UDC 517.948+513.88

MATHEMATICS

V. M. KRASNOSEL’SKII

INVESTIGATION OF THE BIFURCATION OF SMALL EIGENFUNCTIONS IN THE CASE OF MULTIDIMENSIONAL DEGENERACY

(Presented by Academician A. Yu. Ishlinskii, May 22, 1970)

In the present paper the possibility of applying, in the general case, a projection transition to a finite-dimensional equation in the problem of bifurcation points is substantiated.

1. We consider a nonlinear operator (T) acting in a real Banach space (E). It is assumed that the operator (T) is defined in a neighborhood of the zero (0) of the space (E), that (T(0)=0), and

[
T=A+B+\Omega,
\tag{1}
]

where (A) is a linear operator, (B) is a homogeneous Fréchet differentiable operator of order (k) ((k\ge 2)), whose derivative (B'(x)) satisfies the Lipschitz condition

[
|B'(x)-B'(y)|\le c\rho^{k-2}|x-y|\qquad (|x|,\ |y|\le \rho),
\tag{2}
]

and (\Omega) satisfies the condition

[
\lim_{|x|\to\infty}|x|^{-k}\cdot |\Omega x|=0.
]

If, in addition, the condition

[
|\Omega x-\Omega y|\le O(\rho^{k-1})|x-y|,\qquad (|x|,\ |y|\le \rho),
]

is fulfilled, then we shall call the operator (T) smooth.

As is known, bifurcation points of the equation

[
Tx=\lambda x
\tag{3}
]

can only be points of the spectrum of the linear operator (A). Let (\lambda_0) be an isolated nonzero point of the spectrum of the operator (A). We shall assume that (\lambda_0) is a Fredholm point of the spectrum in the sense of M. G. Krein ((^1)). This means that the root subspace (E_0) of the operator (A) corresponding to the eigenvalue (\lambda_0) is finite-dimensional, that the operator (A-\lambda_0 I) is normally solvable, and that the Noether index of the operator (A-\lambda_0 I) is equal to zero. All these properties are satisfied if (A) is completely continuous. We shall assume that the root subspace (E_0) consists only of eigenvectors. Then ((^2)) defines a unique operator (P) of linear projection onto (E_0), commuting with (A). The operator (P) is defined by the simple formula

[
Px=f_1(x)e_1+\cdots+f_n(x)e_n\qquad (x\in E),
\tag{4}
]

where (e_i) form a basis in (E_0), and the functionals (f_i) form a basis in the subspace of zeros of the operator (A^*-\lambda_0 I), these bases being related by the condition (f_i(e_j)=\delta_{ij}) (see, for example, ((^3,^4))).

The principal role in our constructions is played by the nonlinear operator (B_0=PB). This operator is considered only in the (n)-dimensional space (E_0). If (B) is defined by the (k)-th differential of the operator (T), then, for any basis in (E_0), the values of the operator (B_0) have as their components homogeneous polynomials of order (k) in the coordinates of the point (x). We note,

that in all the statements given below it could be assumed that inequality (2) is satisfied not for the operator (B), but for the operator (B_0) in the space (E_0).

The ray (t y_0) ((t>0,\ |y_0|=1,\ y_0\in E_0)) will be called a simple eigenray of the operator (B_0) if (B_0y_0=\gamma y_0), (\gamma\ne0), and (\gamma) is not an eigenvalue of the linear operator (B_0'(y_0)). The latter condition means that the determinant of the matrix (B_0'(y_0)-\gamma I) is different from zero.

Theorem 1. Suppose that the operator (B_0) has at least one simple eigenray, and suppose that the operator (T) is smooth.

Then (\lambda_0) is a bifurcation point of equation (3).

Theorem 2. Suppose that the operator (B_0) has at least one simple eigenray, and suppose that the operator (T) is smooth.

Then (\lambda_0) is a bifurcation point of equation (3).

2. We shall say that the set (\mathfrak R) of nonzero solutions of equation (3) is a branch of eigenvectors of the operator (T), directed along the vector (y_0), corresponding to the bifurcation point (\lambda_0), if (\mathfrak R) contains points of arbitrarily small norm, if

[
\lim_{|x|\to0}\left|\frac{x}{|x|}-y_0\right|=0,\qquad (x\in\mathfrak R),
]

if, for (x\in\mathfrak R), the difference (\lambda(x)-\lambda_0) preserves a constant sign and

[
\lim_{|x|\to0}\lambda(x)=\lambda_0
]

and, finally, if the values (\lambda(x)) ((x\in\mathfrak R)) fill some interval adjoining the point (\lambda_0). In this definition, (\lambda(x)) denotes the value of the parameter (\lambda) for which the point (x\in\mathfrak R) is a solution of equation (3).

A directed branch (\mathfrak R) is called a proper directed branch if (\mathfrak R) is the set of values of a single-valued continuous function (x(\lambda)).

