A. E. GELMAN

ON ANALYTIC SOLUTIONS OF A CLASS OF OPERATOR EQUATIONS

(Presented by Academician V. I. Smirnov on 8 VII 1963)

In this paper we study equations obtained by perturbing multivaluedly solvable linear equations. For the case of a linear perturbation, analytic in the parameter, conditions are indicated that are necessary and sufficient for the existence of a unique small* analytic solution. For this case, as well as for the case of a nonlinear perturbation analytic in the parameter and in the unknown, majorant series of solutions are constructed. This makes it possible to obtain estimates for the radii of convergence and for the remainders of the solution series; they are attained in the class of equations under consideration.

All estimates of the radii of convergence of solution series of nonlinear integral equations in the branching case known to us \((^1)\) follow from the estimates of the present paper. This also applies to the few works of the same character on differential and operator equations \((^{2,3})\). The results of the paper make it possible to study a number of uninvestigated and little-investigated questions in the theory of nonlinear integral equations that arise when the branching equation is multidimensional; new results have also been obtained in the theory of differential equations. (For example, in finding periodic solutions in the case when the Poincaré determinant vanishes.)

Let us introduce notation.

\(X, Y, Y'\) are complex \(B\)-spaces, \(X\) finite-dimensional; \(\lambda\) is a complex number; \(X(\lambda), Y(\lambda), Z(\lambda)\) are spaces of series, convergent in some neighborhood of the point \(O\), of the form

\[ x(\lambda)=\sum_{k=0}^{\infty}\lambda^k x_k,\quad x_k\in X;\qquad y(\lambda)=\sum_{k=0}^{\infty}\lambda^k y_k,\quad y_k\in Y; \]

\[ y'(\lambda)=\sum_{k=0}^{\infty}\lambda^k y'_k,\quad y'_k\in Y'. \]

\(Z=X+Y,\quad Z'=X+Y'.\) Accordingly \(Z(\lambda)=X(\lambda)+Y(\lambda);\)
\(Z'(\lambda)=X'(\lambda)+Y'(\lambda)\).

\[ C(\lambda)=\sum_{k=0}^{\infty}\lambda^k C_k, \]
where \(\{C_k\}\) is a set of linear operators defined on \(Z\) with range in \(Z'\) and such that the series \(C(\lambda)\) converges for sufficiently small \(\lambda\). Obviously,

\[ C(\lambda)=[Z(\lambda)]\subset Z'(\lambda). \]

* By a small analytic solution we mean here a solution analytic in the parameter in a neighborhood of the point \(O\) and vanishing at this point.

The linear operator \(A\) is defined on \(Z\) and has the following properties:
a) \(A(z)=Y'\); b) \(A(x)=0\); c) there exists a bounded inverse \(A^{-1}\) from \(Y'\) into \(Y\): \(A^{-1}(Y')=Y\).

The operator \(\alpha\) projects \(Z'\) into \(X\), i.e. \(\alpha(x+y')=x\) for any \(x\in X\) and \(y'\in Y'\).

The operator \(\alpha_1\) projects \(Z\) into \(X\).

Theorem 1. Let \(X(\lambda)\) be some subset containing \(0\), and let \(\widetilde Z(\lambda)=X(\lambda)+Y'(\lambda)\). In order that the equation

\[ A(z)=\lambda^n[\widetilde z(\lambda)+C(\lambda)z] \tag{1} \]

have a unique small analytic solution for every \(\widetilde z(\lambda)\in \widetilde Z(\lambda)\), it is necessary and sufficient that the equation

\[ \alpha C(\lambda)x=\lambda^n\alpha C(\lambda)y(\lambda)+\widetilde x(\lambda) \]

have a unique small analytic solution \(x(\lambda)\in X(\lambda)\) for any \(y(\lambda)\in Y(\lambda)\), \(\widetilde x(\lambda)\in \widetilde X(\lambda)\).

Introduce into consideration the operator \(C_1(\lambda)=\alpha C(\lambda)\alpha_1\) from \(Z(\lambda)\) into \(X(\lambda)\) and the operator \(C_2(\lambda)=C(\lambda)-C_1(\lambda)\) from \(Z(\lambda)\) into \(Z'(\lambda)\). Obviously, \(C_1(\lambda)\alpha_1=\alpha C_1(\lambda)\) and \(\alpha C_2(\lambda)X=0\).

