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This paper first shows that for any p ∈ (1, 2) there exists a continuously differentiable function f on lp (and Lp) 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 lp and LP 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 lp and Lp 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. [ABSTRACT FROM AUTHOR]

K-UR, K-LUR and K-R are the generalizations of UR, LUR and R respectively, which are of great significance in Banach space theory. While in Orlicz–Lorentz function space Λ φ , ω ∘ [ 0 , γ) equipped with the Orlicz norm, the research methods of K-UR, K-LUR and K-R are much more complicated than those of UR, LUR and R. In this paper we obtain some criteria of K-UR, K-LUR and K-R of Λ φ , ω ∘ [ 0 , γ) by means of the norm of dual space and Hμ property of Λ φ , ω ∘ [ 0 , γ) . [ABSTRACT FROM AUTHOR]

In this correspondence, we establish mean convergence theorems for the maximum of normed double sums of Banach space valued random elements. Most of the results pertain to random elements which are M-dependent. We expand and improve a number of particular cases in the literature on mean convergence of random elements in Banach spaces. One of the main contributions of the paper is to simplify and improve a recent result of Li, Presnell, and Rosalsky [Journal of Mathematical Inequalities, 16, 117–126 (2022)]. A new maximal inequality for double sums of M-dependent random elements is proved which may be of independent interest. The sharpness of the results is illustrated by four examples. [ABSTRACT FROM AUTHOR]

In order to study the boundedness of some operators in general function spaces which include Lorentz spaces and Orlicz spaces as special examples, Lorentz introduced a new space called rearrangement invariant Banach function spaces, denoted by RIBFS. It is shown in this paper that variation operators of singular integrals and their commutators are bounded on RIBFS whenever the kernels satisfy the Lr-Hörmander conditions. Moreover, we obtain some quantitative weighted bounds in the quasi-Banach spaces and modular inequalities for above variation operators and their commutators. [ABSTRACT FROM AUTHOR]

The goal of this paper is to deal with a new dynamic system called a differential evolution hemivariational inequality (DEHVI) which couples an abstract parabolic evolution hemivariational inequality and a nonlinear differential equation in a Banach space. First, by applying surjectivity result for pseudomonotone multivalued mappins and the properties of Clarke's subgradient, we show the nonempty of the solution set for the parabolic hemivariational inequality. Then, some topological properties of the solution set are established such as boundedness, closedness and convexity. Furthermore, we explore the upper semicontinuity of the solution mapping. Finally, we prove the solution set of the system (DEHVI) is nonempty and the set of all trajectories of (DEHVI) is weakly compact in C(I, X). [ABSTRACT FROM AUTHOR]

For bounded linear operators A, B, C and D on a Banach space X, we show that if BAC = BDB and CDB = CAC then I — AC is generalized Drazin—Riesz invertible if and only if I — BD is generalized Drazin—Riesz invertible, which gives a positive answer to Question 4.9 in Yan, Zeng and Zhu [Complex Anal. Oper. Theory 14, Paper No. 12 (2020)]. In particular, we show that Jacobson's lemma holds for generalized Drazin—Riesz inverses. [ABSTRACT FROM AUTHOR]

Let X and Y be Banach spaces. For A ∈ L(X), B ∈ L(Y), C ∈ L(Y, X), let MC be the operator matrix defined on X ⊕ Y by M C = ( A C 0 B ) ∈ L (X ⊕ Y) . In this paper we investigate the decomposability for MC. We consider Bishop's property (β), decomposition property (δ) and Dunford's property (C) and obtain the relationship of these properties between MC and its entries. We explore how σ*(MC) shrinks from σ*(A) ∪ σ*(B), where σ* denotes σβ,σδ,σC, σdec. In particular, we develop some sufficient conditions for equality σ*(MC) = σ*(A) ∪σ*(B). Besides, we consider the perturbation of these properties for MC and show that in perturbing with certain operators C the properties for MC keeps with A, B. Some examples are given to illustrate our results. Furthermore, we study the decomposability for ( 0 A B 0 ) . Finally, we give applications of decomposability for operator matrices. [ABSTRACT FROM AUTHOR]

