Mem. Differential Equations Math. Phys. 21(2000), 138–141

T. Tadumadze

LOCAL REPRESENTATIONS FOR THE VARIATION OF SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS

(Reported on March 13, 2000)

1. Let J = [a, b] be a finite interval; O ⊂ ℝ^{n} be an open set; E be a space of functions f : J × O^{s} → ℝ^{n} satisfying the conditions:

1) for a fixed t ∈ J the function f(t, x_{1}, …, x_{s}) is continuously differentiable with respect to (x_{1} …, x_{s}) ∈ O^{s}; 2) for a fixed (x_{1}, …, x_{s}) ∈ O^{s} the functions f, f_{x_{i}}, i = 1, …, s are measurable with respect to t ∈ J; for an arbitrary compact K ⊂ O there exists a function m_{f,K}(·) ∈ L(J, ℝ_{0}^{+}), ℝ_{0}^{+} = [0, ∞), such that

|f(t, x_{1}, …, x_{s})| + ∑_{i=1}^{s} |f_{x_{i}}(·)| ≤ m_{f,K}(t), ∀(t, x_{1}, …, x_{s}) ∈ J × K^{s}.

Let now τ_{i}(t), i = 1, …, s, t ∈ J, be absolutely continuous functions satisfying the conditions: τ_{i}(t) ≤ t, τ̇_{i}(t) > 0; Δ be a space of piecewise continuous functions φ : J_{1} = [τ, b] → O, τ = min{τ_{1}(a), …, τ_{s}(a)}, with a finite number of discontinuity points of the first kind, satisfying the conditions: cl{φ(t) : t ∈ J_{1}} is a compact lying in O; ∥φ(t)∥ = sup{|φ(t)| : t ∈ J_{1}}.

To every element μ = (t_{0}, x_{0}, φ, f) ∈ A = [a, b) × O × Δ × E there corresponds the delay differential equation

ẋ(t) = f(t, x(τ_{1}(t), …, x(τ_{s}(t))),  (1)

with the initial condition

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

Definition 1. The function x(t) = x(t; μ) ∈ O, t ∈ [τ, t_{1}], t_{1} ∈ (a, b], t_{0} < t_{1} is said to be a solution corresponding to the element μ ∈ A, defined on [τ, t_{1}], if the function x(t) on the interval [τ, t_{0}] satisfies the condition (2), while on the interval [t_{0}, t_{1}] it is absolutely continuous and satisfies the equation (1) almost everywhere.

Introduce the set V = {δμ = (δt_{0}, δx_{0}, δφ, δf) ∈ A − μ : |δt_{0}| ≤ c = const, |δx_{0}|‘ec, ∥ δφ∥ ≤ c, δf = ∑_{i=1}^{k} λ_{i}δf_{i}, |λ_{i}| ≤ c, i = 1, …, k}, where μ = (t_{0}, x_{0}, φ, f) ∈ A; δf_{i} ∈ E − f̃, i = 1, …, k are fixed points. By a standard way it is proved that if x(t) is the solution corresponding to the element μ̃, defined on [τ, t̃_{1}], t̃_{1} < b. Then there exist numbers ε_{0} > 0, δ_{0} > 0 such that for an arbitrary (ε, δμ) ∈ [0, ε] × V to the element μ̃ + εδμ ∈ A there corresponds the solution x(t; εδμ) defined on [τ, t̃_{1} + δ_{0}] ⊂ J_{1}. It is obvious that the solution x(t; 0), t ∈ [τ, t̃_{1} + δ_{0}] is a continuation of the solution x̃(t) in the sequel assumed to be defined on the whole interval [τ, t̃_{1} + δ_{0}].

The above presented discussion allows us to introduce the function

Δx(t; εδμ) = x(t; εδμ) − x̃(t), (t, ε, δμ) ∈ [τ, t̃_{1} + δ_{0}] × [0, ε_{0}] × V.

2000 Mathematics Subject Classification. 34K15.

Key words and phrases. Delay differential equations, variation of solution.

