ISSN 10645624, Doklady Mathematics, 2011, Vol. 84, No. 2, pp. 607–612. © Pleiades Publishing, Ltd., 2011. Original Russian Text © B.V. Loginov, I.V. Konopleva, Yu.B. Rousak, 2011, published in Doklady Akademii Nauk, 2011, Vol. 440, No. 1, pp. 15–20.
MATHEMATICS
Implicit Operator Theorems under Group Symmetry Conditions B. V. Loginova, I. V. Konoplevaa, and Yu. B. Rousakb Presented by Academician E.I. Moiseev January 14, 2011 Received April 5, 2011
DOI: 10.1134/S1064562411060056
Since the 1960s, the modern symmetry branching theory of solutions to nonlinear equations has evolved into a separate discipline within nonlinear functional analysis. Symmetry methods in branching theory were first applied by V.I. Yudovich (1967) and then by B.V. Loginov and V.A. Trenogin (1971) and D. Ruelle (1973). In the 1980–1990s, monographs were pub lished covering applications of the Lyapunov– Schmidt method (D. Sattinger (1979); A. Vanderbau whede (1982); Loginov (1985); and M. Golubitsky, D. Schaeffer, and I. Stewart (1984–1986)) and center manifold methods (A. Mielke (1971), J. Iooss and M. Adelmeyer (1982), and J. Iooss and P. Chossat (1994)) in the case of group symmetry. Vanderbau whede (1980) and N. Dancer (1980) proved a Ginvariant infinitedimensional implicit operator theorem in the general case of a noninvariant kernel assuming that the allowed group is compact. General results concerning finitedimensional reductions were obtained for the variational formulation of nonlinear problems. Specifically, Trenogin and Sidorov (1992) proved a general theorem on the potentiality of a branching equation (BE) with the application of the Morse–Conley theory, and Yu.I. Sapronov (1991, 1996) obtained finitedimensional reductions in smooth extremal problems. The case of a linearized operator with a noninvari ant kernel arises in the problem of asymmetric local ized wave structures in a stratified fluid [1, 2], where, in the variational case with a functional invariant under noncompact symmetry groups, it was proved that the Lyapunov–Schmidt branching equation can be reduced by the group action to a system of lower dimension. Theorems on the inheritance of group
symmetry by BEs (branching equations in root sub spaces (BERs)) in general nonvariational nonlinear stationary and nonstationary problems with a linear ized operator having a noninvariant kernel were proved in [3, 4] ([5–7], respectively). Based on these theorems, reduction results for variational BEs mov ing along the orbit of a branch point were obtained in [4–8]. The goal of this paper is to derive a Ginvariant implicit operator theorem in stationary and nonsta tionary problems on the basis of the general symmetry inheritance theorem for BEs and BERs without assuming that the allowed continuous group G is com pact. In [5], for the stationary bifurcation F(x, ε) = 0, F(x0, ε) ≡ 0 and for the dynamic bifurcation F(p, x, ε) = 0, p = dx , F(0, x0, ε) ≡ 0 in Banach spaces E1 and dt E2, the corresponding Lyapunov and Schmidt BERs moving along the orbit of the branch point x0 were proved to inherit the group symmetry of the nonlinear operators F. If nonlinear equations in Banach spaces possess continuous group symmetry, the Lie group Gl = Gl(a), where a = (a1, a2, …, al) are its essential parameters, is assumed to be an ldimensional differentiable mani fold satisfying the following conditions [1, 2]: (c1) The map a 哫 Lg(a)x0 acting from a neighbor hood of the unit element of Gl(a) to E1 belongs to the class C1, so that Xx0 ∈ E1 for all the infinitesimal oper –1
ators Xx = lim t [Lg(a(t))x – x] in the tangent manifold t→0
l
T g ( a ) to Lg(a). a Ulyanovsk State Technical University, ul. Severny Venets 32,
Ulyanovsk, 432027 Russia email:
[email protected],
[email protected] b University of Canberra, Canberra, Australia email:
[email protected]
(c2) The stationary subgroup of x0 ∈ E1 determines the representation L(Gs) of the local Lie group Gs ⊂ Gl, s
s < l, with an sdimensional subalgebra T g ( a ) of infini tesimal operators. For the stationary (nonstationary) bifurcation, this means that, the elements Xkx0, Xk ∈ 607
608
LOGINOV et al. pk + 1 – l
l
T g ( a ) , form a κ = (l – s) (2κ = 2(l – s))dimensional subspace of the zerosubspace of the linearized operator and the bases in the zerosubspace and in the algebra l T g ( a ) can be ordered so that X k x 0 = ξ k ϕ k ( ξ k ϕ k + ξ k ϕ k ),
(j) 〈 ϕi ,
(l) γk 〉
= δ ik δ jl ,
(l) γk
(j) 〈 zi ,
(l) ψk 〉
∑
=
(p + 2 – l – s)
B *s ψ k k
,
s=1
1 ≤ k ≤ κ,
pk + 1 – j
(j)
zi =
X j x 0 = 0 for j ≥ κ + 1.
