UDC 517.948 + 519.50

MATHEMATICS

Yu. G. BORISOVICH, B. D. GELMAN, E. MUKHAMADIEV, V. V. OBUKHOVSKII

ON THE ROTATION OF MULTIVALUED VECTOR FIELDS

(Presented by Academician P. S. Aleksandrov, 30 XII 1968)

1. A mapping \(F\) acting in a Banach space \(X\) is called multivalued if to each point \(x \in X\) there corresponds a subset \(F(x) \subset X\). A point \(x_0 \in X\) is called a fixed point of the multivalued mapping \(F\) if \(x_0 \in F(x_0)\). The problem of the existence of fixed points for multivalued mappings is of interest for various branches of mathematics (dynamical systems without uniqueness, game theory, etc.). A number of works are devoted to it \((^{1-5})\).

In the present article, for multivalued mappings with convex images, we introduce a new topological invariant—the notion of rotation of the corresponding vector field, closely connected with fixed-point theory. A number of properties analogous to the properties of the rotation of single-valued vector fields are established (on these questions see, for example, \((^{4})\)); applications to fixed-point theorems are given.

2. We first consider the finite-dimensional space \(X = R_n\). Let \(\overline{G}\) be the closure of an open set \(G\) with triangulable boundary \(\Gamma\). Suppose that a mapping \(F\) with convex images is defined on \(\Gamma\) and has the following properties: 1) it is closed (i.e., the graph
\[ \bigcup_{x \in \Gamma} (x, F(x)) \]
is closed in \(X \times X\)); 2) it is compact (i.e.,
\[ \overline{\bigcup_{x \in \Gamma} F(x)} \]
is compact). Such mappings will be called completely continuous mappings, bearing in mind that this definition is to be extended below also to mappings in Banach spaces.

A multivalued mapping defines a multivalued vector field on the boundary \(\Gamma\) by the formula \(\Phi(x) = x - F(x)\) (in the sense of the algebraic difference of two sets), which we shall also call completely continuous. If \(x_0 \in F(x_0)\), then the image \(\Phi(x_0)\) contains \(\Theta\). Such a point \(x_0\) will be called a special point of the multivalued vector field.

Lemma 1. Let \(F\) be a completely continuous multivalued mapping \(\Gamma \to R_n\), and let \(x_j \to x\), \(x_j \in \Gamma\). Then for every \(\varepsilon > 0\) there is an \(n_0\) such that \(F(x_j) \subset F^\varepsilon(x)\), \(j \ge n_0\), where \(F^\varepsilon(x)\) is the \(\varepsilon\)-neighborhood of \(F(x)\).

We make a simplicial subdivision of the boundary \(\Gamma\) and choose an arbitrary simplex \(S_p\) of this subdivision. Consider the \(F\)-images \(F(y_p^1), \ldots, F(y_p^k)\) of its vertices. We fix arbitrarily points \(z^r \in F(y_p^r)\) \((r = 1, \ldots, k)\) and span over them the convex hull \(\Delta\). Define a continuous single-valued mapping \(f: S_p \to \Delta\) by the equality
\[ f\left(\sum_{r=1}^{k} \lambda_r y_p^r\right) = \sum_{r=1}^{k} \lambda_r z^r \qquad \left(\lambda_r \ge 0,\ \sum_{r=1}^{k} \lambda_r = 1\right). \]

Thus we have constructed a continuous single-valued approximation \(f: \Gamma \to R_n\) of the multivalued mapping \(F\).

Lemma 2. Under the condition that \(\Phi x\) has no special points on \(\Gamma\), and for sufficiently fine simplicial subdivisions of the boundary \(\Gamma\), the vector field \(\varphi(x) = x - f(x)\) also has no special points on \(\Gamma\).

