ISSN 10634541, Vestnik St. Petersburg University. Mathematics, 2015, Vol. 48, No. 1, pp. 9–17. © Allerton Press, Inc., 2015. Original Russian Text © Yu.A. Iljin, 2015, published in Vestnik SanktPeterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2015, No. 1, pp. 37–47.
MATHEMATICS
On the C1Equivalence of Essentially Nonlinear Systems of Differential Equations Near an Asymptotically Stable Equilibrium Point Yu. A. Iljin St. Petersburg State University (Petrodvorets Branch), Universitetskii pr. 28, Petergof, 198504 Russia email:
[email protected] Received October 23, 2014
Abstract—A system of differential equations is considered for which the origin is an asymptotically stable equilibrium point and the Taylor expansion in a neighborhood of this point has no linear terms. Under the assumption that the logarithmic norm of the Jacobian matrix is negative definite, it is proved that this system is locally C1 equivalent to any of its perturbations of sufficiently high order of vanishing. Key words: smooth equivalence, smooth conjugacy, essentially nonlinear system, logarithmic norm. DOI: 10.3103/S1063454115010057
1. STATEMENT OF THE PROBLEM This paper considers the systems of differential equations (1) x· = F ( x ) and (2) y· = F ( y ) + G ( y ), k in which x, y ∈ ⺢ , F(0) = G(0) = 0, and the function G is of higher order of vanishing than F at zero. Clearly, zero is an equilibrium point for both systems. We say that systems (1) and (2) are locally C1 equiv alent if, in some neighborhood of zero, there exists a change of variables close to identity, which transforms (2) into (1). By closeness to the identity, we mean that the change has the form (3) x = y + f ( y ), where f(y) is C1 small compared with y. The equivalence problem for systems of equations is classical in the qualitative theory of differential equations. Naturally, the best studied case in the literature is the one where F(x) = Ax is a linear mapping. In this case, the main results on holomorphic equivalence are due to Poincaré and on smooth equivalence, to Sternberg, Chen, and Hartmann (see [7]). This paper is largely devoted to the case where F(x) is an essentially nonlinear function, e.g., F is homo geneous of degree m > 1. Very little is known about such systems. This work is based on Tokarev’s papers [5, 6]. We have succeeded in generalizing Tokarev’s results in two directions. First, as the criterion for the stability of the coefficient, Tokarev used the socalled Wazewski condition, which we replace by a signifi cantly more general condition on the Lozinskii logarithmic norms of the Jacobian matrix F 'y (y). Second, using a more detailed analysis of the properties of the solution, we improve certain estimates of the