Fixed points and quasi-equilibrium problems

A fixed-point theorem for compact acyclic maps defined on convex subsets of notnecessarily locally convex topological vector spaces is applied to the existence of solutions of quasi-equilibrium problems. Such existence theorems extend known ones which were used for unified approaches to quasi-variational inequalities in [1–5] and others.     

Ergodic theorems for noncommutative dynamical systems

Recently, E.C. Lance extended the pointwise ergodic theorem to actions of the group of integers on von Neumann algebras. Our purpose is to extend other pointwise ergodic theorems to von Neumann algebra context: the Dunford-Schwartz-Zygmund pointwise ergodic theorem, the pointwise ergodic theorem for connected amenable locally compact groups, the Wiener's local ergodic theorem for ₊ ^d and for general Lie groups.     

A fixed point theorem for weakly Zamfirescu mappings

In [5], Zamfirescu (1972) gave a fixed point theorem that generalizes the classical fixed point theorems by Banach, Kannan, and Chatterjea. In this paper, we follow the ideas of Dugundji and Granas to extend Zamfirescu’s fixed point theorem to the class of weakly Zamfirescu maps. A continuation method for this class of maps is also given.     

Comments on some fixed point theorems in hyperconvex metric spaces

In this paper, we show that a number of known fixed point theorems for the Fan-Browder type maps or acyclic maps defined on (subsets of) hyperconvex metric spaces are simple consequences of the previously known theorems for corresponding maps defined on generalized convex spaces.     

Fixed-point theorems for families of weakly non-expansive maps

In this paper, we present some fixed-point theorems for families of weakly non-expansive maps under some relatively weaker and more general conditions. Our results generalize and improve several results due to Jungck [G. Jungck, Fixed points via a generalized local commutativity, Int. J. Math. Math. Sci. 25 (8) (2001) 497-507], Jachymski [J. Jachymski, A generalization of the theorem by Rhoades and Watson for contractive type mappings, Math. Japon. 38 (6) (1993) 1095-1102], Guo [C. Guo, An extension of fixed point theorem of Krasnoselski, Chinese J. Math. (P.O.C.) 21 (1) (1993) 13-20], Rhoades [B.E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977) 257-290], and others.     

SST空间中压缩映射的几个不动点定理

本文给出了SST概率度量空间中压缩型映射的几个不动点定理,改进并推广了张石生[2],方锦暄[3]及陈肇姜[4]中的许多重要结果.     

集值变分不等式问题的例外簇

本文首先在Banach空间中证明了零调集值映射的一个Leray-Schauder型不动点定理,然后在Hilbert空间中定义了零调集值映射的变分不等式的例外簇,利用本文给出的不动点定理给出了无界集上的变分不等式问题存在解的一个充分条件．此条件弱于许多已知的关于变分不等式问题的解的存在性条件,并由此得到Hilbert空间中几个变分不等式约解的存在性定理．     

A measurable selection theorem for compact-valued maps

The main theorem of this paper establishes the existence of measurable selections for compact-valued multifunctions whose range space is a regular Hausdorff space which need neither be metrizable nor satisfy any restriction on its weight. It is shown that the selection theorems of Sion [16], Hasumi [10], and one of the author (cf. [8]) are immediate consequences of this general result. Moreover some new results concerning Borel and Baire property selections for upper semi-continuous compact-valued maps are deduced.     

Continuity and Schatten–von Neumann Properties for Localization Operators on Modulation Spaces

We use sharp convolution estimates for weighted Lebesgue and modulation spaces to obtain an extension of the celebrated Cordero-Gröchenig theorems on boundedness and Schatten–von Neumann properties of localization operators on modulation spaces. We also give a new proof of the Weyl connection based on the kernel theorem for Gelfand–Shilov spaces.     

On general best proximity pairs and equilibrium pairs in free abstract economies

In this paper, using Lassonde’s fixed point theorem for Kakutani factorizable multifunctions and Park’s fixed point theorem for acyclic factorizable multifunctions, we will prove new existence theorems for general best proximity pairs and equilibrium pairs for free abstract economies, which generalize the previous best proximity theorems and equilibrium existence theorems due to Srinivasan and Veeramani [P.S. Srinivasan, P. Veeramani, On best approximation pair theorems and fixed point theorems, Abstr. Appl. Anal. 2003 (1) (2003) 33–47; P.S. Srinivasan, P. Veeramani, On existence of equilibrium pair for constrained generalized games, Fixed Point Theory Appl. 2004 (1) (2004) 21–29], and Kim and Lee [W.K. Kim, K.H. Lee, Existence of best proximity pairs and equilibrium pairs, J. Math. Anal. Appl. 316 (2006) 433–446] in several aspects.     

Substochastic matrices and von Neumann majorization

Several classes of substochastic matrices are introduced in higher dimensions, and some theorems about their extreme points are presented. An extension of a theorem of von Neumann concerning doubly substochastic matrices is discussed, and the classes for which this extension remains valid are determined.     

