UDC 519.50+519.54

MATHEMATICS

N. I. GLEBOV

ON A GENERALIZATION OF KAKUTANI’S FIXED POINT THEOREM

(Presented by Academician L. V. Kantorovich, 2 IX 1968)

In the work \((^1)\), I. L. Glicksberg gave a generalization of Kakutani’s fixed point theorem for a closed mapping of a convex compact set to the case of a locally convex linear topological space. In the present note this theorem is extended to a broader class of mappings.

Let \(X\) be a Hausdorff linear topological space and let \(S\) be a subset of \(X\). A mapping \(\Phi:S\to X\) which carries points into nonempty sets is called closed if its graph
\[ \bigcup_{x\in S}(x,\Phi(x)) \]
is closed in \(X\times X\). In terms of generalized sequences this definition is equivalent to one of the following assertions \((^1)\):

a) if \(x_\delta\to x_0,\ y_\delta\in\Phi(x_\delta),\ y_\delta\to y_0\), then \(y_0\in\Phi(x_0)\);

b) if \(x_\delta\to x_0,\ y_\delta\in\Phi(x_\delta)\), and \(y_0\) is a cluster point of the generalized sequence \(\{y_\delta\}\), then \(y_0\in\Phi(x_0)\).

Glicksberg’s result is formulated as follows. If \(\Phi\) is a closed mapping in a locally convex Hausdorff linear topological space \(X\), carrying points of a compact convex set \(S\subset X\) into convex subsets of \(S\), then \(\Phi\) has a fixed point.

Everywhere below \(X\) is a locally convex Hausdorff linear topological space. As a fundamental system of neighborhoods of a point \(x\in X\) we shall consider the system
\[ \mathcal V(x)=\{x+V\mid V\in\mathcal V(0)\}, \]
where \(\mathcal V(0)\) is the system of all open, convex, and symmetric neighborhoods of zero. In those cases where \(\mathcal V(x)\) (respectively \(\mathcal V(x)\times\mathcal V(y)\)) is considered as a directed set \((^2)\), it is assumed that \(U_1\le U_2,\ U_i\in\mathcal V(x),\ i=1,2\) (respectively \((U_1,V_1)\le (U_2,V_2),\ (U_i,V_i)\in\mathcal V(x)\times\mathcal V(y),\ i=1,2\)), if and only if \(U_2\subset U_1\) (respectively \(U_2\subset U_1\) and \(V_2\subset V_1\)).

Lemma 1. Suppose that for every \(x\in S\) the set \(\Phi(x)\) is closed and, moreover, for an arbitrary neighborhood of zero \(U\) there exists a neighborhood \(V\subset\mathcal V(x)\) such that
\[ \Phi(V\cap S)\subset \Phi(x)+U. \]
Then the mapping \(\Phi\) is closed.

Proof. Let \(x_\delta\to x,\ y_\delta\in\Phi(x_\delta),\ y_\delta\to y\), and \(y\notin\Phi(x)\). By the closedness of \(\Phi(x)\), there is a neighborhood of zero \(U\in\mathcal V(0)\) such that
\[ (y+U)\cap\Phi(x)=\varnothing \]
or
\[ (y+\tfrac12 U)\cap(\Phi(x)+\tfrac12 U)=\varnothing. \]
On the other hand, there exists a neighborhood \(V\in\mathcal V(x)\) such that from \(x_\delta\in V\cap S\) it follows that
\[ \Phi(x_\delta)\subset\Phi(x)+\tfrac12 U, \]
whence \(y_\delta\notin y+\tfrac12 U\). Consequently, the point \(y\) cannot be the limit of the generalized sequence \(\{y_\delta\}\). The lemma is proved.

In what follows, the \(\Phi\) under consideration will be assumed to be a mapping from \(S\) into \(S\), where \(S\) is a convex compact set, and by \(\overline A\) and \(\overline{\operatorname{co}}A\) we shall mean the closure and the convex closure of the set \(A\).