behav ior of solutions and, as a result, sharpen bounds for the constants and for the order of the perturbation G in the statements of theorems (relation (4)). The main result of this paper is the following theorem. Theorem. Suppose that, for ||y|| ≤ Δ < 1, systems (1) and (2) satisfy the following conditions: (a) F, G ∈ C2; (b) γ( F 'y (y)) ≤ –λ0||y||m – 1; (c) || F 'y (y)|| ≤ K||y||m – 1; (d) ||G(y)|| ≤ c0||y||2m – 2 + n; 9
10
ILJIN
(e) || C 'y (y)|| ≤ c1||y||m – 1 + n, where λ0 > 0, m ≥ 2, K > 0, c0 > 0, c1 > 0, n ≥ max{N, 1}, and the constant N is defined by m
2 Km N = + 1 – m. (4) λ0 Then, in some neighborhood of the origin, there exists a C1 change in the variables of the form (3) that trans forms (2) into (1) and such that m–1+n n f(y ) = O( y ), f y'( y ) = O ( y ). (5) In the statement of the theorem, ||⋅|| denotes any suitable vector norm on ⺢k and the associated matrix norm. We use γ(A) to denote the upper logarithmic norm of a matrix A generated by these norms (see Sec tion 2). Briefly, the scheme of the proof is as follows. The sought change must satisfy a certain system of partial differential equations, which can be replaced by an integral equation. The solution of the latter is found by the method of successive approximations. 2. AUXILIARY INFORMATION AND RESULTS We begin with a definition and a brief list of properties of logarithmic norms, which we use in the proof (details can be found in [1–3]). Definition. The upper logarithmic norm of a matrix A generated by a given vector norm ||⋅|| is defined as the limit def E + hA – 1 , γ ( A ) = lim h → +0 h where E is the identity matrix. This limit exists for any matrix A. The logarithmic norm has the following properties: (1) γ(A + B) ≤ γ(A) + γ(B); (2) γ(αA) = αγ(A), α ≥ 0; (3) |γ(A)| ≤ ||A||; (4) |γ(A) – γ(B)| ≤ ||A – B||; (5) |γ(A)| is a continuous function of A. The following lemma contains the main estimate on the growth of the norm of solutions, which is obtained by using logarithmic norms (see [2]). Lemma 1. Suppose that x(t) is a solution of a system of differential equations x· = P ( t, x )x + q ( t, x ), where P(t, x) is a continuous (k × k)matrix function and q(t, x) is a continuous vector function. Let ||⋅|| be any vector norm and let γ be the upper logarithmic norm generated by it. Then, d+ x ( t ) ≤ γ ( P ( t, x ( t ) ) ) x ( t ) + q ( t, x ( t ) ) . dt d The symbol + denotes the right derivative. Note that γ is called “norm” conventionally. For example, dt it may be negative, and this property is very valuable from the point of view of the estimate in Lemma 1. Finally, the proof uses the following inequality (see [2]). Lemma 2. Let A(θ) be a continuous matrix function defined in a finite interval 〈a, b〉, and let γ(A(θ)) ≤ 0. Then, b