∑
= δ ik δ jl ,
( pi + 2 – j – s )
Bs ϕi
,
(s)
(s)
ϕi
= ϕ i ( x 0 ),
(4)
s=1
Φ = Φ ( x0 ) 1. STATIONARY PROBLEMS IN BRANCHING THEORY In real Banach spaces E1 and E2, we consider the general stationary branching problem F ( x, ε ) = 0, F ( x 0, 0 ) = 0 (1) assuming that the nonlinear equation (1) can be lin earized in a neighborhood of the branch point (x0; 0) so that B x0 ( x – x 0 ) = B x0 ( ε ) ( x – x 0 ) + C x0 ( ε ) + ρ ( x 0, x – x 0, ε ) ≡ B x0 ( ε ) ( x – x 0 ) + R ( x 0, x – x 0, ε ), (2) ρ ( x 0, 0, ε ) ≡ 0,
(1)
n
P x0 =
pi
∑ ∑ 〈 ·, γ
(j)
n
Q x0 =
with ϕi = ϕi(x0) is the zerosubspace (kernel) of B x0 , n
N*( B x0 ) = span{ψi } 1 with ψi = ψi(x0) is the subspace of n
n
defect functionals, {γi } 1 , γi = γi(x0) ∈ E 1* , and {zi } 1 with zi = zi(x0) are the corresponding biorthogonal systems 〈ϕi, γj〉 = δij and 〈zi, ψj〉 = δij, B x0 (ε) is a linear operator that is sufficiently smooth in ε, and the nonlinear operator ρ is continuously differentiable with respect to x0 and x – x0 and is continuous in ε. (s)
Definition 1. Elements ϕ k , where s = 1, 2, …, pk and k = 1, 2, …, n, form a complete canonical gener alized Jordan set (GJS ≡ B(ε) – JS) of the operator function B – B(ε): E1 → E2 if =
∑
B ( ε ) = B 1 ε + B 2 ε + …, s = 2, 3 , … , p k ,
pk
D p = det
∑ 〈B ϕ j
( pk + 1 – j ) , k
(1)
(5)
(j)
(j)
= 〈 ·, Ψ〉 Z: E 2 → E 2, K = span { z i ( x 0 ) }, are defined and generate the following direct sum decompositions of E1 and E2 corresponding to x0: . ∞–K K E 1 = E 1 ( x 0 ) + E 1 ( x 0 ), (6) . E 2 = E 2, K ( x 0 ) + E 2, ∞ – K ( x 0 ). The operator B0 = B x0 intertwines the projectors P x0 and Q x0 : B x0 Pu = QB x0 on D Bx , 0
B *x0 Ψ = ᑛ 0 γ,
E1 (3)
(1)
ψ l 〉 ≠ 0,
j=1 (1) ϕk ,
∑∑
(j)
〈 ·, ψ i ( x 0 )〉 z i ( x 0 )
i = 1j = 1
∞–K
γ l〉 = 0,
pi
B x0 Φ = ᑛ 0 Z,
(7)
ᑛ 0 = diag ( A 1, A 2, …, A n ),
where Ai are (pi × pi) matrices with ones on the secondary subdiagonal and with zeros outside it, and B0: D B0 ∩
2
j=1 (s) 〈 ϕk ,
K
= 〈 ·, γ〉 Φ: E 1 → E 1
= K ( B x0, B x0 ( ε ) ) = span { ϕ i ( x 0 ) },
n
(s – j) Bj ϕk ,
(j) (j) i ( x 0 )〉 ϕ i ( x 0 )
i = 1j = 1
0
s–1
(p )
(2)
B x0 (ε) has a complete tricanonical GJS, then the projec tors
D Bx ⊂ D Bx ( ε ) , that is dense in E1, N( B x0 ) = span{ϕi } 1
(s) Bϕ k
(1)
Lemma 1. If the Fredholm operator function B x0 –
where B x0 is a Fredholm operator with a domain 0
(p )
(2)
= ( ϕ 1 , ϕ 1 , …, ϕ 1 1 , …, ϕ n , ϕ n , …, ϕ n n ). The vectors γ = γ(x0), Ψ = Ψ(x0), and Z = Z(x0) are defined in a similar manner. For the linear operator function B – εB1, a tricanonical GJS can always be chosen. K = p1 + ... + pn is the root number.
