### 1.1 Modification of the Lyapunov-Schmidt method

In the Hilbert space

${H}_{T}$ defined above, we consider the boundary-value problem

$\frac{d\phi (t)}{dt}=-i{H}_{0}\phi (t)+\epsilon Z(\phi (t),t,\epsilon )+f(t),$

(10)

$\phi (0,\epsilon )-\phi (w,\epsilon )=\alpha .$

(11)

We seek a generalized solution $\phi (t,\epsilon )$ of the boundary-value problem (10), (11) that becomes one of the solutions of the generating equation (1), (2) ${\phi}_{0}(t,\overline{c})$ in the form (6) for $\epsilon =0$.

To find a necessary condition for the operator function

$Z(\phi ,t,\epsilon )$, we impose the joint constraints

$Z(\cdot ,\cdot ,\cdot )\in C([0;w],{H}_{T})\times C[0,{\epsilon}_{0}]\times C[\parallel \phi -{\phi}_{0}\parallel \le q],$

where *q* is a positive constant.

The main idea of the next results was used in [15] for the investigation of bounded solutions.

Let us show that this problem can be solved with the use of the following operator equation for generating amplitudes:

$F(\overline{c})={U}_{0}(w){\int}_{0}^{w}{U}^{-1}(\tau )Z({\phi}_{0}(\tau ,\overline{c}),\tau ,0)\phantom{\rule{0.2em}{0ex}}d\tau =0.$

(12)

**Theorem 2** (Necessary condition)

*Suppose that the nonlinear boundary*-*value problem* (10), (11) *has a generalized solution* $\phi (\cdot ,\epsilon )$ *that becomes one of the solutions* ${\phi}_{0}(t,\overline{c})$ *of the generating equation* (1), (2) *with constant* $\overline{c}={c}^{0}$ *and* $\phi (t,0)={\phi}_{0}(t,{c}^{0})$ *for* $\epsilon =0$. *Then this constant must satisfy the equation for generating amplitudes* (12).

*Proof* If the boundary-value problem (10), (11) has classical generalized solutions, then, by Lemma 1, the following solvability condition must be satisfied:

${U}_{0}(w)(\alpha +{\int}_{0}^{w}{U}^{-1}(\tau )\{f(\tau )+\epsilon Z(\phi (\tau ,\epsilon ),\tau ,\epsilon )\}\phantom{\rule{0.2em}{0ex}}d\tau )=0.$

(13)

By using condition (5), we establish that condition (13) is equivalent to the following:

${U}_{0}(w){\int}_{0}^{w}{U}^{-1}(\tau )Z(\phi (\tau ,\epsilon ),\tau ,\epsilon )\phantom{\rule{0.2em}{0ex}}d\tau =0.$

Since $\phi (t,\epsilon )\to {\phi}_{0}(t,{c}^{0})$ as $\epsilon \to 0$, we finally obtain [by using the continuity of the operator function $Z(\phi ,t,\epsilon )$] the required assertion.

To find a sufficient condition for the existence of solutions of the boundary-value problem (10), (11), we additionally assume that the operator function $Z(\phi ,t,\epsilon )$ is strongly differentiable in a neighborhood of the generating solution ($Z(\cdot ,t,\epsilon )\in {C}^{1}[\parallel \phi -{\phi}_{0}\parallel \le q]$).

This problem can be solved with the use of the operator

${B}_{0}=\frac{dF(\overline{c})}{d\overline{c}}{|}_{\overline{c}={c}_{0}}={U}_{0}(w){\int}_{0}^{w}{U}^{-1}(t){A}_{1}(t)\phantom{\rule{0.2em}{0ex}}dt:H\to H,$

where ${A}_{1}(t)={Z}^{1}(v,t,\epsilon ){|}_{v={\phi}_{0},\epsilon =0}$ (Fréchet derivative). □

**Theorem 3** (Sufficient condition)

*Suppose that the operator* ${B}_{0}$ *satisfies the following conditions*:

- (1)
*The operator* ${B}_{0}$ *is Moore*-*Penrose pseudoinvertible*;

- (2)
${\mathcal{P}}_{N({B}_{0}^{\ast})}{U}_{0}(w)=0$.

*Then*, *for an arbitrary element* $c={c}^{0}\in {H}_{T}$ *satisfying the equation for generating amplitudes* (12), *there exists at least one solution of* (10), (11).

