Three-circles theorems

Journal d'Analyse Mathématique -     

Coincidence theorems and minimax theorems

In the present paper a few coincidence theorems and minimax theorems are given so as to unify and strengthen the corresponding results in [3], [4] and [5].The Project Supported by National Natural Science Foundation of China.     

Coincidence theorems and minimax theorems (II)

A coincidence theorem for a single-valued mapping in [3] is generalized as one for a strongly decomposable multivalued mapping. Project supported by the National Natural Science Foundation of China     

Tauberian theorems

Tauberian constants and estimates are calculated for the difference of two linear transforms from the form (1.1) of the same function satisfying Tauberian conditions. Applications for number-sequences, and connections with previous results are shown.     

Decomposition theorems

The countable-decomposition theorem for linear functionals has become a useful tool in the theory of representing measures (see [4–7]). The original proof of this theorem was based on a rather involved study of extreme points in the state space of a convex cone. Recently M. Neumann [9] gave an independent proof using a refined form of Simons convergence lemma and Choquet's theorem. In this paper a (relatively) short proof of an extension (to a more abstract situation) of the countable-decomposition theorem is given. Furthermore a decomposition criterion is obtained which even works in the case when not all states are decomposable. All the work is based on a complete characterization of those states which are partially decomposable with respect to a given sequence of sublinear functionals.     

Generalized KKM theorems and common fixed point theorems

In this paper, we first prove a generalized KKM theorem, and then use this generalized KKM theorem to establish the generalized equi-KKM theorem, common fixed point theorems for a family of multivalued maps, and the Kakutani-Fan-Glicksberg fixed point theorem. We also show that an existence theorem of the common fixed point theorem is equivalent to the Kakutani-Fan-Glicksberg fixed point theorem.     

Separation theorems and minimax theorems for fuzzy sets

In this paper, we prove some theorems on fuzzy sets. We first show that, in order to demonstrate that the equality of shadows ofA andB implies the equality ofA andB, it is necessary to assume thatA andB are closed and thatS _H (A)=S _H (B) for any closed hyperplane hyperplaneH. We also obtain a separation theorem for two convex fuzzy sets in a Hilbert space. Finally, we investigate results relating to minimax theorems for fuzzy sets. We obtain a necessary and sufficient condition for compactness.The authors wish to express their sincere thanks to Professor Hisaharu Umegaki for his invaluable suggestions and advice.     

Multiplier theorems

We state and prove a multiplier theorem for a central element A of ZG, the group ring over Z of a group G. This generalizes most previously known multiplier theorems for difference sets and divisible difference sets. We also provide applications to show that our theorem provides new multipliers and establish the nonexistence of a family of divisible difference sets which correspond to elliptic semiplanes admitting a regular collineation group. © 1995 John Wiley & Sons, Inc.     

Comparison theorems and existence theorems for ordinary differential equations

We are concerned with uniqueness and existence theorems for two point boundary value problems for the nonlinear differential equation Ly = f(x, y), where L is the classical nth order linear differential operator. In proving our results interesting comparison theorems are proven for linear differential equations.     

Abstract Kuhn-Tucker theorems

In this note, we discuss abstract versions of the Kuhn-Tucker theorem for constrained convex minimization problems, in which we relax both the conditions of convexity and also the requirement that the number of constraints be finite.     

Measurable extension theorems

This paper concerns the use of measurable selection techniques to obtain some measurable extension theorems.     

Nonconventional limit theorems

The polynomial ergodic theorem (PET) which appeared in Bergelson (Ergod. Th. Dynam. Sys. 7, 337–349, 1987) and attracted substantial attention in ergodic theory studies the limits of expressions having the form ${1/N\sum_{n=1}^NT^{q_1(n)}f_1 \cdots T^{q_\ell (n)} f_\ell}$ where T is a weakly mixing measure preserving transformation, f _i ’s are bounded measurable functions and q _i ’s are polynomials taking on integer values on the integers. Motivated partially by this result we obtain a central limit theorem for even more general expressions of the form ${1/\sqrt{N}\sum_{n=1}^N (F(X_0(n),X_1(q_1(n)),X_2(q_2(n)), \ldots, X_\ell(q_\ell(n)))-\overline F)}$ where X _i ’s are exponentially fast ψ-mixing bounded processes with some stationarity properties, F is a Lipschitz continuous function, ${\overline F=\int Fd(\mu_0\times\mu_1\times \cdots\times\mu_\ell)}$ , μ _j is the distribution of X _j (0), and q _i ’s are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when q _i ’s are polynomials of growing degrees. When F(x ₀, x ₁, . . . , x _? ) = x ₀ x ₁ x ₂ . . . x _? exponentially fast α-mixing already suffices. This result can be applied in the case when X _i (n) = T ⁿ f _i where T is a mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well, as in the case when X _i (n) = f _i (ξ _n ) where ξ _n is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.