MATHEMATICS

A. M. Rodnyanskii

On Completely Continuous Vector Fields in a Banach Space

(Presented by Academician P. S. Aleksandrov on 10 IV 1957)

In this note (R) is a real Banach space; (\Lambda) is the empty set; (E \subset R); (E_i = (E)_i) is the set of interior points of (E); (E_g = (E)_g = \overline{E}\setminus E_i) is the boundary of (E); (e) is the identity mapping of (R) onto itself. A mapping (f) of a set (E) into (R) is called a completely continuous vector field defined in (E) if (e-f) is a completely continuous mapping of (E) into (R). A mapping (f) of a set (E) into (R) will be called a local vector field defined on (E) if for every point (x_0\in E) there exists its neighborhood (O^x=O^x(x_0)) in (R) such that (f) is a completely continuous vector field defined on (O^x\cap E). A mapping (f) of a set (E) into (R) will be called a completely continuous vector field defined inside (E), if for every closed bounded (E_1\subseteq E) the mapping (f) is a completely continuous vector field defined on (E_1). By (m(E)) we denote the cardinality of the set (E).

§ 1. In this paragraph (G) is an open subset of (R); (f) is a local vector field defined in (G); (x_0\in G), (y_0=f(x_0)); by (O^x=O^x(x_0)), (O^y=O^y(y_0)) we denote bounded neighborhoods of the points (x_0,y_0) in (R), and we always assume that (\overline{O^x}\subseteq G). The point (x_0) will be called an isolated point of the field (f) if (f(x_0+h)\ne f(x_0)) for all sufficiently small (h\ne 0). If (x_0) is an isolated point of the field (f), then the local degree (\gamma(f,x_0)) is defined. An isolated point (x_0) of the field (f) will be called a regular point of the field (f) if (\gamma(f,x_0)\ne 0). The point (x_0) will be called a point of regular differentiability of the field (f) if (f) is differentiable at the point (x_0) in the sense of Fréchet and (df(x_0,h)\ne 0) for all (h\ne 0). We denote the sets of isolated points, regular points, and points of regular differentiability of the field (f), respectively, by (G^i), (G^r), (G^{rd}). Finally, put

[
G^+ = {x:x\in G^i,\ \gamma(f,x)>0},\qquad
G^- = {x:x\in G^i,\ \gamma(f,x)<0},
]

[
G^0 = {x:x\in G^i,\ \gamma(f,x)=0},\qquad
G^1 = {x:x\in G^i,\ |\gamma(f,x)|=1}.
]

It is known or obvious that

[
G^{rd}\subseteq G^1\subseteq G^r=G^+\cup G^-\subseteq G^i=G^+\cup G^-\cup G^0.
]

The point (x_0) will be called a point of local topologicity of the field (f) if there exist (O^x=O^x(x_0)), (O^y=O^y(y_0)) such that (f) maps (O^x) topologically onto (O^y). The point (x_0) will be called a point of openness of the field (f) if for every (O^x=O^x(x_0)) we have (y_0\in (fO^x)_i). The field (f) is called locally topological (open) in (G) if every point of (G) is a point of local topologicity (openness) of the field (f).

It is obvious that if (f) is open in (G), (O\subseteq G), and (O) is open in (R), then (fO) is open in (R).

The following assertions hold:

1) Let (x_0\in G^r), (O^x=O^x(x_0)). Then for every sufficiently small (connected) (O^y=O^y(y_0)) there exists (one and only one connected) (O_1^x=O_1^x(x_0)), contained in (O^x), such that

[
\overline{O}_1^x\cap f^{-1}y_0={x_0},\quad fO_1^x=O^y,\quad f(O_1^x)_g=(O^y)_g.
]

2) If (x_0\in G^r), then (x_0) is a point of openness of the field (f).

3) Let (G=G^r). Then (f) is open in (G), and the set of points of local topologicity of the field (f) is open in (R) and everywhere dense in (G).

4) Let (G=G^{rd}). Then (f) is locally topological in (G), and (\gamma(f,x)) is constant on each component of the set (G).

5) Let, in addition, (G) be bounded, and let (f) be a completely continuous vector field given in (\overline G). Then (f\overline G=\overline{fG}), and the degree (\gamma(G,f,y)) is defined and is constant on each component of the set (R\setminus fG_g).