ϕk = ψ l = ψ l , k, l = 1, 2, …, n. This set is bicanonical if the GJS of the adjoint opera n tor function B* – B*(ε) and elements {ψl } 1 is canoni cal as well, and it is tricanonical if, additionally,
(x0) → E2, ∞ – K(x0) is an isomorphism.
Corollary 1. The εlinear operator function B0 – εB1 has a tricanonical GJS, and properties (7) are supple mented with the following ones: B 1 P = QB 1 on D B1 , B 1 Φ = ᑛ 1 Z, (8) B 1* Ψ = ᑛ 1 γ, where ᑛ1 = diag(A1, A2, …, An) is a block diagonal matrix and Ai are (pi × pi) matrices with ones on the sec ondary diagonal and with zeros outside it. Thus, the DOKLADY MATHEMATICS
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IMPLICIT OPERATOR THEOREMS
t s1 ( x 0, v ( x 0, ξ ), ε )
operators B0 and B1 act in the invariant pairs of sub K E 1 , E2, K
spaces
and B0: D B0 ∩ isomorphisms.
∞–K E 1 , E2, ∞ – K
and
∞–K E1
corresponding to x0,
→ E2, ∞ – K and
K B1: E 1
→ E2, K are
n
≡–
∑ξ
j1 〈 ( I
(1)
–1
– 〈 ( I – B x0 ( ε )Γ 0 ) R ( x 0, w ( x 0, v ( x 0, ξ ), ε )
Proof. According to (6), setting x = u + v, where u(x0) ∈
(1)
–1
– B x0 ( ε )Γ 0 ) B x0 ( ε )ϕ j ( x 0 ), ψ s ( x 0 )〉
j=1
Theorem 1. If there exists a complete tricanonical GJS, the problem of finding small solutions of Eq. (2) in a neighborhood of x0 is equivalent to finding small solu tions of Lyapunov BER (9) and Schmidt BER (10). v = v(x0, ξ) =
609
∑ξ
∞–K E 1 (x0),
(k) ik ϕ i (x0)
K
= ξ · Φ ∈ E 1 (x0) and u =
(1)
+ v ( x 0, ξ ), ε ), ψ s ( x 0 )〉 = 0, t sσ ( x 0, v ( x 0, ξ ), ε ) n
≡ ξ sσ –
∑ξ
j1 〈 ( I
(1)
–1
– B x0(ε)Γ 0 ) B x0(ε)ϕ j (x 0),
(10)
j=1
( ps + 2 – σ )
ψs
we rewrite Eq. (2) in projections:
(x 0)〉
–1
– 〈 ( I – B x0 ( ε )Γ 0 ) R ( x 0, w ( x 0, v ( x 0, ξ ), ε )
( I – Q x0 )B x0 u = ( I – Q x0 )B x0 ( ε ) ( u + v )
( ps + 2 – σ )
+ v ( x 0, ξ ), ε ), ψ s
+ ( I – Q x0 )R ( x 0, v ( x 0, ξ ) + u ( x 0 ), ε ),
( x 0 )〉 = 0,
σ = 2 , 3 , …, p s .