Proof. Let \(\varepsilon > 0\) be the smallest distance between the points \(x\) and their \(F\)-images, existing by Lemma 1. Let \(V(\Theta)=\)

neighborhood of zero of radius \(\eta<\varepsilon/2\). Then \(V(x)\cap V(F(x))=\varnothing\) for every \(x\in\Gamma\). By \(U(x)\) we denote the largest neighborhood of the point \(x\) such that \(F(y)\subset V(F(x))\), if \(y\in U(x)\) (its existence follows from Lemma 1). Let \(\delta\) be a Lebesgue number of the cover \(\{U(x)\cap V(x)\}\) of the boundary \(\Gamma\); choose a simplicial subdivision of \(\Gamma\) with the diameter of the simplexes less than \(\delta/2\). Then each simplex \(S_p\) belongs to some neighborhood \(U(x)\cap V(x)\), and the \(F\)-images of its vertices lie in \(V(F(x))\). Consequently, the \(f\)-image of the simplex \(S_p\) lies in \(V(F(x))\), whence the assertion of the lemma follows.

Below it is assumed that the space \(R_n\) is oriented. We formulate the basic

Definition 1. The rotation \(\gamma(\Phi,\Gamma)\) of a multivalued field \(\Phi\) on the boundary \(\Gamma\) will mean \(\gamma(\varphi,\Gamma)\) of a single-valued vector field \(\varphi\), where \(\varphi\) is constructed on a sufficiently fine simplicial subdivision of \(\Gamma\).

It can be shown that the definition does not depend on the choice of the simplicial subdivision, the points \(z^r\), and the radius of the separating neighborhood \(V\).

Let us note that another approach is also possible, based on the notion of the degree of a multivalued mapping of the boundary \(\Gamma\) into the sphere \(S^{n-1}\). First define the degree of such a mapping \(\Psi\), which is assumed to be closed and for which the spherical convex hull of each image \(\Psi(x)\) does not contain diametrically opposite points. In this case one can construct a single-valued approximation of the mapping \(\Psi\). To this end, for each point \(x\in\Gamma\) we fix a neighborhood \(U(x)\) so that the \(\Psi\)-images of the points \(y\in U(x)\) belong to the \(\varepsilon_x\)-inflation of the image \(\Psi(x)\), where \(\varepsilon_x\) is so small that the \(\varepsilon_x\)-inflation contains no diametrically opposite points. The cover \(\{U(x)\}\) determines a Lebesgue number \(\delta\), and, if the triangulation is sufficiently fine, then a single-valued approximation \(\psi\) is constructed, as in the first case, affinely on each simplex.

Definition 2. The degree of the mapping \(\psi:\Gamma\to S^{n-1}\) under a sufficiently fine triangulation of \(\Gamma\) will be called the degree of the multivalued mapping \(\Psi:\Gamma\to S^{n-1}\).

It is not difficult to show that this definition is correct.

Definition 3. The rotation of a multivalued vector field \(\Phi\) is called the degree of the multivalued mapping

\[ \Psi(x)=\left\{y:\ y=\frac{z}{\|z\|},\ z\in\Phi(x)\right\} \]

of the boundary \(\Gamma\) into \(S^{n-1}\).

Definitions 1 and 3 are equivalent.

1. A number of properties known for the rotation of a single-valued field \((^4)\) carry over to the notion of the rotation of a multivalued vector field.

Multivalued vector fields \(\Phi_0\) and \(\Phi_1\) will be called homotopic on \(\Gamma\) if on \(\Gamma\times I\), where \(I=[0,1]\), there exists a completely continuous multivalued vector field \(\Phi(x,t)\) such that \(\Theta\in\Phi(x,t)\), \(x\in\Gamma\), \(t\in I\), \(\Phi(x,0)=\Phi_0(x)\), \(\Phi(x,1)=\Phi_1(x)\).

Theorem 1. If \(\Phi_0\) is homotopic to \(\Phi_1\) on \(\Gamma\), then \(\gamma(\Phi_0,\Gamma)=\gamma(\Phi_1,\Gamma)\).

The assertion of the theorem follows from the fact that in a neighborhood of each point \(t_0\in I\) one can construct a common single-valued approximation for \(\Phi(x,t)\).

The following proposition may be regarded as a general criterion for the existence of a fixed point.

Theorem 2. Let a multivalued completely continuous mapping \(F\) be given on the closed domain \(\overline G\), with no fixed points on the boundary \(\Gamma\), and let \(\gamma(\Phi,\Gamma)\ne0\), where \(\Phi\) is the corresponding vector field. Then there exists a fixed point \(x_0\in G\) of the mapping \(F\).