6) Let (R\ne R^1), and let (O^x=O^x(x_0)) be such that ((O^x\setminus{x_0})\subseteq G^{rd}). Then (x_0\in G^r), and for every sufficiently small connected (O^y=O^y(y_0)) there exists one and only one connected (O_1^x=O_1^x(x_0)\subseteq O^x) such that (fO_1^x=O^y), (f(O_1^x)_g=(O^y)_g), (m(O_1^x\cap f^{-1}y_0)=1), (m(O_1^x\cap f^{-1}y)=|\gamma(f,x_0)|=\text{const}>0) for all (y\in O^y\setminus{y_0}).

7) Let (R\ne R^1), and let (G\setminus G^{rd}) be an isolated set. Then:

7,1) (G=G^r), in particular (see 3)), (f) is open in (G);

7,2) (\operatorname{sign}\gamma(f,x)) is constant on each component of the set (G);

7,3) the set (G^1) coincides with the set of points of local topologicity of the field (f), and hence (see 7,1), 3)), (G^1) is open in (R) and everywhere dense in (G);

In assertions 7,4)—7,10) we assume that (f) is a completely continuous vector field given inside (G); (G\setminus G^{rd}) is an isolated set.

7,4) the set (G\cap f^{-1}y) is isolated and at most countable for any (y\in R);

7,5) if (E=F_\sigma(R)\subseteq G), (E_i=\Lambda), then (fE=F_\sigma(R)), ((fE)_i=\Lambda);

7,6) ((f(G\cap(G\setminus G^{rd})))_i=\Lambda);

7,7) if (E=F_\sigma(R)\subseteq G), (E_i=\Lambda), (O) is open in (R) and bounded, (O\subseteq G), (O_g\subseteq E), then the set (R\setminus E) is disconnected;

7,8) if (F) is closed in (R), (F\subseteq G), (F_i=\Lambda), and (O) is a bounded component of the set (R\setminus F), (O\subseteq G), then the set (R\setminus fF) is disconnected;

7,9) if (F) is closed in (R), (F\subseteq G), (F_i=\Lambda), and moreover the cardinality of the system of bounded components of the set (R\setminus F) is greater than the cardinality of the system of bounded components of the set (R\setminus G), then the set (R\setminus fF) is disconnected;

7,10) if (O) is open in (R), (\pi(O)<1), (\pi(G)<\tau), then (\pi(G\cap f^{-1}O)<\tau) (see below).

Remark 1. Let (O) be open in (R). The indicated system ({O_\alpha}) of pairwise disjoint bounded sets (O_\alpha) open in (R) will be called a system of holes in the set (O) if ((O_\alpha)g\subseteq O), (O\alpha\cap(R\setminus O)\ne\Lambda) for every (\alpha). We shall write (\pi(O)<\tau) if every system of holes in the set (O) has cardinality (<\tau).

8) Let (R\ne R^1) and, in addition, let (G) be bounded, and let (f) be a completely continuous vector field given in (\overline G). Let, further, (G\setminus G^{rd}) be an isolated set. Finally, let (\Gamma) be a component of the set (R\setminus fG_g). Then: either (\Gamma\cap fG=\Lambda), or (\Gamma\subset fG). There exist an integer nonnegative number (\beta=\beta(\Gamma)) and a set (B(\Gamma)\subseteq\Gamma), open in (R) and everywhere dense in (\Gamma), such that for any (y\in B(\Gamma)) we have (m(G\cap f^{-1}y)=\beta(\Gamma)).

If (\Gamma) is unbounded, then (\Gamma\cap fG=\Lambda). If (U) is a component of the set (G\setminus f^{-1}fG_g), then (fU) is a component of the set (R\setminus fG_g).

§ 2. In this section (X) is a locally linearly connected topological space, (Z=[X,R]) is the topological product of (X) by (R); (x\in X); (y,u\in R); (O^x=O^x(x_0)), (O^y=O^y(y_0)), (O^u=O^u(u_0)), (O^z=O^z(x_0,y_0)) are absolute neighborhoods, respectively, of the points (x_0,y_0,u_0,(x_0,y_0)) in the spaces (X,R,R,Z); (\pi_x,\pi_y) are the projections of (Z) onto (X,R), respectively; (G\subseteq Z), (G) is open in (Z), ((x_0,y_0)\in G); (M\subseteq R). We further set

[
M(x)={y:(x,y)\in M},\qquad
\widetilde M={(x,y):(x,y)\in \overline M,\ x\in \pi_x M}.
]

(M) is bounded with respect to (y) if (\pi_y M) is bounded in (R); (M) is connected with respect to (y) if (M(x)) is connected for every (x\in \pi_x M).

(\varphi) is a continuous mapping of (G) into (R) such that for every ((x_0,y_0)\in G) there exists (O^z=O^z(x_0,y_0)) such that (\varphi O^z) is relatively compact in (R); (f) is a mapping of (G) into (R), given by the formula (f(x,y)=y-\varphi(x,y)); (f_x) is a mapping of (G(x)) into (R), given by the formula (f_x(y)=f(x,y)) ((y\in G(x))).

