intial condition

z(t, x) = φ(t, x)  on (−∞, 0] × ℝ^{n}.  (2)

We will deal with Carathéodory solutions of problem (1) - (1.2). A function ū : (−∞, b]× ℝ^{n} → B where 0 < b ≤ a is a solution of the above problem provided:

(i) ū is continuous on [0, b] × ℝ^{n} and the derivatives D_{t}ū(t, x) and D_{x}ū(t, x) exist for almost all (t, x) ∈ [0, b] × ℝ^{n}.

(ii) ū satisfies equation (1.1) almost everywhere on [0, b] × ℝ^{n} and condition (1.2) holds.

We adopt the following notations. If z : (−∞, b]×ℝ^{n} → B (0 < b ≤ a) is a function such that z is continuous on [0, b] × ℝ^{n}, then we put for (t, x) ∈ [0, b] × ℝ^{n}

\begin{aligned}∥z∥_{[0,t;x]} & = max \{∥z(τ, s)∥ : (τ, s) ∈ [0, t] × [x − r, x + r]\} \\ ∥z∥_{[0,t;ℝ^{n}]} & = sup \{∥z(τ, s)∥ : (τ, s) ∈ [0, t] × ℝ^{n}\} \end{aligned}

and

Lip z|_{[0,t;x]} = sup \{\frac{∥z(τ, s) − z(τ, s̄)∥}{∥s − s̄∥} : (τ, s), (τ, s̄) ∈ [0, t] × [x − r, x + r]\} .

The fundamental axioms assumed on X are the followings.

Hyperbolic Functional-Differential Equations 99

Assumption H[X]. Suppose the following:

1) (X, ∥ · ∥_{X}) is a Banach space.

2) If z : (−∞, b] × ℝ^{n} → B (0 < b ≤ a) is a function such that z_{(0,x)} ∈ X for x ∈ ℝ^{n} and z is continuous on [0, b] × ℝ^{n}, then z_{(t,x)} ∈ X for (t, x) ∈ (0, b] × ℝ^{n} and

(i) for (t, x) ∈ [0, b] × ℝ^{n} we have ∥z_{(t,x)}∥_{X} ≤ K ∥z∥_{[0,t;x]} + L ∥z_{(0,x)}∥_{X} where K, L ∈ R_{+} are constant independent on z

(ii) the function (t, x) → z_{(t,x)} is continuous on [0, b] × ℝ^{n}.

3) The linear subspace X_{L} ⊂ X is such that

(i) X_{L} endowed with the norm ∥ · ∥_{X_{L}} is a Banach space (ii) if z : (−∞, b] × ℝ^{n} → B (0 < b ≤ a) is a function such that z_{(0,x)} ∈ X_{L}

for x ∈ ℝ^{n}, z is continuous on [0, b] × ℝ^{n} and z(t, ·) : ℝ^{n} → B satisfies the Lipschitz condition with a constant independent on t (t ∈ [0, b]), then

(α) z_{(t,x)} ∈ X_{L} for (t, x) ∈ (0, b] × ℝ^{n} (β) for (t, x) ∈ [0, b] × ℝ^{n} we have

∥z_{(t,x)}∥_{X_{L}} ≤ K_{0}(∥z∥_{[0,t;x]} + Lip z|_{[0,t;x]}) + L_{0}∥z_{(0,x)}∥_{X_{L}}

where K_{0}, L_{0} ∈ ℝ_{+} are constants independent on z.

Examples of phase spaces are given in Section 4. Let us denote by L([α, β], ℝ) ([α, β] ⊂ ℝ) the class of functions

L([α, β], ℝ) = {μ : [α, β] → ℝ : μ integrable on [α, β]}.

Further, we will use the symbol Θ to denote the set of functions

Θ = \{γ : [0, a] × ℝ_{+} → ℝ_{+} | \begin{matrix}γ(t, ·) is non-decreasing for a.a. t ∈ [0, a] \\ γ(·, τ) ∈ L([0, a], ℝ_{+}) for all τ ∈ ℝ_{+} \end{matrix} \} .