Q x0 B x0 v = Q x0 B x0 ( ε ) ( u + v )
Corollary 2. Let B x0 (ε) = εB1. Then the Schmidt BER becomes
+ Q x0 R ( x 0, v ( x 0, ξ ) + u ( x 0 ), ε ). By the implicit operator theorem and Lemma 1, u = u(x0) = u(x0, v(x0, ξ), ε) is uniquely determined by the first equation. Substituting it into the second equa tion gives the Lyapunov BER
p
εs t s1 ( x 0, v ( x 0, ξ ), ε ) ≡ – ξ p s1 1–ε s –1
– 〈 ( I – εB 1 Γ 0 ) R ( x 0, w ( x 0, v ( x 0, ξ ), ε ) + v ( x 0, ξ ), ε ), (1)
f ( x 0, v ( x 0, ξ ), ε ) ≡ ᑛ 0 ξ
ψ s ( x 0 )〉 = 0,
– 〈B x0 ( ε ) ( v ( x 0, ξ ) + u ( x 0, v ( x 0, ξ ), ε ) ) + R(x 0 ,v ( x 0, ξ ) + u ( x 0, v ( x 0, ξ ), ε )), Ψ ( x 0 )〉 = 0. (9)
ε t sσ ( x 0, v ( x 0, ξ ), ε ) ≡ ξ sσ – ξ p s1 1–ε s
By introducing the Schmidt regularizer Γ x0 = Γ0 =
– 〈 ( I – εB 1 Γ 0 ) R ( x 0, w ( x 0, v ( x 0, ξ ), ε ) + v ( x 0, ξ ), ε ),
n
(
(
–1 B x0 ,
B x0 = B x0 +
∑ 〈· ,
(1) (1) γ j (x0)〉 z j (x0),
σ–1
–1
( ps + 2 – σ )
Eq. (2) is
i=1
ψs
( x 0 )〉 = 0,
σ = 2, 3, …, p s .
(
reduced to the system B x0 (x – x0) = B x0 (ε)(x – x0) + n
R(x0, x – x0, ε) +
∑ξ
(1) i1 z i (x0),
(σ)
ξsσ = 〈x – x0, γ s 〉,
i=1
σ = 1, 2, …, ps, s = 1, 2, …, n with a solution of the form (
x – x0 = w + ξ · Φ(x0) = w + v(x0, ξ). Then B x0 w + n
pj
∑∑ξ
(k) jk B x 0 ϕ j (x0)
∑∑ξ
(k) jk B x 0 (ε) ϕ j (x0)
j = 1k = 2 pj n
= B x0 (ε)w
+
j = 1k = 2
the formulas for the transformation of elements of generalized Jordan chains by the operator Γ0 give the Schmidt BER Vol. 84
No. 2
K g F ( x, ε ) = F ( L g x, ε ).
(11)
The branch point (x0, 0) moves along the trajectory Lgx0 of x0, and the following relations [5, 6] hold for linearization (2) of Eq. (1): K g B x0 = B Lg x0 L g and K g B x0 ( ε ) = B Lg x0 ( ε )L g , K g R ( x 0, x – x 0, ε ) = F ( L g x, ε ) – F ( L g x 0, ε )
+ R(x0, w + v(x0, ξ), ε) and
DOKLADY MATHEMATICS
In what follows, we assume that the operator F allows a group G; i.e., G has a representation Lg in E1 and a representation Kg in E2 that intertwine F:
2011
– ( B Lg x0 – B Lg x0 ( ε ) )L g ( x – x 0 )
(12)
= R ( L g x 0, L g ( x – x 0 ), ε ), ϕ i ( L g x 0 ) = L g ϕ i ( x 0 ),
–1
γ j ( L g x 0 ) = L *g γ i ( x 0 ),
i, j = 1, 2, …, n,
610
LOGINOV et al.