b
∫
∫
⎛ ⎞ γ ⎜ A ( θ ) dθ⎟ ≤ γ ( A ( θ ) ) dθ. ⎝ ⎠ a
a
We begin with the derivation of two estimates for the solutions of system (2). Let ϕ(t, y0) denote the solution of (2) that satisfies the initial condition ϕ(0, y0) = y0. Lemma 3. Suppose that, for ||y|| ≤ Δ < 1, system (2) satisfies the following conditions: (a) F, G ∈ C2; (b) γ( F 'y (y)) ≤ –λ0||y||m – 1; VESTNIK ST. PETERSBURG UNIVERSITY. MATHEMATICS
Vol. 48
No. 5
2015
ON THE C1EQUIVALENCE OF ESSENTIALLY NONLINEAR SYSTEMS
11
(c) ||G(y)|| ≤ c0||y||m + a; (d) || C y' (y)|| ≤ c1||y||m – 1 + b, where λ0 > 0, m ≥ 2, c0 > 0, c1 > 0, a ≥ 1, and b ≥ 1. Then there exist constants 0 < Δ0 ≤ Δ and λ > 0 such that, for t ≥ 0 and ||y0|| ≥ Δ0, y0 ϕ ( t, y 0 ) ≤ , 1 ( 1 + λt y 0
(6)
m–1 m–1
)
and ϕ y' ( t, y 0 ) ≤ 1 . 1 ( 1 + λt y 0
(7)
m–1 m–1
)
1 Proof. Since F(y) = ⎛ F 'y ( θy ) dθ⎞ y , we can write system (2) in the form ⎝ 0 ⎠
∫
1
⎛ ⎞ y· = ⎜ F 'y ( θy ) dθ⎟ y + G ( y ). ⎝ ⎠
∫ 0
Lemma 1 implies 1
⎛ ⎞ d + ϕ ( t, y 0 ) ≤ γ ⎜ F 'y ( θϕ ( t, y 0 ) ) dθ⎟ ϕ ( t, y 0 ) + G ( ϕ ( t, y 0 ) ) . dt ⎝ ⎠
∫ 0
It follows from Lemma 2 and condition (b) that 1
1
∫
∫
⎛ ⎞ γ ⎜ F y' ( θϕ ( t, y 0 ) ) dθ⎟ ≤ – λ 0 θϕ ( t, y 0 ) ⎝ ⎠ 0
m–1
λ m dθ = – 0 ϕ ( t, y 0 ) . m
0
Taking into account condition (c), we obtain d + ϕ ( t, y 0 ) λ ≤ – 0 ϕ ( t, y 0 ) m dt
m
+ c 0 ϕ ( t, y 0 )
m+a
λ a m = ⎛ – 0 + c 0 ϕ ( t, y 0 ) ⎞ ϕ ( t, y 0 ) . ⎝ m ⎠
(8)
1
⎧ λ0 ⎞ a ⎫ We set Δ1 = min ⎨ Δ, ⎛ and take y0 with ||y0|| ≤ Δ1. Let us show that ||ϕ(t, y0)|| < Δ1 for all t > 0. For ⎝ 2mc 0⎠ ⎬ ⎩ ⎭ t = 0, we have d + ϕ ( t, y 0 ) dt
λ a ≤ ⎛ – 0 + c 0 y 0 ⎞ y 0 ⎝ m ⎠ t=0
m
λ ≤ – 0 y 0 2m
m
< 0.
Therefore, the inequality ||ϕ(t, y0)|| < 1 holds in some interval (0, δ). Suppose that the assertion being proved is false. Let t1 > 0 be the first moment of time at which ||ϕ(t1, y0)|| = Δ1. According to (8), we have d + ϕ ( t, y 0 ) dt
λ a ≤ ⎛ – 0 + c 0 ϕ ( t 1, y 0 ) ⎞ ϕ ( t 1, y 0 ) ⎝ ⎠ m t = t1
m
λ λ m a m = ⎛ – 0 + c 0 Δ 1⎞ Δ 1 ≤ – 0 Δ 1 < 0. ⎝ m ⎠ 2m
Therefore, for t < t1, the inequality ||ϕ(t, y0)|| > ||ϕ(t1, y0)|| = Δ1 must hold, which contradicts the choice of t1. Thus, ||ϕ(t, y0)|| < Δ1 for all t > 0. Taking into account this estimate and (8), we obtain d + ϕ ( t, y 0 ) ⎛ λ 0 a ≤ – + c 0 ϕ ( t, y 0 ) ⎞ ϕ ( t, y 0 ) ⎝ ⎠ m dt
m
λ a = ⎛ – 0 + c 0 Δ 1⎞ ϕ ( t, y 0 ) ⎝ m ⎠
VESTNIK ST. PETERSBURG UNIVERSITY. MATHEMATICS
Vol. 48
No. 5
m
λ m ≤ – 0 ϕ ( t, y 0 ) . 2m 2015
12
ILJIN
Let us integrate this inequality by using the comparison theorem (see [7]). The solution of the comparison λ y m , where equation u· (t) = – 0 u ( t ) , u(0) = ||y0||, has the form u(t) = 0 2m 1 m–1 m–1 ( 1 + λt y 0 ) def λ 0 ( m – 1 ) . λ = 2m
(9)
Therefore, y0 ϕ ( t, y 0 ) ≤ u ( t ) ≤ . 1 ( 1 + λt y 0