Further, write

\begin{aligned}X[κ] & = \{w ∈ X : ∥w∥_{X} ≤ κ\} \\ X_{L}[κ] & = \{w ∈ X_{L} : ∥w∥_{X_{L}} ≤ κ\} \end{aligned}

where κ ∈ ℝ_{+}.

2. Bicharacteristics of functional-differential equations

We start with assumptions on the initial function φ.

Assumption H_{0}. Suppose that φ : (−∞, 0] × ℝ^{n} → B and
(i) φ_{(0,x)} ∈ X_{L} for x ∈ ℝ^{n}

(ii) there are L̃, L ∈ ℝ_{+} such that ∥φ(0, x)∥ ≤ L̃ for x ∈ ℝ^{n} and ∥φ_{(0,x)}−φ_{(0,x̄)}∥_{X} ≤ L∥x − x̄∥ for x, x̄ ∈ ℝ^{n}.

Assumption H[ϱ]. Suppose the following:

1) The function ϱ(·, x, w) : [0, a] → ℝ^{n} is measurable for (x, w) ∈ ℝ^{n} × X and ϱ(t, ·) : ℝ^{n} × X → ℝ^{n} is continuous for almost all t ∈ [0, a].

2) There exist α, β ∈ Θ such that ∥ϱ(t, x, w)∥ ≤ α(t, κ) for (x, w) ∈ ℝ^{n} × X[κ] almost everywhere on [0, a] and

∥ϱ(t, x, w) − ϱ(t, x̄, w̄)∥ ≤ β(t, κ)[∥x − x̄∥ + ∥w − w̄∥_{X}]  (3)

for (x, w), (x̄, w̄) ∈ ℝ^{n} × X_{L}[κ] almost everywhere on [0, a].

Suppose that φ : (−∞, 0] × ℝ^{n} → B and φ_{(0,x)} ∈ X_{L} for x ∈ ℝ^{n}. Let c ∈ [0, a], d = (d_{0}, d_{1}) ∈ ℝ_{+}^{2} and ω ∈ L([0, a], ℝ_{+}). The symbol Y_{c.φ}[ω, d] denotes the function class

\begin{matrix}Y_{c,φ}[ω, d] = \\ \{z : (−∞, c] × ℝ^{n} → B | \begin{matrix}z(t, x) = φ(t, x) on (−∞, 0] × ℝ^{n} \\ ∥z(t, x)∥ ≤ d_{0} on [0, c] × ℝ^{n} \\ ∥z(t, x) − z(t̄, x̄)∥ ≤ | ∫ t̄ ; tω(τ) dτ| + d_{1}∥x − x̄∥ \\ for (t, x), (t̄, x̄) ∈ [0, c] × ℝ^{n}. \end{matrix} \} \end{matrix}

For the above φ and for z ∈ Y_{c.φ}[ω, d] consider the Cauchy problem

\begin{matrix}η′(τ) & = ϱ(τ, η(τ), z_{(τ,η(τ))}) \\ η(t) & = x \end{matrix} \}  (4)

where (t, x) ∈ [0, c] × ℝ^{n}. We consider Carathéodory solutions of problem (4). Denote by g[z](·, t, x) the solution of the above problem. The function g[z] is the bicharacteristic of equation (1) corresponding to z ∈ Y_{c.φ}[ω, d].

Let Δ_{c} = [0, c] × [0, c] × [0, c]. For φ satisfying Assumption H_{0} define

∥φ∥_{(X,∞)} = sup {∥φ_{(0,x)}∥_{X} : x ∈ ℝ^{n}}.