The function Δx(t; εδμ) is called the variation of the solution x̃(t). In order to formulate the main results, we will need the following notation:

\begin{matrix}ω_{i}^{−} = (t̃_{0}, \underbrace{x̃_{0}, …, x̃_{0}}_{i-times}, \underbrace{φ̃(t̃_{0}−) …, φ̃(t̃_{0}−)}_{(p-i)-times}, φ̃(τ_{p+1}(t̃_{0}−)), …, φ̃(τ_{s}(t̃_{0}−))), \\ i = 0, …, p; \\ ω_{i}^{−} = (γ_{i}, x̃(τ_{1}(γ_{i})), …, x̃(τ_{i−1}γ_{i})), x̃_{0}, φ̃(τ_{i+1}(γ_{i}−)), …, φ̃(τ_{s}(γ_{i}−))), \\ ω^{∘}_{i}^{−}= (γ_{i}, x̃(τ_{1}(γ_{i})), …, x̃(τ_{i−1}(γ_{i})), φ̃(t̃_{0}−), φ̃(τ_{i+1}(γ_{i}−)), …, φ̃(τ_{s}(γ_{i}−))), \\ i = p + 1, …, s; γ_{i} = γ_{i}(t̃_{0}), γ̇_{i}^{−} = γ̇_{i}(t̃_{0}−), i = 1, …, s; \end{matrix} \} (3)

γ_{i}(t) is the function inverse to τ_{i}(t).

\begin{matrix}lim_{ω→ω_{i}^{−}}f̃(ω) = f_{i}^{−}, ω = (t, x_{1}, …, x_{s}) ∈ R− ; t̃_{0} × O^{s}, i = 0, …, p, \\ R− ; t̃_{0} = (−∞, t̃_{0}]; lim_{(ω_{1},ω_{2})→(ω_{i}^{−},ω^{∘}_{i}^{−})}[f̃(ω_{1}) − f̃(ω_{2})] = f_{i}^{−}, \\ ω_{1}, ω_{2} ∈ R_{γ_{i}}^{−} × O^{s}, i = p + 1, …, s. \end{matrix} \}  (4)

Theorem 1. Let γ_{i} = t̃_{0}, i = 1, …, p, t̃_{0} < γ_{p+1} < … < γ_{s} < t̃_{1}, there exist the finite limits: f_{i}^{−}, i = 0, …, s; γ̇_{i}^{−}, i = 1, …, s, there exist a left semi-neighborhood V^{−}(t̃_{0}) of the point t̃_{0} such that

t ≤ γ_{1}(t) ≤ … ≤ γ_{(}t), ∀t ∈ V^{−}(t̃_{0}).  (5)

Then there exist numbers ε_{1} ∈ (0, ε_{0}], δ_{1} ∈ (0, dl_{0}] such that for an arbitrary (t, ε, δμ) ∈ [t̃_{1} − δ_{1}, t̃_{1} + δ_{1}] × V^{−}; V^{−} = δμ ∈ V : δt_{0} ≤ 0, the formula

Δx(t; εδμ) = εδx(t; δμ) + o(t; εδμ),  (6)

is valid, where

δx(t; δμ) =

{Y (t̃_{0}; t) ∑_{i=0}^{p}(γ̂_{i+1}^{−} − γ̂_{i}^{−})f_{i}^{−} − ∑_{i=p+1}^{s} Y (γ_{i}; t)f_{i}^{+}γ̇_{i}^{+}}δt_{0} + α(t; δμ),

γ̂_{0}^{−} = 1, γ̂_{i}^{−} = γ̇_{i}^{−}, i = 1, …, p, γ̇_{p+1}^{−} = 0,

α(t; δμ) = Y (t̃_{0}; t)δx_{0} + ∑_{i=p+1}^{s} ∫_{τ_{i}(t̃_{0})}^{tlt_{0}} Y (γ_{i}(ξ); t)f̃_{x_{i}}[γ_{i}(ξ)]γ̇_{i}(ξ)δφ(ξ)dξ +

+ ∫_{t̃_{0}}^{t} Y (ξ; t)δf[ξ]dξ, f̃_{x_{i}}[ξ] = f̃_{x_{i}}(ξ, x̃(τ_{1}(ξ)), …, x̃(τ_{s}(ξ))),