In this work, we give some criteria of the weakly compact sets and a representation theorem of Riesz's type in Musielak sequence spaces using the ideas and techniques of sequence spaces and Musielak function. Finally, as an immediate consequence of the criteria considered in this paper, the criteria of the weakly compact sets of Orlicz sequence spaces are deduced. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to investigate the existence of solutions to functional differential inclusions with infinite delay in Banach spaces. A relevant set of phase space axioms is proposed. The main tools used in this paper are certain fixed point theorems based on the set-contraction theory. [ABSTRACT FROM AUTHOR]

In this paper, we give a number of characterizations for a Banach space X which is isometric to a subspace of c0, or, c0(Γ), successively, in terms of extreme points of its dual unit ball BX*, Fréchet and Gâteaux derivatives of its norm, or, in terms of w*-strongly exposed points and w*-exposed points of BX*. [ABSTRACT FROM AUTHOR]

Let G be a nonempty closed subset of a Banach space X. Let ℬ (X) be the family of nonempty bounded closed subsets of X endowed with the Hausdorff distance and ℬ G (X) = { A ∈ ℬ (X) : A ∩ G ∅ } ¯ , where the closure is taken in the metric space (ℬ (X) , H) . For x ∈ X and F ∈ ℬ G (X) , we denote the nearest point problem inf{∥x − g∥: g ∈ G} by min(x, G) and the mutually nearest point problem inf{∥f − g∥: f ∈ F,g ∈ G} by min(F, G). In this paper, parallel to well-posedness of the problems min(x, G) and min(F, G) which are defined by De Blasi et al., we further introduce the weak well-posedness of the problems min(x, G) and min(F, G). Under the assumption that the Banach space X has some geometric properties, we prove a series of results on weak well-posedness of min(x, G) and min(F, G). We also give two sufficient conditions such that two classes of subsets of X are almost Chebyshev sets. [ABSTRACT FROM AUTHOR]

In this paper, we show that every super weakly compact convex subset of a Banach space is an absolute uniform retract, and that it also admits the uniform compact approximation property. These can be regarded as extensions of Lindenstrauss and Kalton's corresponding results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and solve the following additive (ρ1,ρ2)-functional inequalities ‖ f (x + y + z) − f (x) − f (y) − f (z) ‖ ≤ ‖ ρ 1 (f (x + z) − f (x) − f (z)) ‖ + ‖ ρ 2 (f (y + z) − f (y) − f (z)) ‖ , where ρ1 and ρ2 are fixed nonzero complex numbers with ∣ρ1∣ + ∣ρ2∣ < 2. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the above additive (ρ1,ρ2)-functional inequality in complex Banach spaces. Furthermore, we prove the Hyers-Ulam stability of hom-derivations in C*-ternary algebras. [ABSTRACT FROM AUTHOR]

The non-uniqueness on geodesics and geodesic disks in the universal asymptotic Teichmüller space AT(D) are studied in this paper. It is proved that if μ is asymptotically extremal in [[ μ]] with h *( μ) < h*( μ) for some point ζ ∈ ∂D, then there exist infinitely many geodesic segments joining [[0]] and [[ μ]], and infinitely many holomorphic geodesic disks containing [[0]] and [[ μ]] in AT(D). [ABSTRACT FROM AUTHOR]

Let X, Y be Banach spaces and M be a linear subspace in X × Y = {{ x, y}| x ∈ X, y ∈ Y}. We may view M as a multi-valued linear operator from X to Y by taking M( x) = { y|{ x, y} ∈ M}. In this paper, we give several criteria for a single-valued operator from Y to X to be the metric generalized inverse of the multi-valued linear operator M. The principal tool in this paper is also the generalized orthogonal decomposition theorem in Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we continue to discuss the properties of iterates generated by a strict pseudocontraction or a finite family of strict pseudo-contractions in a real q-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [Marino, G., Xu, H. K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl., 324, 336–349 (2007)]. In order to get a strong convergence theorem, we modify the normal Mann’s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C. This result extends a recent result of Kim and Xu [Kim, T. H., Xu, H. K.: Strong convergence of modified Mann iterations. Nonl. Anal., 61, 51–60 (2005)] both from nonexpansive mappings to λ-strict pseudo-contractions and from Hilbert spaces to q-uniformly smooth Banach spaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the weak convergence problems of the implicity iteration process for Lipschitzian pseudocontraction semigroups in general Banach spaces. The results presented in this paper extend and improve the corresponding results of Zhou [ Nonlinear Anal., 68, 2977–2983 (2008)], Chen, et al. [ J. Math. Anal. Appl., 314, 701–709 (2006)], Xu and Ori [ Numer. Funct. Anal. Optim, 22, 767–773 (2001)] and Osilike [ J. Math. Anal. Appl., 294, 73–81 (2004)]. [ABSTRACT FROM AUTHOR]