Lemma 2.1. Suppose that Assumptions H[X] and H[ϱ] are satisfied and

1) the functions φ, φ̄ : (−∞, 0] × ℝ^{n} → B satisfy Assumption H_{0}
2) c ∈ [0, a], z ∈ Y_{c.φ}[ω, d] and z̄ ∈ Y_{c,φ̄}[ω, d].

Then the solutions g[z](·, t, x) and g[z̄](·, t, x) are defined on [0, c] and they are unique. Moreover, we have the estimates on Δ_{c}

\begin{matrix}∥g[z](τ, t, x) − g[z](τ, t̄, x̄)∥ ≤ \\ [∥x − x̄∥ + |∫_{t}^{t̄} α(ξ, κ̄) dξ| ] & exp & [d̄|∫_{t}^{τ} β(ξ, κ_{0}) dξ| ] \end{matrix}  (5)

and

\begin{matrix}∥g[z](τ, t, x) − g[z̄](τ, t, x)∥ ≤ \\ |∫_{t}^{τ} β(ξ, κ_{0})[K∥z − z̄∥_{[0,ξ;ℝ^{n}]} + M∥φ − φ̄∥_{(X,∞)}] dξ exp [d̄∫_{t}^{τ} β(ξ, κ_{0}) dξ]| \end{matrix} (6)

where

\begin{matrix}d̄ & = 1 + Kd_{0} + ML \\ κ̄ & = Kd_{0} + M ∥φ∥_{(X,∞)} \\ κ_{0} & = K_{0}(d_{0} + d_{1}) + L_{0} sup \{∥φ_{(0,x)}∥_{X_{L}} : x ∈ ℝ^{n}\}. \end{matrix} \}

Proof. Suppose that (ξ, η), (ξ, η̄) ∈ [0, c]×ℝ^{n} and a function z̃: (−∞, c]×ℝ^{n} → B is defined by

z̃(τ, s) = z(τ, s + η̄ − η) ((τ, s) ∈ (−∞, 0] × ℝ^{n}).

Then z̃_{(ξ,η)} = z_{(ξ,η̄)}. It follows from Assumptions H[X] and H_{0} that

∥z_{(ξ,η)} − z_{(ξ,η̄)}∥_{X} = ∥(z − z̃)_{(ξ,η)}∥_{X} ≤ (Kd_{0} + ML) ∥η − η̄∥.

The existence and uniqueness of the solutions of problem (4) follows from classical theorems. On this purpose, note that the right-hand side of the differential system satisfies the Carathéodory assumptions, and the Lipschitz condition

∥ϱ(τ, η, z_{(τ,η)}) − ϱ(τ, η̄, z_{(τ,η̄)}∥ ≤ d̄β(τ, κ_{0}) ∥η − η̄∥

holds on [0, c] × ℝ^{n}. The function g[z](·, t, x) satisfies the integral equation

g[z](τ, t, x) = x + ∫_{t}^{τ} ϱ(ξ, g[z](ξ, t, x), z_{(ξ,g[z](ξ,t,x))}) dξ.

For (τ, t, x), (τ, t̄, x̄) ∈ Δ_{c} we have

\begin{matrix}∥z_{(τ,g[z](τ,t,x))}∥_{X} & ≤ κ̄ \\ ∥z_{(τ,g[z](τ,t,x))}∥_{X_{L}} & ≤ κ_{0} \end{matrix} \}  (7)

and

∥z_{(τ,g[z](τ,t,x))} − z_{(τ,g[z](τ,t̄,x̄))}∥_{X} ≤ (Kd_{1} + ML)∥g[z](τ, t, x) − g[z](τ, t̄, x̄)∥. (8)

It follows from Assumption H[X] and from (7) - (8) that the integral inequality

\begin{matrix}∥g[z](τ, t, x) − g[z](τ, t̄, x̄)∥ \\ ≤ ∥x − x̄∥ + |∫_{t}^{t̄} α(ξ, κ̄) dξ| + d̄|∫_{t}^{τ} β(ξ, κ_{0}) ∥g[z](ξ, t, x) − g[z](ξ, t̄, x̄)∥dξ| \end{matrix}

is sastisfied. Now we obtain (5) by the Gronwall inequality.

