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V. B. MELAMED
ON THE PROBLEM OF BRANCHING OF SOLUTIONS OF A NONLINEAR ANALYTIC EQUATION
(Presented by Academician S. L. Sobolev on 10 III 1962)
In the complex Banach space \(E\) consider the equation
\[ \varphi = A\varphi, \tag{1} \]
where the operator \(A\) is analytic and completely continuous in some open domain \(G\) of the space \(E\). From the results of Cronin (see \((^1)\), Theorem 5.1, pp. 228—230, and \((^2)\), Theorem B, pp. 177—180) the following assertion follows immediately:
Theorem 1. Let \(\varphi_0\) \((\varphi_0 \in G)\) be an isolated solution of equation (1), and let \(\gamma\) be the index of this solution. Then
\[ \gamma > 0, \tag{2} \]
and in the case when unity is an eigenvalue of the Fréchet derivative of the operator \(A\) at the point \(\varphi_0\),
\[ \gamma > 1. \tag{3} \]
In the present article Theorem 1 is applied to the study of the question of branching of solutions of the equation
\[ \varphi = A_\mu \varphi, \tag{4} \]
where the operator \(A_\mu \varphi\), analytic and completely continuous with respect to \(\varphi\), depends analytically on the complex parameter \(\mu\).
- Suppose that \(A0 = 0\) \((0 \in G)\), and consider the question of bifurcation points of the operator \(A\).
Recall (see \((^3)\)) that the number \(\mu_0\) is called a bifurcation point of the operator \(A\) if, for every \(\varepsilon > 0\), there is a characteristic number \(\mu\) of the operator \(A\) such that \(|\mu - \mu_0| < \varepsilon\), and to this characteristic number there corresponds at least one eigenfunction \(\varphi\): \(\varphi = \mu A\varphi\), with norm less than \(\varepsilon\).
Denote by \(B\) the Fréchet derivative of the operator \(A\) at the point 0. It is not difficult to see that the bifurcation points of the operator \(A\) can only be the characteristic numbers of the operator \(B\) (see \((^3)\), pp. 195—196).
M. A. Krasnosel’skii showed in \((^3)\) that a characteristic number \(\mu_0\) of the operator \(B\) is a bifurcation point of the operator \(A\) (not necessarily analytic) if the zero solution of the equation
\[ \varphi = \mu_0 A\varphi \tag{5} \]
is isolated and the index of this solution is not equal in absolute value to unity.
From this result of M. A. Krasnosel’skii and Theorem 1 it follows:
Theorem 2. The bifurcation points of the operator \(A\) coincide with the characteristic numbers of the operator \(B\).
We note that for nonanalytic operators Theorem 2 is not true.
- In this and the following sections it is assumed that the space \(E\) is a complex Banach space with a basis (for example, a Hilbert space).
Let \(D\) be some bounded connected domain of the space \(E\); \(L\) the boundary of \(D\), and \(D+L \in G\). The theorem below for the finite-dimensional case was proved in \((^2)\).
Theorem 3. Suppose the vector field \(\varphi-A\varphi\) does not vanish on \(L\).
Then the rotation \(\gamma\) of this field on \(L\) is nonnegative.
From Theorems 1 and 3 it follows that inside \(L\) there can be no more than \(\gamma\) isolated solutions of equation (1). It turns out that equation (1) has no non-isolated solutions in the domain \(D\).
Theorem 4. Suppose the conditions of Theorem 3 are satisfied.
Then equation (1) has a finite number of solutions in the domain \(D\).
The proof is based on reduction to the \(n\)-dimensional case.
For \(n=2\), Theorem 4, in a somewhat different formulation, is presented in \((^4)\) (see p. 151).
- Let \(\varphi_0\) (\(\varphi_0\in G\)) be an isolated solution of equation (1). Denote by \(S\) such a sphere with center at the point \(\varphi_0\) that in the ball \(T\) bounded by it equation (1) has no solutions other than \(\varphi_0\).
Let
\[ \alpha=\min_{\varphi\in S}\|\varphi-A\varphi\|. \tag{6} \]
From Theorems 1 and 4 it obviously follows that
Theorem 5. If a completely continuous operator \(A_1\), analytic in the ball \(T\), satisfies, for \(\varphi\in S\), the inequality
\[ \|A_1\varphi-A\varphi\|<\alpha, \tag{7} \]
then the equation
\[ \varphi=A_1\varphi \tag{8} \]
has in the ball \(T\) a finite (but nonzero) number of solutions.
