1.1 Modification of the Lyapunov-Schmidt method
In the Hilbert space defined above, we consider the boundary-value problem
We seek a generalized solution of the boundary-value problem (10), (11) that becomes one of the solutions of the generating equation (1), (2) in the form (6) for .
To find a necessary condition for the operator function , we impose the joint constraints
where q is a positive constant.
The main idea of the next results was used in  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:
Theorem 2 (Necessary condition)
Suppose that the nonlinear boundary-value problem (10), (11) has a generalized solution that becomes one of the solutions of the generating equation (1), (2) with constant and for . 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:
By using condition (5), we establish that condition (13) is equivalent to the following:
Since as , we finally obtain [by using the continuity of the operator function ] 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 is strongly differentiable in a neighborhood of the generating solution ().
This problem can be solved with the use of the operator
where (Fréchet derivative). □
Theorem 3 (Sufficient condition)
Suppose that the operator satisfies the following conditions:
The operator is Moore-Penrose pseudoinvertible;
Then, for an arbitrary element 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:
1.2 Relationship between necessary and sufficient conditions
First, we formulate the following assertion:
Corollary Suppose that a functional has the Fréchet derivative for each element of the Hilbert space H satisfying the equation for generating constants (12). If has a bounded inverse, then the boundary-value problem (10), (11) has a unique solution for each .
Remark 2 If the assumptions of the corollary are satisfied, then it follows from its proof that the operators and are equal. Since the operator is invertible, it follows that assumptions 1 and 2 of Theorem 3 are necessarily satisfied for the operator . In this case, the boundary-value problem (10), (11) has a unique bounded solution for each satisfying (12). Therefore, the invertibility condition for the operator expresses the relationship between the necessary and sufficient conditions. In the finite-dimensional case, the condition of invertibility of the operator is equivalent to the condition of simplicity of the root of the equation for generating amplitudes .
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) .
We now illustrate the obtained assertion. Consider the following differential equation in a separable Hilbert space H:
where T is an unbounded operator with compact . Then there exists an orthonormal basis such that and , . 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 ():
We find the solutions of these equations in the space that, for , turn into one of the solutions of the generating equation. Consider the critical case , . Let . In this case, the set of all periodic solutions of (16), (17) has the form
for all pairs of constants , . The equation for generating amplitudes (12) is equivalent in this case to the following countable systems of algebraic nonlinear equations:
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 that becomes one of the solutions of the generating equations with pairs of constants , . Then only a finite number of these pairs are not equal to zero. Moreover, if , , then these constants lie on an N-dimensional torus in the infinite-dimensional space of constants:
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
in a Hilbert space H with the boundary condition
where, for each , the unbounded operator has the form , is an unbounded self-adjoint operator with domain , and the mapping is strongly continuous. The operator Q is linear and bounded and acts from the Hilbert space H to . As in , we define the operator-valued function
In this case, admits the Dyson representation [, p.311]; denote its propagator by . If , then is a weak solution of (14) with the condition in the sense that, for any , the function is differentiable and
A detailed study of the boundary-value problem (18), (19) will be given in a separate paper.