To guarantee every real-valued convex function bounded above on a set is continuous, how ”thick” should the set be? For a symmetric set A in a Banach space E, the answer of this paper is: Every real-valued convex function bounded above on A is continuous on E if and only if the following two conditions hold: i) spanA has finite co-dimentions and ii) co A has nonempty relative interior. This paper also shows that a subset A ⊂ E satisfying every real-valued convex function bounded above on A is continuous on E if (and only if) every real-valued linear functional bounded above on A is continuous on E, which is also equivalent to that every real-valued convex function bounded on A is continuous on E. [ABSTRACT FROM AUTHOR]

When A ∈ B(H) and B ∈ B(K) are given, we denote by M C an operator acting on the Hilbert space H ⊕ K of the form $$ M_{C} = {\left( {\begin{array}{*{20}c} {A} & {C} \\ {0} & {B} \\ \end{array} } \right)} .$$ In this paper, first we give the necessary and sufficient condition for M C to be an upper semi-Fredholm (lower semi–Fredholm, or Fredholm) operator for some C ∈ B(K,H). In addition, let $$ \sigma _{{SF_{ + } }} $$ ( A) ={λ ∈ ℂ : A − λI is not an upper semi-Fredholm operator} be the upper semi–Fredholm spectrum of A ∈ B(H) and let $$ \sigma _{{SF_{ - } }} $$ ( A) = {λ ∈ ℂ : A − λI is not a lower semi–Fredholm operator} be the lower semi–Fredholm spectrum of A. We show that the passage from $$ \sigma _{{SF_{ ±} }} {\left( A \right)} \cap \sigma _{{SF_{ ±} }} {\left( B \right)}\;{\text{to}}\;\sigma_{{SF_{±} }} {\left( {M_{C} } \right)} $$ is accomplished by removing certain open subsets of $$ \sigma _{{SF_{ - } }} {\left( A \right)} \cap \sigma_{{SF_{ + } }} {\left( B \right)} $$ from the former, that is, there is an equality where $${\fancyscript G}$$ is the union of certain of the holes in $$ \sigma_{{SF_{± } }} {\left( {M_{C} } \right)} $$ which happen to be subsets of $$ \sigma_{{SF_{ - } }} {\left( A \right)} \cap \sigma_{{SF_{ + } }} {\left( B \right)} .$$ Weyl's theorem and Browder's theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl's theorem, Browder's theorem, a–Weyl's theorem and a–Browder's theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space. [ABSTRACT FROM AUTHOR]

Let X, Y be reflexive strictly convex Banach spaces, let T, δT: X → Y be bounded linear operators with closed range R( T). Put $$\overline T = T + \delta T$$. In this paper, by using the concept of quasiadditivity and the so called generalized Neumman lemma, we will give some error estimates of the bounds of $$\left\| {{{\overline T }^M}} \right\|$$. By using a relation between the concepts of the reduced minimum module and the gap of two subspaces, some new existence characterization of the Moore-Penrose metric generalized inverse $${\overline T ^M}$$ of the perturbed operator $$\overline T $$ will be also given. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study a new viscosity iterative algorithm based on a generalized contraction for finding a common element of the set of solutions of a general variational inequality problem for finite inversely strongly accretive mappings and the set of common fixed points for a countable family of strict pseudo-contractions in uniformly smooth Banach spaces. We prove some strong convergence theorems under some suitable conditions. The results obtained in this paper improve and extend the recent ones announced by many others in the literature. [ABSTRACT FROM AUTHOR]