For z ∈ Y_{c.φ}[ω, d] and z̄ ∈ Y_{c.φ̄}[ω, d] we have the estimate

\begin{matrix}∥z_{(ξ,g[z](ξ,t,x))} − z̄_{(ξ,g[z̄](ξ,t,x))}∥_{X} \\ ≤ (Kd_{1} + ML̃) ∥g[z](ξ, t, x) − g[z̄](ξ, t, x)∥ \\ + K∥z − z̄∥_{[0,ξ;ℝ^{n}]} + M ∥φ − φ̄∥_{(X,∞)}. \end{matrix}  (9)

It follows from Assumption H[X] and from (9) that the integral inequality

∥g[z](τ, t, x) − g[z̄](τ, t, x)∥

≤ |∫_{t}^{τ} β(ξ, κ_{0})[∥z − z̄∥_{[0,ξ;ℝ^{n}]} + M∥φ − φ̄∥_{(X,∞)}]dξ|

+ d̄|∫_{t}^{τ} β(ξ, κ_{0})∥g[z](ξ, t, x) − g[z̄](ξ, t, x)∥dξ|

is satisfied. Now we obtain (6) by the Gronwall inequality. This completes the proof of the lemma

3. Existence and uniqueness of solutions

Now we construct an integral operator corresponding to problem (1) - (2). Suppose that the function φ satisfies Assumption H_{0}, c ∈ (0, a], z ∈ Y_{c.φ}[ω, d] and g[z](·, t, x) is the bicharacteristic corresponding to z. Let us define the operator U_{φ} for all z ∈ Y_{c.φ}[ω, d] be the formulas

U_{φ}z(t, x) = φ(0, g[z](0, t, x)) + ∫_{0}^{t} f(τ, g[z](τ, t, x), z_{(τ,g[z](τ,t,x))}) dτ  (10)

where (t, x) ∈ [0, c] × ℝ^{n} and

U_{φ}z(t, x) = φ(t, x)  on (−∞, 0] × ℝ^{n}.  (11)

Remark 3.1. The operator U_{φ} is obtained by integration of equation (1) along bicharacteristics.

Now we give sufficient conditions for the solvability of the equation z = U_{φ}z on Y_{c.φ}[ω, d].

Assumption H[f]. Suppose the following:

1) The function f(·, x, w) : [0, a] → B is measurable for (x, w) ∈ ℝ^{n} × X and f(t, ·) : ℝ^{n} × X → B is continuous for almost all t ∈ [0, a].

2) For (x, w) ∈ ℝ^{n} × X[κ] and for almost all t ∈ [0, a] we have

∥f(t, x, w)∥ ≤ α(t, κ).  (12)

3) For (x, w), (x̄, w̄) ∈ ℝ^{n} × X_{L}[κ] and for almost all t ∈ [0, a] we have

∥f(t, x, w) − f(t, x̄, w̄)∥ ≤ β(t, κ) [∥x − x̄∥ + ∥w − w̄∥_{X}].

Remark 3.2. We prove a theorem on the existence and uniqueness of solutions of problem (1) - (2). For simplicity of notations, we have assumed the same estimation for ϱ and for f. We have assumed also the Lipschitz condition for these functions with the same coefficient.

Lemma 3.3. Suppose that Assumptions H[X], H_{0}, H[ϱ] and H[f] are satisfied. Then there are (d_{0}, d_{1}) = d ∈ ℝ_{+}^{2}, c ∈ (0, a] and ω ∈ L([0, c], ℝ_{+}) such that U_{φ} : Y_{c.φ}[ω, d] → Y_{c.φ}[ω, d].

