广义高继常数与广义光滑模的一些性质
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摘要
本文主要研究了广义高继常数与广义光滑模的一些性质.本文组织如下：
首先,讨论了广义高继常数E(a,X)的一些性质,进而得到了Banach空间X具有一致正规结构的两个充分条件为
·若存在a∈[0,1]使得E(a,X)<2(1+a)2(1+(1-a)/(J(a,X)+2a)2,则X具有一致正规结构.
·若存在a∈[0,1]使得E(a,X)<2(1+4a+(1+a)/(μ(X))2,则X具有一致正规结构.这两个结果推广了高继的原有结论.
接着,受广义凸性模δ(λ)(ε)的启发,本文引入了广义光滑模ρ(λ)(ε),利用此模得到了Banach空间X一致光滑的两个充要条件.特别地计算出了Hilbert空间,lp(p＞>1)空间和无穷序列空间上赋予新范数后组成的空间Xp(p≥1)中广义光滑模的精确值,即ρH(λ)(ε)=1-(?)1-(ε2)/4,ρlp(λ)(ε)=ρXp(λ)(ε)=1+[1-(ε/2)p]1/p此外,利用广义凸性模和广义光滑模定义了空间X的广义形变模d(A)(ε)和广义形变Gx,通过这两个参数来刻画空间在广义模下的形变程度.
最后,在高继引入带参数t的高继常数E(t,X)和带参数ξ,η的常数Eξ,η(X)的基础上讨论了带参数t的Boronti常数Az(t,X)和带参数ξ,η的Alonso-Llonso-Fuster常数Tξ,η(X),利用这两个常数得到了Banach空间具有一致正规结构的几个充分条件,这些条件对J.Gao的一些结论进行了适当推广.特别地,巧妙利用Hanner不等式本文计算出了一些具体空间中常数Tξ,η(X)的精确值,但并未给出当1≤p≤2时空间Xp,λ上常数Tξ,η(X)的精确值.
This thesis is devoted to study several properties of generalized James constant and generalized modulus of smoothness. The paper organized as follows:
Firstly, several geometric properties of generalized Gao's constant is discussed. More-over, We get two sufficient conditions for space X to have uniformly normal structure
(?) If exist a∈[0,1] satisfy E(a,X)< 2(1+a)2(1+(1-a)/(J(a,X)+2a)2, therefore X have uniformly normal structure.
(?) If exist a a∈[0,1] satisfy E(a,X)< 2(1+4a+(1-a)/(μ(X)))2, therefore X have uniformly normal structure. These results generalized the results of J.Gao.
Next, Similar to modulus of convexityδ(λ)(ε), we introduce the generalized modulus of smoothnessρ(λ)(ε). Using the modulus, this paper get two necessary and sufficient conditions for uniformly smoothness. Specifically, it calculates the Hilbert spaces H, space lp(p>1) and renormed endless sequence spaces Xp(p>1), namely In addition, using generalized modulus of convexity and generalized modulus of smoothness, we give the difinition of generalized modulus of deformation and the generalized deformation on spaces, these two definition can depict the deformation degree of spaces X under the generalized modulus.
Finally, the with parameter t constant named Boronti constant A2(t,X) and with parameterξ,ηconstant named Alonso-Llonso-Fuster constant Tξ,η(X) are given. These two constants are based on the with parameter t constant E(t,X) and with parameterξ,ηconstant Eξ,η(X) who introduced by J.Gao. Using the two constants, we get several sufficient conditions for Banach space X to have uniformly normal structure, these con-clusions promotion the results of J.Gao. Specifically, ingenious use Hanner inequality this paper calculates the value of constant Tξ,η(X) on some specific space, when 1≤p≤2, it did not give a precise value of Tξ,η(X) on spaces Xp,λ.
首先,讨论了广义高继常数E(a,X)的一些性质,进而得到了Banach空间X具有一致正规结构的两个充分条件为
·若存在a∈[0,1]使得E(a,X)<2(1+a)2(1+(1-a)/(J(a,X)+2a)2,则X具有一致正规结构.
·若存在a∈[0,1]使得E(a,X)<2(1+4a+(1+a)/(μ(X))2,则X具有一致正规结构.这两个结果推广了高继的原有结论.
