Mem. Differential Equations Math. Phys. 10(1997), 145–149

I. Vitrichenko

CRITICAL CASE OF MULTIPLE PAIRS OF PURE IMAGINARY ROOTS OF A NONAUTONOMOUS ESSENTIALLY NONLINEAR n-th

ORDER EQUATION

(Reported on October 14, 1996)

We investigate the asymptotic stability in the Lyapunov sense as t ↑ ω of the zero solution of a differential equation of the form

y^{(n)} + ∑_{k=1}^{n−1} p_{k}(t) · y^{(n−k)} + p_{n}(t) · y = F(t, y, y′, …, y^{(n−1)}),  (1)

where t ∈ Δ ≡ [a_{0}, ω[, −∞ < a_{0} < ω ≤ +∞, p_{s} : Δ → R, R ≡] − ∞, +∞[, p_{s} ≡ π^{s} · a_{s}, π : Δ → R_{+}, R_{+} =]0, +∞[, a_{s} ≡ a_{s0} + o_{s}(1), a_{s}^{(1)} = o_{sl}(1) as t ↑ ω, a_{s0} ∈ R, l ∈ \{\overline{1, h}\}, h ∈ N, N ≡ \{1, 2, … \}, s = \overline{1, n}, and the following conditions are fulfilled:

(1) the equation P_{n}(λ) ≡ λ^{n} + ∑_{s=1}^{n} a_{s0} · λ^{n−s} = 0 possesses 2 · n_{0}, 1 ≤ n_{0} ≤ [\frac{1}{2} · n] roots λ_{0} satisfying Re λ_{0} = 0; the remaining roots λ possess the property Re λ < 0;

(2) F (t, X) ≡ ∑_{|Q|=2}^{m} F_{Q}(t)·X^{Q}+R_{m}(t, X), X ≡ (x_{1}, …, x_{n}), F : Δ×S(X, r) → R,

S(X, r) ≡ \{X, X^{T} : ∥X∥ ≤ r\}, r ∈ R_{+}, Q = (q_{1}, …, q_{n}), q_{k} ∈ \{0, N\}, k = \overline{1, n}, ∥Q∥ = ∑_{k=1}^{n} q_{k}, X^{Q} ≡ ∏_{k=1}^{n} x_{k}^{q_{k}}, F_{Q} ∈ C_{Δ}^{k}, h ∈ N, ∥Q∥ = \overline{2, m}, m ∈ N \ \{1\},

|R_{m}| ≤ L · ( ∑_{k=1}^{n} |x_{k}|)^{m+a}, L : Δ → [0, +∞[, L ∈ C_{Δ}, a ∈ R_{+}.

Below the use is made of the following definitions and notation:

Definition 1. The differential equation (1) possesses the property St as t ↑ ω if for any arbitrarily small ε ∈ R_{+} there exist δ_{ε} ∈]0, ε] and T_{ε} ∈ Δ such that any solution y = y(t) of (1) satisfying |y(T_{ε})| < δ_{ε} ·π(T_{ε}), |y^{(s−1)}(T_{ε})| < δ_{ε} ·π^{s}(T_{ε}), s = \overline{2, n}, possesses the property |y(t)| < ε_{ε} · π, |y^{(s−1)}(t)| < ε · π^{s} for all t ∈ [T_{ε}, ω[, s = \overline{2, n}.

Definition 2. The differential equation (2) possesses the property AsSt as t ↑ ω if Definition 1 is fulfilled, and π^{−1} · y(t) = o(1) and π^{−s} · y^{(s−1)}(t) = o(1) as t ↑ ω, s = \overline{2, n}.

Definition 1′. The differential system

Y′ = f(t, Y), Y ≡ col(y_{1}, …, y_{n}), f(t, \overline{0}) ≡ \overline{0}, \overline{0} ≡ col(0, …, 0),  (2)

possesses the property St as t ↑ ω if for any arbitrarily small ε ∈ R there exist δ_{ε} ∈]0, ε] and T_{ε} ∈ Δ such that any solution Y = Y (t) of the differential system (2) with the condition ∥Y (T_{ε})∥ < δ_{ε} possesses the property ∥Y (t)∥ < ε for all t ∈ [T_{ε}, ω[.