By (K(y_0;\varepsilon)) we denote the cone in the space (E) consisting of those elements (x) for which, for some (t\ge0), the inequality (|x-ty_0|\le t\varepsilon) holds. The set of nonzero solutions of equation (3) lying in (K(y_0;\varepsilon)) and having norm not greater than (\rho) will be denoted by (\mathfrak R(y_0,\varepsilon,\rho)).

Theorem 3. Suppose the conditions of Theorem 1 are satisfied. Suppose that (\varepsilon) and (\rho) are sufficiently small.

Then (\mathfrak R(y_0,\varepsilon,\rho)) is a branch of eigenvectors of the operator (T), directed along the vector (y_j), corresponding to the bifurcation point (\lambda_0).

Theorem 4. Suppose the conditions of Theorem 2 are satisfied. Suppose that (\varepsilon) and (\rho) are sufficiently small.

Then (\mathfrak R(y_0,\varepsilon,\rho)) is a proper branch of eigenvectors of the operator (T), directed along the vector (y_0), corresponding to the bifurcation point (\lambda_0).

Under the conditions of Theorems 3 and 4, the sign of the difference (\lambda(x)-\lambda_0) ((x\in\mathfrak R(y_0,\varepsilon,\rho))) coincides with the sign of (\gamma). The same can also be said as follows: the points (x\in\mathfrak R(y_0,\varepsilon,\rho)) are solutions (x=x(\lambda)) of equation (3) for values of (\lambda) sufficiently close to (\lambda_0) such that (\gamma(\lambda-\lambda_0)>0). The function (x(\lambda)) is single-valued only under the conditions of Theorem 4.

Theorem 5. Suppose the conditions of Theorem 3 or of Theorem 4 are satisfied. Then the solutions (x(\lambda)\in\mathfrak R(y_0,\varepsilon,\rho)) have the asymptotic representation

[
x(\lambda)=\left(\frac{\lambda-\lambda_0}{\gamma}\right)^{1/(k-1)}y_0+h(\lambda),
\tag{5}
]

where (h(\lambda)=O\left(|\lambda-\lambda_0|^{1/(k-1)}\right)) and

[
|h(\lambda)-Ph(\lambda)|\le c|\lambda-\lambda_0|^{k/(k-1)}.
\tag{6}
]

1. We shall say that (\lambda_0) is a point of simple bifurcation if there exist positive numbers (\varepsilon_0) and (\rho_0) such that the nonzero solutions of equation (3) for (|\lambda-\lambda_0|<\varepsilon_0), lying in the ball (|x|<\rho), form a finite number of directed branches of eigenvectors. If all these directed branches are regular, then (\lambda_0) is a point of regular bifurcation.

We shall call the operator (B_0) regular if each of its eigenvectors (y_0\in E_0) ((By_0=\gamma y_0)) determines a simple eigenray (t y_0) ((t>0)).

Let (Tx_n=\lambda_n x_n), (\lambda_n\to\lambda_0) as (n\to\infty). Then the sequence (|x|^{-1}x_n) is compact and its limit points can only be eigenvectors of the operator (B_0). Hence, from Theorems 3 and 4 the following assertions follow.

Theorem 6. Let the operator (B_0) be regular, and let the operator (T) be completely continuous.

Then (\lambda_0) is a point of simple bifurcation of equation (3).

Theorem 7. Let (B_0) be regular, and let (T) be smooth.

Then (\lambda_0) is a point of regular bifurcation of equation (3).

1. If (E_0) is two-dimensional, the operator (B_0) has the form

[
B_0(\xi e_1+\eta e_2)=\varphi(\xi,\eta)e_1+\psi(\xi,\eta)e_2,
\tag{7}
]

where (e_1) and (e_2) are a basis in (E_0), and (\varphi(\xi,\eta)) and (\psi(\xi,\eta)) are homogeneous functions (the order of homogeneity is equal to (k)). Suppose that the vector (e_2) is not an eigenvector of the operator (B_0). Then the eigenvectors may be sought in the form (y_0=e_1+\eta_0 e_2), and (\eta_0) is determined from the equation

[
\eta\varphi(1,\eta)-\psi(1,\eta)=0.
\tag{8}
]

In ordinary cases (8) is an algebraic equation of order (k+1). A root (\eta_0) of equation (8) is called simple if

[
\varphi(1,\eta_0)+\eta_0\varphi'{\eta}(1,\eta_0)-\psi'(1,\eta_0)\ne 0.
]

Theorem 8. The vectors (t e_1+t\eta_0 e_2) ((t>0)) and (-t e_1-t\eta_0 e_2) ((t>0)) form simple eigenrays of the operator (7) if and only if (\eta_0) is a simple root of equation (8).

