Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 20, 2005, 79–91

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

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

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].

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 ${\sigma }_A\left(B\right)$ 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 ${\sigma }_A\left(B\right)$ 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 ${\sigma }_A^0\left(B\right)$ 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, $A:E_1\supset D_A\to E_2$, $B:E_1\supset D_B\to E_2$ be densely defined closed linear Fredholm operators, where $D_B\subset D_A$ and $A$ is subordinated to $B$ (i.e. $\parallel Ax\parallel \le \parallel Bx\parallel +\parallel x\parallel$ on $D_B$), or $D_A\subset D_B$ and $B$ is subordinated to $A$ (i.e. $\parallel Bx\parallel \le \parallel Ax\parallel +\parallel x\parallel$ on $D_A$). The differential equation

with sufficiently smooth operator $R$ is considered.

It is supposed the nontriviality of the the zero-subspaces $\mathcal{N}\left(A\right)=\text{span}\left\{{\phi }_1,\dots ,{\phi }_m\right\}$, $\mathcal{N}\left(B\right)=\text{span}\left\{{\varphi }_1,\dots ,{\varphi }_n\right\}$ with non-degeneracy condition $\mathcal{N}\left(A\right)\cap \mathcal{N}\left(B\right)=\left\{0\right\}$ and the defect-subspaces ${\mathcal{N}}^{\ast }\left(A\right)=\text{span}\left\{{\widehat{\psi }}_1,\dots ,{\widehat{\psi }}_m\right\}$, ${\mathcal{N}}^{\ast }\left(B\right)=\text{span}\left\{{\psi }_1,\dots ,{\psi }_n\right\}$. The corresponding biorthogonal systems ${\left\{{\vartheta }_j\right\}}_1^m$, , , , are introduced in [19]. For the reader convenience here some auxiliary results from [11, 12, 17, 19] are given.

This GJS is called bicanonical if the corresponding $B^{\ast }$-JS ($A^{\ast }$-JS) of the adjoint operator $A^{\ast }$ ($B^{\ast }$) 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 root-subspace $K\left(A;B\right)$ ($K\left(B;A\right)$) of the Fredholm point $\lambda =0\in {\sigma }_B\left(A\right)$ ($\mu =0\in {\sigma }_A\left(B\right)$) of the operator-function $A-\lambda B$ ($B-\mu A$), where $k_A=dimK\left(A;B\right)=\sum _{i=1}^m{q}_i$ ($k_B=dimK\left(B;A\right)=\sum _{i=1}^n{p}_i$) is called the root-number of the Fredholm point.

Elements of $B$ and $B^{\ast }$-Jordan sets ($A$-and $A^{\ast }$-Jordan sets) of the operator-functions $A-\lambda B$ and $A^{\ast }-\lambda B^{\ast }$ ($B-\mu A$ and $B^{\ast }-\mu A^{\ast }$) can be chosen so that the following biorthogonality conditions hold true:

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

(where $\phi =\left({\phi }_1^{\left(1\right)},\cdots ,{\phi }_1^{\left(q_1\right)},\cdots ,{\phi }_m^{\left(1\right)},\cdots ,{\phi }_m^{\left(q_m\right)}\right)$, and the vectors $\vartheta$, $\hat{\psi }$, $\zeta$, $\varphi$, $\gamma$, $\psi$, $z$ are defined in the same way) generating the following direct sums expansions

The intertwining relations are realized

with cell-diagonal matrices ${\mathfrak{A}}_A=\left(A_1,\dots ,A_m\right)$, ${\mathfrak{A}}_B=\left(B_1,\dots ,B_m\right)$ (${\mathcal{A}}_B=\left(B^1,\dots ,B^n\right)$,${\mathcal{A}}_A=\left(A^1,\dots ,A^n\right)$), where the $q_i\times q_i$-cells ($p_i\times p_i$-cells) have the forms

($B^i$’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:

$\stackrel{\sqcap }{A}=A{\mid }_{E_1^{\infty -k_A}\cap D_A},\stackrel{\sqcap }{B}=B{\mid }_{E_1^{\infty -k_A}\cap D_B}$, and the mappings $B:E_1^{k_A}\to E_{2,k_A},\stackrel{\sqcap }{A}:E_1^{\infty -k_A}\cap D_A\to E_{2,\infty -k_A}$ 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 $E_1^{\infty -k_B}$, $E_{2,\infty -k_B}$ and also $\stackrel{\bigsqcup }{B}=B{\mid }_{E_1^{\infty -k_B}\cap D_B}:E_1^{\infty -k_B}\cap D_B\to E_{2,\infty -k_B}$, $A:E_1^{k_B}\to E_{2,k_B}$ are isomorphisms.