m–1 m–1
)
This proves inequality (6). Let us prove (7). It is known from a general course in differential equations that ϕ 'y (t, y0) is a funda mental matrix Φ(t) of the variational system z· = ( F 'y ( ϕ ( t, y 0 ) ) + G 'y ( ϕ ( t, y 0 ) ) )z which satisfies the initial condition Φ(0) = E. Applying Lemmas 1 and 2, property (3) of the logarithmic norm, and conditions (b) and (d) in Lemma 3, we obtain d+ Φ ( t ) m–1 m–1+b + c 1 ϕ ( t, y 0 ) ) Φ ( t ) . (10) ≤ γ ( F 'y ( ϕ ( t, y 0 ) ) + G 'y ( ϕ ( t, y 0 ) ) ) Φ ( t ) ≤ ( – λ 0 ϕ ( t, y 0 ) dt 1
⎧ λ0 ⎞ b ⎫ Inequality (6) implies that ||ϕ(t, y0)|| ≤ ||y0||. Let Δ0 = min ⎨ Δ 1, ⎛ . Then, for ||y0|| ≤ Δ0, we have ⎝ 2mc 1⎠ ⎬ ⎩ ⎭ λ ≤ – 0 ϕ ( t, y 0 ) 2 Substituting this relation into (10) and taking into account estimate (6), we arrive at λ y0 d+ Φ ( t ) λ m–1 Φ ( t ) ≤ – 0 Φ(t) . ≤ – 0 ϕ ( t, y 0 ) 1 2 2 dt m–1 m–1 ( 1 + λt y 0 ) The comparison equation λ y u· ( t ) = – 0 0 u ( t ), u ( 0 ) = Φ ( 0 ) = 1 1 2 m – 1 m–1 ( 1 + λt y 0 ) – λ 0 ϕ ( t, y 0 )
m–1
+ c 1 ϕ ( t, y 0 )
has the solution u(t) = ( 1 + λt y 0 obtain
m–1
)
m–1+b
λ – 0 2λ ( 9 )
b
≤ – ( λ 0 + c 1 y 0 ) ϕ ( t, y 0 )
= ( 1 + λt y 0
m–1
)
– m m–1
ϕ 'y ( t, y 0 ) ≤ u ( t ) = ( 1 + λt y 0 This proves inequality (7) and, thus, Lemma 3.
m–1
m–1
.
. Applying the comparison theorem, we
m–1
)
m – m–1
.
3. PROOF OF THE MAIN THEOREM As in Lemma 3, let ϕ(t, y0) denote a solution of system (2). First, note that system (2) satisfies the con ditions of Lemma 3 for a = m + n – 2 ≥ 1 and b = n ≥ 1. Take Δ0 from this lemma, so that, for ||y0|| ≤ Δ0, estimates (6) and (7) hold. Let x = y + f(y) be the required change of variables (3). Let us differentiate it subject to systems (1) and (2): x· = ( E + f ' ( y ) )y· ⇔ F ( x ) = ( E + f ' ( y ) ) ( F ( y ) + G ( y ) ) ⇔ f ' ( y ) ( F ( y ) + G ( y ) ) (11) = F ( y + f ( y ) ) – F ( y ) – G ( y ). VESTNIK ST. PETERSBURG UNIVERSITY. MATHEMATICS
Vol. 48
No. 5
2015
ON THE C1EQUIVALENCE OF ESSENTIALLY NONLINEAR SYSTEMS
13
Thus, f(y) must be a solution of the system of partial differential equations (11), and it must satisfy condi tions (5). This solution can be found as follows. Let us substitute the solution ϕ(t, y) for y in (11): f ' ( ϕ ( t, y ) ) ( F ( ϕ ( t, y ) ) + G ( ϕ ( t, y ) ) ) = F ( ϕ ( t, y ) + f ( ϕ ( t, y ) ) ) – F ( ϕ ( t, y ) ) – G ( ϕ ( t, y ) ). Note that we have d f ( ϕ ( t, y ) ) on the lefthand side so that dt d f ( ϕ ( t, y ) ) = F ( ϕ ( t, y ) + f ( ϕ ( t, y ) ) ) – F ( ϕ ( t, y ) ) – G ( ϕ ( t, y ) ). dt Integrating this relation from 0 to +∞ and taking into account the conditions f(0) = 0, ϕ(0, y) = y, and ϕ(+∞, y) = 0, we obtain +∞
f(y) = –
∫ ( F ( ϕ ( t, y ) + f ( ϕ ( t, y ) ) ) – F ( ϕ ( t, y ) ) – G ( ϕ ( t, y ) ) ) dt.