接着,受广义凸性模δ(λ)(ε)的启发,本文引入了广义光滑模ρ(λ)(ε),利用此模得到了Banach空间X一致光滑的两个充要条件.特别地计算出了Hilbert空间,lp(p＞>1)空间和无穷序列空间上赋予新范数后组成的空间Xp(p≥1)中广义光滑模的精确值,即ρH(λ)(ε)=1-(?)1-(ε2)/4,ρlp(λ)(ε)=ρXp(λ)(ε)=1+[1-(ε/2)p]1/p此外,利用广义凸性模和广义光滑模定义了空间X的广义形变模d(A)(ε)和广义形变Gx,通过这两个参数来刻画空间在广义模下的形变程度.
最后,在高继引入带参数t的高继常数E(t,X)和带参数ξ,η的常数Eξ,η(X)的基础上讨论了带参数t的Boronti常数Az(t,X)和带参数ξ,η的Alonso-Llonso-Fuster常数Tξ,η(X),利用这两个常数得到了Banach空间具有一致正规结构的几个充分条件,这些条件对J.Gao的一些结论进行了适当推广.特别地,巧妙利用Hanner不等式本文计算出了一些具体空间中常数Tξ,η(X)的精确值,但并未给出当1≤p≤2时空间Xp,λ上常数Tξ,η(X)的精确值.
This thesis is devoted to study several properties of generalized James constant and generalized modulus of smoothness. The paper organized as follows:
Firstly, several geometric properties of generalized Gao's constant is discussed. More-over, We get two sufficient conditions for space X to have uniformly normal structure
(?) If exist a∈[0,1] satisfy E(a,X)< 2(1+a)2(1+(1-a)/(J(a,X)+2a)2, therefore X have uniformly normal structure.
(?) If exist a a∈[0,1] satisfy E(a,X)< 2(1+4a+(1-a)/(μ(X)))2, therefore X have uniformly normal structure. These results generalized the results of J.Gao.
Next, Similar to modulus of convexityδ(λ)(ε), we introduce the generalized modulus of smoothnessρ(λ)(ε). Using the modulus, this paper get two necessary and sufficient conditions for uniformly smoothness. Specifically, it calculates the Hilbert spaces H, space lp(p>1) and renormed endless sequence spaces Xp(p>1), namely In addition, using generalized modulus of convexity and generalized modulus of smoothness, we give the difinition of generalized modulus of deformation and the generalized deformation on spaces, these two definition can depict the deformation degree of spaces X under the generalized modulus.
Finally, the with parameter t constant named Boronti constant A2(t,X) and with parameterξ,ηconstant named Alonso-Llonso-Fuster constant Tξ,η(X) are given. These two constants are based on the with parameter t constant E(t,X) and with parameterξ,ηconstant Eξ,η(X) who introduced by J.Gao. Using the two constants, we get several sufficient conditions for Banach space X to have uniformly normal structure, these con-clusions promotion the results of J.Gao. Specifically, ingenious use Hanner inequality this paper calculates the value of constant Tξ,η(X) on some specific space, when 1≤p≤2, it did not give a precise value of Tξ,η(X) on spaces Xp,λ.
引文
[1]J. Gao and S. Saejung, Some geometric measures of spheres in Banach spaces, Applied Mathematics and Computation,2009,214:102-107.
[2]J. Gao, On some geometric parameters in Banach spaces, J. Math. Anal. Appl.2007, 334:114-122.
[3]J. A. Clarkson, Uniformly convex spaces, Trans. Amth. Soc.1936,40:396-414.
[4]J. A. Clarkson, The Von Neumann-Jordan constant for the Lebesgue space, Ann. of Math.1937,38:114-115.
[5]M.Kato and Y.Takahashi, On the Von Neumann-Jordan constant for Banach spaces, Proc. Amer. Math. Soc.1997,125:1055-1062.
[6]M. Kato, L. Maligranda and Y. Takahashi, On James, Von Neumann-Jordan constants and the normal structure coefficients of Banach spaces, Studia. Math.2001,144:275-295.
[7]Y.Takahashi, Some geometric constants of Banach spaces, A unified approach, Banach and Function Spaces Ⅱ,2007:191-220.
[8]J. Gao and K. S. Lau, On two classes of Banach spaces with uniform normal structure, Studia Math.1991,99:41-56.