Proof. Suppose that the constants (d_{0}, d_{1}) = d and c ∈ (0, a] and the function ω ∈ L([0, c], ℝ_{+}) satisfy the conditions

\begin{matrix}d_{0} & ≥ L̃ + ∫_{0}^{c} α(τ, κ̄) dτ \\ d_{1} & ≥ Γ_{c} \\ ω(t) & ≥ (1 + Γ_{c}) α(t, κ̄) \end{matrix} \}

where

Γ_{c} = [L + d̄∫_{0}^{c} β(τ, κ_{0}) dτ] exp[d̄∫_{0}^{c} β(τ, κ_{0}) dτ] .  (13)

Suppose that z ∈ Y_{c.φ}[ω, d]. Then we have

∥U_{φ}z(t, x)∥ ≤ L̃ + ∫_{0}^{c} α(τ, κ̄) dτ ≤ d_{0}  on [0, c] × ℝ^{n}.  (14)

If (t, x), (t̄, x̄) ∈ [0, c] × ℝ^{n}, then using Lemma 2.1 and (10) we obtain

\begin{matrix}∥U_{φ}z(t, x) − U_{φ}z(t̄, x̄)∥ \\ ≤ ∥φ(0, g[z](0, t, x)) − φ(0, g[z](0, t̄, x̄))∥ + |∫_{t}^{t̄} α(τ, κ̄) dτ| \end{matrix}

\begin{aligned}+ & ∫_{0}^{t} β(τ, κ_{0})[∥g[z](τ, t, x) − g[z](τ, t̄, x̄)∥ + ∥z_{(τ,g[z](τ,t,x))} − z_{(τ,g[z](τ,t̄,x̄))}∥_{X}]dτ \\ & ≤ Γ_{c} [∥x − x̄∥ + |∫_{t}^{t̄} α(τ, κ̄) dτ| ] + |∫_{t}^{t̄} α(τ, κ̄) dτ| . \end{aligned}

Thus we see that

∥U_{φ}z(t, x) − U_{φ}z(t̄, x̄)∥ ≤ d_{1}∥x − x̄∥ + |∫_{t}^{t̄} ω(τ) dτ| .  (15)

It follows from (14) and (15) that U_{φ}z ∈ Y_{c.φ}[ω, d] which completes the proof of the lemma

Next we will show that there exists exactly one solution of problem (1) - (2). The solution is local with respect to t.

Theorem 3.4. Suppose that Assumptions H[X], H_{0}, H[ϱ] and H[f] are satisfied. Then there are (d_{0}, d_{1}) = d ∈ ℝ_{+}^{2}, c ∈ (0, a] and ω ∈ L([0, c], ℝ_{+}) such that problem (1) − (2) has exactly one solution u ∈ Y_{c.φ}[ω, d].

If φ̄ : (−∞, 0] × ℝ^{n} → B satisfies Assumption H_{0} and ū ∈ Y_{c.φ}[ω, d] is a solution of equation (1) with the initial condition z = φ̄ on (−∞, 0] × ℝ^{n}, then there is Λ_{c} ∈ ℝ_{+} such that

∥u − ū∥_{[0,t;ℝ^{n}]} ≤ Λ_{c}[∥φ − φ̄∥_{(X,∞)} + sup_{y∈ℝ^{n}} ∥φ(0, y) − φ̄(0, y)∥]  (16)

where t ∈ [0, c].

Proof. Lemma 3.3 shows that there are (d_{0}, d_{1}) = d, c ∈ (0, a] and ω ∈ L([0, c], ℝ_{+}) such that U_{φ} : Y_{c.φ}[ω, d] → Y_{c.φ}[ω, d]. Write

λ_{c} = K(1 + Γ_{c}) ∫_{0}^{c} β(τ, κ_{0}) dτ

where Γ_{c} is given by (13). Let c ∈ (0, a] be such a constant that λ_{c} < 1. Now we prove that U_{φ} is a contraction on Y_{c.φ}[ω, d]. If z, z̃ ∈ Y_{c.φ}[ω, d], then

∥U_{φ}z(t, x) − U_{φ}z̃(t, x)∥ ≤ L ∥g[z](0, t, x) − g[z̃](0, t, x)∥
+ ∫_{0}^{t} β(τ, κ_{0})[∥g[z](τ, t, x) − g[z̃](τ, t, x)∥
+ ∥z_{(τ,g[z](τ,t,x))} − z̃_{(τ,g[z](τ,t,x))}∥_{X}]dτ.