(12)
0
Thus, f(y) must satisfy the integral equation (12). The converse is also true: If f is a smooth solution of the integral equation (12) satisfying conditions (5), and such that differentiation under the integral sign is allowed, then change (3) is as required. Indeed, substituting ϕ(s, y) for y into (12), we obtain +∞
f ( ϕ ( s, y ) ) = –
∫ ( F ( ϕ ( t, ϕ ( s, y ) ) + f ( ϕ ( t, ϕ ( s, y ) ) ) ) – F ( ϕ ( t, ϕ ( s, y ) ) ) – G ( ϕ ( t, ϕ ( s, y ) ) ) ) dt. 0
Let us apply the identity ϕ(t, ϕ(s, y)) = ϕ(t + s, y)) and make the change of variable t + s = τ under the integral sign: +∞
f ( ϕ ( s, y ) ) = –
∫ ( F ( ϕ ( τ, y ) + f ( ϕ ( τ, y ) ) ) – F ( ϕ ( τ, y ) ) – G ( ϕ ( τ, y ) ) ) dτ. s
Differentiating this identity with respect to s and replacing ϕ· (s, y) by the righthand side of Eq. (2), whose solution is ϕ, we obtain f ' ( ϕ ( s, y ) ) ( F ( ϕ ( s, y ) ) + G ( ϕ ( s, y ) ) ) = F ( ϕ ( s, y ) + f ( ϕ ( s, y ) ) ) – F ( ϕ ( s, y ) ) – G ( ϕ ( s, y ) ). Finally, substituting s = 0, we arrive at f ' ( y ) ( F ( y ) + G ( y ) ) = F ( y + f ( y ) ) – F ( y ) – G ( y ). Therefore, f satisfies (11) and the change defined by (3) transforms system (2) into (1). Thus, we have reduced the proof of the theorem to proving the existence of a solution for the integral equation (12) that satisfies conditions (5) and allows differentiation under the integral sign. For this pur pose, we apply the method of successive approximations. First, note that, thanks to condition (a) in the theorem, there exists an L > 0 such that, for any y1, 2 in the Δneighborhood of the origin, F' ( y 1 ) – F' ( y 2 ) ≤ y 1 – y 2 . Next, we introduce the constants m
2 Km μ = , ( m + n – 1 )λ 0
(13)
2mc 0 σ 0 = , ( m + n – 1 )λ 0
(14)
VESTNIK ST. PETERSBURG UNIVERSITY. MATHEMATICS
Vol. 48
No. 5
2015
14
ILJIN
and ⎧ 2mc 1 8mLc 0 ⎫ σ 1 = max ⎨ , ⎬, m ⎩ ( m + n )λ 0 2 Kλ 0 ⎭
(15)
where λ0 is the same as in Lemma 3. Note that, since n ≥ N, where N is defined by (4), it follows that μ < 1. Finally, we narrow the neighborhood of the origin so that, for any y in this neighborhood and any n ≥ N, we have m+n–2
σ0 y ≤ 1, 1–μ
n
σ1 y ≤ 1. 1–μ
1
(16)
1
– μ⎞ m + N – 2, ⎛ 1 – μ⎞ N ⎫ and assume hereafter that ||y|| ≤ Δ ˆ = min ⎧ Δ , ⎛ 1 ˆ. For this purpose, we set Δ ⎨ 0 ⎝ ⎠ ⎝ ⎠ ⎬ σ σ 0 1 ⎩ ⎭ We define successive approximations as def
f 0 ( y ) = 0, +∞ def
fi + 1 ( y ) = –