For ω < +∞, the property St of the differential system (2) is defined by a rephrasing of this property for ω = +∞ [2, p. 168].

Definition 2′. The differential system (2) possesses the property AsSt as t ↑ ω if Definition 1′ is fulfilled, and ∥Y (t)∥ = o(1) as t ↑ ω.

1991 Mathematics Subject Classification. 34B05.

Key words and phrases. Nonautonomous differential equation, asymptotic stability, pure imaginary roots.

E_{k}, H_{k} are respectively the unit and the displacement matrices of the dimension k×k; Y_{k} is a column vector of the dimension k;

Y^{−1} ≡ col(y_{1}^{−1}, …, y_{n}^{−1}), Z = col(z_{1}, …, z_{n}) = col(Z_{n_{1}}, …, Z_{n_{k}}),

Y · Z ≡ col(y_{1} · z_{1}, …, y_{n} · z_{n}), < Y, Z >≡ ∑_{k=1}^{n} y_{k}z_{k},

∥Y ∥^{2} ≡ ∑_{k=1}^{n} |y_{k}|^{2}, grad V (t, Y) ≡ col (\frac{∂V}{∂y_{1}} , …, \frac{∂V}{∂y_{n}} ),

E_{k}^{T} ≡ (0, …, 0, 1_{k}, 0, …, 0); L_{Δ} ≡ {f : Δ → R, ∫^{ω} |f| · dt < +∞};

Λ ≡ max i\{f_{s} : Δ → R; s = \overline{1, n}\}, for Λ : Δ → R_{+},

Λ^{−1} · f_{s} = c_{s} + o_{s}(1) as t ↑ ω, c_{s} ∈ R, ∑_{s=1}^{n} |c_{s}| > 0.

The results of this paper are effectively applied to the differential equation (1) whose coefficients are slowly varying functions, i.e., the functions whose derivatives are small as t ↑ ω in comparison with the functions themselves. For example,

p_{k} ≡ t^{k·β} · [a_{k0} + b_{k} · t^{−a_{k}} · (ln t)^{β_{k}} · sin t^{γ_{k}}],
F_{Q} ≡ t^{γ·∥Q∥} · [F_{Q0} + g_{Q} · t^{−a_{Q}} · (ln t)^{β_{Q}} · sin t^{γ_{Q}}],
k, β, a_{k0}, b_{k0}, β_{k}, γ, F_{Q0}, g_{Q}, β_{Q} ∈ R, a_{k}, a_{Q} ∈ {0, R_{+}},
γ_{k}, γ_{Q} ∈]0, 1], k = \overline{1, n}, ∥Q∥ = \overline{2, m}.

Lemma. If π′ · π^{−2} = o(1) as t ↑ ω, then the transformation y = π · y_{1}, y^{(s)} = π^{s+1} · y_{s+1}, s = \overline{1, n − 1}, reduces the differential equation (1) to that of the kind

Y′ = π · P · Y + G,  (3)

where P = ∥p_{sk}∥, s, k = \overline{1, n}, p_{ss} ≡ −s · π′ · π^{−2}, s = \overline{1, n − 1}, p_{nn} ≡ −a_{1} − n · π′ · π^{−2}, p_{s,s+1} ≡ 1, s = \overline{1, n − 1}, p_{sk} ≡ 0,

s = \overline{1, n − 2}, k = \overline{s + 2, n}, p_{sk} ≡ 0, s = \overline{2, n − 1}, k = \overline{s − 1, n − 2},

p_{nk} ≡ −a_{n−k+1}, k = \overline{1, n − 1}, G ≡ col(\overline{0}, G_{n}).

G_{n} ≡ ∑_{|Q|=2}^{m} f_{Q} · π^{−n+∑_{s=1}^{n} s·q_{s}} · Y^{Q} + R_{m}^{∗},

|R_{m}^{∗}| ≤ ( ∑_{k=1}^{n} π^{k})^{m+a} · π^{−n} · L · ( ∑_{k=1}^{n} |y_{k}|)^{m+a},

det[P (ω) − λ · E_{n}] ≡ P_{n}(λ).

The proof of the lemma is obvious.

