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
This paper through discussing subdifferentiability and convexity of convex functions shows that a Banach space admits an equivalent uniformly [locally uniformly, strictly] convex norm if and only if there exists a con...This paper through discussing subdifferentiability and convexity of convex functions shows that a Banach space admits an equivalent uniformly [locally uniformly, strictly] convex norm if and only if there exists a continuous uniformly [locally uniformly, strictly] convex function on some nonempty open convex subset of the space and presents some characterizations of super-reflexive Banach spaces.展开更多
This paper gives some relations and properties of several kinds of generalized convexity in Banach spaces. As a result, it proves that every kind of uniform convexity implies the Banach-Sakes property, and several not...This paper gives some relations and properties of several kinds of generalized convexity in Banach spaces. As a result, it proves that every kind of uniform convexity implies the Banach-Sakes property, and several notions of uniform convexity in literature are actually equivalent.展开更多
Let X and Y be real Banach spaces. The stability of Hyers Ulam Rassias approximate isometries on restricted domains S (unbounded or bounded) for into mapping f: S→Y satisfying ‖ f(x)-f(y)‖-‖x-y‖≤ε(x,y) for al...Let X and Y be real Banach spaces. The stability of Hyers Ulam Rassias approximate isometries on restricted domains S (unbounded or bounded) for into mapping f: S→Y satisfying ‖ f(x)-f(y)‖-‖x-y‖≤ε(x,y) for all x,y∈S is studied in case that the target space Y is uniformly convex Banach space of the modulus of convexity of power type q ≥2 or Y is the L q(Ω,,μ) (1<q <+∞) space or Y is a Hilbert space. Furthermore, the stability of approximate isometries for the case that (x,y)=‖x‖ p+‖y‖ p or (x,y)=‖x-y‖ p for p ≠1 is investigated.展开更多
Let C be a bounded convex subset in a uniformly convex Banach space X, x 0, u n∈C , then x n+1 =S nx n, where S n=α n0 I+α n1 T+α n2 T 2+…+α nk T k+γ nu n, α ni ≥0, 0<α≤α ...Let C be a bounded convex subset in a uniformly convex Banach space X, x 0, u n∈C , then x n+1 =S nx n, where S n=α n0 I+α n1 T+α n2 T 2+…+α nk T k+γ nu n, α ni ≥0, 0<α≤α n0 ≤b<1, ∑ki=0α ni +γ n=1, and n≥1. It is proved that x n converges to a fixed point on T if T is a nonexpansive mapping.展开更多
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This paper through discussing subdifferentiability and convexity of convex functions shows that a Banach space admits an equivalent uniformly [locally uniformly, strictly] convex norm if and only if there exists a continuous uniformly [locally uniformly, strictly] convex function on some nonempty open convex subset of the space and presents some characterizations of super-reflexive Banach spaces.
基金Supported b-y National Natural Science Foundation of China （Grant No. 10926042 and 11001231）, China Postdoctoral Science Foundation （Grant No. 20090460356）, RFDP （Grant No. 200803841018）Acknowledgements The authors would like to thank Professor Cheng Lixin and Professor Bu Shangquan for many helpful conversations on this paper, and also thank the referee for many valuable suggestions.
文摘This paper gives some relations and properties of several kinds of generalized convexity in Banach spaces. As a result, it proves that every kind of uniform convexity implies the Banach-Sakes property, and several notions of uniform convexity in literature are actually equivalent.
文摘Let X and Y be real Banach spaces. The stability of Hyers Ulam Rassias approximate isometries on restricted domains S (unbounded or bounded) for into mapping f: S→Y satisfying ‖ f(x)-f(y)‖-‖x-y‖≤ε(x,y) for all x,y∈S is studied in case that the target space Y is uniformly convex Banach space of the modulus of convexity of power type q ≥2 or Y is the L q(Ω,,μ) (1<q <+∞) space or Y is a Hilbert space. Furthermore, the stability of approximate isometries for the case that (x,y)=‖x‖ p+‖y‖ p or (x,y)=‖x-y‖ p for p ≠1 is investigated.
文摘Let C be a bounded convex subset in a uniformly convex Banach space X, x 0, u n∈C , then x n+1 =S nx n, where S n=α n0 I+α n1 T+α n2 T 2+…+α nk T k+γ nu n, α ni ≥0, 0<α≤α n0 ≤b<1, ∑ki=0α ni +γ n=1, and n≥1. It is proved that x n converges to a fixed point on T if T is a nonexpansive mapping.