An important characteristic of the operator (7) is the Poincaré index (\gamma_0) of its singular point. If the functions (\varphi(\xi,\eta)) and (\psi(\xi,\eta)) are homogeneous polynomials of order (k), then, as is known, in the case of even (k), (\gamma_0) can take the values (0,\pm2,\ldots,\pm k), and for odd (k) the values (\pm1,\pm3,\ldots,\pm k). In particular, for (k=2) the number (\gamma_0) may take one of three values: (0,2), or (-2). The computation of (\gamma_0) is carried out ((^5)) with the aid of the generalized Sturm sequence (T_0(\eta),T_1(\eta),\ldots,T_l(\eta)), where (T_0(\eta)=\varphi(1,\eta)), (T_1(\eta)=\psi(1,\eta)), (T_{i-1}(\eta)=\varepsilon_i(\eta)T_i(\eta)-T_{i+1}(\eta)) ((i=1,\ldots,l)), and (T_{l-1}(\eta)=\varepsilon_l(\eta)T_l(\eta)). If the polynomial (T_l(\eta)) has no real roots, then the operator (B_0) is nondegenerate and (\gamma_0=s(+\infty)-s(-\infty)), where (s(-\infty)) denotes the number of sign changes in the sequence of values of the generalized Sturm sequence for negative values of (\eta) large in absolute value, and (s(+\infty)) is the number of sign changes for large positive (\eta). From Theorem 8 it follows that

Theorem 9. Let (E_0) be two-dimensional and let (\gamma_0=-k), where (k) is the order of homogeneity of the operator (B_0).

Then the operator (B_0) is regular and has exactly (2k+2) simple eigenrays.

1. The theorems of this paper are applicable to the investigation, for example, of nonlinear integral equations. Here we present one unexpected assertion which we encountered in the study of the integral operator

[
Tx(t)=\int_a^b G(t,s)f[s,x(s)]\,ds.
\tag{9}
]

We shall consider this operator in the space (C) of functions continuous on ([a,b]). For simplicity we shall assume that the kernel (G(t,s)) is continuous, symmetric, and positive definite. The function (f(s,u)) ((a \leq s \leq b;\ |u| \leq \rho)) will be assumed continuous in the aggregate of its variables and of the form

[
f(s,u)=u+g(s)u^2+o(u^2).
]

Then (T) can be represented in the form (1), where

[
Ax(t)=\int_a^b G(t,s)x(s)\,ds,\qquad
Bx(t)=\int_a^b G(t,s)g(s)x^2(s)\,ds.
]

If (\lambda_0) is an eigenvalue of the operator (A) of multiplicity two, to which there correspond two orthogonal and normalized eigenfunctions (e_1(t)) and (e_2(t)), then the operator (B_0) is determined by the formula

[
\begin{aligned}
B_0(\xi e_1+\eta e_2)
&=e_1(t)\int_a^b g(s)[\xi e_1(s)+\eta e_2(s)]^2 e_1(s)\,ds+\
&\quad +e_2(t)\int_a^b g(s)[\xi e_1(s)+\eta e_2(s)]^2 e_2(s)\,ds.
\end{aligned}
\tag{10}
]

Theorem 10. The Poincaré index (\gamma_0), defined by the operator (10), when this operator is nondegenerate, is equal either to zero or to (-2).

6. Simple proofs of the main theorems of the article were obtained by us using some ideas of A. E. Gelman and V. A. Trenogin, developed by them in the theory of branching of solutions; an extensive bibliography is contained in the handbook (6). It seems to us that the rules set forth above are convenient for applications, since they do not require the construction and investigation of branching equations (see (6)).

The results of this work substantially develop the theorems of our article (7): we have succeeded in freeing ourselves from all three principal restrictions of the work (7); in it it was assumed that (n=2), (k=2), and that for all points (\mu) of the spectrum of the operator (A) not coinciding with (\lambda_0), the inequality (|\mu|<\lambda_0) holds.

The present work was carried out in connection with the remarks and wishes of G. E. Shilov concerning the work (7); I am sincerely grateful to him.

Institute for Problems of Control
(Automation and Telemechanics)
Moscow

Received
10 III 1970

CITED LITERATURE

1. I. Ts. Gokhberg, M. G. Krein, UMN, 12, no. 2 (1957).
2. N. Dunford, J. T. Schwartz, Linear Operators, General Theory, IL, 1962.
3. A. N. Kolmogorov, S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, “Nauka,” 1968.
4. L. A. Lyusternik, V. I. Sobolev, Elements of Functional Analysis, “Nauka,” 1965.
5. M. A. Krasnosel’skii, A. I. Perov et al., Vector Fields in the Plane, 1963.
6. P. P. Zabreiko et al., Integral Equations, “Nauka,” 1968.
7. V. M. Krasnosel’skii, DAN, 180, no. 1 (1968).