which show that the Fredholm operator B x0 is sym metric only with respect to the stationary subgroup of x0. The ranges of the operators B x0 and B x0 (ε) satisfy the relations ( B x0 ) = (Kg B x0 L g ) = Kg( B x0 ). Then, for the zerosubspace of the adjoint operator B *x0 , we have –1
2. DYNAMIC BRANCHING PROBLEMS In real Banach spaces E1 and E2, we consider Poincaré–Andronov–Hopf bifurcation for a differen tial equation unsolved for the derivative: F ( p, x, ε ) = 0,
(13) –1 –1 ⇒ N* ( B Lg x0 ) = span { K *g ψ 1 ( x 0 ), …, K *g ψ n ( x 0 ) }, j = 1, 2, …, n,
(s)
ϕ k ( L g x 0 ) = L g ϕ k ( x 0 ); (s) ψk ( Lg x0 )
F 'x ( 0, x 0, 0 ) = – B x0 = – B 0 ,
–1 ( s ) K *g ψ k ( x 0 );
(19)
F 'p ( 0, x 0, ε ) = A 0 + A x0 ( ε ) = A ( ε ),
and it can be proved that the elements of GJS chains arranged in order of increasing lengths are trans formed by the formulas (s)
F ( 0, x 0, ε ) ≡ 0,
F 'p ( 0, x 0, 0 ) = A x0 = A 0 ,
N* ( B x0 ) = span { ψ 1 ( x 0 ), …, ψ n ( x 0 ) }
z j ( L g x 0 ) = K g z j ( x 0 ),
p = dx , dt
F x' ( 0, x 0, ε ) = – B 0 + B x0 ( ε ),
D B0 = E 1 ,
D A0 ⊂ D ( A ( ε ) ). Suppose that the A0spectrum σ A0 (B0) of the Fred
(14)
holm operator B0 decomposes into two parts: σ A0 (B0)
At the same time, the generalized Jordan sets at points of the orbit satisfy biorthogonality conditions (4). Relations (12)–(14) imply the following result.
contained strictly in the left halfplane and σ A0 (B0) consisting of the eigenvalues ±iα of multiplicity n cor responding to the eigenvectors uj = u1j ± iu2j and the eigenvectors vj = v1j ± iv2j of the adjoint operator with generalized Jordan chains of lengths pj. This means the
=
(s)
(s)
z k ( L g x 0 ) = K g z k ( x 0 ).
0
Lemma 2. If there exists a tricanonical GJS, then projectors (5) satisfy the intertwining properties P Lg x0 =
–1 L g P x0 L g
or L g P x0 = P Lg x0 L g ,
–1
(15)
and the generate direct sum decompositions (6) of E1 and E2. The bases in N( B x0 ) and N*( B x0 ) and in the
that
(1) uj
= uj, (k)
(k – 1)
⎯ A *0 v j
(1) vj
= vj, (B0 – (k – 1)
= –A0 u j
(k)
(k)
and v j , v j (k) iαA0) u j
=
such
(k – 1) A0 u j , (k)
; ( B *0 + iα A *0 ) v j (k)
, and ( B *0 – iα A *0 ) v j
(k – 1)
= A *0 v j
=
. The
0
K
∞–K
〈A0 u s , v σ σ
corresponding root subspaces and E2, K(x0) can be chosen so that . ∞–K K E 1 = E 1 ( L g x 0 ) + E 1 ( L g x 0 ), E 1 ( L g x 0 ) = L g E 1 ( x 0 ), ∞–K
(k)
general case of the spectrum σ A0 (B0) consisting of a finitely many nonzero points ±iαr, αr = krα of multi plicity nr, r = 1, 2, …, ν, where kr are integers without nontrivial common divisors, encounters technical dif ficulties. The A0Jordan set can always be chosen to be tricanonical. By the GJS biorthogonality lemma,
K E 1 (x0)
( L g x 0 ) = L g E 1 ( x 0 ), . E 2 = E 2, K ( L g x 0 ) + E 2, ∞ – K ( L g x 0 ), E1
(k)
existence of elements u j , u j
(B0 + iαA0) u j
Q Lg x0 = K g Q x0 K g or K g Q x0 = Q Kg x0 K g
K
–
(k)
Assuming that F is sufficiently smooth, Eq. (19) becomes dx dx A 0 = B 0 ( x – x 0 ) – A x0 ( ε ) – B x0 ( ε ) ( x – x 0 ) dt dt
E 2, ∞ – K ( L g x 0 ) = K g E 2, ∞ – K ( x 0 ). Theorem 2. If there exists a tricanonical GJS, then Lyapunov BER (9) and Schmidt BER (10) inherit the group symmetry of Eq. (2): f ( L g x 0, L g v ( x 0, ξ ), ε ) = f ( L g x 0, v ( L g x 0, ξ ), ε ) (17)
t ( L g x 0, L g v ( x 0, ξ ), ε ) = t ( L g x 0, v ( L g x 0, ξ ), ε ) = L g t ( x 0, v ( x 0, ξ ), ε ).