It is easy to see that for every (x) the set (G(x)) is open in (R), and (f_x) is a completely continuous vector field defined in (G(x)). Therefore we may set

[
G^r={(x,y):y\in (G(x))^r},\qquad
G^{rd}={(x,y):y\in (G(x))^{rd}},
]

[
G^+={(x,y):y\in (G(x))^+},\qquad
G^-={(x,y):y\in (G(x))^-}.
]

Finally, we set (u_0=f(x_0,y_0)=f_{x_0}(y_0)).

The following assertions hold:

9) Let ((x_0,y_0)\in G^r) and let (O^z=O^z(x_0,y_0)) be given. Then for every sufficiently small connected (O^u=O^u(u_0)) there is a neighborhood (O_1^z=O_1^z(x_0,y_0)\subseteq O^z), connected with respect to (y), bounded with respect to (y), and connected, such that:

9,1) (O_1^z\subseteq G);

9,2) (\overline{O_1^z}(x_0)\cap f_{x_0}^{-1}u_0={y_0});

9,3) (f_x O_1^z(x)=O^u) for any (x\in \pi_x O_1^z);

9,4) (f_x(O_1^z(x))_g=(O^u)_g) for any (x\in \pi_x O_1^z);

9,5) For any (x\in \pi_x O_1^z), (u\in O^u), the degree (\gamma(O_1^z(x),f_x,u)) is defined and
[
\gamma(O_1^z(x),f_x,u)=\gamma(f_{x_0},u_0).
]

10) Let (G=G^{rd}). Then each of the sets (G^+), (G^-) is open in (Z). Hence it follows that (\gamma(f_x,y)) is constant on each component of the set (G).

11) Let (O^z=O^z(x_0,y_0)\subseteq G^{rd}). Then for every sufficiently small connected (O^u=O^u(u_0)) there is a neighborhood (O_1^z=O_1^z(x_0,y_0)), connected, bounded with respect to (y), and connected with respect to (y), contained in (O^z) and such that the mapping (\psi), defined on the set ([\pi_x O_1^z,O^u]) by the formula

[
\psi(x,u)=O_1^z(x)\cap f_x^{-1}u \quad ((x,u)\in [\pi_x O_1^z,O^u])
]

is a single-valued continuous open mapping of the set ([\pi_x O_1^z,O^u]) into the space (R_y), and we have:

11,1) for every (x\in \pi_x O_1^z), the mapping (f_x) is a topologically regularly differentiable mapping of (O_1^z(x)) onto (O^u), and the mapping (\psi_x)

((\psi_x(u)=\psi(x,u))) is a topological regularly differentiable mapping of (O^z) onto (O_1^z(x));

(11,2)) for every ((x,u)\in[\pi_x O_1^z, O^u]);

(11,2,1)) ((x,\psi(x,u))\in O_1^z);

(11,2,2)) (f(x,\psi(x,u))=u).

(12)) Let (\Phi) be a bicompact set (\subset G=G^{rd}). Then

[
\sup_x\left(\sup_n m\left(\Phi(x)\cap f_x^{-1}u\right)\right)<+\infty .
]

Remark 2. In the special case (R=R^n), part of the results of this note was obtained in papers ((^{1-10})), as a rule, under stronger (and sometimes under weaker) restrictions on the mappings and sets. Some of the results are essentially new even in the case (R=R^n).

Moscow Physico-Technical
Institute

Received
10 III 1957

CITED LITERATURE

(^{1}) A. M. Rodnyanskii, DAN, 72, No. 1 (1950).
(^{2}) A. M. Rodnyanskii, DAN, 91, No. 5 (1953).
(^{3}) A. M. Rodnyanskii, Matem. sborn., 36 (78), 2 (1955).
(^{4}) A. M. Rodnyanskii, Matem. sborn., 37 (79), 1 (1955).
(^{5}) A. M. Rodnyanskii, Tr. Mosk. aviatsion. inst., 1956.
(^{6}) A. M. Rodnyanskii, DAN, 115, No. 4 (1957).
(^{7}) L. D. Kudryavtsev, Matem. sborn., 32 (74), 3 (1953).
(^{8}) L. D. Kudryavtsev, Usp. matem. nauk, 9, 3 (61) (1954).
(^{9}) L. D. Kudryavtsev, DAN, 95, No. 5 (1954).
(^{10}) L. D. Kudryavtsev, DAN, 104, No. 1 (1955).