MATHEMATICS

V. B. MELAMED

ON BIFURCATION POINTS OF A CERTAIN CLASS OF EQUATIONS

(Presented by Academician M. A. Lavrent’ev, 18 IV 1963)

In a complex Banach space \(E\), consider the equation

\[ \varphi=\lambda B\varphi+D(\varphi,\lambda), \tag{1} \]

where \(\lambda\) is a complex parameter; \(B\) is a linear completely continuous operator; the operator \(D(\varphi,\lambda)\) is analytic jointly in \(\varphi,\lambda\) for \(\varphi\) belonging to some neighborhood of zero in the space \(E\) and \(\lambda\) belonging to some open domain \(M\) of the complex plane, and satisfies the condition

\[ \lim_{\|\varphi\|\to 0}\|D(\varphi,\lambda)\|\,\|\varphi\|^{-1}=0 \qquad (\lambda\in M). \]

In the present note, the law of linearization is proved for the problem of bifurcation points of equation (1)*.

Theorem. In order that \(\lambda_0\) \((\lambda_0\in M)\) be a bifurcation point of equation (1), it is necessary and sufficient that \(\lambda_0\) be a characteristic value of the operator \(B\).

The necessity of the condition of the theorem follows directly from the Hildebrandt–Graves theorem on the implicit function \((^1)\).

We give a brief exposition of the proof of sufficiency.

\(1^\circ\). Let \(\lambda_0\) be a characteristic value of the operator \(B\). Denote \(\lambda-\lambda_0=\mu\) and write equation (1) in the form

\[ \varphi=(\lambda_0+\mu)B\varphi+\sum_{k+j\ge 2}^{\infty}\mu^j D_{kj}\varphi^k, \tag{2} \]

where \(D_{kj}\varphi^k\) is a homogeneous form of degree \(k\).

To investigate the question of small solutions of equation (2) for small values of the parameter \(\mu\), we apply to this equation the Lyapunov–Schmidt method of transition to branching equations.

Let \(e_1,\ldots,e_n\) be a complete system of linearly independent eigenvectors of the operator \(B\) corresponding to the eigenvalue \(1/\lambda_0\), and let \(q_1,\ldots,q_n\) be such linear functionals that \((q_i,e_j)=\delta_{ij}\). Let \(\psi_1,\ldots,\psi_n\) be linearly independent eigenfunctionals of the adjoint operator \(B^*\), corresponding to the same eigenvalue, and let \(p_1,\ldots,p_n\) be such elements of \(E\) that \((\psi_i,p_j)=\delta_{ij}\). Then for each small solution \(\varphi\) of equation (2), the numbers \(c_i=(q_i,\varphi)\), \(i=1,\ldots,n\), satisfy the branching equations

\[ c_i-(q_i,f(c_1,\ldots,c_n,\mu))=0 \qquad (i=1,\ldots,n), \tag{3} \]

where \(f(c_1,\ldots,c_n,\mu)\) is a small solution of the equation

\[ \varphi=\lambda_0B\varphi-\sum_{i=1}^{n}(q_i,\varphi)p_i+\mu B\varphi +\sum_{k+j\ge 2}^{\infty}\mu^jD_{kj}\varphi^k+\sum_{i=1}^{n}c_ip_i. \tag{4} \]

* The same result was obtained in \((^4)\) for the equation \(\varphi=\lambda A\varphi\), where \(A\varphi\) is analytic and completely continuous and \(A0=0\).

Let us note that equation (4), for small values of the parameters \(c_1,\ldots,c_n,\mu\), has a unique small solution, and this solution can be represented in the form of a series in integral nonnegative powers of \(c_1,\ldots,c_n,\mu\).

\(2^\circ\). Consider system (3) for \(\mu=0\). It is known \({}^{(2)}\) that the functions \(c_i-(q_i,f(c_1,\ldots,c_n,0))\), \(i=1,\ldots,n\), are of order of smallness no lower than the second relative to \(c_1,\ldots,c_n\). Therefore, from the results of Cronin (see \({}^{(3)}\), pp. 177–180) it follows that, in the case when the zero solution of the system

\[ c_i-(q_i,f(c_1,\ldots,c_n,0))=0 \qquad (i=1,\ldots,n) \tag{5} \]

is isolated, the index of this solution is greater than one (in the case of non-isolation of the zero solution of system (5), the zero solution of equation (2) for \(\mu=0\) is also non-isolated).

\(3^\circ\). Now consider system (3) for small \(\mu\), different from zero, and show that the index of the zero solution of this system is equal to 1. Obviously, it is sufficient to prove that the system

\[ c_i-(q_i,f(c_1,\ldots,c_n,\mu))=\alpha_i \qquad (i=1,\ldots,n) \tag{6} \]

for small \(\alpha_1,\ldots,\alpha_n\) has a unique small solution depending continuously on \(\alpha_1,\ldots,\alpha_n\). To prove the latter assertion, consider the equation

\[ \varphi=(\lambda_0+\mu)B\varphi+ \sum_{k+j\geq 2}^{\infty}\mu^j D_{kj}\varphi^k+ \sum_{i=1}^{n}\alpha_i p_i . \tag{7} \]

This equation is equivalent to system (6) in the sense that to every small solution \(\overline{\varphi}\) of equation (7) there corresponds, by the formulas

\[ \overline{c}_i=(q_i,\overline{\varphi})+\alpha_i \qquad (i=1,\ldots,n), \]

a certain small solution \(\overline{c}_1,\ldots,\overline{c}_n\) of system (6), and conversely, if \(\overline{c}_1,\ldots,\overline{c}_n\) is a small solution of system (6), then

\[ \overline{\varphi}=f(\overline{c}_1,\ldots,\overline{c}_n,\mu) \]

is a small solution of equation (7).

Since equation (7) has a unique small solution and this solution depends analytically on \(\alpha_1,\ldots,\alpha_n\), system (6) also has a unique small solution depending analytically on \(\alpha_1,\ldots,\alpha_n\).

\(4^\circ\). From the discrepancy between the indices of the zero solution of system (3) for \(\mu=0\) and for small \(\mu\) different from zero, it obviously follows that \(\mu=0\) is a bifurcation point of system (3). But then for equation (2) as well, \(\mu=0\) is a bifurcation point, as was required to prove.

Omsk Institute of Railway Transport Engineers

Received
9 IV 1963

CITED LITERATURE

\({}^{1}\) T. N. Hildebrandt, L. M. Graves, Trans. Am. Math. Soc., 29, 127 (1927).
\({}^{2}\) E. Schmidt, Math. Ann., 65, 370 (1908).
\({}^{3}\) J. Cronin, Ann. Math., 58, 175 (1953).
\({}^{4}\) V. B. Melamed, DAN, 145, No. 3, 531 (1962).