As an important type of polynomial approximation, approximation of functions by Bernstein operators is an important topic in approximation theory and computational theory. This paper gives global and pointwise estimat...As an important type of polynomial approximation, approximation of functions by Bernstein operators is an important topic in approximation theory and computational theory. This paper gives global and pointwise estimates for weighted approximation of functions with singularities by Bernstein operators. The main results are the Jackson＇s estimates of functions f∈ （Wwλ）2 andre Cw, which extends the result of （Della Vecchia et al., 2004）.展开更多
Let C be a nonempty bounded closed convex subset of a Banach space X, and T : C → C be uniformly L-Lipschitzian with L ≥ 1 and asymptotically pseudocontractive with a sequence {kn}(?)[1, ∞), limn→∞ kn = 1. Fix u ...Let C be a nonempty bounded closed convex subset of a Banach space X, and T : C → C be uniformly L-Lipschitzian with L ≥ 1 and asymptotically pseudocontractive with a sequence {kn}(?)[1, ∞), limn→∞ kn = 1. Fix u ∈ C. For each n ≥ 1, xn is a unique fixed point of the contraction Sn(x) = (1 - (tn)/(Lkn))u + (tn)/(Lkn)Tnx(?)x ∈ C, where {tn}(?)[0,1). Under suitable conditions, the strong convergence of the sequence{xn}to a fixed point of T is characterized.展开更多
Let M^n be a smooth, compact manifold without boundary, and F0 ： M^n→ R^n＋1 a smooth immersion which is convex. The one-parameter families F（·, t） ： M^n× [0, T） → R^n＋1 of hypersurfaces Mt^n= F（...Let M^n be a smooth, compact manifold without boundary, and F0 ： M^n→ R^n＋1 a smooth immersion which is convex. The one-parameter families F（·, t） ： M^n× [0, T） → R^n＋1 of hypersurfaces Mt^n= F（·,t）（M^n） satisfy an initial value problem dF/dt （·,t） = -H^k（· ,t）v（· ,t）, F（· ,0） = F0（· ）, where H is the mean curvature and u（·,t） is the outer unit normal at F（·, t）, such that -Hu = H is the mean curvature vector, and k 〉 0 is a constant. This problem is called H^k-fiow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type Ⅰ and Type Ⅱ. It is proved that for Type Ⅰ singularity, the limiting hypersurface satisfies an elliptic equation; for Type Ⅱ singularity, the limiting hypersurface must be a translating soliton.展开更多
文摘As an important type of polynomial approximation, approximation of functions by Bernstein operators is an important topic in approximation theory and computational theory. This paper gives global and pointwise estimates for weighted approximation of functions with singularities by Bernstein operators. The main results are the Jackson＇s estimates of functions f∈ （Wwλ）2 andre Cw, which extends the result of （Della Vecchia et al., 2004）.
基金The Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China, and The Dawn Program Fund in Shanghai.
文摘Let C be a nonempty bounded closed convex subset of a Banach space X, and T : C → C be uniformly L-Lipschitzian with L ≥ 1 and asymptotically pseudocontractive with a sequence {kn}(?)[1, ∞), limn→∞ kn = 1. Fix u ∈ C. For each n ≥ 1, xn is a unique fixed point of the contraction Sn(x) = (1 - (tn)/(Lkn))u + (tn)/(Lkn)Tnx(?)x ∈ C, where {tn}(?)[0,1). Under suitable conditions, the strong convergence of the sequence{xn}to a fixed point of T is characterized.
基金supported by the National Natural Science Foundation of China (Nos. 10771189, 10831008)
文摘Let M^n be a smooth, compact manifold without boundary, and F0 ： M^n→ R^n＋1 a smooth immersion which is convex. The one-parameter families F（·, t） ： M^n× [0, T） → R^n＋1 of hypersurfaces Mt^n= F（·,t）（M^n） satisfy an initial value problem dF/dt （·,t） = -H^k（· ,t）v（· ,t）, F（· ,0） = F0（· ）, where H is the mean curvature and u（·,t） is the outer unit normal at F（·, t）, such that -Hu = H is the mean curvature vector, and k 〉 0 is a constant. This problem is called H^k-fiow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type Ⅰ and Type Ⅱ. It is proved that for Type Ⅰ singularity, the limiting hypersurface satisfies an elliptic equation; for Type Ⅱ singularity, the limiting hypersurface must be a translating soliton.