〉 = δsσδkl.
(16)
E 2, K ( L g x 0 ) = K g E 2, K ( x 0 ),
= K g f ( x 0, v ( x 0, ξ ), ε ),
(p + 1 – l)
(18)
dx – R ⎛ x 0, , x – x 0, ε⎞ . ⎝ dt ⎠
(20)
τ By introducing the Poincaré substitutions t = α+μ 2π and x(t) = y(τ), the problem of constructing α+μ periodic solutions (where μ = μ(ε) is a small addition DOKLADY MATHEMATICS
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2011
IMPLICIT OPERATOR THEOREMS
to the oscillation frequency) is reduced to finding 2πperiodic solutions of the equations dy + B ( ε )y ᑜ x0 y = μᏯy + ( α + μ )A x0 ( ε ) x0 dτ dy, y, ε⎞ = μᏯy + ᑬ ⎛ x , dy , y, μ, ε⎞ , + R ⎛ x 0, ( α + μ ) ⎝ 0 dτ ⎠ ⎝ ⎠ dτ (21) dy ᑜ x0 y = ( ᑜ 0 y ) ( τ ) ≡ B 0 y ( τ ) – αA 0 , dτ ( Ꮿy ) ( τ ) ≡ A 0 dy . dτ The Fredholm operator (ᑜ0y)(τ) and operators (21) map the space Y of 2πperiodic continuously differen · tiable functions τ with values in Ᏹ1 = E1 + iE1 to the space Z of 2πperiodic continuous functions τ with · values in Ᏹ2 = E2 + iE2 with functionals of the special 2π
1 form 〈〈y, f 〉〉 = 〈 y(τ), f(τ)〉dτ, y ∈ Y, f ∈ Y*, (y ∈ Z, 2π
∫ 0
f ∈ Z*). The operators ᑜ0 and ᑜ *0 have 2ndimen uj(x0)eiτ;
(1) ϕj }
and N( ᑜ *0 ) =
(1) span{ ψ j
=
(k) v j (x0)eiτ
and = that correspond to formulas (3) and (4) and satisfy the biorthogonality (k) (l) (k) (l) conditions 〈〈 ϕ j , γ s 〉〉 = δjsδkl and 〈〈 z j , ψ s 〉〉 = δjsδkl, ( ps + 1 – l )
where k(l) = 1, 2, …, pj(ps); γ s = A *0 ψ s ( pj + 1 – k )
A0 ϕ j
tors P x0 =
(k)
, zj =
, and j(s) = 1, 2, …, n. We define the projec n
pj
∑ ∑ 〈〈· , γ
(k) (1) j 〉〉ϕ j
= 〈〈·, γ〉〉Φ, P x0 ; Q x0 =
j = 1k = 1 n
pj
∑ ∑ 〈〈· , ψ
(1) (1) j 〉〉 z j
= 〈〈·, Ψ〉〉Z, Q x0 ; ⺠ x0 = P x0 +
j = 1k = 1
P x0 , ⺡ x0 = Q x0 + Q x0 , which generate decomposi tions of the Banach spaces Y and Z into the direct sums · · Y = Y 2n(x0) + Y ∞ – 2n(x0) and Z = Z2n(x0) + Z∞ – 2n(x0). The operators ᑜ0 and A0 are intertwined by the projec tors P x0 and Q x0 , P x0 and Q x0 : ᑜ0 P x0 u = Q x0 ᑜ0u on D B0 , ᑜ0Φ = ᑛ0Z, ᑜ 0* Ψ = ᑛ0γ, ᑛ0 = diag{B1, B2, …, Bn), Bi is the (pi × pi) matrix (7), Ꮿ P x0 u = Q x0 Ꮿu on D A0 , A0Φ = ᑛ1Z, A *0 Ψ = ᑛ1γ, and ᑛ1 is the pi × pi matrix (8). The operators A0 and ᑜ0 act in the invari ∞–K
K
ant pairs of subspaces Y 1 (x0), Z2, K(x0) and Y 1 DOKLADY MATHEMATICS
Vol. 84
→ Z2, K are isomorphisms.
