Banach空间中多类1-集-压缩映象的逼近定理和不动点定理

本文证明了 Ｋｙ Ｆａｎ定理［１］对定义在 Ｂａｎａｃｈ空间中含有内点的有界闭凸子集上的１－集－压缩映象正确．１－集－压缩映象类包含凝聚映象、非扩张映象、半压缩映象、局部几乎非扩张映象和其它映象．作为定理的应用,得到了一些非自映象在许多众知边界条件下的不动点定理,改进和推广了许多作者的最近结果．     

Approximation Theorems and Fixed Point Theorems for Various Classes of 1-set-contractive Mappings in Banach Spaces

In this paper, we will prove that Ky Fan's Theorem (Math. Z. 112(1969), 234–240) is true for 1-set-contractive maps defined on a bounded closed convex subset K in a Banach space with intK≠0. This class of 1-set-contractive maps includes condensing maps, nonexpansive maps, semicontractive maps, LANE maps and others. As applications of our theorems, some fixed point theorems of non-selfmaps are proved under various well-known boundary conditions. Our results are generalizations and improvements of the recent results obtained by many authors. Project supported by the National Natural Science Foundation of China and Natural Science Foundation of Shandong Province of China     

Some random approximations and random fixed point theorems for 1-set-contractive random operators

In this paper, we will prove that the random version of Fan's Theorem (Math. Z. 112 (1969), 234-240) is true for 1-set-contractive random operator , where is a weakly compact separable closed ball in a Banach space and is a measurable space. This class of 1-set-contractive random operator includes condensing random operators, semicontractive random operators, LANE random operators, nonexpansive random operators and others. As applications of our theorems, some random fixed point theorems of non-self-maps are proved under various well-known boundary conditions.

     

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.     

随机算子的若干新结果

本文研究了随机半闭1－集压缩算子和随机凝聚算子的随机不动点问题，推广了Leggett和Williams的若干定理。     

Random fixed points of discontinuous random maps

We prove some random fixed-point theorems for random maps which are not necessarily continuous. This may lead to the discovery of some new results in random fixed-point theory for discontinuous maps.     

Robinson－Ursescu Theorem in p－normed Spaces

For a convex set-valued map between p-normed (0 < p ≤ 1) spaces, we give a criterion for its inverse to be locally Lipschitz of order p. From this we obtain the Robinson-Ursescu Theorem in p-normed spaces and the open mapping and closed graph theorems for closed convex set-valued maps.     

Deterministic and random coincidence point results for f-nonexpansive maps

Some deterministic and random coincidence theorems for f-nonexpansive maps are obtained. As applications, invariant approximation theorems are derived. Our results unify, extend and complement various known results existing in the literature.     

（1.2） INVERSES OF OPERATORS BETWEEN BANACH SPACES AND LOCAL CONJUGACY THEOREM

1.(1.2)InverseandLocalFinePropertyofaFamilyofOperatorsTxLetEandFbebothBanachspaces,andB(E,F)thesetofallboundedlinearoperatorsfromEintoFAnoperatorT B(F,E)issaidtobea(1.2)inverseofTifTT T=TandT TT =T .IfT satisfiesonlythefirstcondition,thenT issaidtobe...     

Uniform random colored complexes

We present here random distributions on (D + 1)‐edge‐colored, bipartite graphs with a fixed number of vertices 2p. These graphs encode D‐dimensional orientable colored complexes. We investigate the behavior of those graphs as p→∞. The techniques involved in this study also yield a Central Limit Theorem for the genus of a uniform map of order p, as p→∞.     

A non-integral-dimensional random walk

We present a space-homogeneous, time-inhomogeneous random walk that behaves as if it were a simple random walk ind dimensions, whered is not necessarily an integer. Analogues of the Local Central Limit Theorem, Zero-One Laws, distance, angle, asymptotics on the Green's function and the hitting probability, recurrence and transience, and results about the intersection behavior of the random walk paths are obtained.     

Continuity of a class of random time transformations

A random time change is defined as a map from one function space to another. The continuity of this map is investigated. Applications are made to weak limit theorems of random processes.     

Anomalous behaviour for the random corrections to the cumulants of random walks in fluctuating random media

The Central Limit Theorem for a model of discrete-time random walks on the lattice ℤ^ν in a fluctuating random environment was proved for almost-all realizations of the space-time nvironment, for all ν > 1 in [BMP1] and for all ν≥ 1 in [BBMP]. In [BMP1] it was proved that the random correction to the average of the random walk for ν≥ 3 is finite. In the present paper we consider the cases ν = 1,2 and prove the Central Limit Theorem as T→∞ for the random correction to the first two cumulants. The rescaling factor for theaverage is for ν = 1 and (ln T), for ν=2; for the covariance it is , ν = 1,2. Received: 25 November 1999 / Revised version: 7 June 2000 / Published online: 15 February 2001     

Limit theorems for random transformations and processes in random environments

I derive general relativized central limit theorems and laws of iterated logarithm for random transformations both via certain mixing assumptions and via the martingale differences approach. The results are applied to Markov chains in random environments, random subshifts of finite type, and random expanding in average transformations where I show that the conditions of the general theorems are satisfied and so the corresponding (fiberwise) central limit theorems and laws of iterated logarithm hold true in these cases. I consider also a continuous time version of such limit theorems for random suspensions which are continuous time random dynamical systems.

     

随机变数配重和的两个收敛定理

本文给出随机变数配重和几乎必然收敛的两个定理,同时讨论了可分赋范线性空间中随机元的相应推广。 定理1 设随机变数序列{x_n}的律一致地以一个随机变数X的律为界,即对所有n和     

Topological degree for 1-set-contractive fields in M-PN spaces and its applications

In this paper, we define the topological degree for 1-set-contractive fields in PN spaces. Based on this, we obtain some new fixed point theorems for 1-set-contractive operators. As an application, we study the existence of solutions for a kind of nonlinear Volterra integral equations in Z-M-PN space.     

Scaling limits of random bipartite planar maps with a prescribed degree sequence

We study the asymptotic behavior of uniform random maps with a prescribed face‐degree sequence, in the bipartite case, as the number of faces tends to infinity. Under mild assumptions, we show that, properly rescaled, such maps converge in distribution toward the Brownian map in the Gromov–Hausdorff sense. This result encompasses a previous one of Le Gall for uniform random q‐angulations where q is an even integer. It applies also to random maps sampled from a Boltzmann distribution, under a second moment assumption only, conditioned to be large in either of the sense of the number of edges, vertices, or faces. The proof relies on the convergence of so‐called “discrete snakes” obtained by adding spatial positions to the nodes of uniform random plane trees with a prescribed child sequence recently studied by Broutin and Marckert. This paper can alternatively be seen as a contribution to the study of the geometry of such trees.     

Absolutely continuous invariant measures for random maps with position dependent probabilities

A random map is discrete-time dynamical system in which one of a number of transformations is randomly selected and applied at each iteration of the process. Usually the map τ_k is chosen from a finite collection of maps with constant probability p_k. In this note we allow the p_k's to be functions of position. In this case, the random map cannot be considered to be a skew product. The main result provides a sufficient condition for the existence of an absolutely continuous invariant measure for position dependent random maps on [0,1]. Geometrical and topological properties of sets of absolutely continuous invariant measures, attainable by means of position dependent random maps, are studied theoretically and numerically.