Suppose that X, Y are two real Banach Spaces. We know that for a standard ∈-isometry f: X → Y, the weak stability formula holds and by applying the formula we can induce a closed subspace N of Y*. In this paper, by using again the weak stability formula, we further show a sufficient and necessary condition for a standard ∈-isometry to be stable in assuming that N is w*-closed in Y*. Making use of this result, we improve several known results including Figiel's theorem in reflexive spaces. We also prove that if, in addition, the space Y is quasi-reflexive and hereditarily indecomposable, then L(f)≡span¯[f(x)] contains a complemented linear isometric copy of X; Moreover, if X = Y, then for every ∈-isometry f : X → X, there exists a surjective linear isometry S : X → X such that f − S is uniformly bounded by 2∈ on X. [ABSTRACT FROM AUTHOR]

This paper gives the concepts of finite dimensional irreducible operators ((FDI) operators) and infinite dimensional irreducible operators ((IDI) operators). Discusses the relationships of (FDI) operators, (IDI) operators and strongly irreducible operators ((SI) operators) and illustrates some properties of the three classes of operators. Some sufficient conditions for the finite-dimensional irreducibility of operators which have the forms of upper triangular operator matrices are given. This paper proves that every operator with a singleton spectrum is a small compact perturbation of an (FDI) operator on separable Banach spaces and shows that every bounded linear operator T can be approximated by operators in (ΣFDI)( X) with respect to the strong-operator topology and every compact operator K can be approximated by operators in (ΣFDI)( X) with respect to the norm topology on a Banach space X with a Schauder basis, where (ΣFDI)( X):= { T ∈ B( X): T = Σ ⊕ T, T ∈ (FDI), k ∈ ℕ}. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to establish necessary and sufficient conditions for a matrix A ∈ (( ℓ), ℓ), where T is an arbitrary triangle, 1 ≤ p ≤ ∞, to be a compact operator. In the past, only sufficient conditions were established in almost all of those cases, by using the Hausdorff measure of noncompactness. We improve those results by applying another method for the characterizations of compact linear operators between BK spaces. [ABSTRACT FROM AUTHOR]

In this paper, the adjoint of a densely defined block operator matrix in a Hilbert space X × X is studied and the sufficient conditions under which the equality holds are obtained through applying Frobenius-Schur factorization. [ABSTRACT FROM AUTHOR]

This paper is concerned with the measure of noncompactness in the spaces of continuous functions and semilinear functional differential equations with nonlocal conditions in Banach spaces. The relationship between the Hausdorff measure of noncompactness of intersections and the modulus of equicontinuity is studied for some subsets related to the semigroup of linear operators in Banach spaces. The existence of mild solutions is obtained for a class of nonlocal semilinear functional differential equations without the assumption of compactness or equicontinuity on the associated semigroups of linear operators. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that there is a gap in the paper [Chidume, C. E., Shahzad, N.: Weak convergence theorems for a finite family of strict pseudo-contractions. Nonlinear Anal., 72, 1257-1265 (2010)], consequently, the main results of the paper do not hold in uniformly smooth Banach spaces. Meanwhile, it is also shown that the main results (Lemma 3.4, Theorems 3.5-3.6, 3.8-3.9) in the paper [Cholamjiak, P., Suantai, S.: Weak convergence theorems for a countable family of strict pseudo-contractions in Banach spaces. Fixed Point Theory Appl., 2010, Article ID 632137, 16 pages (2010)] do not hold in L for p > 3. Finally, some modified results are presented in the setting of uniformly smooth and uniformly convex Banach spaces which include L for p ≥ 2 as special cases. Furthermore, our arguments are also different from the ones given by the authors above. [ABSTRACT FROM AUTHOR]

This paper studies framings in Banach spaces, a concept raised by Casazza, Han and Larson, which is a natural generalization of traditional frames in Hilbert spaces and unconditional bases in Banach spaces. The minimal unconditional bases and the maximal unconditional bases with respect to framings are introduced. Our main result states that, if ( xi, fi) is a framing of a Banach space X, and ( eimin) and ( eimax) are the minimal unconditional basis and the maximal unconditional basis with respect to ( xi, fi), respectively, then for any unconditional basis ( ei) associated with ( xi, fi), there are A,B > 0 such that [Figure not available: see fulltext.]. It means that for any framing, the corresponding associated unconditional bases have common upper and lower bounds. [ABSTRACT FROM AUTHOR]