Theorem 3. Under the assumptions made, the prob lem of finding 2πperiodic solutions of Eq. (21) in a neighborhood of the branch point x0 is equivalent to find ing small solutions of Lyapunov BER (22) and Schmidt BER (23) in the basis {ϕ, ϕ }: ⎧ ᒃ ( x 0, v ( x 0, ξ, ξ ), μ, ε ) = Q x0 ⎨ μᏯ [ … ] ⎩ d + ᑬ ⎛ x 0, [ … ], [ … ], μ, ε⎞ = [ ᑛ 0 – iμᑛ 1 ]ξ ⎝ dτ ⎠ –
d ᑬ ⎛ x 0, [ … ], [ … ], μ, ε⎞ , Ψ ( x 0 ) ⎝ dτ ⎠
No. 2
(x0),
2011
(22)
= 0,
ᒃ ( x 0, v ( x 0, ξ, ξ ), μ, ε ) = 0, [ … ] = u ( v ( x 0, ξ, ξ ), μ, ε ) + v ( x 0, ξ, ξ ); p
– 〈 〈 ( I – μΓ x0 Ꮿ ) ᑬ [ … ], ψ s ( x 0 )〉 〉 = 0,
(k)
(l)
→ E2, ∞ – K and A0:
= ψj(x0, τ) =
(1)
(k) ψj
K Y1
= ϕj(x0, τ) =
vj(x0)eiτ; ψ j } with A0 and A *0 Jordan chains ϕ j (k) u j (x0)eiτ
∞–K
Z2, ∞ – K(x0), while ᑜ0: D ᑜ0 ∩ Y 1
( iμ ) s ᒑ s1 ( x 0, v ( x 0, ξ, ξ), μ, ε ) = – p ξ s1 1 – ( iμ ) s
(1)
sional zerosubspaces N(ᑜ0) = span{ ϕ j
611
(1)
–1
σ–1
( iμ ) ᒑ sσ ( x 0, v ( x 0, ξ, ξ ), μ, ε) = ξ sσ – p ξ s1 1 – ( iμ ) s –1
( ps + 2 – σ )
– 〈 〈 ( I – μΓ x0 Ꮿ ) ᑬ [ … ], ψ s s = 1, 2, …, n,
(23)
( x 0 )〉 〉 = 0,
σ = 1 , 2, …, p s .
Note that, under the group symmetry conditions, since the GJS is tricanonical, the theory developed in Section 1 can be extended to the situation in question, and the following symmetry inheritance theorem for mulated in the basis {ϕ1, ϕ 1 , …, ϕn, ϕ n } holds. Theorem 4. Lyapunov BER (22) and Schmidt BER (23) inherit the group symmetry of Eq. (19): f ( L g x 0, L g v ( x 0, ξ, ξ ), μ, ε ) = f(L g x 0, v(L g x 0, ξ, ξ), μ, ε) = K g f(x 0, v(x 0, ξ, ξ), μ, ε), t ( L g x 0, L g v ( x 0, ξ, ξ ), μ, ε ) = t(L g x 0, v(L g x 0, ξ, ξ), μ, ε) = L g t(x 0, v(x 0, ξ, ξ), μ, ε). 3. IMPLICIT OPERATOR THEOREMS Theorem 5. Suppose that the stationary branching problem (1) is associated with the complete tricanonical GJS of the operator function B x0 = B x0 (ε), and let con dition (c2) be such that κ = n and Gs (s < l) is a normal
612
LOGINOV et al. s
divisor of Gl with a corresponding ideal T g ( a ) of generat ing operators. Then, under the assumptions made about the action smoothness of the continuous group Gl, there exists a con tinuous function v(x0, ξ, ε) = v(x0, ξ) + u(x0, v(x0, ξ),
Corollary 3. Theorems 5 and 6 hold for semisimple branch points, i.e., in the absence of generalized Jordan chains. Remark. By using Theorem 5, stability results for stationary solutions can be obtained on the basis of [9, 10].
n
ε): T g ( a ) x0 × (–δ, δ) → E1 invariant under the factor n
group Gκ = Gn = Cl/Gs on T g ( a )x0 such that n
F ( x 0 + v ( x 0, ξ, ε ) ) = 0 for v ( x 0, ξ ) ∈ T g ( a ) ( x 0 ), ε < δ.
