Lobachevskii Journal of Mathematics

http://ljm.ksu.ru

Vol. 20, 2005, 77–89 \begin{matrix}c \\ ◯ \end{matrix} B. Karasözen, I. V. Konopleva, and B. V. Loginov

B. Karasözen, I. V. Konopleva, and B. V. Loginov

DIFFERENTIAL-ALGEBRAIC EQUATIONS IN THE THEORY OF INVARIANT MANIFOLDS FOR SINGULAR

EQUATIONS

(submitted by A. M. Elizarov)

Abstract. Analogs of Grobman-Hartman theorem on stable and unstable manifolds solutions for differential equations in Banach spaces with degenerate Fredholm operator at the derivative are proved. Jordan chains tools and the implicit operator theorem are used. In contrast to the usual evolution equation here the central manifold appears even for the case of spectrum absence on the imaginary axis. If on the imaginary axis there is only a finite number of spectrum points, then the original nonlinear equation is reduced to two differential–algebraic systems on the center manifold.

1. Introduction

78 B. KARASÖZEN, I. V. KONOPLEVA, AND B. V. LOGINOV

Branching theory of solutions of nonlinear equations has various applications in scientific computing [5, 7, 8]. This is one of the areas in applied mathematics which is intensively developing in last fifty years. The goals of this theory are the qualitative theory of dynamical systems [7], computation of their solutions [4] without assumptions ensuring the uniqueness of solutions. The classical Lyapounov-Schmidt method, even in the modern form [19], is often insufficient for computation of complicated dynamics, like bifurcation to invariant tori. Therefore in the last two decades the center manifold theory [2, 7, 10, 14, 16] and methods are developed. However, no results of this theory concerning evolution equations with degenerate operator at the derivative are known, though these equations have numerous applications in filtration theory [1], nonlinear waves theory (the Boussinesq-Love equation) [22] and motion theory of non-Newtonian fluids [15].

2000 Mathematical Subject Classification. 34G20, 58E07, 34A9.

Key words and phrases. Pseudoparabolic singular differential equations, Grobman–Hartman theorem, center manifold, differential–algebraic systems.

The present work, as an introduction to center manifold methods for evolution equations with Fredholm operator at the derivative, considers invariant manifolds technique on the base of the resolving systems theory [13] developed by authors. It has found some applications to investigation of the bifurcating solutions stability [11].

The second section of this article contains the necessary tools of generalized Jordan chains [19], the third, forth, and fifth ones; some aspects of invariant manifolds theory, and Grobman–Hartman theorem analogs for such equations. Here the nontrivial center manifold arises even in the case when the operator B has no A-spectrum σ_{A}(B) on the imaginary axis.

For the computation of center manifold, in section 3 successive approximation method is suggested. It is considered also the sufficiently general case of σ_{A}(B) presence on imaginary axis (section 4) that will be the subject of our future investigations. Only for representation of the nonlinear equation in the form of two equations system in the direct sum of Banach spaces complete results are obtained. Here, if the spectrum on imaginary axis σ_{A}^{0}(B) is non-empty and it is separated on the other parts of spectrum, then the original nonlinear equation is reduced to two differential-algebraic systems on the center manifold, for solving of which the authors suppose to develop numerical methods.

2. Generalized Jordan chains and sets for Fredholm operators

Let E_{1} and E_{2} be Banach spaces, A : E_{1} ⊃ D_{A} → E_{2}, B : E_{1} ⊃ D_{B} → E_{2} be densely defined closed linear Fredholm operators, where D_{B} ⊂ D_{A} and A is subordinated to B (i.e. ∥Ax∥ ≤ ∥Bx∥ + ∥x∥ on D_{B}), or D_{A} ⊂ D_{B} and B is subordinated to A (i.e. ∥Bx∥ ≤ ∥Ax∥ + ∥x∥ on D_{A}). The differential equation

A\frac{dx}{dt} = Bx − R(x), R(0) = 0, R_{x}(0) = 0  (1)

with sufficiently smooth operator R is considered.

It is supposed the nontriviality of the the zero-subspaces N (A) = span{ϕ_{1}, …, ϕ_{m}}, N (B) = span{φ_{1}, …, φ_{n}} with non-degeneracy condition N (A)∩N (B) = {0} and the defect-subspaces N^{∗}(A) = span{ψ̂_{1}, …, ψ̂_{m}}, N^{∗}(B) = span{ψ_{1}, …, ψ_{n}}. The corresponding biorthogonal systems {ϑ_{j}}_{1}^{m}, ⟨ϕ_{i}, ϑ_{j}⟩ = δ_{ij}; {ζ_{j}}_{1}^{m}, ⟨ζ_{i}, ψ̂_{j}⟩ = δ_{ij}, {γ_{j}}_{1}^{n}, ⟨φ_{i}, γ_{j}⟩ = δ_{ij}; {z_{j}}_{1}^{n}, ⟨z_{i}, ψ_{j}⟩ = δ_{ij} are introduced in [19]. For the reader convenience here some auxiliary results from [11, 12, 17, 19] are given.

