Mem. Differential Equations Math. Phys. 11(1997), 168–169

N. Gorgodze

CONTINUOUS DEPENDENCE OF THE SOLUTION OF A CLASS OF NEUTRAL DIFFERENTIAL EQUATIONS ON THE INITIAL DATA

AND ON THE RIGHT-HAND SIDE

(Reported on March 17–24, 1997)

Let J = [a, b] be a finite interval, ℝ^{n} be an n-dimensional Euclidean space, O ⊂ ℝ^{n} an open set, η : J → ℝ^{1} and τ : J → ℝ^{1} be continuously differentiable functions satisfying respectively the conditions: η(t) < t, η̇(t) > 0; τ(t) ≤ t, τ̇(t) > 0; moreover, let L_{1}(J, ℝ_{+}^{1}) be the space of summable functions m : J → ℝ_{+}^{1}, ℝ_{+}^{1} = [0, +∞), Δ(J, ℝ^{n×n}) be the space of piecewise continuous n × n matrix functions C : J → ℝ^{n×n} with a finite number of points of discontinuity of the first kind, ∥C∥ = sup_{t∈J} |C(t)|, C^{1}(J_{1}, O) be the space of continuously differentiable functions φ : J_{1} → O, J_{1} = [τ, b], τ = min{η(a), τ(a)}, for which ∥φ∥ = |φ(a)| + max_{t∈J_{1}} |φ̇(t)|, and let E_{f} be the space of the functions f : J × O^{2} → ℝ^{n} satisfying the following conditions:

(1) the function f(·, x, y) : J → ℝ^{n} is measurable for every (x, y) ∈ O^{2}:

(2) for any compactum K ⊂ O and any function f ∈ E_{f} there exist m_{f,K}(·), L_{f,K}(·) ∈ L_{1}(J, ℝ_{+}^{1}) such that

|f(t, x, y)| ≤ m_{f,K}(t), ∀(t, x, y) ∈ J × K^{2},

|f(t, x′, y′) − f(t, x′′, y′)| ≤ L_{f,K}(t)(|x′ − x′′| + |y′ − y′′|), ∀(t, x′, x′′, y′, y′′) ∈ J × K^{4}.

Introduce the sets:

V_{1}(K, δ) = {f ∈ E_{f} : max_{(t′,t′′,x,y)∈J^{2}×K^{2}} | ∫_{t′}^{t′′}f(t, x, y) dt| ≤ δ},

V_{2}(K, α) = {f ∈ E_{f} : ∫_{J} [m_{f,K}(t) + L_{f,K}(t)] dt ≤ α},

W (K, δ, α) = V_{1}(K, δ) ∩ V_{2}(K, α),

where K ⊂ O is a compact set, δ > 0 and α > 0 are arbitrary numbers.

To every element σ = (t_{0}, x_{0}, φ, C, f) ∈ Σ = J × O × C^{1}(J_{1}, O) × Δ(J, ℝ^{n×n}) × E_{f} we assign the neutral differential equation

ẋ(t) = C(t) ˙ (x)(η(t)) + f(t, x(t), x(τ(t)))  (1)

with the initial condition

x(t) = φ(t), t ∈ [τ_{0}, t_{0}), x(t_{0}) = x_{0},  (2)

where τ_{0} = min{η(t_{0}), τ(t_{0})}.

1991 Mathematics Subject Classification. 34K40.

Key words and phrases. Neutral equation, continuous dependence of the solution.

Definition. Every function x(t; σ) ∈ O defined on the interval [τ_{0}, t_{1}] ⊂ (τ, b] will be called a solution corresponding to the element σ ∈ Σ if it satisfies on [τ_{0}, t_{0}] the condition (2), is absolutely continuous on [t_{0}, t_{1}] and almost everywhere satisfies the equation (1).

Theorem. Let _{x}(t) = x(t; _{σ}), t ∈ [_{t_{0}},_{t_{1}}], be a solution corresponding to the element _{σ} = (_{t_{0}}, _{x_{0}}, _{φ}, _{C}, _{f}) ∈ Σ and let the compactum K_{1} ⊂ O contain a neighborhood of the set K_{0} = {_{x}(t) : t ∈ [_{τ_{0}},_{t_{1}}]}, where _{τ_{0}} = min{η(_{t_{0}}), τ(_{t_{0}})}. Then for any ε > 0 there exists a number δ = δ(ε) > 0 such that to every element

σ ∈ V (_{σ}, K_{1}, δ, α_{0}) = V (_{t_{0}}, δ) × V (_{x_{0}}, δ) × V (_{φ}, δ) × V (_{C}, δ) × W (K_{1}, δ, α_{0})

there corresponds the solution x(t; σ) defined on the interval [τ_{0},_{t_{0}} + δ]. Moreover, if

σ_{i} = (t_{0}^{i}, x_{0}^{i}, φ_{i}, C_{i}, f_{i}) ∈ V (_{σ}, K_{1}, δ, α_{0}), i = 1, 2, then

|x(t; σ_{1}) − x(t; σ_{2})| ≤ ε, t ∈ [\overline{t}_{0},_{t_{1}} + δ],

where \overline{t}_{0} = max(t_{0}^{1}, t_{0}^{2}), α_{0} > 0 is a fixed number.

Here V (_{t_{0}}, δ), V (_{x_{0}}, δ), V (_{φ_{0}}, δ), V (_{C}, δ) are δ-neighborhoods of the points _{t_{0}}, _{x_{0}}, _{φ}, _{C} in the spaces ℝ^{1}, ℝ^{n}, C^{1}(J_{1}, O), Δ(J, ℝ^{n×n}), respectively.

The above formulated theorem is an analogue of a theorem stated in [1], [2] (see also [3] and [4]). This theorem can be proved by the method described in [2].

In conclusion it should be noted that if the right-hand side of the equation (1) depends nonlinearly on ẋ(η(t)), then the theorem is, generally speaking, invalid. The appropriate example is given in [3].

References

1. R. V. Gamkrelidze and G. L. Kharatishvili, Extremal problems in linear topological spaces. (Russian) Izv. Akad. Nauk SSSR. Ser. Mat. 33(1969), No. 4, 781–839.

2. R. V. Gamkrelidze, Fundamentals of optimal control. (Russian) Tbilisi University Press, Tbilisi, 1977.

3. T. A. Tadumadze, A. V. Tavadze, and F. A. Dvalishvili, On continuous dependence on regular perturbations of the solution of a family of differential equations with a deviating argument. (Russian) In: Qualitative Methods of Function Theory and Differential Equations. (Russian) Moscow, 1990, 79–86.

4. M. Z. Akubardia, Continuity and differentiability of the solution of the neutral differential equation with respect to the initial data. Reports of enlarged session of the seminar of I. N. Vekua Institute of Applied Mathematics 7(1992), No. 3, 5–8.

Author’s address:

Institute of Applied Mathematics

Tbilisi State University

2, University str.

Tbilisi 380043,

Georgia