Let X be a Banach space of dimension ≥ 2 over the real or complex field F and A a standard operator algebra in B( X). A map Φ: A→ A is said to be strong 3-commutativity preserving if [Φ( A), Φ( B)] = [ A, B] for all A, B ∈ A, where [ A, B] is the 3-commutator of A, B defined by [ A, B] = [[[ A, B], B], B] with [ A, B] = AB− BA. The main result in this paper is shown that, if Φ is a surjective map on A, then Φ is strong 3-commutativity preserving if and only if there exist a functional h: A→F and a scalar λ ∈ F with λ = 1 such that Φ( A) = λ A + h( A) I for all A ∈ A. [ABSTRACT FROM AUTHOR]

For a non-trivial Banach space X, let J( X), C ( X), C ( X) respectively stand for the James constant, the von Neumann-Jordan constant and the generalized von Neumann-Jordan constant recently inroduced by Cui et al. In this paper, we discuss the relation between the James and the generalized von Neumann-Jordan constants, and establish an inequality between them: C ( X) ≤ J( X) with p ≥ 2, which covers the well-known inequality C ( X) ≤ J( X). We also introduce a new constant, from which we establish another inequality that extends a result of Alonso et al. [ABSTRACT FROM AUTHOR]

R-duals of certain sequences in Hilbert spaces were introduced by Casazza, Kutyniok and Lammers in 2004 and later generalized to Banach spaces by Xiao and Zhu. In this paper we provide some characterizations of R-dual sequences in Banach spaces. [ABSTRACT FROM AUTHOR]

By a ball-covering B of a Banach space X, wemean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere of X; and X is said to have the ball-covering property provided it admits a ball-covering of countably many balls. This paper shows that universal finite representability and B-convexity of X can be characterized by properties of ball-coverings of its finite dimensional subspaces. [ABSTRACT FROM AUTHOR]

In the present paper we obtain and extend the boundedness property of the Adams type for multilinear fractional integral operators. Also, we deal with the Olsen type inequality. [ABSTRACT FROM AUTHOR]

The paper investigates L convergence and Marcinkiewicz-Zygmund strong laws of large numbers for random elements in a Banach space under the condition that the Banach space is of Rademacher type p, 1 < p < 2. The paper also discusses L convergence and L bound for random elements without any geometric restriction condition on the Banach space. [ABSTRACT FROM AUTHOR]

The purpose of this paper is by using the modified block iterative method to propose an algorithm for finding a common element in the intersection of the set of common fixed points of an infinite family of quasi- ϕ-asymptotically nonexpansive and the set of solutions to an equilibrium problem and the set of solutions to a variational inequality. Under suitable conditions some strong convergence theorems are established in 2-uniformly convex and uniformly smooth Banach spaces. As applications we utilize the results presented in the paper to solving the convex feasibility problem (CFP) and zero point problem of maximal monotone mappings in Banach spaces. The results presented in the paper improve and extend the corresponding results announced by many authors. [ABSTRACT FROM AUTHOR]

In this paper, continuous homogeneous selections for the set-valued metric generalized inverses T∂ of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces. Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T∂ are given. The results are an answer to the problem posed by Nashed and Votruba. [ABSTRACT FROM AUTHOR]

Let D and D′ be domains in real Banach spaces of dimension at least 2. The main aim of this paper is to study certain arc distortion properties in the quasihyperbolic metric defined in real Banach spaces. In particular, when D′ is a QH inner ψ-uniform domain with ψ being a slow (or a convex domain), we investigate the following: For positive constants c,h,C,M, suppose a homeomorphism f: D → D′ takes each of the 10-neargeodesics in D to ( c, h)-solid in D′. Then f is C-coarsely M-Lipschitz in the quasihyperbolic metric. These are generalizations of the corresponding result obtained recently by Väisälä. [ABSTRACT FROM AUTHOR]

This paper proves the existence and uniqueness of solutions in a Banach space for the generalized stochastic Ginzburg-Landau equation with a multiplicative noise in two spatial dimensions. The noise is white in time and correlated in spatial variables. The condition on the parameters is the same as in the deterministic case. The Banach contraction principle and stochastic estimates in Banach spaces are used as the main tool. [ABSTRACT FROM AUTHOR]