Let
\[ \overline{\Phi}(x)=\bigcap_{V\in\mathcal V(x)}\overline{\Phi(V)} \]
and
\[ F(x)=\overline{\operatorname{co}}\Phi(x), \]
where
\[ \Phi(V)=\{y\mid y\in\Phi(x),\ x\in V\cap S\}. \]

Lemma 2. The mapping \(F\) has a fixed point.

Proof. By Glicksberg’s theorem, it suffices to show that the mapping \(F\) is closed. Choose arbitrarily a point \(x \in S\) and an (open) neighborhood \(U \in \mathcal{V}(0)\). There exists an (open) neighborhood \(V_0 \in \mathcal{V}(x)\) such that
\[ \Phi(V_0)\subset \Phi(x)+\tfrac12 U. \]
If this were not so, then the system consisting of the closed sets
\[ S\setminus \bigl(\overline{\Phi(x)+\tfrac12 U}\bigr) \]
and \(\overline{\Phi(V)}\), \(V\in\mathcal{V}(x)\), would be centered, which is impossible by compactness of \(S\) and the absence of a common point for the sets of this system. Let \(y\in V_0\), \(V\in\mathcal{V}(y)\), and \(V\subset V_0\). Then
\[ \Phi(y)\subset \Phi(V)\subset \Phi(V_0)\subset \Phi(x)+\tfrac12 U\subset \overline{\operatorname{co}\Phi(x)}+\tfrac12 U. \]
Since the closure of the convex set
\[ \overline{\operatorname{co}\Phi(x)}+\tfrac12 U \]
is a convex set contained in
\[ \overline{\operatorname{co}\Phi(x)}+U, \]
we have
\[ F(y)=\overline{\operatorname{co}\Phi(y)}\subset \overline{\operatorname{co}\Phi(x)}+U=F(x)+U. \]
Therefore, by Lemma 1, the mapping \(F\) is closed. The lemma is proved.

For an arbitrary point \(x\in X\) and a set \(B\subset X\), by \(L(x,B)\) we shall denote the set of all points \(y\in X\) representable in the form
\[ y=x+\lambda(z-x),\qquad z\in B,\quad \lambda\geq 0, \]
i.e.,
\[ L(x,B)=\bigcup_{\lambda\geq 0}[x+\lambda(B-x)]. \]
It is obvious that if \(B\subset A\), then \(L(x,B)\subset L(x,A)\).

Definition. A mapping \(\Phi:S\to S\) will be called partially closed if from \(x_\delta\to x\), \(y_\delta\in\Phi(x_\delta)\), \(y_\delta\to y\) it follows that
\[ L(x,y)\cap \Phi(x)\neq \varnothing. \]

Since for any cluster point \(y\) of the generalized sequence \(\{y_\delta\}\) one can select from \(\{y_\delta\}\) a generalized subsequence converging to \(y\), in the preceding definition \(y\) may be regarded as a cluster point of \(\{y_\delta\}\). In this way we obtain an equivalent definition of a partially closed mapping.

It is clear that every closed mapping is also partially closed.

Theorem. Every partially closed mapping \(\Phi\), which maps the points of a compact convex set \(S\subset X\) into convex subsets of \(S\), has a fixed point.