Theorem 2. Suppose that the condition of Theorem 1 is satisfied. Then:

1. The operator \(C_1(\lambda)\) has an inverse \(C_1^{-1}(\lambda)\) from \(X\) into \(X\), representable in the form of a Laurent series converging in some neighborhood of the point \(0\) \((\lambda\ne0)\):

\[ C_1^{-1}(\lambda)=\sum_{k=-r}^{+\infty}\lambda^k\widetilde C_k \qquad (\widetilde C_{-r}\ne0). \]

1. The operators \(C_1^{-1}(\lambda)\) and \(C_1^{-1}(\lambda)\lambda^n\alpha C_2(\lambda)\) map, respectively, \(\widetilde X(\lambda)\) and \(Y'(\lambda)\) into the space \(\lambda X(\lambda)\).

2. The solution \(z(\lambda)\) of equation (1) will be majorized by the series \(\overline z(\lambda)\)

\[ z(\lambda)\preccurlyeq \overline z(\lambda) = \frac{p(\lambda)\overline x(\lambda)+r(\lambda)\overline y'(\lambda)} {1-q(\lambda)\lambda^n}, \]

where \(\overline x(\lambda)\) and \(\overline y'(\lambda)\) are majorants of the elements \(\widetilde x(\lambda)\) and \(y'(\lambda)\), and \(p(\lambda)\), \(r(\lambda)\), and \(q(\lambda)\) are majorants of the operators \(C_1^{-1}(\lambda)\), \([\lambda^n+C_1^{-1}(\lambda)\lambda^n\alpha C_2(\lambda)]A^{-1}\), and \([\lambda^n+C_1^{-1}(\lambda)\lambda^n\alpha C_2(\lambda)]A^{-1}[C(\lambda)-\alpha C(\lambda)]\), respectively.*

Corollary. If \(\Lambda\) denotes the unique positive root of the equation \(\lambda^n q(\lambda)=1\), and \(R\) is the radius of convergence of the series \(z(\lambda)\), then \(R\ge \Lambda\).

Theorem 3. Suppose:

1. The operator \(C_1(\lambda)\) has an inverse \(C_1^{-1}(\lambda)\) from \(X\) into \(X\), representable in the form of a Laurent series:

\[ C_1^{-1}(\lambda)=\sum_{k=-r}^{+\infty}\lambda^k\widetilde C_k. \]

* By the word “majorants” in the formulation of the theorem are meant not simply majorant series, but somewhat more complicated objects. Suppose, for example, that \(\widetilde x(\lambda)=x_1(\lambda)+\lambda^{r+1}x_2(\lambda)\), where \(x_1(\lambda)\) is a polynomial of degree not higher than \(r\), and \(x_2(\lambda)\) is a power series. Then by a majorant of the element \(x(\lambda)\) is meant a power series \(\overline x(\lambda)\) that is majorant for \(x_1(\lambda)\) and \(x_2(\lambda)\) simultaneously. By a majorant of the operator \(C_1^{-1}(\lambda)\) is meant a majorant series of the operator \(\lambda^{r+1}C_1^{-1}(\lambda)\). The majorants of other objects and operators are defined in the same way.

1. The inclusion holds
   \[ \alpha C_2(\lambda)Y(\lambda)\subset \lambda^m X(\lambda), \]
   where \(m>r-n\).

2. \(\Omega(t,\lambda)\) is an operator analytic in both variables in a neighborhood of the point \((0,0)\), with range in \(Z'(\lambda)\), and satisfying the conditions
   \[ \Omega(0,\lambda)=\Omega_z'(0,\lambda)=0. \]

Then:

1. The equation
   \[ A(z)=\lambda^n\bigl[\lambda^{r+1}x(\lambda)+y'(\lambda)+C(\lambda)z+\lambda^k\Omega(z,\lambda)\bigr] \]
   for \(k\ge r\) has a unique small analytic solution \(z(\lambda)\) for arbitrary \(x(\lambda)\) and \(y'(\lambda)\).

2. \(z(\lambda)\) is majorized by the unique small analytic root of the equation
   \[ \bar z= \frac{ p(\lambda)\bigl[\lambda^{r+1}\bar x(\lambda)+\lambda^k\bar\Omega(\bar z,\lambda)\bigr] + r(\lambda)\bigl[\bar y'(\lambda)+\lambda^k\bar\Omega(\bar z,\lambda)\bigr]^* }{ 1-\lambda^n q(\lambda) }. \]

Let us illustrate Theorem 1 by two examples (from the theory of differential and integral equations).