In general, Banach space-valued Riemann integrable functions defined on [0, 1] (equipped with the Lebesgue measure) need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space L of all Bochner integrable functions from [0, 1] to the Banach space X. We show that L has the weak Lebesgue property whenever X has the Radon-Nikodým property and X* is separable. This generalizes the result by Chonghu Wang and Kang Wan [ Rocky Mountain J. Math., 31(2), 697–703 (2001)] that L1[0, 1] has the weak Lebesgue property. [ABSTRACT FROM AUTHOR]

In this paper, we prove the generalized Hyers-Ulam stability of homomorphisms in quasi-Banach algebras associated with the following Pexiderized Jensen functional equation This is applied to investigating homomorphisms between quasi-Banach algebras. The concept of the generalized Hyers-Ulam stability originated from Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297–300 (1978). [ABSTRACT FROM AUTHOR]

In this paper, we prove that every operator in a class of contraction operators on a Banach space whose spectrum contains the unit circle has a nontrivial hyperinvariant subspace. [ABSTRACT FROM AUTHOR]

This paper considers the isometric extension problem concerning the mapping from the unit sphere S 1( E) of the normed space E into the unit sphere S 1( l ∞(Γ)). We find a condition under which an isometry from S 1( E) into S 1( l ∞(Γ)) can be linearly and isometrically extended to the whole space. Since l ∞(Γ) is universal with respect to isometry for normed spaces, isometric extension problems on a class of normed spaces are solved. More precisely, if E and F are two normed spaces, and if V 0: S 1( E) → S 1( F) is a surjective isometry, where c 00(Γ) ⊆ F ⊆ l ∞(Γ), then V 0 can be extended to be an isometric operator defined on the whole space. [ABSTRACT FROM AUTHOR]

In this paper we give the existence of mild solutions for semilinear Cauchy problems u'( t) = Au( t) + f( t, u( t)), t ∈¸ I, a.e. with nonlocal initial condition u(0) = g( u) + u 0 when the map g loses compactness in Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundary dilatation of quasiconformal mappings in the unit disc. By using Strebel mapping by heights theory we show that an asymptotical Hamilton sequence is determined by a quasisymmetric function [ABSTRACT FROM AUTHOR]

In this paper, we .rst derive the representation theorem of onto isometric mappings in the unit spheres ofl1(G) type spaces, and then conclude that such mappings can be extended to the whole space as real linear isometries by using a previous result of the author. [ABSTRACT FROM AUTHOR]

In this paper, two new functions are introduced to depict the Jamison weighted sum of random variables instead using the common methods, theft properties and relationships are systematically discussed. We also analysed the implication of the conditions in previous papers. Then we apply these consequences to B-valued random variables, and greatly improve the original results of the strong convergence of the general Jamison weighted sum. Furthermore, our discussions are useful to the corresponding questions of real-valued random variables. [ABSTRACT FROM AUTHOR]

Let S be the familiar class of normalized starlike functions of order α in the unit disk. In this paper, we establish the Fekete and Szegö inequality for the class S , and then we generalize this result to the unit ball in a complex Banach space or on the unit polydisk in C. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the estimates of all homogeneous expansions for a subclass of biholomorphic mappings which have parametric representation on the unit ball of complex Banach spaces. Meanwhile, we also establish the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in C. Especially, the above estimates are only sharp for biholomorphic starlike mappings and starlike mappings of order α under restricted conditions. Our derived results generalize many known results. [ABSTRACT FROM AUTHOR]

Let X denote a compact metric space with distance d and F : X × ℝ → X or F : X → X denote a C -flow. From the point of view of ergodic theory, all important dynamical behaviors take place on a full measure set. The aim of this paper is to introduce the notion of Banach upper density recurrent points and to show that the closure of the set of all Banach upper density recurrent points equals the measure center or the minimal center of attraction for a C -flow. Moreover, we give an example to show that the set of quasi-weakly almost periodic points can be included properly in the set of Banach upper density recurrent points, and point out that the set of Banach upper density recurrent points can be included properly in the set of recurrent points. [ABSTRACT FROM AUTHOR]

In this paper, we show that a closed convex subset C of a Banach space is strongly proximinal (proximinal, resp.) in every Banach space isometrically containing it if and only if C is locally (weakly, resp.) compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonempty set-valued and upper semi-continuous. [ABSTRACT FROM AUTHOR]