*This solution can be found by using the following iterative process*:

$\begin{array}{c}{\overline{v}}_{k+1}(t,\epsilon )=\epsilon G[Z({\phi}_{0}(\tau ,{c}^{0})+{v}_{k},\tau ,\epsilon ),\alpha ](t),\hfill \\ {c}_{k}=-{B}_{0}^{+}{U}_{0}(w){\int}_{0}^{w}{U}^{-1}(\tau )\{{A}_{1}(\tau ){\overline{v}}_{k}(\tau ,\epsilon )+\mathcal{R}({v}_{k}(\tau ,\epsilon ),\tau ,\epsilon )\}\phantom{\rule{0.2em}{0ex}}d\tau ,\hfill \\ \mathcal{R}({v}_{k}(t,\epsilon ),t,\epsilon )=Z({\phi}_{0}(t,{c}^{0})+{v}_{k}(t,\epsilon ),t,\epsilon )-Z({\phi}_{0}(t,{c}^{0}),t,0)-{A}_{1}(t){v}_{k}(t,\epsilon ),\hfill \\ \mathcal{R}(0,t,0)=0,\phantom{\rule{2em}{0ex}}{\mathcal{R}}_{x}^{1}(0,t,0)=0,\hfill \\ {v}_{k+1}(t,\epsilon )=U(t){U}_{0}(w){c}_{k}+{\overline{v}}_{k+1}(t,\epsilon ),\hfill \\ {\phi}_{k}(t,\epsilon )={\phi}_{0}(t,{c}^{0})+{v}_{k}(t,\epsilon ),\phantom{\rule{1em}{0ex}}k=0,1,2,\dots ,\phantom{\rule{2em}{0ex}}{v}_{0}(t,\epsilon )=0,\phi (t,\epsilon )=\underset{k\to \mathrm{\infty}}{lim}{\phi}_{k}(t,\epsilon ).\hfill \end{array}$

### 1.2 Relationship between necessary and sufficient conditions

First, we formulate the following assertion:

**Corollary** *Suppose that a functional* $F(\overline{c})$ *has the Fréchet derivative* ${F}^{(1)}(\overline{c})$ *for each element* ${c}^{0}$ *of the Hilbert space* *H* *satisfying the equation for generating constants* (12). *If* ${F}^{1}(\overline{c})$ *has a bounded inverse*, *then the boundary*-*value problem* (10), (11) *has a unique solution for each* ${c}^{0}$.

**Remark 2** If the assumptions of the corollary are satisfied, then it follows from its proof that the operators ${B}_{0}$ and ${F}^{(1)}({c}^{0})$ are equal. Since the operator ${F}^{(1)}(\overline{c})$ is invertible, it follows that assumptions 1 and 2 of Theorem 3 are necessarily satisfied for the operator ${B}_{0}$. In this case, the boundary-value problem (10), (11) has a unique bounded solution for each ${c}^{0}\in {H}_{T}$ satisfying (12). Therefore, the invertibility condition for the operator ${F}^{1}(\overline{c})$ expresses the relationship between the necessary and sufficient conditions. In the finite-dimensional case, the condition of invertibility of the operator ${F}^{(1)}(\overline{c})$ is equivalent to the condition of simplicity of the root ${c}^{0}$ of the equation for generating amplitudes [5].

In this way, we modify the well-known Lyapunov-Schmidt method. It should be emphasized that Theorems 2 and 3 give us a condition for the chaotic behavior of (10) and (11) [16].

### 1.3 Example

We now illustrate the obtained assertion. Consider the following differential equation in a separable Hilbert space

*H*:

$\ddot{y}(t)+Ty(t)=\epsilon (1-{\parallel y(t)\parallel}^{2})\dot{y}(t),$

(14)

$y(0)=y(w),\phantom{\rule{2em}{0ex}}\dot{y}(0)=\dot{y}(w),$

(15)

where

*T* is an unbounded operator with compact

${T}^{-1}$. Then there exists an orthonormal basis

${e}_{i}\in H$ such that

$y(t)={\sum}_{i=1}^{\mathrm{\infty}}{c}_{i}(t){e}_{i}$ and

$Ty(t)={\sum}_{i=1}^{\mathrm{\infty}}{\lambda}_{i}{c}_{i}(t){e}_{i}$,

${\lambda}_{i}\to \mathrm{\infty}$. In this case, the operator system (10), (11) for the boundary-value problem (14), (15) is equivalent to the following countable system of ordinary differential equations (

${c}_{k}(t)={x}_{k}(t)$):

$\begin{array}{c}{\dot{x}}_{k}(t)=\sqrt{{\lambda}_{k}}{y}_{k}(t),\phantom{\rule{1em}{0ex}}k=1,2,\dots ,\hfill \\ {\dot{y}}_{k}(t)=-\sqrt{{\lambda}_{k}}{x}_{k}(t)+\epsilon \sqrt{{\lambda}_{k}}(1-\sum _{j=1}^{\mathrm{\infty}}{x}_{j}^{2}(t)){y}_{k}(t),\hfill \end{array}$

(16)

${x}_{k}(0)={x}_{k}(w),\phantom{\rule{2em}{0ex}}{y}_{k}(0)={y}_{k}(w).$

(17)