Definition 1. [19] The elements ϕ_{i}^{(s)}, s = 1, …, q_{i}, ϕ_{i}^{(1)} = ϕ_{i}, i = 1, …, m (φ_{i}^{(s)} , s = 1, …, p_{i} , φ_{i}^{(1)} = φ_{i} , i = 1, …, n) form the complete canonical generalized Jordan set (GJS ≡ B-JS) relative to the operatorfunction A − λB (B − μA, respectively) if

Aϕ_{i}^{(s)} = Bϕ_{i}^{(s−1)}, ⟨ϕ_{i}^{(s)}, ϑ_{j}⟩ = 0, s = 2, … q_{i}, i, j = 1, …, m;

(Bφ_{i}^{(s)} = Aφ_{i}^{(s−1)}, ⟨φ_{i}^{(s)}, γ_{j}⟩ = 0, s = 2, …, p_{i}, i, j = 1, …, n)

D_{q} ≡ det [⟨Bϕ_{i}^{(q_{i})}, ψ̂_{j}⟩] ≠ 0, (D_{p} ≡ det [⟨Aφ_{i}^{(p_{i})}, ψ_{j}⟩] ≠ 0).

This GJS is called bicanonical if the corresponding B^{∗}-JS (A^{∗}-JS) of the adjoint operator A^{∗} (B^{∗}) is also canonical.

The conditions in definition 1 determine the B-JS (A-JS) uniquely. Its elements are linearly independent and form a basis for the rootsubspace K(A; B) (K(B; A)) of the Fredholm point λ = 0 ∈ σ_{B}(A) (μ = 0 ∈ σ_{A}(B)) of the operator-function A − λB (B − μA), where k_{A} = dim K(A; B) = ∑_{i=1}^{m} q_{i} (k_{B} = dim K(B; A) = ∑_{i=1}^{n} p_{i}) is called the

root-number of the Fredholm point.

Elements of B and B^{∗}-Jordan sets (A-and A^{∗}-Jordan sets) of the operator-functions A − λB and A^{∗} − λB^{∗} (B − μA and B^{∗} − μA^{∗}) can be chosen so that the following biorthogonality conditions hold true:

\begin{matrix}⟨ϕ_{i}^{(j)}, ϑ_{k}^{(l)}⟩ = δ_{ik}δ_{jl}, ⟨ζ_{i}^{(j)}, ψ̂_{k}^{(l)}⟩ = δ_{ik}δ_{jl}, j(l) = 1, …, q_{i}(q_{k}), \\ ϑ_{k}^{(l)} = B^{∗}ψ̂_{k}^{(q_{k}+1−l)}, ζ_{i}^{(j)} = Bϕ_{i}^{(q_{i}+1−j)}, i, k = 1, …, m \end{matrix} (2)

\begin{matrix}⟨φ_{i}^{(j)}, γ_{k}^{(l)}⟩ = δ_{ik}δ_{jl}, ⟨z_{i}^{(j)}, ψ_{k}^{(l)}⟩ = δ_{ik}δ_{jl}, j(l) = 1, …, p_{i}(p_{k}), \\ γ_{k}^{(l)} = A^{∗}ψ_{k}^{(p_{k}+1−l)}, z_{i}^{(j)} = Aφ_{i}^{(p_{i}+1−j)}, i, k = 1, …, n. \end{matrix} (3)

The relations (2)–(3) allow to introduce the projectors [19]

p = ∑_{i=1}^{m} ∑_{j=1}^{q_{i}} ⟨·, ϑ_{i}^{(j)}⟩ ϕ_{i}^{(j)} = ⟨·, ϑ⟩ ϕ : E_{1} → E_{1}^{k_{A}} = K(A, B),

q = ∑_{i=1}^{m} ∑_{j=1}^{q_{i}} ⟨·, ψ̂_{i}^{(j)}⟩ ζ_{i}^{(j)} = ⟨·, ψ̂⟩ ζ : E_{2} → E_{2,k_{A}} = span{ζ_{i}^{(j)}},

P = ∑_{i=1}^{n} ∑_{j=1}^{p_{i}} ⟨·, γ_{i}^{(j)}⟩ φ_{i}^{(j)} = ⟨·, γ⟩ φ : E_{1} → E_{1}^{k_{B}} = K(B; A),

Q = ∑_{i=1}^{n} ∑_{j=1}^{p_{i}} ⟨·, ψ_{i}^{(j)}⟩ z_{i}^{(j)} = ⟨·, ψ⟩ z : E_{2} → E_{2,k_{B}} = span{z_{i}^{(j)}}

(4)

