THREE KINDS OF DENTABILITIES IN BANACH SPACES AND THEIR APPLICATIONS

In this paper,we study some dentabilities in Banach spaces which are closely related to the famous Radon-Nikodym property.We introduce the concepts of the weak^(*)-weak denting point and the weak^(*)-weak^(*)denting point of a set.These are the generalizations of the weak^(*)denting point of a set in a dual Banach space.By use of the weak^(*)-weak denting point,we characterize the very smooth space,the point of weak^(*)-weak continuity,and the extreme point of a unit ball in a dual Banach space.Meanwhile,we also characterize an approximatively weak compact Chebyshev set in dual Banach spaces.Moreover,we define the nearly weak dentability in Banach spaces,which is a generalization of near dentability.We prove the necessary and sufficient conditions of the reflexivity by nearly weak dentability.We also obtain that nearly weak dentability is equivalent to both the approximatively weak compactness of Banach spaces and the w-strong proximinality of every closed convex subset of Banach spaces.

K-very Smooth Banach Spaces

We introduce the notion of K-very smoothness which is a generalization of very smoothness in Banach spaces. A necessary and sufficient condition for a Banach space to be K-very smooth is obtained. We also consider some relations between K-very smoothness and other geometrical notions.

ITERATIVE APPROXIMATION OF FIXED POINTS FOR MULTIVALUED OPERATORS OF THE MONOTONE TYPE IN UNIFORMLY SMOOTH BANACH SPACES

In this paper, let K be a nonempty subset of a uniformly smooth Banach space X, and T:K→2~k be a multivalued operator of the monotone type. The iterative sequence which converges strongly to the unique fixed point of T is given. Our results are the extension and improvements of the results obtained previously by several authors including Dunn, Chidume, Deng and Ding.

A NOTE ON SINGULAR INTEGRALS WITH DOMINATING MIXED SMOOTHNESS IN TRIEBEL-LIZORKIN SPACES

Let h, be a measurable function defined on R^＋ ×R^＋. Let Ω ∈ L（log L^＋）^υq （S^n1-1 × S^n2-1） （1≤ υq ≤ 2） be homogeneous of degree zero and satisfy certain cancellation conditions. We show that the singular integral Tf（x1,x2）=p.v.∫∫R^n1＋n2 Ω（y′1,y′2）h（｜y1｜,｜y2｜）/｜y1｜^n1｜y2｜^n2 f（x1-y1,x2-y2）dy1dy2maps from Sp,q^α1,α2F（R^n1×R^n2）boundedly to itself for 1 〈 p, q 〈 ∞, α1, α2 ∈R.

PRECISE SMOOTHING EFFECTS OF HOMOGENEOUS LANDAU EQUATION IN SOBOLEV SPACES

In this work, we study the smoothing effect of Cauchy problem in Sobolev space for the spatially homogeneous Landau equation in the Maxwellian case. We obtain a precise estimate with respect to time variable, which implies the ultra-analytic effect of weak solutions.

ISHIKAWA ITERATIVE PROCESS IN UNIFORMLY SMOOTH BANACH SPACES

Let E be a uniformly smooth Banach space, K be a nonempty closed convex subset of E, and suppose: T: K --> K is a continuous Phi-strongly pseudocontractive operator with a bounded range. Using a new analytical method, under general cases, the Ishikawa iterative process {x(n)} converges strongly to the unique fixed point x* of the operator T were proved. The paper generalizes and extends a lot of recent corresponding results.

A New System of Generalized Variational Inclusions Involving H-η-monotone Operators in Uniformly Smooth Banach Spaces

In this paper, we introduce and study a new system of generalized vari- ational inclusions involving H-η-monotone operators in uniformly smooth Banach spaces. Using the resolvent operator technique associated with H-η-monotone opera- tors, we prove the approximation solvability of solutions using an iterative algorithm. The results in this paper extend and improve some known results from the literature.

CHARACTERIZATION OF HLDER SPACES CORRESPONDING TO DITZIAN-TOTIK MODULUS OF SMOOTHNESS

For the step-weight function , we prove that the Holder spaces ∧a,p on the interval [-1,1], defined in terms of moduli of smoothness with the step-weight function ,are linearly isomorphic to some sequence spaces, and the isomorphism is given by the cofficients of function with respect to a system of orthonormal splines with knots uniformly distributed according to the measure with density . In case ∧a,p is contained in the space of continuous functions, we give a discrete characterization of this space, using only values of function at the appropriate knots. Application of these results to characterize the order of polynomial approximation is presented.

Notes of k-smoothness in Orlicz sequence spaces

The criterion for k-smooth points of the Orlicz sequence space endowed with the Orlicz norm is proved. The necessary and sufficient conditions of k-smoothness of l M and l （M ） are obtained, respectively. Finally, we give the counterexamples which show that previous results are not true.

Iterative Approximation of Fixed Points for Uniformly Continuous and Strongly Pseudocontractive Mappings in Smooth Banach Spaces

With an inequality and some analysis techniques,iterative approximation of fixed points for uniformly continuous and strongly pseudocontractive mappings in smooth Banach spaces is studied,and the recent corresponding results of Chidume are improved.

