We present a short and simple proof of the recent result of Yang and Wang [12]. Stimulated by their idea, two geometric parameters Uax（ε） and βx （ε）, both related to Gao＇s modulus of U-convexity of a Banach sp...We present a short and simple proof of the recent result of Yang and Wang [12]. Stimulated by their idea, two geometric parameters Uax（ε） and βx （ε）, both related to Gao＇s modulus of U-convexity of a Banach space X, are introduced. Their properties and the relationships with normal structure are studied. Some existing results involving normal structure and fixed points for non-expansive mappings in Banach spaces are improved.展开更多
In this article, the authors study a generalized modulus of convexity, δ（α） （ε）. Certain related geometrical properties of this modulus are analyzed. Their main result is that Banach space X has uniform normal ...In this article, the authors study a generalized modulus of convexity, δ（α） （ε）. Certain related geometrical properties of this modulus are analyzed. Their main result is that Banach space X has uniform normal structure if there exists ε, 0 ≤e≤ 1, such that δ^（α）（1＋ε ） 〉 （1- α）ε.展开更多
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, whe...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.展开更多
The open question raised by Reich is studied in a Banach space with uniform normal structure, whose norm is uniformly Gateaux differentiable. Under more suitable assumptions imposed on an asymptotically nonexpansive m...The open question raised by Reich is studied in a Banach space with uniform normal structure, whose norm is uniformly Gateaux differentiable. Under more suitable assumptions imposed on an asymptotically nonexpansive mapping, an affirmative answer to Reich＇ s open question is given. The results presented extend and improve Zhang Shisheng＇ s recent ones in the following aspects ： （i） Zhang＇ s stronger condition that the sequence of iterative parameters converges to zero is removed; （ii） Zhang＇ s stronger assumption that the asymptotically nonexpansive mapping has a fixed point is removed; （iii） Zhang＇ s stronger condition that the sequence generated by the Banach Contraction Principle is strongly convergent is also removed. Moreover, these also extend and improve the corresponding ones obtained previously by several authors including Reich, Shioji, Takahashi,Ueda and Wittmann.展开更多
In this note, we investigate the generalized modulus of convexity δ（λ） and the generalized modulus smoothness p（λ）. We obtain some estimates of these moduli for X=lp. We obtain inequalities between WCS coeffici...In this note, we investigate the generalized modulus of convexity δ（λ） and the generalized modulus smoothness p（λ）. We obtain some estimates of these moduli for X=lp. We obtain inequalities between WCS coefficient of a KSthe sequence space X and δX（λ）. We show that, for a wide class of class the sequence spaces X, if for some ε∈（0, 9/10] holds δx（e） 〉 1/3（1- √3/2）ε, then X has normal structure.展开更多
In this paper, we introduce a new geometric constant C;（a, X） of a Banach space X, which is closely related to the generalized von Neumann-Jordan constant and analyze some properties of the constant. Subsequently, w...In this paper, we introduce a new geometric constant C;（a, X） of a Banach space X, which is closely related to the generalized von Neumann-Jordan constant and analyze some properties of the constant. Subsequently, we present several sufficient conditions for normal structure of a Banach space in terms of this new constant, the generalized James constant, the generalized Garc′?a-Falset coefficient and the coefficient of weak orthogonality of Sims. Our main results of the paper generalize some known results in the recent literature.展开更多
Let X be a Banach space with a weak uniform normal structure and C a non–empty convex weakly compact subset of X. Under some suitable restriction, we prove that every asymptotically regular semigroup T = {T(t) : t ∈...Let X be a Banach space with a weak uniform normal structure and C a non–empty convex weakly compact subset of X. Under some suitable restriction, we prove that every asymptotically regular semigroup T = {T(t) : t ∈? S} of selfmappings on C satisfying展开更多
文摘We present a short and simple proof of the recent result of Yang and Wang [12]. Stimulated by their idea, two geometric parameters Uax（ε） and βx （ε）, both related to Gao＇s modulus of U-convexity of a Banach space X, are introduced. Their properties and the relationships with normal structure are studied. Some existing results involving normal structure and fixed points for non-expansive mappings in Banach spaces are improved.
基金Supported by Education Foundation of Henan Province(2003110006)
文摘In this article, the authors study a generalized modulus of convexity, δ（α） （ε）. Certain related geometrical properties of this modulus are analyzed. Their main result is that Banach space X has uniform normal structure if there exists ε, 0 ≤e≤ 1, such that δ^（α）（1＋ε ） 〉 （1- α）ε.
文摘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.
文摘The open question raised by Reich is studied in a Banach space with uniform normal structure, whose norm is uniformly Gateaux differentiable. Under more suitable assumptions imposed on an asymptotically nonexpansive mapping, an affirmative answer to Reich＇ s open question is given. The results presented extend and improve Zhang Shisheng＇ s recent ones in the following aspects ： （i） Zhang＇ s stronger condition that the sequence of iterative parameters converges to zero is removed; （ii） Zhang＇ s stronger assumption that the asymptotically nonexpansive mapping has a fixed point is removed; （iii） Zhang＇ s stronger condition that the sequence generated by the Banach Contraction Principle is strongly convergent is also removed. Moreover, these also extend and improve the corresponding ones obtained previously by several authors including Reich, Shioji, Takahashi,Ueda and Wittmann.
基金supported by National Fund for Scientific Research of the Bulgarian Ministry of Education and Science, Contract MM-1401/04
文摘In this note, we investigate the generalized modulus of convexity δ（λ） and the generalized modulus smoothness p（λ）. We obtain some estimates of these moduli for X=lp. We obtain inequalities between WCS coefficient of a KSthe sequence space X and δX（λ）. We show that, for a wide class of class the sequence spaces X, if for some ε∈（0, 9/10] holds δx（e） 〉 1/3（1- √3/2）ε, then X has normal structure.
文摘In this paper, we introduce a new geometric constant C;（a, X） of a Banach space X, which is closely related to the generalized von Neumann-Jordan constant and analyze some properties of the constant. Subsequently, we present several sufficient conditions for normal structure of a Banach space in terms of this new constant, the generalized James constant, the generalized Garc′?a-Falset coefficient and the coefficient of weak orthogonality of Sims. Our main results of the paper generalize some known results in the recent literature.
基金supported both by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE,P.R.C.by the National Natural Science Foundation 19801023
文摘Let X be a Banach space with a weak uniform normal structure and C a non–empty convex weakly compact subset of X. Under some suitable restriction, we prove that every asymptotically regular semigroup T = {T(t) : t ∈? S} of selfmappings on C satisfying