(where ϕ = (ϕ_{1}^{(1)}, ⋯, ϕ_{1}^{(q_{1})}, ⋯, ϕ_{m}^{(1)}, ⋯, ϕ_{m}^{(q_{m})}), and the vectors ϑ, ψ̂, ζ, φ, γ, ψ, z are defined in the same way) generating the following direct sums expansions

\begin{matrix}E_{1} = E_{1}^{k_{A}} ˙+E_{1}^{∞−k_{A}}, E_{2} = E_{2,k_{A}} ˙+E_{∞−k_{A}}, \\ E_{1} = E_{1}^{k_{B}} ˙+E_{1}^{∞−k_{B}}, E_{2} = E_{2,k_{B}} ˙+E_{2,∞−k_{B}}. \end{matrix}  (5)

The intertwining relations are realized

Ap = qA on D_{A}, Bp = qB on D_{B},
(BP = QB on D_{B}, AP = QA on D_{A}),
Aϕ = 𝔄_{A}ζ, Bϕ = 𝔄_{B}ζ, B^{∗}ψ̂ = 𝔄_{B}ϑ,
(Bφ = A_{B}z, Aφ = A_{A}z, A^{∗}ψ = A_{A}γ),

(6)

with cell-diagonal matrices 𝔄_{A} = (A_{1}, …, A_{m}), 𝔄_{B} = (B_{1}, …, B_{m}) (A_{B} = (B^{1}, …, B^{n}), A_{A} = (A^{1}, …, A^{n})), where the q_{i}×q_{i}-cells (p_{i}×p_{i}- cells) have the forms

A_{i} = ( \begin{matrix}0 & 0 & 0 & … & 0 & 0 \\ 0 & 0 & 0 & … & 0 & 1 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & 1 & … & 0 & 0 \\ 0 & 1 & 0 & … & 0 & 0 \end{matrix} ) , B_{i} = ( \begin{matrix}0 & 0 & 0 & … & 0 & 1 \\ 0 & 0 & 0 & … & 1 & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 1 & 0 & … & 0 & 0 \\ 1 & 0 & 0 & … & 0 & 0 \end{matrix} )

(B^{i}’s have the same form as the A_{i}’s, correspondingly A^{i}’s have also the same form as the B_{i}’s). The following relations for the operators A and B hold:

\begin{matrix}N (A) ⊂ E_{1}^{k_{A}}, AE_{1}^{k_{A}} ⊂ E_{2,k_{A}}, A(E_{1}^{∞−k_{A}} ∩ D_{A}) ⊂ E_{2,∞−k_{A}}, \\ N (B) ⊂ E_{1}^{∞−k_{A}}, BE_{1}^{k_{A}} ⊂ E_{2,k_{A}}, B(E_{1}^{∞−k_{A}} ∩ D_{B}) ⊂ E_{2,∞−k_{A}}. \end{matrix} (7)

A^{⊓}= A|_{E_{1}^{∞−k_{A}}∩D_{A}}, B^{⊓}= B|_{E_{1}^{∞−k_{A}}∩D_{B}}, and the mappings B : E_{1}^{k_{A}} → E_{2,k_{A}}, A^{⊓}:

E_{1}^{∞−k_{A}} ∩ D_{A} → E_{2,∞−k_{A}} are one-to-one. In the same way, the operators B and A act in invariant pairs of the subspaces E_{1}^{k_{B}}, E_{2,k_{B}} and

E_{1}^{∞−k_{B}}, E_{2,∞−k_{B}} and also B^{⊔}= B|_{E_{1}^{∞−k_{B}} ∩D_{B}} : E_{1}^{∞−k_{B}} ∩ D_{B} → E_{2,∞−k_{B}},

A : E_{1}^{k_{B}} → E_{2,k_{B}} are isomorphisms.

3. Grobman–Hartman theorem analogs when σ_{A}^{0}(B) = ø

We suppose that, for the A-spectrum σ_{A}(B) of the operator B, Re σ_{A}(B) ≠ 0 and the spectral sets σ_{A}^{−}(B) = {μ ∈ σ_{A}(B)| Re μ < 0} and σ_{A}^{+}(B) = {μ ∈ σ_{A}(B)| Re μ > 0} are distant from the imaginary axis on some distance 0 < d < ∞.

All solutions of the corresponding to (1) linear Cauchy problem

A\frac{dx}{dt} = Bx,  x(0) = x_{0}  (8)

belong to E_{1}^{∞−k_{A}} and (8) is solvable if and only if x_{0} ∈ E_{1}^{∞−k_{A}}. In

fact, one sets x(t) = x_{0} + v(t) + w(t), v(t) = ∑_{i=1}^{m} ∑_{s=1}^{q_{i}} ξ_{is}(t)ϕ_{i}^{(s)} ∈ E_{1}^{k_{A}},

w(t) ∈ E_{1}^{∞−k_{A}}. Then (8) splits into the system

\begin{matrix}A^{⊓} \frac{dw}{dt} =B^{⊓} w + (I − q)Bx_{0}, \frac{dξ_{is}(t)}{dt} = ξ_{i,s−1}, \\ s = 2, …, q_{i}, i = 1, …, m, ξ_{iq_{i}} = 0. \end{matrix}  (9)