Variational Analysis Based on Proximal Subdifferential on Smooth Banach Spaces

This paper first shows that for any p∈(1,2)there exists a continuously differentiable function f on l^(p)(and L^(p))such that the proximal subdifferential of f is empty everywhere,and hence it is not suitable to develop theory on proximal subdifferential in the classical Banach spaces l^(P)and L^(P) with p∈(1,2).On the other hand,this paper establishes variational analysis based on the proximal subdifferential in the framework of smooth Banach spaces of power type 2,which conclude all Hilbert spaces and all the classical spaces l^(P)and L^(P)with p∈(2,+∞).In particular,in such a smooth space,we provide the proximal subdifferential rules for sum functions,product functions,composite functions and supremum functions,which extend the basic results on the proximal subdifferential established in the framework of Hilbert spaces.Some of our main results are new even in the Hilbert space case.As applications,we provide KKT-like conditions for nonsmooth optimization problems in terms of proximal subdifferential.

CARLESON MEASURES AND THE GENERALIZED CAMPANATO SPACES OF VECTOR-VALUED MARTINGALES

In this paper, the so-called（p,Ф）-Carleson measure is introduced and the rela-tionship between vector-valued martingales in the general Campanato spaces Lp,Ф（X） and the （p, Ф）-Carleson measures is investigated. Specifically, it is proved that for q ∈ [2, ∞）, the measure d# ：-=││ dfk││^qdP dm is a （q, Ф）-Carleson measure on Ω × N for every f ∈ Lq,Ф（X） if and only if X has an equivalent norm which is q-uniformly convex; while for p C （1, 2], the measure dμ ：=││dfk││^pP dm is a （p, Ф）-Carleson measure on Ω ×N implies that f ∈ Lp,Ф（X） if and only if X admits an equivalent norm which is p-uniformly smooth. This result extends an earlier result in the literature from BMO spaces to general Campanato spaces.

STRONG CONVERGENCE OF APPROXIMATED SEQUENCES FOR NONEXPANSIVE MAPPINGS IN BANACH SPACES

This paper studies the convergence of the sequence defined by x0 ∈ C, xn＋l =αnu＋（1-αn）Txn, n=0, 1,2,..., where 0 ≤αn ≤ 1, limn→∞ αn = 0, ∑n=0^∞ αn = ∞, and T is a nonexpansive mapping from a nonempty closed convex subset C of a Banach space X into itself. The iterative sequence {xn} converges strongly to a fixed point of T in the case when X is a uniformly convex Banach space with a uniformly Gateaux differentiable norm or a uniformly smooth Banach space only. The results presented in this paper extend and improve some recent results.

A New Class of Banach Spaces with Uniform Normal Structure

Let X be a Banach space, S(X) be the unit sphere of X, φ be a function: S(X)→ S(X *) such that φ(x)∈ x, and v φ(ε) =inf 1-12x+y: x,y∈S(X), and 〈φ(x), x-y 〉≥ε, 0≤ε≤2, where x is the set of norm 1 supporting functionals of S(X) at x. A geometric concept, modulus of V convexity V(ε)= sup {V φ(ε), for all φ: S(X)→S(X *)}, is introduced; the properties of V(ε) and the relationship between V(ε) and other geometric concepts are discussed. The main result is that V12>0 implies normal structure.

Strong convergence theorems for nonexpansive semi-groups in Banach spaces

Some strong convergence theorems of explicit composite iteration scheme for nonexpansive semi-groups in the framework of Banach spaces are established. Results presented in the paper not only extend and improve the corresponding results of ShiojiTakahashi, Suzuki, Xu and Aleyner-Reich, but also give a partially affirmative answer to the open questions raised by Suzuki and Xu.

MODE SPACE SMOOTHING ALGORITHM TO ESTIMATE DOA OF COHERENT SIGNALS IMPINGING ON UNIFORM CIRCULAR ARRAY

A novel method to estimate DOA of coherent signals impinging on a uniform circular array( UCA) is presented in this paper. A virtual uniform linear array (VULA) is first derived by using spatial DFT technique, transforming the UCA from element space to phase mode space to obtain the properties of ordinary ULA, and then the well known spatial smoothing technique is applied to the VULA so that the lost rank of covariance matrix due to signal coherence can be retrieved. This method makes it feasible to use the simple MUSIC algorithm to estimate DOA of coherent signals impinging on a UCA without heavy computation burden. Simulation results strongly verify the effectiveness of the algorithm.

GENERALIZED BESOV SPACES AND TRIEBEL-LIZORKIN SPACES

In this paper the classical Besov spaces B^sp.q and Triebel-Lizorkin spaces F^sp.q for s∈R are generalized in an isotropy way with the smoothness weights { ｜2j｜^α→ln }7=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by B^α→p.q and F^α→p.q for α^→ E Nk and k ∈N, respectively, keep many interesting properties, such as embedding theorems （with scales property for all smoothness weights）, lifting properties for all parameters 5, and duality for index 0 〈 p 〈∞ By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between B^sp.q and ∪t〉s B^tp.q, and between F^sp.q and ∪t〉s F^tp.q, respectively.

A New System of Variational Inclusions with （H,η）-accretive Operators in Banach Spaces

在这份报纸，我们首先介绍说出的概括 accretive 操作员的一个新班( H ，)在学习性质的 Banach space.By 的 -accretive ( H ，) -accretive,we 扩大与 m-accretive 操作员联系到的溶剂操作员的概念新( H ，)新溶剂操作员技术的 -accretive operators.In 术语，我们证明变化 inclusions.We 的这个新系统的解决方案的存在和唯一也为接近构造一个新算法这个系统和 dis

A system of general nonlinear variational inclusions in Banach spaces

A new system of general nonlinear variational inclusions is introduced and studied in Banach spaces. An iterative algorithm is developed and analyzed by use of the resolvent operator techniques to find the approximate solutions of the system of general nonlinear variational inclusions involving different nonlinear operators in uniformly smooth Banach spaces.