The estimate

\begin{matrix}∥z_{(τ,g[z](τ,t,x))} − z̃_{(τ,g[z](τ,t,x))}∥_{X} \\ ≤ (Kd_{1} + ML̃) ∥g[z](τ, t, x) − g[z̃](τ, t, x)∥ + K∥z − z̃∥_{[0,τ;ℝ^{n}]} \end{matrix}

and Lemma 2.1 imply

∥U_{φ}z(t, x) − U_{φ}z̃(t, x)∥ ≤ K(1 + Γ_{c}) ∫_{0}^{t} β(τ, κ_{0})∥z − z̃∥_{[0,τ;ℝ^{n}]}dτ

for all (t, x) ∈ [0, c] × ℝ^{n}, and consequently

∥U_{φ}z − U_{φ}z̃∥_{[0,c;ℝ^{n}]} ≤ λ_{c}∥z − z̃∥_{[0,c;ℝ^{n}]}.

By the Banach fixed point theorem there exists a unique solution u ∈ Y_{c.φ}[ω, d] of the equation z = U_{φ}z.

Now we prove that u is a solution of (1). We have proved that

u(t, x) = φ(0, g[u](0, t, x)) + ∫_{0}^{t} f(τ, g[u](τ, t, x), u_{(τ,g[u](τ,t,x))})dτ  (17)

on [0, c] × ℝ^{n}. For given x ∈ ℝ^{n} let us put η = g[u](0, t, x). It follows that g[u](τ, t, x) = g[u](τ, 0, η) for τ ∈ [0, c] and that x = g[u](t, 0, η). The relations η = g[u](0, t, x) and x = g[u](t, 0, η) are equivalent for x, η ∈ ℝ^{n}. It follows from (17) that

u(t, g[u](t, 0, η)) = φ(0, η) + ∫_{0}^{t} f(τ, g[u](τ, 0, η), u_{(τ,g[u](τ,0,η))})dτ  (18)

where (t, η) ∈ [0, c] × ℝ^{n}. By differentiating (18) with respect to t and by using the transformation η = g[u](0, t, x) which preserves sets of measure zero, we obtain that u satisfies equation (1) for almost all (t, x) ∈ [0, c] × ℝ^{n}. It follows from (11) that u satisfies also condition (2).

Now we prove relation (16). If u = U_{φ}u and ū = Uφ̄ū, then ∥u(t, x) − ū(t, x)∥

≤ sup_{y∈ℝ^{n}} ∥φ(0, y) − φ̄(0, y)∥ + L ∥g[u](0, t, x) − g[ū](0, t, x)∥

+ ∫_{0}^{t} β(τ, κ_{0})[d̄∥g[u](τ, t, x) − g[ū](τ, t, x)∥

+ K ∥u − ū∥_{[0,τ;ℝ^{n}]} + M∥φ − φ̄∥_{(X,∞)}]dτ

where (t, x) ∈ [0, c] × ℝ^{n}. Put

A_{c} = (1 + Γ_{c})M ∫_{0}^{t} β(τ, κ_{0}) dτ  and  γ(t) = K(1 + Γ_{c}) β(t, κ_{0}).