Consequently ξ_{is}(t) ≡ 0, x_{0} ∈ E_{1}^{∞−k_{A}}, Bx_{0} =B^{⊓} x_{0} ∈ E_{2,∞−k_{A}} and the

solution of (8) takes the form

x(t) = x_{0} + ∫_{0}^{t}[exp(A^{⊓}^{−1}B^{⊓} (t − s)] A^{⊓}^{−1}B^{⊓} x_{0}ds = exp(A^{⊓}^{−1}B^{⊓} t)x_{0}, (10)

Thus one has σ_{A}(B) = σ(A^{⊓}^{−1}B^{⊓}). Here the function exp(A^{⊓}^{−1}B^{⊓} t) has the

form of the contour integral \frac{1}{2πi} ∫_{γ}(μI− A^{⊓}^{−1}B^{⊓})^{−1}e^{μt} dt at the assumption

about sectorial property [7] of the operator A^{⊓}^{−1}B^{⊓} (or, that is the same, about A-sectorial property of the operator B [18]) with some special contour γ belonging to sector S_{α,θ}(B) in A-resolvent set of the operator

B [18]. Moreover, this is true when the operator A^{⊓}^{−1}B^{⊓} is bounded.

For the generalization of the Grobman-Hartman theorem we will follow the work [6]. Let us define the spaces D_{k}, k = 1, 2 with graphs norms:

(1) D_{1} = D_{B} ⊂ D_{A} with the norm ∥x∥_{1} = ∥x∥_{E_{1}} + ∥Bx∥_{E_{2}}, x ∈ D_{1}, if A is subordinated to B,

(2) D_{2} = D_{A} ⊂ D_{B} with the norm ∥x∥_{2} = ∥x∥_{E_{1}} + ∥Ax∥_{E_{2}}, x ∈ D_{2}, if B is subordinated to A, and introduce the spaces X_{k0}, X_{k1}, X_{k2}, Y_{k0}, Y_{k1}, Y_{k2} consisting of the bounded uniformly continuous functions f(t) on [0, ∞) with their values correspondingly in D_{k}, D_{k} ∩ E_{1}^{∞−k_{A}}, E_{1}^{k_{A}}, E_{2}, E_{2,∞−k_{A}}, E_{2,k_{A}} with supremum norms on the relevant spaces, and the spaces

X_{ks}^{1} = {f(t) ∈ X_{ks}|ḟ(t) ∈ X_{ks}}, ∥f(t)∥_{X_{ks}^{1}} = max{∥f(t)∥_{X_{ks}}, ∥ḟ(t)∥_{X_{ks}}}.

Everywhere below the operator A^{⊓}^{−1}B^{⊓} is supposed to be bounded in X_{k1} (for the case k=1 it is evident) and the operator R be sufficiently smooth in a small neighborhood of zero in D_{k}.

Then the operator

Ax = Aẋ − Bx  (11)

acting from X_{k0}^{1} to Y_{k0} is linear, continuous and vanishes on some set X̃_{k2} ⊂ N (A) dense in X_{k2}.

Let be D_{k} ⊃ S_{k} = {initial values of solutions of the equation (8), which are defined and remain in a small neighborhood of zero in D_{k} for t ∈ [0, +∞)} and U_{k} = {initial values of solutions of (8), which are defined and remain in a small neighborhood of zero in D_{k} for t ∈ (−∞, 0]}. From (11) it follows that S_{k} ˙+U_{k} = E_{1}^{∞−k_{A}} ∩ D_{k}. Then the equality σ_{A}(B) =

σ(A^{⊓}^{−1}B^{⊓}) allows to define the projectors P^{−}u = \frac{1}{2πi} ∫_{γ_{−}}(μI_{E_{1}^{∞−k_{A}}}− A^{⊓}^{−1}B^{⊓}

)^{−1}udμ (γ_{−} is the contour in ρ_{A}(B) surrounding the points μ ∈ σ_{A}(B) with Re μ < 0), and P^{+} = I_{E_{1}^{∞−k_{A}}} − P^{−}. Whence D_{k} = D_{k}^{−} ˙+D_{k}^{0} ˙+D_{k}^{+}, D_{k}^{0} = E_{1}^{k_{A}}, D_{k}^{±} = P^{±}D_{k}. Operator A is Noetherian [19] with R(A) = Y_{k1} and

N (A) = {f(t) ∈ X_{k0}^{1}|f(t) = exp(A^{⊓}^{−1}B^{⊓} t)P^{−}f(0) ∈ D_{k}^{−}} ˙+{f(t) ∈ D_{k}^{0}}
= N_{1}(A) ˙+N_{2}(A) for t ≥ 0
(N (A) = {f(t) ∈ X_{k0}^{1}|f(t) = exp(A^{⊓}^{−1}B^{⊓} t)P^{+}f(0) ∈ D_{k}^{+}} ˙+{f(t) ∈ D_{k}^{0}}
for t ≤ 0).