Assume first that using the generalized ”shearing” [3], ”frozen” [4] and K.P. Persidsky’s methods of transformations, we can construct a nondegenerate substitution y_{s} = h_{s}(t, Z), h_{s}(t, \overline{0}) ≡ 0, s = \overline{1, n}, which reduces the differential system (3) to that of the special form

\begin{cases}Z′_{n_{s}} = π_{s} · (i · μ_{s} · E_{n_{s}} + Ω_{n_{s}}) · Z_{n_{s}} + Φ_{n_{s}}, \\ i^{2} = −1, s = \overline{1, k_{0}}, ∑_{s=1}^{k_{0}} n_{s} = n_{0} \\ Z′_{n−2·n_{0}} = π · P_{n−2·n_{0}} · Z_{n−2·n_{0}} + Φ_{n−2·n_{0}}, \end{cases}  (4)

where π_{s} : Δ → R_{+}, μ_{s} ∈ R_{+}, s = \overline{1, k_{0}} are known numbers, ∥Ω_{n_{s}}∥ = o(1) as t ↑ ω, s = \overline{1, k_{0}}, the roots μ of the equation det(P_{n−2·n_{0}} −μ·E_{n−2·n_{0}}) = 0 possess the property Re μ : Δ →] − ∞, −γ], y ∈ R_{+}; s = \overline{1, k_{0}}, Φ_{n−2·n_{0}} are small in a sense.

Theorem 1. Let the differential equation (1) be such that

(1) π′ · π^{−2} = o(1) as t ↑ ω, and the transformation y = π · h_{1}(t, Z), y^{(s)} = π^{s+1} × h_{s+1}(t, Z), s = \overline{1, n − 1}, reduces the differential equation (1) to the differential system (4) with π_{s} · ∥ Re Ω_{n_{s}}∥ ∈ L_{Δ}, s = \overline{1, k_{0}};

(2) for all Z ∈ S(Z, r), it holds ∥Φ_{n_{s}}∥, ∥Φ_{n−2·n_{0}}∥ ∈ L_{D}, s = \overline{1, k_{0}}, h_{k}(t, Z) = o(1) as t ↑ ω, k = \overline{1, n}.

Then the differential equation (1) possesses the property AsSt as t ↑ ω.

Proof. Consider the differential system (4) in terms of the quasi-linear differential system and apply the results of [6].

Assume now that by the methods of ”shearing” [3] and “frozen” [4] transformations we can construct a nondegenerate change of variables

y_{k} = f_{s}(t, Z) ≡ ∑_{|Q|=2}^{m} f_{sQ} · Z^{Q}, s = \overline{1, n},

reducing the differential system (3) to that of the special form

\begin{cases}Z′_{n_{s}} = π_{s} · (i · μ_{s} · E_{n_{s}} + H_{n_{s}}) · Z_{n_{s}}+ \\ + ∑_{∥Q_{ns} +L_{ns} ∥=2}^{m} g_{n_{s},Q_{ns} ,L_{ns}} · Z_{n_{s}}^{Q_{ns}} · \overline{Z}_{n_{s}}^{l_{ns}} + Θ_{n_{s}}, \\ i^{2} = −1, ∑_{k=1}^{k_{0}} μ_{k} · π_{k} · (∥Q_{n_{k}} − L_{n_{k}} ∥) + μ_{s} · π_{s} = 0, s = \overline{1, k_{0}}, ∑_{s=1}^{k_{0}} n_{s} = n_{0}, \\ Z′_{n−2·n_{0}} = π · P_{n−2·n_{0}} · Z_{n−2·n_{0}}+ \\ +Z_{n−2·n_{0}} · ∑_{∥Q_{ns} ∥=1}^{m−1} g_{n−2·n_{0}} · Z_{n_{s}}^{Q_{ns}} \overline{Z}_{n_{s}}^{Q_{ns}} + Θ_{n−2·n_{0}}, \end{cases} (5)

where π_{s} : Δ → R_{+}, μ_{s} ∈ R_{+}, g_{n_{s},Q_{ns} ,L_{ns}}, g_{n−2·n_{0},Q_{ns}} are known values, the equation det(P_{n−2·n_{0}} − μ · E_{n−2·n_{0}}) = 0 possesses only the roots μ with the property Re μ : Δ → ] − ∞, γ], Θ_{n_{s}}, Θ_{n−2·n_{0}} are small in a sense.