We find the solutions of these equations in the space

${W}_{2}^{1}([0;w])$ that, for

$\epsilon =0$, turn into one of the solutions of the generating equation. Consider the critical case

${\lambda}_{i}=4{\pi}^{2}{i}^{2}/{w}^{2}$,

$i\in N$. Let

$w=2\pi $. In this case, the set of all periodic solutions of (16), (17) has the form

$\begin{array}{c}{x}_{k}(t)=cos(kt){c}_{1}^{k}+sin(kt){c}_{2}^{k},\hfill \\ {y}_{k}(t)=-sin(kt){c}_{1}^{k}+cos(kt){c}_{2}^{k}\hfill \end{array}$

for all pairs of constants

${c}_{1}^{k},{c}_{2}^{k}\in R$,

$k\in N$. The equation for generating amplitudes (12) is equivalent in this case to the following countable systems of algebraic nonlinear equations:

$\begin{array}{c}{\left({c}_{1}^{k}\right)}^{3}+2\sum _{j=1,j\ne k}({c}_{1}^{k}{\left({c}_{1}^{j}\right)}^{2}+{c}_{1}^{k}{\left({c}_{2}^{j}\right)}^{2})+{c}_{1}^{k}{\left({c}_{2}^{k}\right)}^{2}-4{c}_{1}^{k}=0,\hfill \\ {\left({c}_{2}^{k}\right)}^{3}+2\sum _{j=1,j\ne k}({c}_{2}^{k}{\left({c}_{1}^{j}\right)}^{2}+{c}_{2}^{k}{\left({c}_{2}^{j}\right)}^{2})+{\left({c}_{1}^{k}\right)}^{2}{c}_{2}^{k}-4{c}_{2}^{k}=0,\phantom{\rule{1em}{0ex}}k\in N.\hfill \end{array}$

Then we can obtain the next result.

**Theorem 4** (Necessary condition for the van der Pol equation)

*Suppose that the boundary*-

*value problem* (16), (17)

*has a bounded solution* $\phi (\cdot ,\epsilon )$ *that becomes one of the solutions of the generating equations with pairs of constants* $({c}_{1}^{k},{c}_{2}^{k})$,

$k\in N$.

*Then only a finite number of these pairs are not equal to zero*.

*Moreover*,

*if* $({c}_{1}^{{k}_{i}},{c}_{2}^{{k}_{i}})\ne (0,0)$,

$i=\overline{1,N}$,

*then these constants lie on an* *N*-

*dimensional torus in the infinite*-

*dimensional space of constants*:

${\left({c}_{1}^{{k}_{i}}\right)}^{2}+{\left({c}_{2}^{{k}_{i}}\right)}^{2}={\left(\frac{2}{\sqrt{2N-1}}\right)}^{2},\phantom{\rule{1em}{0ex}}i=\overline{1,N}.$

**Remark** Similarly, we can study the Schrödinger equation with a variable operator and more general boundary conditions (as noted in the introduction).

Consider the differential Schrödinger equation

$\frac{d\phi (t)}{dt}=-iH(t)\phi (t)+f(t),\phantom{\rule{1em}{0ex}}t\in J$

(18)

in a Hilbert space

*H* with the boundary condition

$Q\phi (\cdot )=\alpha ,$

(19)

where, for each

$t\in J\subset R$, the unbounded operator

$H(t)$ has the form

$H(t)={H}_{0}+V(t)$,

${H}_{0}={H}_{0}^{\ast}$ is an unbounded self-adjoint operator with domain

$D=D({H}_{0})\subset H$, and the mapping

$t\to V(t)$ is strongly continuous. The operator

*Q* is linear and bounded and acts from the Hilbert space

*H* to

${H}_{1}$. As in [

12], we define the operator-valued function

$\tilde{V}(t)={e}^{it{H}_{0}}V(t){e}^{-it{H}_{0}}.$

In this case,

$\tilde{V}(t)$ admits the Dyson representation [[

12], p.311]; denote its propagator by

$\tilde{U}(t,s)$. If

$U(t,s)={e}^{-it{H}_{0}}\tilde{U}(t,s){e}^{is{H}_{0}}$, then

${\psi}_{s}(t)=U(t,s)\psi $ is a weak solution of (14) with the condition

${\phi}_{s}(s)=\psi $ in the sense that, for any

$\eta \in D({H}_{0})$, the function

$(\eta ,{\psi}_{s}(t))$ is differentiable and

$\frac{d}{dt}(\eta ,{\psi}_{s}(t))=-i({H}_{0}\eta ,{\psi}_{s}(t))-i(V(t)\eta ,{\psi}_{s}(t))+(f(t),{\psi}_{s}(t)),\phantom{\rule{1em}{0ex}}t\in J.$

A detailed study of the boundary-value problem (18), (19) will be given in a separate paper.