∫ ( F ( ϕ ( t, y ) + f ( ϕ ( t, y ) ) ) – F ( ϕ ( t, y ) ) – G ( ϕ ( t, y ) ) ) dt. i
0
ˆ neighborhood of the origin, the following conditions hold: Let us check by induction that, in the Δ (I) fi + 1 is defined and continuously differentiable, and its derivative can be found by differentiation under the integral sign; i
(II) f i + 1 ( y ) – f i ( y ) ≤ σ 0 μ y i
m+n–1
;
n
(III) f i '+ 1 ( y ) – f i ' ( y ) ≤ σ 1 μ y . +∞
Let i = 0. Then, f1(y) =
∫
G ( ϕ ( t, y ) ) dt . Condition (d) and estimate (6) imply
0 +∞
∫
f1 ( y ) – f0 ( y ) =
+∞ (d)
G ( ϕ ( t, y ) ) dt ≤
0
∫
+∞
c 0 ϕ ( t, y )
2m + n – 2
(6)
dt ≤
0
( m – 1 )c 0 = y ( m + n – 1 )λ
∫ 0
2mc 0 = y ( m + n – 1 )λ 0
m + n – 1 (9)
2m + n – 2
c0 y dt 2m + n – 2 ( 1 + λt y
m + n – 1 (14)
0
= σ0 μ y
m–1
)
m+n–1
m–1
.
It follows that f1 is defined and (II) holds: ⎛ f 1' ( y ) – f 0' ( y ) = ⎜ ⎝ +∞ (6)
≤
∫ 0
+∞
∫ 0
⎞' G ( ϕ ( t, y ) ) dt⎟ = ⎠y m+n–1
+∞
∫
+∞
G 'y ( ϕ ( t, y ) )ϕ 'y ( t, y ) dt
0
c1 y ( m – 1 )c 1 dt = 1 y m m + n – ( m + n )λ + m–1 m–1 m–1 ( 1 + λt y )
(e), (7)
≤
∫ 0
m+n–1
c 1 ϕ ( t, y ) dt m ( 1 + λt y
m–1 m–1
)
2mc 1 n (15) 0 n = y ≤ σ1 μ y . ( m + n )λ 0
n (9)
The integrals converge uniformly in y, which proves the possibility of differentiating under the integral sign and the smoothness of f1. Let us prove the induction step i
i + 1.
VESTNIK ST. PETERSBURG UNIVERSITY. MATHEMATICS
Vol. 48
No. 5
2015
ON THE C1EQUIVALENCE OF ESSENTIALLY NONLINEAR SYSTEMS
15
We have +∞
∫ ( F ( ϕ ( t, y ) + f ( ϕ ( t, y ) ) ) – F ( ϕ ( t, y ) + f
fi + 1 ( y ) – fi ( y ) =
i
i – 1 ( ϕ ( t,
y ) ) ) ) dt =
0
(for brevity, we set ϕ = ϕ(t, y) and fi = fi(ϕ(t, y)) and apply the finite increment formula) ∞1
∫ ∫ F' ( ϕ + θf + ( 1 – θ )f
≤
y
i
i – 1 ) dθ ( f i
– f i – 1 ) dt ≤
00
(we apply (c) and the induction hypothesis (II)) ∞1
∫ ∫ K ϕ + θf + ( 1 – θ )f
≤
i
m–1 i–1
dθ f i – f i – 1 dt
00
∞
∫
≤ Kσ 0 μ
i–1
ϕ ( t, y )
m + n – 1⎛
1
∫
m–1
⎜ ϕ + θf i + ( 1 – θ )f i – 1 ⎝
0
0
⎞ dθ⎟ dt. ⎠
Let us estimate the inner integral separately. Taking into account (16) and the inequality μ < 1, we obtain (II)
f i ≤ f i – f i – 1 + f i – 1 – f i – 2 + … + f 1 – f 0 ≤ σ 0 ϕ ( t, y ) = σ 0 ϕ ( t, y )
m+n–1
(μ
m+n–1
i–1
+μ
i–2
+ … + 1)
m+n–2
i (16) σ0 y – μ σ 0 ϕ ( t, y ) ≤ ϕ ( t, y ) ≤ ϕ ( t, y ) . ≤ 1–μ 1–μ 1–μ
m + n – 11
Similarly, ||fi – 1|| ≤ ||ϕ||, and, therefore, 1
1
∫ ϕ + θf + ( 1 – θ )f i
m–1 i–1
∫
dθ ≤ ( ϕ + θ ϕ + ( 1 – θ ) ϕ )
0
m–1
dθ ≤ ( 2 ϕ )
m–1
.