Common fixed point theorems of Gregus type for weakly compatible mappings satisfying contractive conditions of integral type

We prove a common fixed point theorem of Gregus type for four mappings satisfying a contractive condition of integral type in metric spaces using the concept of weak compatibility which generalizes Theorem 2 of [A. Djoudi, L. Nisse, Gregus type fixed points for weakly compatible mappings, Bull. Belg. Math. Soc. 10 (2003) 369-378] and other papers. We prove also common fixed point theorems of Gregus type using a strict contractive condition of integral type, a property (E.A) and a common property (E.A) introduced by [M. Aamri, D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002) 181-188] and [W. Liu, J. Wu, Z. Li, Common fixed points of single-valued and multi-valued maps, Int. J. Math. Math. Sci. 19 (2005) 3045-3055], respectively.     

Calderón-Zygmund operaors on the Hardy spaces of weighted Herz type

Applying the decomposition theorems in [1] and [2], we obtain the boundedness theorem of Calderón-Zygmund operator of type δ on the Hardy spaces of weighted Herz type and establish interpolation theorem of linear operators on the weighted Herz spaces. Supported by NSF of China and the Fund of Doctoral Program of N.E.C.     

Produit croisé d'une algèbre de von Neumann par une algèbre de Kac

We introduce a notion of action of a Kac algebra (see [2, 9]) on a von Neumann algebra, and the cross-product of a von Neumann algebra by a Kac algebra with respect to an action α. The results of Takesaki [11, Chaps. 3 and 4] are then generalized, particularly the theorem of the double cross-product.     

Proximity maps, best approximations and fixed points

In [Canada Math. Bull. 22, pp. 513–515, 1979], the author extended an interesting (best approximation) theorem of Ky Fan [see Theorem 2, Math. Z. 112, pp. 234–240] to condensing maps defined on a closed ball in a Banach space and defined on a closed convex subset in a Hilbert space. In [J. Approximation Theory 52, pp. 141–148, 1988], the author and Yen further extended it to 1-set-contractive maps in a Hilbert space. In this paper, we will address several related questions. First, we extend Lin and Yens’ result for 1-set-contractive maps but to satisfy weaker condition, which allows us to include generalized contraction maps. Second, we give ways to construct the best approximation (not just the existence results) for nonexpansive maps. Third, we give conditions on when the best approximation {u_n|n=1,2,…} of a sequence of contractions {f_n|n=1,2,…} will converge to the best approximation u of a contraction f. Fourth, common best approximation theorems for nonexpansive maps are proved. Applications to fixed point theorems for weakly inward maps and others are given for all those four topics.     

Proximity maps, best approximations and fixed points

In [Canada Math. Bull. 22, pp. 513–515, 1979], the author extended an interesting (best approximation) theorem of Ky Fan [see Theorem 2, Math. Z. 112, pp. 234–240] to condensing maps defined on a closed ball in a Banach space and defined on a closed convex subset in a Hilbert space. In [J. Approximation Theory 52, pp. 141–148, 1988], the author and Yen further extended it to 1-set-contractive maps in a Hilbert space. In this paper, we will address several related questions. First, we extend Lin and Yens’ result for 1-set-contractive maps but to satisfy weaker condition, which allows us to include generalized contraction maps. Second, we give ways to construct the best approximation (not just the existence results) for nonexpansive maps. Third, we give conditions on when the best approximation {u_n|n=1,2,…} of a sequence of contractions {f_n|n=1,2,…} will converge to the best approximation u of a contraction f. Fourth, common best approximation theorems for nonexpansive maps are proved. Applications to fixed point theorems for weakly inward maps and others are given for all those four topics.     

The matching theorems in G-convex spaces with applications

The purpose of this paper is to establish some new matching theorems in G-convex spaces and, as applications, to obtain some new fixed point theorems, section theorems and a minimax theorem in G-convex spaces. The results presented in this paper improve and generalize the corresponding results in [1], [2], [3], [4], [5], [7], [8], [9], [10], [11] and [12].     

Random fixed point theorems for composites of acyclic multifunctions

In this paper we obtain random versions of Kakutani–Fan type fixed point theorems for a class V _c ⁺ of multifunctions which contains Kakutani factorizable maps and composites of acyclic maps. As applications, we derive some random approximation theorems.     

A robust von Neumann minimax theorem for zero-sum games under bounded payoff uncertainty

The celebrated von Neumann minimax theorem is a fundamental theorem in two-person zero-sum games. In this paper, we present a generalization of the von Neumann minimax theorem, called robust von Neumann minimax theorem, in the face of data uncertainty in the payoff matrix via robust optimization approach. We establish that the robust von Neumann minimax theorem is guaranteed for various classes of bounded uncertainties, including the matrix 1-norm uncertainty, the rank-1 uncertainty and the columnwise affine parameter uncertainty.     

Anmerkungen zu einem lemma von hoàng tuy

In [1], [2] HOÀNG TUY gave an approach to the main theorems of the convex analysis and the convex optimization being based on a lemma; he has proved it by means o induction. In [1] the equivalence of the main theorems of convex optimization given in [1], [2] does not use a separation theorem or equivalent statements. In this note the author has proved that the lemma of HOÀNG TUY can be characterized as a special separation theorem and be obtained from a separation theorem of Eidelheit. That means that the lemma is equivalent to the theorem of Hahn-Banach.