MATHEMATICS

A. E. GEL’MAN

THE METHOD OF A SMALL PARAMETER FOR OPERATOR EQUATIONS

(Presented by Academician V. I. Smirnov on 11 VII 1958)

Lemma. Let
[
F(\lambda,x)=a+\lambda\sum_{i,j=0}^{\infty} a_{ij}x^i\lambda^j,
]
where (a,a_{ij}\geqslant 0); (\rho, R) are positive radii of convergence with respect to (\lambda) and (x). Then, if (a<R), there exists a unique root of the equation
[
x=F(\lambda,x), \tag{1}
]
representable in the form of the series
[
x(\lambda)=\sum_{k=0}^{\infty} x_k\lambda^k .
]
Moreover (x_k\geqslant 0), and the radius of convergence of the series (x(\lambda)) (if (F(\lambda,0)\ne 0))* is determined by the formula
[
\Lambda=\sup_{0<x<R}\lambda,
]
where (\lambda) and (x) are connected by equation (1).

If, moreover, the function (F(\lambda,x)) is nonlinear with respect to (x), then the series (x(\lambda)) also converges at the point (\lambda=\Lambda) (in this case (\Lambda) is finite)**.

Let (Y) be a space of type (B), and let (Y_\lambda) be the linear system of all formally constructed power series of the form
[
y(\lambda)=\sum_{k=0}^{\infty} y_k\lambda^k,
]
where (y_k\in Y). We shall write
[
y(\lambda)=\sum_{k=0}^{\infty} y_k\lambda^k \ll \sum_{k=0}^{\infty} x_k\lambda^k,
]
if
[
|y_k|\leqslant x_k.
]
In the case where the series (\sum_{k=0}^{\infty} x_k\lambda^k) has a nonzero radius of convergence, we shall denote its sum by (x(\lambda)),
[
x(\lambda)=\sum_{k=0}^{\infty} x_k\lambda^k
]
and write
[
y(\lambda)\ll x(\lambda).
]

* If (F(\lambda,0)=0), then the trivial case (x(\lambda)=0) occurs (i.e. (x_k=0)).

** This fact is extremely important for estimating the remainder term of the series (x(\lambda)).

Theorem 1. Let the operator (\Omega_\lambda) satisfy the following conditions:

1) (\Omega_\lambda) maps (Y_\lambda) into itself, and

[
\Omega_\lambda[y(\lambda)]
=\Omega_\lambda(y_0+y_1\lambda+y_2\lambda^2+\cdots)
=\Omega_0(0)+\sum_{k=1}^{\infty}\lambda^k\omega_k(y_0,y_1,\ldots,y_{k-1}),
]

where (\omega_k) is an operator mapping the set of (k)-dimensional vectors with components from (Y) into (Y) (i.e., if (y_0,y_1,\ldots,y_{k-1}\in Y), then (\omega_k(y_0,y_1,\ldots,y_{k-1})\in Y)).

2) There exists a multiple power series

[
\overline{\Omega}(\lambda,x)=a+\lambda\sum_{i,j=0}^{\infty}a_{ij}x^i\lambda^j
]

with positive radii of convergence such that, for (y(\lambda)\preccurlyeq x(\lambda)), (x(0)<R), where (R) is the radius of convergence with respect to (x) of the series (\overline{\Omega}(\lambda,x)), the relation

[
\Omega_\lambda[y(\lambda)]\preccurlyeq \overline{\Omega}[\lambda,x(\lambda)]
]

holds.

3) (|\Omega_0(0)|<R).

Then:

a) The equation

[
y=\Omega_\lambda(y)
]

has a unique solution (y(\lambda)) belonging to (Y_\lambda):

[
y(\lambda)=y_0+y_1\lambda+y_2\lambda^2+\cdots,
\tag{2}
]

where (y_k\in Y).*

b) The series (2) converges for (|\lambda|<\Lambda)**, where (\Lambda=\sup_{0<x<R}\lambda), and (\lambda) and (x) are connected by the equation

[
x=\overline{\Omega}(\lambda,x).
\tag{3}
]

c) The relation

[
y(\lambda)\preccurlyeq x(\lambda)
]

holds, where (x(\lambda)) is the unique root, analytic with respect to (\lambda), of equation (3).

Assertion a) of this theorem is obvious; assertions b) and c) are proved by constructing a majorant series and using the lemma.

Theorem 2. Let the operators (L) and (\omega_\lambda) satisfy the following conditions:

1) (L) and (\omega_\lambda) map some linear subsystem (\widetilde{Y}\lambda) of the system (Y\lambda) into (Y_\lambda).***

2) The operator (L) has an inverse (L^{-1}), mapping (Y_\lambda) into (\widetilde{Y}_\lambda). Moreover, there exists a power series (\overline{L^{-1}}(x)) with positive radius of convergence such that, for (y(\lambda)\preccurlyeq x(\lambda)) and (x(0)<R_1), where (R_1) is the radius of convergence of (\overline{L^{-1}}(x)), the relation

[
L^{-1}[y(\lambda)]\preccurlyeq \overline{L^{-1}}[x(\lambda)]
]

holds.

3) The operator (\Omega_\lambda=\omega_\lambda L^{-1}) satisfies all the conditions of Theorem 1.

* Assertion a) follows only from condition (1).

** If the function (\overline{\Omega}(\lambda,x)) is essentially nonlinear with respect to (x), the series also converges for (\lambda=\Lambda) (in this case (\Lambda) is finite).

*** The theorem remains valid also in the case when (\omega_\lambda) maps (\widetilde{Y}\lambda) into (Y\lambda).

Then:

a) The equation (L(y)=\omega_\lambda(y)) has a unique solution (y(\lambda)\in \widetilde{Y}_\lambda):

[
y(\lambda)=y_0+y_1\lambda+y_2\lambda^2+\cdots .
\tag{4}
]

b) The series (4) converges for (|\lambda|<\Lambda), where

[
\Lambda=\sup_{0