(24)
Proof. On the basis of intertwining properties (15), (16) of the projectors P x0 and Q x0 , there exists a linear isomorphism ∞–K
B 0 = B x0 : D B0 ∩ E 1
( x 0 ) → E 2, ∞ – K ,
(25)
intertwined by the projectors P x0 and Q x0 ; i.e., B0 P x0 x = Q x0 B0x for x ∈ DB and B Lg x0 P Lg x0 x = Q Lg x0 B Lg x0 x for x ∈ Lg D B0 . The Jacobian of BERs (9) and (10) with respect to ξjs (s = 1, 2, …, pj; j = 1, 2, …, n) is nonzero for ε ≠ 0 and the theorem on group symmetry inherit ance by the BERs returns us to the Ginvariance of Eq. (24). Theorem 6. In the dynamic branching problem (19), the A0Jordan set can always be chosen to be tricanoni cal. Let condition (c2) be such that κ = n and Gs (s < l) is s
a normal divisor of Gl with a corresponding ideal T g ( a ) of generating operators. Then, under the assumptions made about the action smoothness of the continuous group Gl, there exists a con tinuous function v(x0, ξ, ξ , μ, ε) = v(x0, ξ, ξ ) + u(x0, v(x0, ξ, ξ ), μ, ε): T g ( a ) x0 × (–δ, δ) → Ᏹ1 invariant 2n
2n
under the factor group Gκ = Gn = Gl/Gs on T g ( a ) x0 such that F(x0 + v(x0, ξ, ξ , μ, ε)) = 0 for v(x0, ξ, ξ ) ∈ 2n
T g ( a ) x0, |ε| < δ.
ACKNOWLEDGMENTS This work was supported by the Federal Targeted Program “Scientific and Pedagogical Staff for Innova tive Russia” (state contract no. P1122) and (project no. 2.1.2/11180) by the Program “Development of the Scientific Potentials of Higher Educational Institu tions” of the Ministry for Education and Science of the Russian Federation. REFERENCES 1. N. I. Makarenko, Dokl. Akad. Nauk 348, 302–304 (1996). 2. N. I. Makarenko, in Proceedings of International School–Seminar on Application of Symmetry and Cosymmetry in the Theory of Bifurcations and Phase Transitions, Sochi (RostovonDon Univ., 2001), pp. 109–120. 3. B. V. Loginov, Uzbek. Mat. Zh., No. 1, 38–44 (1991). 4. B. V. Loginov, I. V. Konopleva, and Yu. B. Rousak, Izv. Vyssh. Uchebn. Zaved., Mat. 4 (527), 30–40 (2006). 5. I. V. Konopleva, B. V. Loginov, and Yu. B. Rousak, Izv. Vyssh. Uchebn. Zaved. SeveroKavkaz. Reg. Estestv. Nauki. Spets. Vyp., 115–124 (2009). 6. I. V. Konopleva, B. V. Loginov, and Yu. B. Rousak, in Proceedings of International Conference on Analytical Methods of Analysis and Differential Equations AMADE, Minsk, September 14–19, 2009, Vol. 1: Mathematical Analysis (Minsk, 2009), pp. 90–95. 7. B. V. Loginov, O. V. Makeev, I. V. Konopleva, and Yu. B. Rousak, ROMAI J. 3 (1), 151–173 (2007). 8. I. V. Konopleva and B. V. Loginov, Dokl. Math. 80, 541–546 (2009). 9. V. A. Trenogin, B. V. Loginov, and L. R. KimTyan, Zh. Srednevolzh. Mat. O–va 12 (3), 8–17 (2010). 10. B. V. Loginov and Yu. B. Rousak, Nonlinear. Anal. TMA 17 (1), 219–231 (1991).
DOKLADY MATHEMATICS
Vol. 84
No. 2
2011