0
Let us apply this to the initial estimate: ∞
∫
f i + 1 ( y ) – f i ( y ) ≤ Kσ 0 μ
i–1 m–1
2
ϕ ( t, y )
2m + n – 2
dt
0
(6)
≤ Kσ 0 μ
i–1 m–1
2
∞⎛
⎞ 0 ⎜ y ⎟ 1 ⎟ ⎜ m – 1 m – 1⎠ 0⎝ ( 1 + λt y 0 )
2m + n – 2
∫
i–1 m–1
2 m (9) Kσ 0 μ dt = y ( m + n – 1 )λ 0
m + b – 1 (13)
i
= σ0 μ y
m+b–1
.
Thus, we have proved (II). Let us prove (III). We have ∞
f i '+ 1 ( y ) – f i ' ( y ) =
∫ ( F ( ϕ ( t, y ) + f ( ϕ ( t, y ) ) ) – F ( ϕ ( t, y ) + f i
i – 1 ( ϕ ( t,
y ) ) ) ) 'y dt
0
∞
=
∫ ( F' ( ϕ + f ) ( E + f ' )ϕ' – F' ( ϕ + f y
i
i
y
y
i – 1 )( E
+ f i '– 1 )ϕ 'y + F 'y ( ϕ + f i ) ( E + f i '– 1 )ϕ 'y – F 'y ( ϕ + f i ) ( E + f i '– 1 )ϕ 'y ) dt
0
∞
∫
≤ ( F 'y ( ϕ + f i ) f i ' – f i '– 1 + F 'y ( ϕ + f i ) – F 'y ( ϕ + f i – 1 ) E + f i '– 1 ) ϕ 'y dt 0
(a)(c)(III)
≤
∞
∫ (K ϕ + f
m–1 i
σ1 μ
i–1
n ϕ + L f i ' – f i '– 1 E + f i '– 1 ) ϕ 'y dt.
0
VESTNIK ST. PETERSBURG UNIVERSITY. MATHEMATICS
Vol. 48
No. 5
2015
16
ILJIN
ˆ , ||f (ϕ(t, y))|| ≤ ||ϕ(t, y)||. We esti In the proof of (II), we have already shown that, for t ≥ 0 and ||y|| ≤ Δ i mate ||E + f i '– 1 || in a similar way as shown below: (III)
n
E + f i '– 1 ≤ E + f i '– 1 – f i '– 2 + … + f 1' – f 0' ≤ σ 1 ϕ ( μ n
k–2
+μ
k–3
+ … + μ + 1) + 1
n
(16) σ 1 ϕ ( t, y ) σ1 y ≤ + 1 ≤ + 1 ≤ 2. 1–μ 1–μ
Thus, ∞
∫
f i '+ 1 ( y ) – f i ' ( y ) ≤ ( K2
m–1
ϕ
m–1
σ1 μ
i–1
n
ϕ + 2L f i – f i – 1 ) ϕ 'y dt
0
(II)(7)
≤
∞
∫
μ
i–1
( σ1 2
m–1
m+n–1
∞
m–1 m–1
0
m+n–1
(6) i–1 m–1 ϕ ( t, y ) y K + 2Lσ 0 ) dt ≤ μ ( σ 1 2 K + 2Lσ 0 ) dt m m m+n–1
( 1 + λt y
0
i–1
)
∫
m–1
( 1 + λt y 0
μ ( σ 1 2 K + 2Lσ 0 ) ( m – 1 ) = y ( m + n )λ (14)
= μ
i–1
+ m–1 m–1 m–1
)
m
μ ( σ 1 2 K + 4Lσ 0 )m = y ( m + n )λ 0
n (9)
n
i – 1⎛
m 2 m m ⎞ n(15) σ 1 2 Km i–1 2 Km 8Lm 2 Lm + 2⎟ y ≤ σ 1 μ ⎛ + ⎞ y ⎜ ⎝ ( m + n )λ ( m + n )λ ( m + n ) ( m + n – 1 )λ 0⎠ ⎝ 0 0 ( m + n ) ( m + n – 1 )λ 0⎠
= σ1 μ
i–1
m
2 Km y ( m + n – 1 )λ 0
n (13)
ii
n
n
= σ1 μ y .