δf[ξ] = δf(ξ, x̃(τ_{1}(ξ)), …, x̃(τ_{s}(ξ))),

lim_{ε→0} \frac{|o(t;εδμ)|}{ε} = 0, uniformly with recpect to (t, δμ) ∈ [t̃_{1} − δ_{1}, t̃_{1} + δ_{1}] × V^{−} and Y (ξ; t) is a matrix function satisfying the equation

\frac{∂Y (ξ; t)}{∂ξ} = − ∑_{i=1}^{s} Y (γ_{i}(ξ); t)f̃_{x_{i}}[γ_{i}(ξ)]γ̇_{i}(ξ), ξ ∈ [t̃_{0}, t],

and the condition

Y (ξ; t) = \begin{cases}I, s = t, \\ Θ, s > t, \end{cases}

I is the identity matrix, Θ is the zero matrix.

Remark 1. If φ̃(t̃_{0}−) = x̃_{0}, then f_{0}^{−} = ⋯ = f_{p}^{−}, f_{i}^{−} = 0, i = p + 1, …, s. If γ̇_{p}^{−} < ⋯ < γ̇_{1}^{−} < 1, then the condition (5) is fulfilled.

Theorem 2. Let γ_{i} = t̃_{0}, i = 1, …, p, t̃_{0} < γ_{p+1} < ⋯ < γ_{s} < t̃_{1}, there exist the finite limits f_{i}^{+}, i = 0, …, s; γ̇_{i}^{+}, i = 1, …, s (see (3), (4)), and there exist a right-hand semi-neighborhood V^{+}(t̃_{0}) of the point t̃_{0} such that

t < γ_{1}(t) ≤ ⋯ ≤ γ_{p}(t), ∀t ∈ V^{+}(t̃_{0}).  (7)

Then there exist numbers ε_{1} ∈ (0, ε_{0}], δ_{1} ∈ (0, ϵ_{0}], such that for an arbitrary (t, ε, δμ) ∈ [t̃_{1} − δ_{1}, t̃_{1} + δ_{1}] × [0, ε_{1}] × V^{+} = {δμ ∈ V : δt_{0} ≥ 0} the formula (6) is valid, where

δx(t; δμ) =

{Y (t̃_{0}; t) ∑_{i=0}^{p}(γ̂_{i+1}^{+} − γ̂_{i}^{+})f_{i}^{+} − ∑_{i=p+1}^{s} Y (γ_{i}; t)f_{i}^{+}γ̇_{i}^{+}}δt_{0} + α(t; δμ),

γ̂_{0}^{+} = 1, γ̂_{i}^{+} = γ̇_{i}^{+}, i = 1, …, p, γ̂_{p+1}^{+} = 0.

Remark 2. If φ̃(t̃_{0}+) = x̃_{0}, then f_{0}^{+} = ⋯ = f_{p}^{+}, f_{i}^{+} = o, i = p + 1, …, s. If 1 < γ̇_{1}^{+} < ⋯ < γ̇_{p}^{+}, then the condition (7) is fulfilled.

Theorem 3. Let the assumptions of Theorems 1, 2 are fulfilled and

∑_{i=0}^{p}(γ̂_{i+1}^{−} − γ̂_{i}^{−})f_{i}^{−} = ∑_{i=0}^{p}(γ̂_{i+1}^{+} − γ̂_{i}^{+})f_{i}^{+} = f_{0};

f_{i}^{−}γ̇_{i}^{−} = f_{i}^{+}γ̇_{i}^{+} = f_{i}, i = p + 1, …, s.

Then there exist numbers ε_{1} ∈ (0, ε_{0}], δ_{1} ∈ (0, δ_{0}], such that for an arbitrary (t, ε, δμ) ∈ [t̃− δ_{1}, t̃+ δ_{1}] × [0, ε_{0}] × V the formula (6) is valid, where

δx(t, δμ) = [Y (t̃_{0}; t)f_{0} − ∑_{i=p+1}^{s} Y (γ_{i}; t)f_{i}]δf_{0} + α(t; δμ).

For the case s = 2, τ_{1}(t) ≡ t analogous theorems are proved in [1].