- We now consider the equation depending on the complex parameter \(\mu\),
\[ \varphi=B\varphi+\mu f+\sum_{l+k\ge 2}^{\infty}\mu^l A_{lk}\varphi^k, \tag{9} \]
where \(f\) is some element of \(E\); \(B\) is a linear completely continuous operator, \(A_{lk}\varphi^k\) is a homogeneous form of degree \(k\), and the series \(\sum_{l+k\ge 2}^{\infty}\mu^l A_{lk}\varphi^k\) converges for \(|\mu|\le \varepsilon_1\), \(\|\varphi\|\le \varepsilon_2\), uniformly with respect to \(\mu,\varphi\).
In seeking small solutions of equation (9), two cases are possible:
1) Unity is not an eigenvalue of the operator \(B\). Then, for sufficiently small values of \(\mu\), equation (9) has a unique small solution, and it is representable in the form
\[ \varphi=\sum_{i=1}^{\infty}\mu^i\varphi_i\quad(\varphi_1,\varphi_2,\ldots\in E). \tag{10} \]
2) Unity is an eigenvalue of the operator \(B\) of multiplicity \(n\). Then to equation (9) we apply the Lyapunov–Schmidt method (see, for example, \((^5)\)) of reduction to the branching equations
\[ \psi_i(c_1,\ldots,c_n,\mu)=0\quad(i=1,\ldots,n;\ c_1,\ldots,c_n\text{ are numerical parameters}), \tag{11} \]
and every small solution of equation (9) can be obtained from some series
\[ \varphi = \sum_{r_0+r_1+\cdots+r_n\geq 1}^{\infty} h_{r_0,r_1,\ldots,r_n}\mu^{r_0}c_1^{r_1}\cdots c_n^{r_n} \quad (h_{r_0,r_1,\ldots,r_n}\in E) \tag{12} \]
by substituting into it a small solution \(c_1,\ldots,c_n,\mu\) of system (11).
Denote by \(H\) the manifold of solutions of system (11). The manifold \(H\) is a certain analytic manifold in the \((n+1)\)-dimensional complex space \(R_{n+1}\) with coordinates \(c_1,\ldots,c_n,\mu\). In some neighborhood of the origin of the coordinates of the space \(R_{n+1}\), the manifold \(H\) can be represented uniquely in the form of the sum of a finite number of irreducible manifolds analytic at the origin of coordinates (see \(\left({}^{6}\right)\)):
\[ H=H_1+\cdots+H_m. \tag{13} \]
Suppose further that for \(\mu=0\) the zero solution of equation (9) is isolated. In this case, by means of Theorem 5 it is established that the dimension of each manifold \(H_j\) of the decomposition (13) is equal to one. Thus, the coordinates \(c_1,\ldots,c_n,\mu\) of the points of \(H_j\) \((j=1,\ldots,m)\) are analytic functions of some parameter \(t\):
\[ c_1=c_{1j}(t),\ldots,c_n=c_{nj}(t),\quad \mu=\mu_j(t)\quad (c_{ij}(0)=\mu_j(0)=0,\ \mu_j(t)\neq 0). \tag{14} \]
From formulas (12), (14) it follows that the small solutions of equation (9) are representable by the series
\[ \varphi=\varphi_j(\mu)=\sum_{j=1}^{\infty}\varphi_{ij}(\mu)^{i/s_j} \quad (j=1,\ldots,m;\ \varphi_{ij}\in E;\ s_j\text{ are natural numbers}). \tag{15} \]
Let us formulate the result obtained:
Theorem 6. Suppose that for \(\mu=0\) the zero solution of equation (9) is isolated.
Then there exist positive numbers \(\delta_1,\delta_2\) such that, for \(|\mu|<\delta_1\), equation (9) has in the ball of radius \(\delta_2\) with center at zero a finite number of solutions.
Each of these solutions is representable in the form (15).
We note that in the case where unity is a simple eigenvalue of the operator \(B\), the representability of small solutions of equation (9) by series (15) was proved by V. V. Pokornyi \(\left({}^{7}\right)\).
For the case where unity is an eigenvalue of the operator \(B\) of multiplicity 2, the assertion of Theorem 6 was obtained by another method jointly by P. P. Rybin and the author (unpublished).
The author expresses gratitude to M. A. Krasnosel’skii for a number of suggestions.
Received
27 II 1962
CITED LITERATURE
\({}^{1}\) J. Cronin, Trans. Am. Math. Soc., 69, 2, 208 (1950).
\({}^{2}\) J. Cronin, Ann. of Math., 58, 175 (1953).
\({}^{3}\) M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, 1956.
\({}^{4}\) B. A. Fuks, Theory of Analytic Functions of Several Complex Variables, 1948.
\({}^{5}\) E. Schmidt, Math. Ann., 65, 370 (1908).
\({}^{6}\) S. Bochner, W. T. Martin, Functions of Several Complex Variables, 1951.
\({}^{7}\) V. V. Pokornyi, Proceedings of the Seminar on Functional Analysis, Voronezh State University, 2, 39 (1956).