Select from the differential system (5) that of the form

\begin{cases}Z′_{n_{s}} = π_{s} · (i · μ_{s} · E_{n_{s}} + H_{n_{s}}) · Z_{n_{s}} + ∑_{∥Q_{ns} +L_{ns} ∥=2}^{m} g_{n_{s},Q_{ns} ,L_{ns}} · Z_{n_{s}}^{Q_{ns}} · \overline{Z}_{n_{s}}^{l_{ns}} , \\ ∑_{k=1}^{k_{0}} μ_{k} · π_{k} · (∥Q_{n_{k}} ∥ − ∥L_{n_{k}} ∥) + μ_{s} · π_{s} = 0, s = \overline{1, k_{0}}, ∑_{s=1}^{k_{0}} n_{s} = n_{0}. \end{cases} _{(6)}

Assume that the differential system (6) can be substituted by an equivalent 2 · n_{0}-th order differential equation with respect to one of the components of the vector col(Z_{n_{s}}, …, Z_{n_{k0}} ). Then, using a method presented in [7], we can obtain asymptotic representation of all proper solutions of the above obtained differential equation. Denote by Ψ_{n_{s}} = Ψ_{n_{s}}(t), s = \overline{1, k_{0}}, an asymptotic representation of one of the proper solutions of the differential system (6).

Theorem 2. Let the differential equation (1) be such that

(1) π′ · π^{−2} = o(1) as t ↑ ω, and the transformation y = π · f_{1}(t, Z), y^{(s)} = π^{s+1} · f_{s+1}(t, Z), s = \overline{1, n − 1}, reduces the differential equation (1) to (5) with ∥P′_{n−2·n_{0}}∥ · π^{−1} = o(1) as t ↑ ω;

(2) there exists an asymptotic representation of one of the proper solutions of the differential system (6), Ψ_{n_{s}} = Ψ_{n_{s}}(t), such that ∥Ψ_{n_{s}}∥ = o(1) and ∥Ψ′_{n_{s}} · Ψ_{n_{s}}^{−1}∥ · π_{s}^{−1} = o(1) as t ↑ ω, s = \overline{1, k_{0}};

(3) there exist positive definite Lyapunov functions V = V_{n_{s}}(Z_{n_{s}}) such that for all t ∈ Δ and all col(Z_{n_{s}}, …, Z_{n_{s}}, \overline{O}) ∈ S(Z, r)

Re < grad V_{n_{s}}(Z_{n_{s}}), π_{s} · [(i · μ_{s} − Ψ′_{n_{s}} · Ψ_{n_{s}}^{−1} · π_{s}^{−1}) · E_{n_{s}} + H_{n_{s}}] · Z_{n_{s}} +

+ ∑_{∥Q_{ns} +L_{ns} ∥=2}^{m} g_{n_{s},Q_{ns} ,L_{ns}} · Ψ_{n_{s}}^{Q_{ns} ,−E_{ns}^{T}} · \overline{Ψ}_{n_{s}}^{L_{ns}} · Z_{n_{s}}^{Q_{ns}} · \overline{Z}_{n_{s}}^{L_{ns}} >≡

≡ Λ_{s} · [W_{0s}(Z_{n_{s}}) + W_{1s}(t, Z_{n_{s}})],

Λ_{s} ≡ max \{∥g_{n_{s},Q_{ns} ,L_{ns}} · Ψ_{n_{s}}^{Q_{ns} ,−E_{ns}^{T}} · \overline{Ψ}_{n_{s}}^{L_{ns}} ∥, ∑_{k=1}^{k_{0}} μ_{k} · π_{k} · (∥Q_{n_{k}} ∥ − ∥L_{n_{k}} ∥) +

+μ_{s} · π_{s} = 0, ∥Q_{n_{s}} + L_{n_{s}}∥ = \overline{2, m}\},
W_{0s}(Z_{n_{s}}) < 0, Z_{n_{s}} ≠ \overline{0}, W_{0s}(\overline{0}) = 0,
W_{1s}(t, Z_{n_{s}}) = o(1), Λ_{s}^{−1} · π_{s} = o(1),