[9]S. Dhompongsa, A. Kaewkhao and S. Tasena, On a generalized James constant, J. Math. Anal. Appl.2003,285:419-435.
[10]S. Dhompongsa, P. Piraisangjun and S. Saejung, On a generalized Jordan-Von Neumann constants and uniform normal structure, Bull. Austral. Math. Soc.2003,67:225-240.
[11]J. Gao and S. Saejung, Normal structure and the generalized James ang Zbaganu con-stants, Nonlinear Analysis,2009,71:3047-3052.
[12]A. G. Aksoy and M. A. Khamsi, Nonstandard methods in fixed point theory, Springer-Verlag, Heideberg,1990.
[13]J. Gao, A pythagorean approach in Banach spaces, J. Inequal. Appl.2006:1-11.
[14]H. Cui and F. Wang, Gaos constants of Lorentz sequence spaces, Soochow Journal of Mathematics,2007,33(4):707-717.
[15]王丰辉,Banach空间的模与常数及其在不动点理论中的应用,河南师范大学硕士毕业论文,2006.
[16]M. Kato and Y. Takahashi, Some results on Von Neumann-Jordan type constants of Banach spaces, Depermental Bulletin,2007,1570:118-122.
[17]C. Yang and F. Wang, On a generalized modulus of convexity, Acta. Math. Sci.2007, 27B(4):838-844.
[18]S. Saejung, Sufficient conditions for uniform normal structure of Banach spaces and their duals, J. Math. Anal. Appl.2007,330:597-604.
[19]S. Saejung, The characteristic of convexity of a Banach space and normal structure, J. Math. Anal. Appl.2008,337:123-129.
[20]M. M. Day, Uniform convexity in factor and conjugate spaces, Ann. of Math.1940,45: 375-385.
[21]J. Banas, On moduli of smoothness of Banach Spaces, Bull. Pol. Acad. Sci. Math.1986, 34:287-293.
[22]M. Baronti and P. L. Papini, Convexity, smoothness and moduli, Nonlinear Analysis, 2009,70:2457-2465.
[23]N. I. Gurarii and Y. U. Sozonov, Normed spaces in which the unit sphere has no bias, Mat. Zametkit,1970,3:307-309.(Russian)
[24]J. Banas and K. Fraczek, Deformation of Banach spaces, Comment. Math. Univ. Corolin. 1993,34:47-53.
[25]J. Gao, Normal structure and some parameters in Banach spaces, Nonlinear Func. Anal. Appl.2005,10(2):299-310.
[26]J. Gao, On the generalized pythagorean parameters and the applications in Banach spaces, Discrete contin. Dyn. Syst. Ser. B 8(3),2007:557-567.
[27]J. Alonso and E. L. Fuster, Geometric mean and triangles inscribed in a semicircle in Banach spaces,2008,340:1271-1283.
[28]R. C. James, Uniformly nonsquare Banach spaces, Annals of Math.1964,80:542-550.
[29]W. A. Kirk, A fixed point theorem for mappings which do not increase distance, Amer. Math. Monthly,1965,72:1004-1006.
[30]C. Yang and F. Wang, On estimaties of the generalized Jordan-Von Neumann constant of Banach spaces, J. Inequal. Pure. Appl. Math.2006,7(1):Art.18.
[31]杨长森,杜艳霞,陈利,广义James常数与一致正规结构.河南师范大学学报(自然科学版),2008,36(3)：7-9.
[32]杨长森,赵峻峰,凸性模估计定理的推广,数学学报,1998,4(1)：81-86.
[33]杨长森,左红亮,Hann-Banach定理在凸性模定义中的应用,数学物理学报,2001,21(1):133-137.
[34]刘培德,鞅与Banach空间几何理论,武汉,武汉大学出版社,1993.
[2]J. Gao, On some geometric parameters in Banach spaces, J. Math. Anal. Appl.2007, 334:114-122.
[3]J. A. Clarkson, Uniformly convex spaces, Trans. Amth. Soc.1936,40:396-414.
[4]J. A. Clarkson, The Von Neumann-Jordan constant for the Lebesgue space, Ann. of Math.1937,38:114-115.
[5]M.Kato and Y.Takahashi, On the Von Neumann-Jordan constant for Banach spaces, Proc. Amer. Math. Soc.1997,125:1055-1062.