Now setting x = y + z + v, z ∈ D_{k}^{+}, v ∈ D_{k}^{0} = E_{1}^{k_{A}}, y ∈ D_{k}^{−}, one can write (1) in the form (w = y + z in (9))

Az = R(z + y + v) (Ay = R(y + z + v))  (12)

and apply the implicit operator theorem to (12) regarding y, v (z, v) as functional parameters (see the relevant theorems 22.1 and 22.2 in [19] for continuous and analytic operator R, respectively). It follows that (12) has a sufficiently smooth or analytic (according to the properties of the operator R) solution in some neighborhoods of zero values of parameters y, v (z, v):

z = z(y + v), z(0) = 0 = Dz(0) (y = y(z + v), y(0) = 0 = Dy(0)) (13)

Thus we get the following Grobman–Hartman theorem [6] analog asserting that the local solutions behavior for nonlinear equation in hyperbolic equilibrium neighborhood is the same that for its linearization.

Theorem 1. There exist a neighborhood ω^{−}(ω^{+}) of zero in D_{k}^{0} ˙+D_{k}^{−} (in D_{k}^{0} ˙+D_{k}^{+}) and a sufficiently smooth mapping z_{R} = z_{R}(ξ, η) = z_{R}(ξ·ϕ+η) : ω^{−} → D_{k}^{+}, η ∈ D_{k}^{−}(y_{R} = y_{R}(ξ, ζ) = y_{R}(ξ · ϕ + ζ) : ω^{+} → D_{k}^{−}, ζ ∈ D_{k}^{+}), such that a) z_{R}(0, 0) = 0, D_{ξ}z_{R}(0, 0) = 0, D_{η}z_{R}(0, 0) = 0 (y_{R}(0, 0) = 0, D_{ξ}y_{R}(0, 0) = 0, D_{ζ}y_{R}(0, 0) = 0), b) for any solution x(t) of (1) with initial data x(0) = ξ·ϕ+η+z_{R}(ξ·ϕ+η) (x(0) = ξ·ϕ+y_{R}(ξ·ϕ+ζ)+ζ) one has z(t) = z_{R}(ξ(t)·ϕ+y(t)) ∈ D_{k}^{+} for t ≥ 0 (y(t) = y_{R}(ξ(t)·ϕ+z(t)) ∈ D_{k}^{−} for t ≤ 0), c) any solution x(t) of (1) with initial data from b) takes the form x(t) = ξ(t) · ϕ + y(t) + z_{R}(ξ(t) · ϕ + y(t)) (x(t) = ξ(t) · ϕ + y_{R}(ξ(t) · ϕ + z(t)) + z(t)) and tends to zero when t → +∞(t → −∞), and belongs, consequently, to local stable manifold S_{k}(R) (local unstable manifold U_{k}(R)).

Proof. We give here the proof for the function z_{R} and the local stable manifold S_{k}(R), the proof of the second part is analogous. Define the

projector P̃^{−}of X_{k1}^{1} onto N_{1}(A) by the equality (P̃^{−}f)(t) = exp(A^{⊓}^{−1}B^{⊓} t)P^{−}f(0), t ≥ 0. If one sets x(t) = v(t)+y(t)+z(t), v(t) = px(t), v(0) =

ξ · ϕ = ∑_{i=1}^{m} ∑_{s=1}^{q_{i}} ξ_{is} · ϕ_{i}^{(s)}, y(t) = P̃^{−}x(t) = exp(A^{⊓}^{−1}B^{⊓} t)η, η = y(0), z(t) =

(I_{X_{k1}^{1}}−P̃^{−}) x(t), then the Lyapounov–Schmidt method (theorem 27.1 [19] for Noetherian operators with d-characteristic (n, 0) and the indicated above theorems (22.1, 22.2 [19]) implies that there is a unique solution of (12) z = z_{R}(ξ(t)·ϕ+y(t)) ∈ X_{k1}^{1} such that x(0) = ξ·ϕ+η+z_{R}(ξ·ϕ+η), i. e. the unique solution of (1) x(t) = v(t) + y(t) + z_{R}(ξ(t) · ϕ + y(t)), v(t) = ξ(t) · ϕ, in a sufficiently small semi-neighborhood of t = 0, where the function z_{R}(ξ, η) = z_{R}(ξ · ϕ + η) is sufficiently smooth by ξ, η, and z_{R}(0, 0) = 0, D_{ξ}z_{R}(0, 0) = 0, D_{η}z_{R}(0, 0) = 0.