Then we get the integral inequality

\begin{matrix}∥u − ū∥_{[0,t;ℝ^{n}]} \\ ≤ sup_{y∈ℝ^{n}} ∥φ(0, y) − φ̄(0, y)∥ + A_{c}∥φ − φ̄∥_{(X,∞)} + ∫_{0}^{t} γ(τ)∥u − ū∥_{[0,τ;ℝ^{n}]}dτ \end{matrix}

for all t ∈ [0, c]. It follows from the Gronwall inequality that we have estimate (16) for Λ_{c} = exp[ ∫_{0}^{c} γ(τ) dτ]. This completes the proof of the theorem

4. Phase spaces

We give examples of spaces X satisfying Assumption H[X].

Example 4.1. Let X be the class of all function w : (−∞, 0]×[−r, +r] → B which are uniformly continuous and bounded on (−∞, 0] × [−r, +r]. For w ∈ X we write

∥w∥_{X} = sup {∥w(τ, s)∥ : (τ, s) ∈ (−∞, 0] × [−r, +r]}.

Let X_{L} ⊂ X denote the set of all w ∈ X such that

|w|_{L} = sup \{\frac{∥w(τ, s) − w(τ, s̄)∥}{∥s − s̄∥} : (τ, s), (τ, s̄) ∈ (−∞, 0] × [−r, +r]\} < +∞. (19)

Write ∥w∥_{X_{L}} = ∥w∥_{X} + |w|_{L} where w ∈ X_{L}. Then Assumption H[X] is satisfied.

Example 4.2. Let X be the class of all functions w : (−∞, 0] × [−r, +r] → B such that

(i) w is continuous and bounded on (−∞, 0] × [−r, +r]

(ii) the limit lim_{t→−∞} w(t, x) exists uniformly with respect to x ∈ [−r, +r]. Let

∥w∥_{X} = sup {∥w(τ, s)∥ : (τ, s) ∈ (−∞, 0] × [−r, +r]}.

Let X_{L} ⊂ X denote the class of all w ∈ X such that the Lipschitz condition (19) is satisfied. Write ∥w∥_{X_{L}} = ∥w∥_{X} + |w|_{L} where w ∈ X_{L}. Then Assumption H[X] is satisfied.

Example 4.3. Let γ : (−∞, 0] → (0, +∞) be a continuous function. Assume also that γ is non-increasing on (−∞, 0]. Let X be the space of continuous functions w : (−∞, 0] × [−r, +r] → B for which

lim_{τ→∞} \frac{∥w(τ, x)∥}{γ(τ)} = 0 (x ∈ [−r, +r]).

Put

∥w∥_{X} = sup \{\frac{∥w(τ, s)∥}{γ(τ)} : (τ, s) ∈ (−∞, 0] × [−r, +r]\} .

Denote by X_{L} ⊂ X the set of all w ∈ X such that

|w|_{γ.L} = sup \{\frac{∥w(τ, s) − w(τ, s̄)∥}{γ(τ)∥s − s̄∥} : (τ, s), (τ, s̄) ∈ (−∞, 0] × [−r, +r]\} < +∞.

For w ∈ X_{L} put ∥w∥_{X_{L}} = ∥w∥_{X} + |w|_{γ.L}. Then Assumption H[X] is satisfied.

Example 4.4. Let δ ∈ ℝ_{+} and p ≥ 1 be fixed. Denote by X the class of all functions w : (−∞, 0] × [−r, +r] → B such that

(i) w is continuous on [−δ, 0] × [−r, +r]

(ii) for x ∈ [−r, +r] we have ∫_{−∞}^{−δ} ∥w(τ, x∥^{p} dτ < +∞

(iii) w(t, ·) : [−r, +r] → B is continuous for t ∈ (−∞, −δ].

Write

\begin{aligned}∥w∥_{X} & = sup \{∥z(τ, s)∥ : (τ, s) ∈ [−δ, 0] × [−r, +r]\} \\ & + sup \{( ∫_{−∞}^{−δ} ∥w(τ, x∥^{p}dτ)^{1/p} : x ∈ [−r, +r]\}. \end{aligned}

Let X_{L} ⊂ X be the set of functions w ∈ X such that the Lipschitz condition (19) is satisfied. Write ∥w∥_{X_{L}} = ∥w∥_{X} + |w|_{L} where w ∈ X_{L}. Then Assumption H[X] is satisfied.