Proof. Suppose that for some open neighborhood of zero \(V_0\), for all \(x\in S\) one has
\[ (x+V_0)\cap \Phi(x)=\varnothing . \]
Denote by \(A(x)\) the closure of the set \(\Phi(x)\). Obviously,
\[ (x+V_0)\cap A(x)=\varnothing . \]
Let \(z\in V_0\). Choose an open neighborhood of zero \(V\in\mathcal{V}(0)\) such that
\[ z+V\subset V_0. \]
If \(y\in x+\tfrac12 V\), then \(x\in y+\tfrac12 V\), and
\[ x+z+\tfrac12 V\subset y+z+V\subset y+V_0. \]
Consequently,
\[ (x+z+\tfrac12 V)\cap \Phi(x+\tfrac12 V)=\varnothing \]
and
\[ x+z\notin \overline{\Phi(x+\tfrac12 V)}, \]
whence \(x+z\notin \overline{\Phi(x)}\), or
\[ (x+V_0)\cap \overline{\Phi(x)}=\varnothing . \tag{1} \]
We now show that
\[ \overline{\Phi(x)}\subset L(x,\Phi(x)). \tag{2} \]
Indeed, let \(y\in \overline{\Phi(x)}\). For an arbitrary
\[ \delta=(U,V)\in\mathcal{V}(x)\times\mathcal{V}(x) \]
we have \(y\in \Phi(V)\), i.e.,
\[ U\cap \Phi(V)\neq \varnothing . \]
Let \(x_\delta\in V\), \(y_\delta\in U\cap\Phi(x_\delta)\). Then \(x_\delta\to x\), \(y_\delta\to y\), and, by partial closedness of the mapping \(\Phi\),
\[ L(x,y)\cap \Phi(x)=\varnothing, \]
i.e., for some \(\lambda>0\) we shall have
\[ z=x+\lambda(y-x)\in\Phi(x), \]
or
\[ y=x+\frac{1}{\lambda}(z-x)\in L(x,\Phi(x)). \]

Thus, from (1) and (2) we obtain

\[ \overline{\Phi}(x)\subset S\cap L(x,\Phi(x))\setminus (x+V_0). \tag{3} \]

By virtue of the boundedness of the set \(S\), there exists \(\lambda_0>0\) such that \(\lambda_0(S-x)\subset V_0\). Then for \(\lambda<\lambda_0\) we have \(x+\lambda(A(x)-x)\subset x+V_0\) and, taking (1) into account,
\(S\cap\left[x+\frac1\lambda(A(x)-x)\right]=\varnothing\). Consequently,

\[ S\cap L(x,\Phi(x))\setminus (x+V_0)\subset \bigcup_{\lambda_0\leqslant\lambda\leqslant 1/\lambda_0} [x+\lambda(A(x)-x)]=D. \tag{4} \]

Since the set \(A(x)\) is convex, \(D\) is also convex. Moreover, being the image of the compact set \([\lambda_0,1/\lambda_0]\times A(x)\) under the continuous mapping \((\lambda,z)\to x+\lambda(z-x)\), \(D\) is closed. Taking (3) and (4) into account, we have \(\operatorname{co}\overline{\Phi}(x)\subset D\). Since \(x\notin D\), it follows that \(x\notin \operatorname{co}\overline{\Phi}(x)\) for every \(x\in S\), which contradicts Lemma 2.

Suppose now that for every neighborhood of zero \(V\in\mathcal V(0)\) there exists a point \(x_V\in S\) for which

\[ (x_V+V)\cap \Phi(x_V)\ne\varnothing. \]

Let \(y_V=x_V+z_V\in\Phi(x_V)\), \(z_V\in V\). The generalized sequence \(\{x_V\}\) has an accumulation point \(x\in S\). Consider the set
\(\Delta=\{(U,V)\mid x_V\in U,\ (U,V)\in\mathcal V(x)\times\mathcal V(0)\}\). Since \(x\) is an accumulation point for \(\{x_V\}\), \(\Delta\) forms a direction cofinal with \(\mathcal V(x)\times\mathcal V(0)\). For \(\delta=(U,V)\in\Delta\), putting \(x_\delta=x_V\) and \(z_\delta=z_V\), we shall have
\(x_\delta\to x\), \(z_\delta\to0\), and \(y_\delta=x_\delta+z_\delta\to x\). Then, by virtue of the partial closedness of the mapping \(\Phi\), we have \(L(x,x)=x\in\Phi(x)\). The theorem is proved.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
1 VII 1968

REFERENCES

1. I. L. Glicksberg, Proc. Am. Math. Soc., 3, No. 1, 170 (1952); I. L. Glicksberg, in: Infinite Antagonistic Games, Moscow, 1963, p. 497.
2. L. V. Kantorovich, G. P. Akilov, Functional Analysis in Normed Spaces, Moscow, 1959.