From this theorem one easily obtains a generalization of S. Kakutani’s theorem \((^2)\).

Theorem 3. If a multivalued completely continuous mapping \(F\) is given on the closed ball \(T \subset R_n\) and, for every \(x\) belonging to the boundary \(S^{n-1}\) of the ball \(T\), the condition \(F(x)\cap T=\varnothing\) is satisfied, then there exists a fixed point \(x_0\in T\) of the mapping \(F\).

Indeed, in this case the rotation \(\gamma(\Phi,\Gamma)\) of the field \(\Phi\) is equal to one.

Multivalued mappings, besides isolated fixed points, may have connected sets of fixed points. Both will be called fixed elements. A fixed element \(x_*\) will be called isolated if some neighborhood of it does not intersect other fixed elements. The rotation of a multivalued vector field on the boundaries of such neighborhoods is constant; we shall call it the index \(\gamma(x_*)\) of the fixed element.

Theorem 4. If, on the closed domain \(\overline{G}\), a multivalued completely continuous vector field \(\Phi\) has a finite number of isolated fixed elements \(x_1,\ldots,x_k\) and has no special points on the boundary \(\Gamma\), then

\[ \gamma[\Phi,\Gamma]=\sum_{i=1}^{k}\gamma(x_k). \]

4. In conclusion we consider completely continuous multivalued vector fields \(\Phi\) in a Banach space (we retain the definition of Sec. 2). In this case Lemmas 1 and 2 remain valid. We construct a multivalued finite-dimensional approximation \(\Phi_m\) of the field \(\Phi\).

Let \(V\) be the closed neighborhood of zero of radius \(\rho<\varepsilon/2\). Choose in the range \(F(\Gamma)\) a \(\rho\)-net \(\{y_1,\ldots,y_m\}\), and denote by \(K_m\) the convex hull, and by \(E_m\) the linear hull, of the net \(\{y_1,\ldots,y_m\}\). We shall assume that \(G\cap E_m\) has a polyhedral boundary \(\Gamma_m\) for each \(E_m\). Consider the multivalued vector field \(\Phi_m(x)=(\Phi(x)+V)\cap K_m\) on \(\Gamma_m\). This field, as can be shown, has no special points and satisfies the conditions of Sec. 2.

Theorem 5. The rotation \(\gamma[\Phi_m,\Gamma_m]\) does not depend on the choice of the approximating space \(E_m\) or on the choice of \(\rho<\varepsilon/2\).

It is therefore natural to give the following definition:

\[ \gamma\{\Phi,\Gamma\}=\gamma\{\Phi_m,\Gamma_m\}. \]

The theorems proved in Sec. 3 are also valid in the case of a Banach space, in the very same formulations. We give the formulation of a theorem generalizing I. Glicksberg’s theorem \({}^{1}\).

Theorem 6. If a multivalued completely continuous field \(\Phi(x)\) is defined on the closed ball \(T\) of a Banach space, has no special points on the boundary \(\Gamma\), and satisfies the condition \(F(x)\cap T=\varnothing,\ x\in\Gamma\), then \(\gamma[\Phi,\Gamma]=1\); consequently, there exists a fixed point.

Applying the general schemes known in the theory of rotations of single-valued vector fields, one can also prove more general theorems on the existence of fixed points.

Remark. The condition of complete continuity of the mapping \(F\) in the preceding constructions can be somewhat weakened: the assumption of compactness of the image \(F(\Gamma)\) may be replaced by the condition of the existence of a compact set \(K\) intersecting each image \(F(x)\), where \(x\in\overline{G}\).

Voronezh State University

Received
4 XII 1968

References

\({}^{1}\) I. L. Glicksberg, Proc. Am. Math. Soc., 3, 170 (1952).
\({}^{2}\) S. Kakutani, Duke Math. J., 8, 457 (1941).
\({}^{3}\) K. Berge, General Theory of Games of Several Persons, Moscow, 1961.
\({}^{4}\) M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, 1956.
\({}^{5}\) A. D. Myshkis, Matem. sborn., 34, no. 2 (1954).