[6]M. Kato, L. Maligranda and Y. Takahashi, On James, Von Neumann-Jordan constants and the normal structure coefficients of Banach spaces, Studia. Math.2001,144:275-295.
[7]Y.Takahashi, Some geometric constants of Banach spaces, A unified approach, Banach and Function Spaces Ⅱ,2007:191-220.
[8]J. Gao and K. S. Lau, On two classes of Banach spaces with uniform normal structure, Studia Math.1991,99:41-56.
[9]S. Dhompongsa, A. Kaewkhao and S. Tasena, On a generalized James constant, J. Math. Anal. Appl.2003,285:419-435.
[10]S. Dhompongsa, P. Piraisangjun and S. Saejung, On a generalized Jordan-Von Neumann constants and uniform normal structure, Bull. Austral. Math. Soc.2003,67:225-240.
[11]J. Gao and S. Saejung, Normal structure and the generalized James ang Zbaganu con-stants, Nonlinear Analysis,2009,71:3047-3052.
[12]A. G. Aksoy and M. A. Khamsi, Nonstandard methods in fixed point theory, Springer-Verlag, Heideberg,1990.
[13]J. Gao, A pythagorean approach in Banach spaces, J. Inequal. Appl.2006:1-11.
[14]H. Cui and F. Wang, Gaos constants of Lorentz sequence spaces, Soochow Journal of Mathematics,2007,33(4):707-717.
[15]王丰辉,Banach空间的模与常数及其在不动点理论中的应用,河南师范大学硕士毕业论文,2006.
[16]M. Kato and Y. Takahashi, Some results on Von Neumann-Jordan type constants of Banach spaces, Depermental Bulletin,2007,1570:118-122.
[17]C. Yang and F. Wang, On a generalized modulus of convexity, Acta. Math. Sci.2007, 27B(4):838-844.
[18]S. Saejung, Sufficient conditions for uniform normal structure of Banach spaces and their duals, J. Math. Anal. Appl.2007,330:597-604.
[19]S. Saejung, The characteristic of convexity of a Banach space and normal structure, J. Math. Anal. Appl.2008,337:123-129.
[20]M. M. Day, Uniform convexity in factor and conjugate spaces, Ann. of Math.1940,45: 375-385.
[21]J. Banas, On moduli of smoothness of Banach Spaces, Bull. Pol. Acad. Sci. Math.1986, 34:287-293.
[22]M. Baronti and P. L. Papini, Convexity, smoothness and moduli, Nonlinear Analysis, 2009,70:2457-2465.
[23]N. I. Gurarii and Y. U. Sozonov, Normed spaces in which the unit sphere has no bias, Mat. Zametkit,1970,3:307-309.(Russian)
[24]J. Banas and K. Fraczek, Deformation of Banach spaces, Comment. Math. Univ. Corolin. 1993,34:47-53.
[25]J. Gao, Normal structure and some parameters in Banach spaces, Nonlinear Func. Anal. Appl.2005,10(2):299-310.
[26]J. Gao, On the generalized pythagorean parameters and the applications in Banach spaces, Discrete contin. Dyn. Syst. Ser. B 8(3),2007:557-567.
[27]J. Alonso and E. L. Fuster, Geometric mean and triangles inscribed in a semicircle in Banach spaces,2008,340:1271-1283.
[28]R. C. James, Uniformly nonsquare Banach spaces, Annals of Math.1964,80:542-550.
[29]W. A. Kirk, A fixed point theorem for mappings which do not increase distance, Amer. Math. Monthly,1965,72:1004-1006.
[30]C. Yang and F. Wang, On estimaties of the generalized Jordan-Von Neumann constant of Banach spaces, J. Inequal. Pure. Appl. Math.2006,7(1):Art.18.
[31]杨长森,杜艳霞,陈利,广义James常数与一致正规结构.河南师范大学学报(自然科学版),2008,36(3)：7-9.
[32]杨长森,赵峻峰,凸性模估计定理的推广,数学学报,1998,4(1)：81-86.
[33]杨长森,左红亮,Hann-Banach定理在凸性模定义中的应用,数学物理学报,2001,21(1):133-137.
[34]刘培德,鞅与Banach空间几何理论,武汉,武汉大学出版社,1993.