Smoothness of the distribution of the norm in uniformly convex Banach spaces

Let (E, · ) be a uniformly convex Banach space of power type. In this paper we investigate differentiability properties of the distribution function of the norm of a random series with one-dimensional independent components inE.     

Locally uniformly convex norms in Banach spaces and their duals

It is shown that a Banach space with locally uniformly convex dual admits an equivalent norm that is itself locally uniformly convex.     

Absolutely convex,uniformly starlike and uniformly convex harmonic mappings

In this paper, we prove necessary and sufficient conditions for a sense-preserving harmonic function to be absolutely convex in the open unit disc. We also estimate the coefficient bound and obtain growth, covering and area theorems for absolutely convex harmonic mappings. A natural generalization of the classical Bernardi-type operator for harmonic functions is considered and its connection between certain classes of uniformly starlike harmonic functions and uniformly convex harmonic functions is also investigated. At the end, as applications, we present a number of results connected with hypergeometric and polylogarithm functions.     

逼近一致凸Banach空间中渐近非扩张映象的不动点

借助于B ruck′s不等式,研究了一致凸Banach空间中渐近非扩张映象不动点的具误差的Ish ikaw a迭代序列的强收敛定理.所得的结果推广和改进了Schu,Rhoades,周海云,王绍荣等作者的相应结果.     

A RANDOM FIXED POINT ITERATION FOR THREE RANDOM OPERATORS ON UNIFORMLY CONVEX BANACH SPACES

In the present paper we introduce a random iteration scheme for three random operators defined on a closed and convex subset of a uniformly convex Banach space and prove its convergence to a common fixed point of three random operators. The result is also an extersion of a known theorem in the corresponding non-random case.     

Locally uniformly rotund points in convex-transitive Banach spaces

Almost transitive superreflexive Banach spaces have been considered in [C. Finet, Uniform convexity properties of norms on superreflexive Banach spaces, Israel J. Math. 53 (1986) 81–92], where it is shown that they are uniformly convex and uniformly smooth. We characterize such spaces as those convex transitive Banach spaces satisfying conditions much weaker than that of uniform convexity (for example, that of having a weakly locally uniformly rotund point). We note that, in general, the property of convex transitivity for a Banach space is weaker than that of almost transitivity.     

Generalized Arrow-Barankin-Blackwell theorems in locally convex spaces

This paper deals with generalizations of the Arrow-Barankin-Blackwell theorem in locally convex spaces, partially ordered by cones whose duals have nonempty quasi-interiors.This research has been partially supported by the World Laboratory, Lausanne, Switzerland, by the Department of Mathematics, University of Pisa, Pisa, Italy, and by the National Natural Sciences Foundation of China. Useful discussions with Professor F. Ferro are gratefully acknowledged.     

The disintegration of the Lebesgue measure on the faces of a convex function

We consider the disintegration of the Lebesgue measure on the graph of a convex function f:Rn→R w.r.t. the partition into its faces, which are convex sets and therefore have a well defined linear dimension, and we prove that each conditional measure is equivalent to the k-dimensional Hausdorff measure on the k-dimensional face on which it is concentrated. The remarkable fact is that a priori the directions of the faces are just Borel and no Lipschitz regularity is known. Notwithstanding that, we also prove that a Green-Gauss formula for these directions holds on special sets.     

A note on the duality mapping of a locally uniformly convex Banach space

Let X be a real locally uniformly convex Banach space with normalized duality mapping J:X→2^X*. The purpose of this note is to show that for every R>0 and every x₀X there exists a function , which is nondecreasing and such that (r)>0 for r>0,(0)=0 and

for all . Simply, it is shown that the necessity part of the proof of the original analogous necessary and sufficient condition of Prüß, for real uniformly convex Banach spaces, goes over equally well in the present setting. This is a natural setting for the study of many existence problems in accretive and monotone operator theories.     

The structure of functions satisfying the law of large numbers in a class of locally convex spaces

For each function that satisfies the law of large numbers with values in a certain class of locally convex spaces with the Radon-Nikodym property the following decomposition holds: , where is integrable by seminorm, and is a Pettis integrable function which is scalarly 0.

     

On p-adic vector measure spaces

For a separating algebra R of subsets of a set X, E a complete Hausdorff non-Archimedean locally convex space and m:R→E a bounded finitely additive measure, we study some of the properties of the integrals with respect to m of scalar-valued functions on X. The concepts of convergence in measure, with respect to m, and of m-measurable functions are introduced and several results concerning these notions are given.     

The alternating algorithm in a uniformly convex and uniformly smooth Banach space

Let X   be a uniformly convex and uniformly smooth Banach space. Assume that the _Mi$M_i$, i=1,…,r$i=1,\dots ,r$, are closed linear subspaces of X  , _PMi$P_{M_i}$ is the best approximation operator to the linear subspace _Mi$M_i$, and M:=_M1+?+_Mr$M:=M_1+?+M_r$. We prove that if M is closed, then the alternating algorithm given by repeated iterations of

(I−_PMr)(I−_PMr−1)?(I−_PM1)$(I-P_{M_r})(I-P_{M_{r-1}})?(I-P_{M_1})$

applied to any x∈X$x\in X$ converges to x−_PMx$x-P_{M}x$, where _PM$P_M$ is the best approximation operator to the linear subspace M  . This result, in the case r=2$r=2$, was proven in Deutsch [4].     

On order of strongly starlikeness in the class of uniformly convex functions

In this paper we consider the order of strongly starlikeness in the classes uniformly convex functions.     

The structure of measurable mappings with values in locally convex spaces

The purpose of this paper is to show that a theorem of A. Wisniewski remains valid without the approximation property.

     

The order of starlikeness of the class of uniformly starlike functions

We investigate classes of k‐uniformly starlike functions which generalize the class UST introduced by Goodman in 1991. Obtained results enable us to solve the problem of finding the largest such that the class of k‐uniformly starlike functions is contained in the class of starlike functions of order α.     

The sets of monomorphisms and of almost open operators between locally convex spaces

If the set of monomorphisms between locally convex spaces is not empty, then it is an open subset of the space of all continuous and linear operators endowed with the topology of the uniform convergence on the bounded sets if and only if the domain space is normable. The corresponding characterization for the set of almost open operators is also obtained; it is related to the lifting of bounded sets and to the quasinormability of the domain space. Other properties and examples are analyzed.

     

Approximating fixed points of a pair of contractive type mappings in generalized convex metric spaces

In this paper, we consider an Ishikawa type iteration process with errors, which converges to the unique common fixed point of a pair of contractive type mappings in complete generalized convex metric spaces. Furthermore, we obtain the corresponding results in Banach spaces. Our results extend and generalize many known results.