This proves estimate (III), the possibility of differentiating under the integral sign, and the smoothness of fi(y). It follows from estimates (II) and (III) that the sequences of fi(y) and f i ' (y) uniformly converge for ||y|| ≤ ˆ . Clearly, the limit function f(y) is a solution of Eq. (12); according to the Cauchy theorem, it is smooth, Δ f '(y). This also substantiates the possibility of differentiating under the integral sign in the and f i ' (y) integral equation (12). It remains to show that f satisfies (5). For this purpose, consider the series (f1 – f0) + (f2 – f1) + … + (fi + 1 – fi) + …, which converges to f. Applying (II), we obtain f ≤ f1 – f0 + … + fi + 1 – fi + … ≤ σ0 y
m+n–1
σ0 i ( 1 + … + μ + … ) ≤ y 1–μ
m+n–1
.
In a similar way, norm of f ' is estimated. This completes the proof of the theorem. ACKNOWLEDGMENTS This work was supported by the Russian Foundation for Basic Research (project no. 130100624) and by the subject plan of St. Petersburg State University (subject no. 6.0.112.2010). REFERENCES 1. D. Yu. Volkov and Yu. A. Il’in, “On the existence of an invariant torus for an essentially nonlinear system of dif ferential equations,” Vestn. S.Peterb. Univ., Ser. 1: Mat., Mekh., Astron. No. 1, 118–119 (1992). 2. Yu. A. Il’in, “On the Application of Logarithmic Norms in Differential Equations”, in Nonlinear Dynamic Sys tems. Issue 2, Ed. by G. A. Leonov (S.Peterb. Univ, St. Petersburg, 1999), pp. 103–121 [in Russian]. VESTNIK ST. PETERSBURG UNIVERSITY. MATHEMATICS
Vol. 48
No. 5
2015
ON THE C1EQUIVALENCE OF ESSENTIALLY NONLINEAR SYSTEMS
17
3. Yu. A. Il’in, “On the existence of a localintegral manifold of neutral type for an essentially nonlinear system of differential equations,” Vestn. St. Petersburg Univ.: Math. 40, 36–45 (2007). 4. L. A. Lyusternik and I. I. Sobolev, A Short Course of Functional Analysis (Vysshaya Shkola, Moscow, 1982) [in Russian]. 5. S. P. Tokarev, “Smooth conjugation of systems of differential equations in the neigborhood of an asymptotically stable complex singular point,” Differ. Uravn. 13, 766–769 (1977) 6. S. P. Tokarev, “Local C1equivalence of systems of differential equations in the neigborhood of an asymptoti cally stable complex singular point,” Differ. Uravn. 23, 1826–1828 (1987) 7. P. Hartman, Ordinary Differential Equations (Wiley, New York, 1964; Mir, Moscow, 1970).
Translated by O. Sipacheva
VESTNIK ST. PETERSBURG UNIVERSITY. MATHEMATICS
Vol. 48
No. 5
2015