Remark 4.5. Differential equations with a deviated argument and differentialintegral equations can be obtained from equation (1) by specializing operators ϱ and f.

Remark 4.6. It is important in our considerations that we have assumed the Lipschitz condition for given functions on some special function spaces. More precisely, we have assumed that the functions ϱ(t, ·) and f(t, ·) satisfy the Lipschitz condition on the space ℝ^{n} × X_{L} for almost all t ∈ [0, a], and the condition is local with respect to the functional variable.

Let us consider simplest assumption on ϱ and f. Suppose that there is P ∈ ℝ_{+} such that for almost all t ∈ [0, a] we have

∥ϱ(t, x, w) − ϱ(t, x̄, w̄)∥ ≤ P [∥x − x̄∥ + ∥w − w̄∥_{X}]  (20)
∥f(t, x, w) − f(t, x̄, w̄)∥ ≤ P [∥x − x̄∥ + ∥w − w̄∥_{X}]  (21)

where (x, w), (x̄, w̄) ∈ ℝ^{n} × X. Of course, our results are true if we assume (20), (21) instead of (3), (12).

Now we show that formulations (3), (12) are important. We show that there is a class of quasilinear equations satisfying (3), (12) but not satisfying (20), (21). Let X and X_{L} be the spaces given in Example 4.1. Consider the equation with a deviated argument

D_{t}z(t, x) + ∑_{i=1}^{n} ϱ̃_{i}(t, x, z(ψ_{0}(t), ψ(t, x)))D_{x_{i}}z(t, x) = f̃(t, x, z(ψ_{0}(t), ψ(t, x))) (22)

where

\begin{matrix}ϱ̃ = (ϱ̃_{1}, …, ϱ̃_{n}) & : [0, a] × ℝ^{n} × B → ℝ^{n} \\ f & : [0, a] × ℝ^{n} × B → B \\ ψ_{0} & : [0, a] → (−∞, a] \\ ψ & : [0, a] × ℝ^{n} → ℝ^{n}. \end{matrix} \}

We assume that ψ(t) ≤ t and −r ≤ ψ(t, x) − x ≤ +r for (t, x) ∈ [0, a] × ℝ^{n}. We get (22) by putting in (1)

\begin{aligned}ϱ(t, x, w) & = ϱ̃(t, x, w(ψ_{0}(t) − t, ψ(t, x) − x)) \\ f(t, x, w) & = f̃(t, x, w(ψ_{0}(t) − t, ψ(t, x) − x)). \end{aligned}

From now we consider the function ϱ only. Suppose that there are C̄, C̃ ∈ ℝ_{+} such that

\begin{aligned}∥ϱ̃(t, x, ζ) − ϱ̃(t, x̄, ζ̄)∥ & ≤ C̄[∥x − x̄∥ + ∥ζ − ζ̄∥] \\ ∥ψ(t, x) − ψ(t, x̄)∥ & ≤ C̃∥x − x̄∥. \end{aligned}

It is evident that for (x, w), (x̄, w̄) ∈ ℝ^{n} × X_{L}[κ] and for almost all t ∈ [0, a] we have

∥ϱ(t, x, w) − ϱ(t, x̄, w̄)∥ ≤ C̄[1 + κ(1 + C̃)] ∥x − x̄∥ + C̄ ∥w − w̄∥_{X}.

Then condition (3) is satisfied.

We see at once the the function ϱ(t, ·) does not satisfy the global Lipschitz condition (20) for (x, w), (x̄, w̄) ∈ ℝ^{n} × X. Similar consideration apply to f.

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Received 24.08.1998