Some generic properties of α-nonexpansive mappings

Let X be a Banach space, C a bounded closed subset of X, A a convex closed subset of X, $\text{E}$ a complete metric space formed by all α-nonexpansive mappings fC → A and $\text{M}$ a complete metric space formed by α-nonexpansive differentiable mappings fC → X. The following assertions are proved in this paper: (1) Properness of I ? f is a generic property in $\text{E}$ (2)the subset of $\text{E}$ formed by all α-contractive mappings is of Baire first category in $\text{E}$; and (3) for every y?X, the functional equation x ? f(x) = y has generically a finite number of solutions for f in $\text{M}$. Some applications to the fixed point theory and calculation of the topological degree are given.     

Some topological properties of weakly almost periodic mappings

A map φ:X→X induces a linear operator T:C(X)→C(X) by composition: Tf(x)=f°φ(x). T and φ are termed weakly almost periodic if the sequence {Tⁿ} is precompact in the weak operator topology. Using general structure theorems for weakly almost periodic operators, the properties of these point maps are studied from the viewpoint of dynamical systems. The structure of individual minimal sets and of the union M of all minimal sets of φ are investigated. One key result is that, if X is compact, then φ is a strongly almost periodic (i.e., has uniformly equicontinuous iterates) homeomorphisms of M and M is a retract of X. These and other general results are applied to the case where X is a manifold. Several results in which weak implies strong almost periodicity are obtained.     

集值映射Pettis-Aumann积分的一些性质

对集值映射的Pettis-Aumann型积分的性质作了讨论,给出了Pettis-Aumann积分型积分的Fatou引理形式以及其它一些收敛定理,探讨了带参数集值映射的积分性质。     

Some properties of isolated singularities of real polynomial mappings

In this note, one of the assumptions of J. Milnor, concerning bundles associated with isolated singularities of real polynomial mappings, is proved. Some additional properties of such bundles are discovered.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 565–572, April, 1973.In conclusion I express gratitude to V. I. Kuz'minov for his attention to this work.     

Some ergodic and rigidity properties of discrete Heisenberg group actions

The goal of this paper is to study ergodic and rigidity properties of smooth actions of the discrete Heisenberg group \(\mathcal{H}\). We establish the decomposition of the tangent space of any C^∞ compact Riemannian manifold M for Lyapunov exponents, and show that all Lyapunov exponents for the central elements are zero. We obtain that if an \(\mathcal{H}\) action contains an Anosov element, then under certain conditions on the eigenvalues of this element, the action of each central element is of finite order. In particular, there is no faithful codimension one Anosov Heisenberg group action on any compact manifold, and there is no faithful codimension two Anosov Heisenberg group action on tori. In addition, we show smooth local rigidity for higher rank ergodic \(\mathcal{H}\) actions by toral automorphisms, using a generalization of the KAM (Kolmogorov–Arnold–Moser) iterative scheme.     

Some Bernstein-type rigidity theorems

We prove some Bernstein-type rigidity theorems for complete submanifolds in a Euclidean space and space-like submanifolds of a Lorentzian space. In particular, we obtain a Bernstein rigidity theorem for complete minimal submanifolds of arbitrary codimension in Euclidean space.     

Some mappings of pfriodic groups

For the rational integera and the rational primeb, letP(a,b) be the set of primesp such that . A natural integerq satisfies (i) iffq is a power product fromP(a,b). In the (additively written) Abelian groupG, g→ag permutes the elements ofG ^# in cycles whose lengths are multiples ofb, but not ofb ², iffG is aπ-group withπ⊂P (a,b). The casea=−2,b=2 has combinational applications.     

Some relationships between induced mappings

We prove some relationships between f, Cn(f), HSn(f) and f², when f, Cn(f), HSn(f) or f² belong to the following mapping classes: monotone, OM, confluent, semi-confluent, weakly confluent, pseudo-confluent, quasi-monotone, weakly monotone or joining.     

Some results for asymptotically pseudo-contractive mappings and asymptotically nonexpansive mappings

Some convergence theorems of modified Ishikawa and Mann iterative sequences with errors for asymptotically pseudo-contractive and asymptotically nonexpansive mappings in Banach space are obtained. The results presented in this paper improve and extend the corresponding results in Goebel and Kirk (1972), Kirk (1965), Liu (1996), Schu (1991) and Chang et al. (to appear).

     

Limit properties of induced mappings

Given a class of mappings f between continua, near- stands for the class of uniform limits of sequences of mappings from . Let 2^f and C(f) mean the induced mappings between hyperspaces. Relations are studied between the conditions: f near-, 2^f near- and C(f) near-. A special attention is paid to the classes of open and of monotone mappings.