Example 1. Let \(C(\lambda,t)\) and \(\varphi(\lambda,t)\) be, respectively, a square matrix and a vector, \(2\pi\)-periodic, continuous in \(t\), and analytic in \(\lambda\). Let \(C_1(\lambda)\) be the mean value of the matrix, and
\[ C_2(\lambda,t)=C(\lambda,t)-C_1(\lambda). \]
Then the following theorem is valid:

Theorem 4. In order that the linear system of differential equations
\[ \dot V=\lambda^n\bigl[\varphi(\lambda,t)+C(\lambda,t)V\bigr] \]
have a unique small solution analytic in \(\lambda\) and \(2\pi\)-periodic for every \(\varphi(\lambda,t)\) with mean value \(0\), it is necessary and sufficient that the system of equations
\[ C_1(\lambda)V=\lambda^n\int_0^{2\pi} C_2(\lambda,t)\psi(t)\,dt \]
have a unique small analytic solution for every \(2\pi\)-periodic continuous vector-function \(\psi(t)\).

In particular, for example, the system of equations
\[ \begin{aligned} \dot v_1&=\lambda\bigl[\varphi_1(\lambda,t)+(1+\lambda^2)v_1+(1+a\sin t)v_2\bigr],\\ \dot v_2&=\lambda\bigl[\varphi_2(\lambda,t)+(1-\lambda^2)v_1+(1+b\sin t)v_2\bigr] \end{aligned} \]
has a unique small solution analytic in \(\lambda\) and \(2\pi\)-periodic for arbitrary \(\varphi_1(t,\lambda)\) and \(\varphi_2(t,\lambda)\) with mean value \(0\) if and only if \(a=b\).

Example 2. Let the homogeneous integral equation**
\[ u(t)-\int_a^b K(s,t)u(s)\,ds=0 \]

* In this formula, by \(\bar x(\lambda)\), \(\bar y'(\lambda)\), \(p(\lambda)\), \(r(\lambda)\), \(q(\lambda)\) are meant the ordinary majorant series of the objects and operators indicated in Theorem 2; by \(\bar\Omega\), the majorant series of the operator \(\Omega\).

** \(K(s,t)\) is a continuous function and \(K(s,t)=K(t,s)\).

has exactly \(k\) linearly independent orthonormal solutions \(\psi_1(t)\), \(\psi_2(t), \ldots, \psi_k(t)\), and they have the following property: the linear combination
\[ \sum_{i=1}^{k}(\alpha_i+\beta_i f(t))\psi_i(t), \]
where \(\alpha_i\) and \(\beta_i\) are constants and \(f(t)\) is a discontinuous function, vanishes if and only if all the binomials \(\alpha_i+\beta_i f(t)\) are identically zero.

Let \(\varphi(t,\lambda)\) be a function continuous in \(t\) and analytic in \(\lambda\) in a neighborhood of the point \(\lambda=0\). Consider the symmetric matrix
\[ M(\lambda)=(a_{ij}), \]
where
\[ a_{ij}=\int_a^b \varphi(t,\lambda)\psi_i(t)\psi_j(t)\,dt, \]
and let
\[ [M(\lambda)]^{-1}=\sum_{i=-r}^{+\infty}\lambda^i M_i, \]
where \(M_i\) are constant matrices, with \(M_{-r}\ne 0\). Finally, let
\[ \varphi(t,\lambda)-\frac{1}{b-a}\int_a^b \varphi(t,\lambda)\,dt=\lambda^m F(t,\lambda), \qquad \text{where } F(t,0)\ne 0. \]

Theorem 5. In order that the equation
\[ u(t)-\int_a^b K(s,t)u(s)\,ds=\lambda^n[\xi(t,\lambda)+u(t)\varphi(t,\lambda)] \]
have a unique small analytic solution for every function \(\xi(t,\lambda)\) continuous in \(t\), analytic in \(\lambda\), and orthogonal to all \(\psi_i(t)\) \((i=1,2,\ldots,k)\), it is necessary and sufficient that the inequality \(m+n>r\) hold.

Leningrad Electrotechnical Institute
named after V. I. Ulyanov (Lenin)

Received
2 VII 1963

CITED LITERATURE

\(^{1}\) M. M. Vainberg, V. A. Trenogin, UMN, 17, no. 2 (104) (1962).
\(^{2}\) Yu. A. Ryabov, DAN, 118, no. 4 (1958).
\(^{3}\) A. E. Gelman, DAN, 144, no. 1 (1962).