Recently people proved that every f∈C[0, 1] can be uniformly approximated by polynomial sequences {P_n}, {Q_n} such for any x∈[0,1] and n=1,2,…that Q_n(x)<Q_(n+1)(x)<f(x)<P_(n+1)(x)<P_n(x). For example...Recently people proved that every f∈C[0, 1] can be uniformly approximated by polynomial sequences {P_n}, {Q_n} such for any x∈[0,1] and n=1,2,…that Q_n(x)<Q_(n+1)(x)<f(x)<P_(n+1)(x)<P_n(x). For example, Xie and Zhou showed that one can construct such monotone polynomial sequences which do achieve the best uniform approximation rate for a continuous func- tion. Actually they obtained a result as ‖P_n(x)-Q_n(x)‖≤42E_n (f), (1) which essentially improved a conclusion in Gal and Szabados. The present paper, by optimal procedure, improves this inequality to ‖[P_n(x)-Q_n(x)‖≤(18+ε)E_n(f), where εis any positive real number.展开更多
In this article, we study Levitin-Polyak type well-posedness for generalized vector equilibrium problems with abstract and functional constraints. Criteria and characterizations for these types of well-posednesses are...In this article, we study Levitin-Polyak type well-posedness for generalized vector equilibrium problems with abstract and functional constraints. Criteria and characterizations for these types of well-posednesses are given.展开更多
Let E be a real Banach space and K be a nonempty closed convex and bounded subset of E. Let Ti ： K→ K, i=1, 2,... ,N, be N uniformly L-Lipschitzian, uniformly asymptotically regular with sequences {ε^（i）n} and as...Let E be a real Banach space and K be a nonempty closed convex and bounded subset of E. Let Ti ： K→ K, i=1, 2,... ,N, be N uniformly L-Lipschitzian, uniformly asymptotically regular with sequences {ε^（i）n} and asymptotically pseudocontractive mappings with sequences {κ^（i）n}, where {κ^（i）n} and {ε^（i）n}, i = 1, 2,... ,N, satisfy certain mild conditions. Let a sequence {xn} be generated from x1 ∈ K by zn：= （1-μn）xn＋μnT^nnxn, xn＋1 ：= λnθnx1＋ [1 - λn（1 ＋ θn）]xn ＋ λnT^nnzn for all integer n ≥ 1, where Tn = Tn（mod N）, and {λn}, {θn} and {μn} are three real sequences in [0, 1] satisfying appropriate conditions. Then ｜｜xn- Tixn｜｜→ 0 as n→∞ for each l ∈ {1, 2,..., N}. The results presented in this paper generalize and improve the corresponding results of Chidume and Zegeye, Reinermann, Rhoades and Schu.展开更多
文摘Recently people proved that every f∈C[0, 1] can be uniformly approximated by polynomial sequences {P_n}, {Q_n} such for any x∈[0,1] and n=1,2,…that Q_n(x)<Q_(n+1)(x)<f(x)<P_(n+1)(x)<P_n(x). For example, Xie and Zhou showed that one can construct such monotone polynomial sequences which do achieve the best uniform approximation rate for a continuous func- tion. Actually they obtained a result as ‖P_n(x)-Q_n(x)‖≤42E_n (f), (1) which essentially improved a conclusion in Gal and Szabados. The present paper, by optimal procedure, improves this inequality to ‖[P_n(x)-Q_n(x)‖≤(18+ε)E_n(f), where εis any positive real number.
基金supported by the National Science Foundation of China and Shanghai Pujian Program
文摘In this article, we study Levitin-Polyak type well-posedness for generalized vector equilibrium problems with abstract and functional constraints. Criteria and characterizations for these types of well-posednesses are given.
基金Foundation item： the National Natural Science Foundation of China （No. 10771141） the Natural Science Foundation of Zhejiang Province （Y605191） the Natural Science Foundation of Heilongjiang Province （No. A0211） and the Scientific Research Foundation from Zhejiang Province Education Committee （No. 20051897）.
文摘Let E be a real Banach space and K be a nonempty closed convex and bounded subset of E. Let Ti ： K→ K, i=1, 2,... ,N, be N uniformly L-Lipschitzian, uniformly asymptotically regular with sequences {ε^（i）n} and asymptotically pseudocontractive mappings with sequences {κ^（i）n}, where {κ^（i）n} and {ε^（i）n}, i = 1, 2,... ,N, satisfy certain mild conditions. Let a sequence {xn} be generated from x1 ∈ K by zn：= （1-μn）xn＋μnT^nnxn, xn＋1 ：= λnθnx1＋ [1 - λn（1 ＋ θn）]xn ＋ λnT^nnzn for all integer n ≥ 1, where Tn = Tn（mod N）, and {λn}, {θn} and {μn} are three real sequences in [0, 1] satisfying appropriate conditions. Then ｜｜xn- Tixn｜｜→ 0 as n→∞ for each l ∈ {1, 2,..., N}. The results presented in this paper generalize and improve the corresponding results of Chidume and Zegeye, Reinermann, Rhoades and Schu.