3. Grobman–Hartman theorem analogs when ${\sigma }_A^0\left(B\right)=ø$

We suppose that, for the $A$-spectrum ${\sigma }_A\left(B\right)$ of the operator $B$, $\text{Re}{\sigma }_A\left(B\right)\ne 0$ and the spectral sets ${\sigma }_A^{-}\left(B\right)=\left\{\mu \in {\sigma }_A\left(B\right)\mid \text{Re}\mu <0\right\}$ and ${\sigma }_A^{+}\left(B\right)=\left\{\mu \in {\sigma }_A\left(B\right)\mid \text{Re}\mu >0\right\}$ are distant from the imaginary axis on some distance $0<d<\infty$.

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

belong to $E_1^{\infty -k_A}$ and (8) is solvable if and only if $x_0\in E_1^{\infty -k_A}$. In fact, one sets $x\left(t\right)=x_0+v\left(t\right)+w\left(t\right)$, $v\left(t\right)=\sum _{i=1}^m\sum _{s=1}^{q_i}{\xi }_{is}\left(t\right){\phi }_i^{\left(s\right)}\in E_1^{k_A}$, $w\left(t\right)\in E_1^{\infty -k_A}$. Then (8) splits into the system

Consequently ${\xi }_{is}\left(t\right)\equiv 0,x_0\in E_1^{\infty -k_A},B{x}_0=\stackrel{\sqcap }{B}{x}_0\in E_{2,\infty -k_A}$ and the solution of (8) takes the form

Thus one has ${\sigma }_A\left(B\right)=\sigma \left({\stackrel{\sqcap }{A}}^{-1}\stackrel{\sqcap }{B}\right)$. Here the function $exp\left({\stackrel{\sqcap }{A}}^{-1}\stackrel{\sqcap }{B}t\right)$ has the form of the contour integral $\frac{1}{2\pi i}{\int \; <!--nolimits-->}_{\gamma }{\left(\mu I-{\stackrel{\sqcap }{A}}^{-1}\stackrel{\sqcap }{B}\right)}^{-1}{e}^{\mu t}$ $dt$ at the assumption about sectorial property [7] of the operator ${\stackrel{\sqcap }{A}}^{-1}\stackrel{\sqcap }{B}$ (or, that is the same, about $A$-sectorial property of the operator $B$ [18]) with some special contour $\gamma$ belonging to sector $S_{\alpha ,\theta }\left(B\right)$ in $A$-resolvent set of the operator $B$ [18]. Moreover, this is true when the operator ${\stackrel{\sqcap }{A}}^{-1}\stackrel{\sqcap }{B}$ 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. $D_1=D_B\subset D_A$ with the norm $\parallel x{\parallel }_1=\parallel x{\parallel }_{E_1}+\parallel Bx{\parallel }_{E_2},x\in D_1,$ if $A$ is subordinated to $B$,
2. $D_2=D_A\subset D_B$ with the norm $\parallel x{\parallel }_2=\parallel x{\parallel }_{E_1}+\parallel Ax{\parallel }_{E_2},x\in D_2$, 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 $f\left(t\right)$ on $\left[0,\infty \right)$ with their values correspondingly in $D_k$, $D_k\cap E_1^{\infty -k_A}$, $E_1^{k_A},E_2,E_{2,\infty -k_A}$, $E_{2,k_A}$ with supremum norms on the relevant spaces, and the spaces

Everywhere below the operator ${\stackrel{\sqcap }{A}}^{-1}\stackrel{\sqcap }{B}$ 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

acting from $X_{k0}^1$ to $Y_{k0}$ is linear, continuous and vanishes on some set ${\tilde{X}}_{k2}\subset \mathcal{N}\left(A\right)$ dense in $X_{k2}$.