Λ_{s}^{−1} · g_{n−2·n_{0},Q_{ns}} · Ψ_{n_{s}}^{Q_{ns} ,−E_{ns}^{T}} · \overline{Ψ}_{n_{s}}^{L_{ns}} = o(1) as t ↑ ω, s = \overline{1, k_{0}};

(4) there exists ν : Δ → R_{+}, ν ∈ C_{Δ}^{−1}, such that ν = o(1), ν′ · ν^{−1} · π^{−1} = o(1), t ↑ ω and for all Z ∈ S(Z, r) \ \overline{O}

[ ∑_{s=1}^{k_{0}} ∥Θ_{n_{s}}(t, Ψ_{n_{1}} · Z_{n_{1}}, …, Ψ_{n_{k0}} · Z_{n_{k0}} , ν · Z_{n−2·n_{0}}) · Ψ_{n_{s}}^{−1}∥ +

+∥Θ_{n−2·n_{0}}(t, Ψ_{n_{1}} · Z_{n_{1}}, …, Ψ_{n_{k0}} · Z_{n_{k0}} , ν · Z_{n−2·n_{0}}) · v^{−1}∥] ×

×[ ∑_{s=1}^{k_{0}} Λ_{s} · W_{0s}(Z_{n_{s}}) − π · ∥Z_{n−2·n_{0}}∥^{2}]^{−1} = o(1),

f_{s}(t, Ψ_{n_{1}} · Z_{n_{1}}, …, Ψ_{n_{k0}} · Z_{n_{k0}} , v · Z_{n−2·n_{0}}) = o(1) as t ↑ ω, s = \overline{1, k_{0}}.

Then the differential equation (1) possesses the property AsSt as t ↑ ω.

Proof. In the differential system (5), we make the substitution Z_{n_{s}} = Ψ_{n_{s}} ·Y_{n_{s}}, s = \overline{1, k_{0}}, Z_{n−2·n_{0}} = v ·Y_{n−2·n_{0}} and apply to the differential system with respect to Y_{n_{s}}, s = \overline{1, k_{0}}, Y_{n−2·n_{0}} the analogue of the lemma [4] on the stability in a ring-shaped domain involving the origin.

Remark. If the coefficients of the differential equation (1) are slowly varying functions, then applying several times the method of “frozen’ transformations, one can attain for fixed Z that the functions Φ_{n_{s}}, s = \overline{1, k_{0}}, Φ_{n−2·n_{0}} and Θ_{n_{s}}, s = \overline{1, k_{0}}, Θ_{n−2·n_{0}} in the differential systems (4) and (5), respectively, would tend rapidly enough to zero as t ↑ ω.

References

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2. V. V. Nemytskiǐ and V. V. Stepanov, Quantitative theory of differential equations. (Russian) Gostekhizdat, Moscow–Leningrad, 1949.

3. I. E. Vitrichenko and V. V. Nikonenko, On the reduction to a nearly blocktriangular (diagonal) form of a linear non-autonomous system in the case of multiple zero proper value of a limiting matrix of coefficients. Proc. A. Razmadze Math. Inst. 110(1994), 59–67.

4. A. V. Kostin and I. E. Vitrichenko, Generalization of the Liapounov’s theorem on the stability in the case of one zero characteristic exponent for non-autonomous systems. (Russian) Dokl. Akad. Nauk SSSR 264(1982), No. 4, 819–822.

5. K. P. Persidskiǐ, On characteristic numbers of a linear system of differential equations. (Russian) Izv. Akad. Nauk Kaz. SSR, Ser. Matem. i Mekh. 1947, 5–47.

6. I. E. Vitrichenko, To the stability of trivial solution of n-th order non-autonomous quasi-linear equation in a critical case of multiple zero of the limiting characteristic equation. Ukrain. Mat. Zh. 47(1995), No. 8, 1138–1143.

7. A. V. Kostin, Asymptotics of the proper solutions of nonlinear ordinary differential equations. (Russian) Differentsial’nye Uravneniya 23(1987), No. 3, 522–526.

Author’s address:

Chair of Higher Mathematics

Odessa I.I. Mechnikov State University

Odessa 270086

Ukraine