Writing the equation (1) in p, q-projections one can get the system for the determination of ξ_{is}(t) (so-named the resolving system (RS) for the equation (1) [11–13]). Here x(t) = ξ(t) · ϕ + w(t), where w(t) = y(t) + z_{R}(ξ(t) · ϕ + y(t)) for t ≥ 0 and w(t) = y_{R}(ξ(t) · ϕ + z(t)) + z(t) for t ≤ 0

A^{⊓} \frac{dw}{dt} =B^{⊓} w − (I_{D_{k}} − q)R(ξ · ϕ + w)  (14)

\begin{matrix}0 = ξ_{iq_{i}}(t) − ⟨R(ξ(t) · ϕ + w), ψ̂_{i}^{(1)}⟩ , \\ ξ̇_{iq_{i}}(t) = ξ_{i,q_{i}−1}(t) − ⟨R(ξ(t) · ϕ + w), ψ̂_{i}^{(2)}⟩ , \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ ξ̇_{i2}(t) = ξ_{i1}(t) − ⟨R(ξ(t) · ϕ + w), ψ̂_{i}^{(q_{i})}⟩ , \\ ξ_{is}(0) = ξ_{is}, s = 1, …, q_{i}, i = 1, …, m. \end{matrix}  (15)

Consequently, the manifold S_{k}(R) = { initial values of solutions of the equation (1), which are defined and remain in a small neighborhood of 0 ∈ D_{k} for t ∈ [0, +∞)} (the manifold U_{k}(R) = { initial values of solutions (1), which are defined and remain in a small neighborhood of 0 ∈ D_{k} for t ∈ (−∞, 0]}) has the local presentation x(0) = ξ · ϕ + η + z_{R}(ξ · ϕ + η) (x(0) = ξ · ϕ + y_{R}(ξ · ϕ + ζ) + ζ), where η ∈ D_{k}^{−} (ζ ∈ D_{k}^{+}) and ξ are small. □

Remark 1. The invariant manifold 𝔐 determined by the function ξ ·ϕ+ η +z_{R}(ξ ·ϕ+η) for t ≥ 0 (ξ ·ϕ+y_{R}(ξ ·ϕ+ζ)+ζ for t ≤ 0) can be regarded as the center manifold (ξ ·ϕ ∈ D_{k}^{0}), that is nontrivial for the equation (1) even if {μ ∈ σ_{A}(B)|Re μ = 0} = ø. Here {ξ · ϕ} can be called the linear center manifold tangent to 𝔐. One can say that 𝔐 has an hyperbolic structure. Thus the RS (15) represents the differential-algebraic system on 𝔐. Of course, if the operator A is invertible, 𝔐 and the system (15) are absent, i.e. in the Grobman–Hartman theorem z_{R} = z_{R}(η) [6].

Theorem 2. Let the operators A, B and R in (1) be intertwined by the group G representations L_{g} (acting in E_{1}) and K_{g} (acting in E_{2}) and the condition I (direct supplements E_{1}^{∞−m} to N (A) and E_{1}^{∞−n} to N (B) are invariant relative to L_{g}) holds true. Then the center manifold 𝔐 is invariant relative to the operators L_{g}.

Proof. According to [13], projectors p, P(q, Q) commute with the operators L_{g}(K_{g}) and invariant pairs of subspaces reduce the representations L_{g}(K_{g}). □

In the article [11] it is proved that the stability (instability) of the trivial solution (even for non-autonomous) equation (1) at sufficiently general conditions is determined by the RS (15) with corollaries for the investigation of the stability (instability) of bifurcating solutions.

In applications, of interest is the case when σ_{A}^{+}(B) = ø. Then D_{k} = D_{k}^{−} ˙+D_{k}^{0}, x(t) = ξ(t) · ϕ + y(t) and the center manifold has the form ξ(t) · ϕ + y(ξ(t) · ϕ). Here the equation (14) gives

\begin{matrix}A^{⊓} y′(ξ(t) · ϕ)(\frac{dξ}{dt} · ϕ) =B^{⊓} y(ξ(t) · ϕ)+ \\ (I − q)R(ξ(t) · ϕ + y(ξ(t) · ϕ)), \\ y(0) = 0, y′(0) = 0 \end{matrix}  (16)

Combined with (15) this gives a possibility for the determination of center manifold w(ξ(t) · ϕ) = ξ(t) · ϕ + y(ξ(t) · ϕ) by successive approximations in conditions of sufficiently smooth operator y(ξ · ϕ). However on this way essential difficulties arise which are connected with the fact that the system (15) is differential-algebraic, i.e. the differential equations for the functions ξ_{i1}(t), i = 1, …, m, are absent. One can find y(ξ ·ϕ) iteratively at the differentiation of the first equations (15).

Remark 2. Theorem 1 and all corollaries remain true for the parameter depending equation

A\frac{dx}{dt} = Bx − R(x, λ), R(0, λ) ≡ 0, R_{x}(0, 0) = 0,  (17)

(λ ∈ Λ, Λ is some Banach space) in a small neighborhood of λ = 0, when, as above, Re σ_{A}(B) ≠ 0, i.e. λ = 0 is not a bifurcation point. However all functions w, z_{R} and y_{R} will depend on small parameter λ.

4. The case of σ_{A}^{0}(B) ≠ ø

Here we consider the sufficiently general case when σ_{A}^{0}(B) consists of a finite number of eigenvalues with finite multiplicities, but σ_{A}^{h}(B) = σ_{A}^{−}(B) ∪ σ_{A}^{+}(B) is separated from the imaginary axis by the lines Re μ = ±d, 0 < d < ∞. As above, the main assumption consists of the operator

A^{⊓}^{−1}B^{⊓} which is not bounded on X_{k1}. Then σ_{A}(B) = σ_{A^{⊓}}(B^{⊓}) and the Banach space D_{k} = D_{k} ∩ E_{1}^{∞−k_{A}} can be decomposed into the direct sum D_{k} = D_{k}^{0} ˙+D_{k}^{h}, D_{k}^{h} = D_{k}^{+} ˙+D_{k}^{−}. Now, to equation (14) one can apply the theorem on center manifold [7, 20] in order to prove the following statement:

Theorem 3. Let the root number k_{A} be finite, the operator R l-time be differentiable, and the conditions of Section 4 hold true. Then, in a sufficiently small neighborhood Ω in D_{k} there exists the mapping χ ∈ C^{l}(D_{k}^{0}, D_{k}^{h}) such that χ(0) = 0, Dχ(0) = 0 and the graph of χ is a manifold M_{c} having the following properties:

(1) M_{c} is locally invariant under the flow generated by the equation (14) in D_{k},

(2) if σ_{A}^{+}(B) = ø (σ_{A}^{−}(B) = ø), then M_{c} is locally exponentially attracting as t → +∞ (t → −∞).

Remark 3. In applications the case σ_{A}^{+}(B) = ø is interesting. Then theorem 3 reduces the equation (1) to two differential-algebraic systems (resolving systems [13]), one of which is (15) on the manifold 𝔐 and the second one represents the system on the center manifold M_{c}.

Under assumptions of theorem 2 these differential-algebraic resolving systems inherit the group symmetry of the original equation (1). This follows from [13] according to projectors p, P (q, Q)-commutativity with the representation operators L_{g} (K_{g}) and their reducibility by invariant pairs of subspaces. The investigation of connections between M_{c} and 𝔐 and corresponding resolving systems is the subject of our future work.

These questions become clear for the corresponding to σ_{A}^{0}(B) ≠ ø simple case [9] when σ_{A}^{+}(B) = ø, but σ_{A}^{0}(B) = {μ ∈ σ_{A}(B)|Re μ = 0} ≠ ø contains some finite number 2n = 2n_{1} + ⋯ + 2n_{ℓ} A-eigenvalues ±iα_{s} of multiplicities n_{s}, s = 1, …, ℓ, α_{s} = κ_{s}α, α ≠ 0 with coprime κ_{s} > 0 or (and) zero-eigenvalue. Without loss of generality we can suppose that the equation (1) is written in the form of the system

\begin{matrix}A_{1}ẋ = B_{1}x − f(x, y) \\ A_{2}ẏ = B_{2}y − R(x, y), \end{matrix} A = ( \begin{matrix}A_{1} & 0 \\ 0 & A_{2} \end{matrix} ) , B = ( \begin{matrix}B_{1} & 0 \\ 0 & B_{2} \end{matrix} ) , (18)

where the linear operators A_{1}, B_{1} : E_{1}^{k_{B_{1}}} → E_{2,k_{B_{1}}} (k_{B_{1}} = 2n_{1}p_{1} + ⋯ + 2n_{ℓ}p_{ℓ}, p_{s} are A_{1}-Jordan chains lengths for ±iα_{s}, s = 1, …, ℓ) act in the invariant pair of finite dimensional subspaces E_{1}^{k_{B_{1}}}, E_{2,k_{B_{1}}} and A_{2}, B_{2} act in the invariant pair of subspaces E_{1}^{∞−k_{B_{1}}}, E_{2,∞−k_{B_{1}}}. Thus, σ_{A_{1}}(B_{1}) = σ_{A}^{0}(B) and σ_{A_{2}}^{0}(B_{2}) = ø. Here f and R are C^{2}-functions vanishing together with their first derivatives at the origin.

In the simplest case the main assumption is

N (A_{1}) = {0}, N (A_{2}) = span {ϕ_{(2)1}, …, ϕ_{(2)m_{2}}}  (19)

Then, under conditions of section 3, there exists the function y_{R}(ξ_{2}(t) · ϕ_{(2)}, x) vanishing together with its first derivatives at the origin, such that the second equation (18) reduces to the system

A^{⊓}_{2} \frac{dy_{R}}{dt} =B^{⊓}_{2} y_{R} − (I − q_{(2)})R(x, ξ_{2}(t) · ϕ_{(2)} + y_{R}(ξ_{2}(t) · ϕ_{(2)}, x)) (20)

(q_{(2)} = ∑_{i=1}^{m_{2}} ∑_{j=1}^{q_{2,i}} < ·, ψ̂_{(2),i}^{(j)} > ζ_{(2)}^{(j)} : E_{2,∞−k_{B_{1}}} → span{ζ_{(2)i}^{(j)}}, A^{⊓}_{2}, B^{⊓}_{2} act in

invariant pair of subspaces E_{1}^{∞−k_{B_{1}}−k_{A_{2}}}, E_{2,∞−k_{B_{1}}−k_{A_{2}}})

\begin{matrix}0 = ξ_{2iq_{2,i}}(t) − ⟨R(x, ξ_{2}(t) · ϕ_{(2)} + y_{R}(ξ_{2}(t) · ϕ_{(2)}, x)), ψ̂_{(2),i}^{(1)}⟩ , \\ ξ̇_{2iq_{2,i}}(t) = ξ_{2i,q_{2,i}−1}(t) − ⟨R(x, ξ_{2}(t) · ϕ_{(2)} + y_{R}(ξ_{2}(t) · ϕ_{(2)}, x)), ψ̂_{(2),i}^{(2)}⟩ \\ … & … & … & … & … & … & … & … & … & … & … & … & … & … & … & … & … & … \\ ξ̇_{2i2}(t) = ξ_{2i1}(t) − ⟨R(x, ξ_{2}(t) · ϕ_{(2)} + y_{R}(ξ_{2}(t) · ϕ_{(2)}, x)), ψ̂_{(2),i}^{(q_{2,i})}⟩ , \\ ξ_{2iσ}(0) = ξ_{2iσ}, σ = 1, …, q_{2,i}, i = 1, …, m_{2}. \end{matrix} (21)

If the system (18) is equipped with initial values x(0), y(0), then they must satisfy the equality

y(0) = ξ_{2} · ϕ_{(2)} + y_{R}(ξ_{2} · ϕ_{(2)}, x(0)).  (22)

Now one has to solve the problem

A_{1}ẋ = B_{1}x − f(x, ξ_{2}(t) · ϕ_{(2)} + y_{R}(ξ_{2}(t) · ϕ_{(2)}, x))  (23)

at the initial data x(0) satisfying (22).

Thus one has two systems (21) and (23) on the center manifold y = y_{R}(ξ_{2}(t) · ϕ_{(2)}, x), where the differential–algebraic system (23) possesses the properties indicated in theorem 3.

5. Grobman–Hartman theorem analog for maps

According to section 3 the equation (14) can be written in the form

\frac{dw}{dt} =A^{⊓}^{−1}B^{⊓} w− A^{⊓}^{−1} (I_{D_{k}} − q)R(ξ · ϕ + w)  (24)

in the space X_{k1}^{1}. Then the assumption about the boundedness of the op-

erator A^{⊓}^{−1}B^{⊓} in X_{k1} allows to prove Grobman–Hartman theorem analog for maps [21]. In fact, then for small ξ there exists the resolving operator U_{ξ}(t, ·) : X_{k1} → X_{k1}^{1}, w_{0} ↦ w(t) for the problem (24) with the initial value w(0) = w_{0} (at ξ = 0, U_{0}(t) is linear). Thus the following assertion is true:

Theorem 4. For small ξ at σ_{A}^{0}(B) = ø and operator A^{⊓}^{−1}B^{⊓} boundedness assumption there exits the resolving operator U_{ξ}(t, w_{0}) and a homeomorphism Φ_{ξ} : X_{k1}^{1} → X_{k1}^{1}, ||ξ|| ≪ 1, such that for t ∈ R and w_{0} ∈ X_{k1} the following relation

U_{0}(t)Φ_{ξ}(w_{0}) = Φ_{ξ}(U_{ξ}(t, w_{0})) = Φ_{ξ}(w(t))  (25)

is true, where the function w(t) and the initial values w_{0}, ξ_{0} satisfy the initial value problem for differential–algebraic system (15).

Remark 4. The case of σ_{A}^{0}(B) presence on imaginary axis remains unstudied. See on this connection the work [21].

Conclusion and future work. The results of this article remain true for the more general operators subordinateness (A is subordinate to B if on D_{B} ∥Ax∥ ≤ ∥Bx∥ + α∥x∥ , α ≥ 0).

At the usage of the work [3] one can extend our results on partial differential equations in Banach spaces with degenerate operator at the highest differential expression.

The obtained results can serve only as the first step in the center manifold theory and its methods for computation of bifurcation solution asymptotics and their stability investigation. Future work here is the development of qualitative and numerical methods for the investigation of these differential–algebraic systems.

Acknowledgements

This work was supported by the Russian Foundation for Basic Researches, project No 01-01-00019, and the NATO–TÜBITAK PC program.

The authors are thankful to Prof. V.S. Mokeychev (the Kazan State University) for his remarks on improvement of our article and Prof. L.R Volevich (M. V. Keldysh Applied Mathematics Institute RAN) for kind information about his investigation results.

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Ankara Middle–East Technical University, 06531 Ankara, Turkey E-mail address: bulent@metu.edu.tr

Ulyanovsk State Technical University, Severny Venets 32, 432027 Ulyanovsk, Russia E-mail address: i.konopleva@ulstu.ru E-mail address: loginov@ulstu.ru

Received October 29, 2005