Let be $D_k\supset S_k=$ initial values of solutions of the equation (8), which are defined and remain in a small neighborhood of zero in $D_k$ for $t\in \left[0,+\infty \right)$ and $U_k=$ initial values of solutions of (8), which are defined and remain in a small neighborhood of zero in $D_k$ for $t\in \left(-\infty ,0\right]$. From (11) it follows that $S_k\dot{+}{U}_k=E_1^{\infty -k_A}\cap D_k$. Then the equality ${\sigma }_A\left(B\right)=\sigma \left({\stackrel{\sqcap }{A}}^{-1}\stackrel{\sqcap }{B}\right)$ allows to define the projectors $P^{-}u=\frac{1}{2\pi i}{\int \; <!--nolimits-->}_{{\gamma }_{-}}{\left(\mu I_{E_1^{\infty -k_A}}-{\stackrel{\sqcap }{A}}^{-1}\stackrel{\sqcap }{B}\right)}^{-1}ud\mu$ ${\gamma }_{-}$ is the contour in ${\rho }_A\left(B\right)$ surrounding the points $\mu \in {\sigma }_A\left(B\right)$ with Re $\mu <0$, and $P^{+}=I_{E_1^{\infty -k_A}}-P^{-}$. Whence $D_k=D_k^{-}\dot{+}{D}_k^0\dot{+}{D}_k^{+}$, $D_k^0=E_1^{k_A}$, $D_k^{\pm }=P^{\pm }{D}_k$. Operator $A$ is Noetherian [19] with $R\left(A\right)=Y_{k1}$ and

Now setting $x=y+z+v$, $z\in D_k^{+}$, $v\in D_k^0=E_1^{k_A},y\in D_k^{-}$, one can write (1) in the form ($w=y+z$ in (9))

and apply the implicit operator theorem to (12) regarding $y,v$ $\left(z,v\right)$ 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 $y,v$ ($z,v$):

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.

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 ${\sigma }_A^{+}\left(B\right)=ø$. Then $D_k=D_k^{-}\dot{+}{D}_k^0$, $x\left(t\right)=\xi \left(t\right)\cdot \phi +y\left(t\right)$ and the center manifold has the form $\xi \left(t\right)\cdot \phi +y\left(\xi \left(t\right)\cdot \phi \right)$. Here the equation (14) gives

Combined with (15) this gives a possibility for the determination of center manifold $w\left(\xi \left(t\right)\cdot \phi \right)=\xi \left(t\right)\cdot \phi +y\left(\xi \left(t\right)\cdot \phi \right)$ by successive approximations in conditions of sufficiently smooth operator $y\left(\xi \cdot \phi \right)$. 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 ${\xi }_{i1}\left(t\right)$, $i=1,\dots ,m$, are absent. One can find $y\left(\xi \cdot \phi \right)$ iteratively at the differentiation of the first equations (15).

4. The case of ${\sigma }_A^0\left(B\right)\ne ø$

Here we consider the sufficiently general case when ${\sigma }_A^0\left(B\right)$ consists of a finite number of eigenvalues with finite multiplicities, but ${\sigma }_A^h\left(B\right)={\sigma }_A^{-}\left(B\right)\cup {\sigma }_A^{+}\left(B\right)$ is separated from the imaginary axis by the lines $\text{Re}\mu =\pm d,0<d<\infty .$ As above, the main assumption consists of the operator ${\stackrel{\sqcap }{A}}^{-1}\stackrel{\sqcap }{B}$ which is not bounded on $X_{k1}.$ Then ${\sigma }_A\left(B\right)={\sigma }_{\stackrel{\sqcap }{A}}\left(\stackrel{\sqcap }{B}\right)$ and the Banach space ${\mathcal{D}}_k=D_k\cap E_1^{\infty -k_A}$ can be decomposed into the direct sum ${\mathcal{D}}_k={\mathcal{D}}_k^0\dot{+}{\mathcal{D}}_k^h,{\mathcal{D}}_k^h={\mathcal{D}}_k^{+}\dot{+}{\mathcal{D}}_k^{-}.$ Now, to equation (14) one can apply the theorem on center manifold [7, 20] in order to prove the following statement: