The paper by Ky Fan discusses fixed-point and minimax theorems in locally convex topological linear spaces. Fan generalizes both Kakutani's and Tychonoff's fixed-point theorems to a broader class of spaces. Specifically, he considers a compact convex set \( K \) in a locally convex topological linear space \( L \) and an upper semicontinuous point-to-set transformation \( f \) from \( K \) into the family of all closed convex subsets of \( K \). Fan proves that there exists a point \( x_0 \in K \) such that \( x_0 \in f(x_0) \).
The paper also includes applications of this theorem, such as Theorem 2, which extends von Neumann's minimax theorem to locally convex topological linear spaces, and Theorem 3, which generalizes Ville's minimax theorem. Additionally, Fan provides a result on systems of linear equations with dominant diagonal coefficients, showing that if the coefficients satisfy a certain condition, the system has a solution within the interval \([-1, 1]\).
The proofs rely on several lemmas and theorems from the literature, including those by Eckmann, Whitehead, and others, and Fan's own contributions to the field.The paper by Ky Fan discusses fixed-point and minimax theorems in locally convex topological linear spaces. Fan generalizes both Kakutani's and Tychonoff's fixed-point theorems to a broader class of spaces. Specifically, he considers a compact convex set \( K \) in a locally convex topological linear space \( L \) and an upper semicontinuous point-to-set transformation \( f \) from \( K \) into the family of all closed convex subsets of \( K \). Fan proves that there exists a point \( x_0 \in K \) such that \( x_0 \in f(x_0) \).
The paper also includes applications of this theorem, such as Theorem 2, which extends von Neumann's minimax theorem to locally convex topological linear spaces, and Theorem 3, which generalizes Ville's minimax theorem. Additionally, Fan provides a result on systems of linear equations with dominant diagonal coefficients, showing that if the coefficients satisfy a certain condition, the system has a solution within the interval \([-1, 1]\).
The proofs rely on several lemmas and theorems from the literature, including those by Eckmann, Whitehead, and others, and Fan's own contributions to the field.