2. To every element ζ = (t_{0}, φ, f) ∈ A_{1} = [a.b) × Δ × E there eorresponds the delay differential equation (1) with the initial condition x(t) = φ(t), t ∈ [τ, t_{0}].

Introduce the set

V_{1} = {δζ = (δt_{0}, δφ, δf) ∈ A_{1} − ζ̃:

|δt_{0}| ≤ c, δφ = ∑_{i=1}^{k} λ_{i}δf_{i}, |λ_{i}| ≤ c, i = 1, …, k},

where ζ = (t_{0}, φ, f) ∈ A_{1}; δf_{i} ∈ E − f_{,} δφ ∈ Δ − φ̃, i = 1, …, k are fixed points. Analogously we set the function (see Section 1)

Δx(t; εδζ) = x(t; εδζ) − x̃(t), (t, ε, δζ) ∈ [τ, t̃_{1} + δ_{0}] × [0, ε_{0}] × V_{1}.

Theorem 4. Let φ̃(t) be absolutely continuous in a left semi-neighborhood of the point t̃_{0}, there exist the finite limits φ̇^{−} = ˙ φ̃(t̃_{0}−) and

lim_{ω→ω^{∘}^{−}} f̃^{−}(ω) =f^{∘}^{−}, ω ∈ ℝ− ; t̃_{0} × O^{s}, ω^{∘}^{−}= (t̃_{0}, φ̃(τ_{1}(t̃_{0}−)), …, (τ_{s}(t̃_{0}−))).

Then for an arbitrary (t, ε, δ) ∈ [t̃_{0}, t̃_{1} + δ_{0}] × [0, ε_{0}] × V_{1}^{−} = {δζ ∈ V_{1} : δt_{0} ≤ 0} the formula

Δεδζ) = εδx(t; δζ) + o(t; εδζ)  (8)

is valid, where

δx(t; δζ) = Y (t̂_{0}; t)[δφ^{−} + (φ̇^{−}− f^{∘}^{−})δt_{0}] + β(t; δζ), δφ^{−} = δφ(t̃_{0}−),

β(t; δζ) =

= ∑_{i=1}^{s} ∫_{τ_{i}(t̃_{0})}^{t̃_{0}} Y (γ_{i}(ξ); t)f̂_{x_{i}}[γ_{i}(ξ)]γ̇_{i}(ξ)dξ + ∫_{t̃_{0}}^{t} Y (ξ; t)δf[ξ]dξ.

Theorem 5. Let φ̃(t) be absolutely continuous in a right semi-neighborhood of the point t̃_{0}, there exist the finite limits φ̇^{+} = ˙ φ̃(t̃_{0}+) and

lim_{ω→ω^{∘}^{+}} f̃(ω) =f^{∘} ^{−}, ω ∈ ℝ+ ; t̃_{0} × O^{s}, ω^{∘} ^{+} = (t̃_{0}, φ̃(τ_{1}(t̃_{0}+)), …, (τ_{s}(t̃_{0}+))).

Then for each t̄ ∈ (t̃_{0}, t̃_{1}) there exists a number ε_{1} ∈ (0, ε_{0}] such that for an arbitrary (t, ε, δζ) ∈ [t̃_{0}, t̃_{1} + δ_{0}] × [0, ε_{1}] × V_{1}^{+} = {δζ ∈ V_{1} : δt_{0} ≥ 0} the formula (8) is valid, where

δx(t; δζ) = Y (t̂_{0}; t)[δφ^{+} + (φ̇^{+}− f^{∘}^{+})δt_{0}] + β(t; δζ), δφ^{+} = δφ(t̃_{0}+).

Finally we note that the formulas (6), (8) play an important role when invstigating delay optimal problems.

References

1. G. Kharatishvili, T. Tadumadze, and N. Gorgodze, Continuous dependence and differentiability of solution with respect to initial data and right-hand side for differential equations with deviating argument. Mem. Differential Equations Math. Phys. 19(2000), 3–105.

Author’s address:

I. Vekua Institute of Applied Mathematics

I. Javakhishvili Tbilisi State University

2, University St., Tbilisi 380043

Georgia