Hardy-Hilbert型式的不等式和Cauchy加法映射的穩定性

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曾冠逞(Guan-Cheng Zeng) 查詢紙本館藏

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論文名稱

Hardy-Hilbert型式的不等式和Cauchy加法映射的穩定性
(On Hardy-Hilbert Type Inequalities and Stability of Cauchy Additive Mappings)

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摘要(中)

這篇論文研究兩個主題:Hardy-Hilbert型式的積分不等式和Cauchy加法映射的穩定性。下列是主要結果：1) 將B. Yang對某種有界的自伴積分算子T : L2 (0,∞) → L2 (0,∞) 的範數及其應用到Hardy -Hilbert型式的不等式的結果，從 L2 (0,∞)空間推廣到Lp (0,∞) 空間 (p > 1) ; 2) 推廣Rassias關於Cauchy加法映射的穩定性定理; 3) 給予Park等人[6]的定理的一個正確的證明; 4) 以一個唯一的群的同態變換 (或環的同態變換) 去逼近一個特定的向量映射的奇部分。

摘要(英)

This thesis is concerned with two subjects of research; Hardy-Hilbert type inequalities and the stability of Cauchy additive mappings. The following are done : 1) to extend B. Yang’’s result on the norm of a bounded self- adjoint integral operator T : L2 (0,∞) → L2 (0,∞) and its applications to Hardy-Hilbert type integral inequalities from the space L2 (0,∞) to the space Lp (0,∞) with p > 1 ; 2) to generalize Rassias’’s theorem on the stability of Cauchy additive mappings ; 3) to give a correct proof of Park et al’’s theorem in [6]; 4) to approximate the odd part of a certain vector mapping by a unique group homomorphism and ring homomorphism, respectively.

關鍵字(中)

★ 近乎線性映射
★ 穩定性
★ Holder's 不等式
★ Hardy-Hilbert 型式的不等式
★ Norm
★ 積分算子

關鍵字(英)

★ Approximately linear mapping
★ stability
★ Holder's inequality
★ inequality of Hardy-Hilbert type
★ integral operator
★ Norm

論文目次

Abstract..................................................................................................................................1
1. Introduction........................................................................................................................2
Part I
2. Norms of Some Integral Operators and Applications to Hardy- Hilbert Type
Inequalities...........................................................................................................................11
2.1 General results................................................................................................................11
2.2 Applications to some examples of operators..................................................................15
Part II
3. Stability of Cauchy Additive Mappings...........................................................................26
References............................................................................................................................42

參考文獻

[1] R. Badora, On approximate ring homomorphism, J. Math. Anal. Appl., 276 (2002),589-597.
[2] Z. Gajda, On stability of additive mappings, Internat. J. Math. & Math. Sci., 14(1991), 431-434.
[3] G. H. Hardy , J. E. Littlewood and G. Polya, Inequalities, Cambridge UniversityPress , Cambridge, 1952.
[4] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222-224.
[5] Y. Li, Z. Wang, and B. He, Hilbert’s type linear operator and some extensions of Hilbert’s inequality, J. Inequal. Appl. Vol. 2007 (2007), Article ID 82138, 10 pages.
[6] C. Park, Y. Cho and M. Han, Functional inequalities associated with Jordan-Von Neumann type additive functional equations, J. Inequal. Appl. (2007), to appear.
[7] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300.
[8] T. M. Rassias, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc., 114 (1992), 989-993.
[9] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publ., New York, 1960.
[10] Z. Wang, D. Gua., An introduction to special functions, Science Press, Bejing, 1979.
[11] B. Yang, On the norm of a self-adjoint operator and applications to the Hilbert’s type inequalities, Bulletin of the Belgian Mathematical Society, 13 (2006), 577-584.
[12] B. Yang, On the norm of a certain self-adjoint integral operator and applications to bilinear integral inequalities, Taiwanese J. Math., to apprar.
[13] D. H. Zhang and H. X. Cao, Stability of functional equations in several variables, Acta Math. Sinica. (Engl. Ser.) 23 (2007), 321-326.
42

指導教授

林欽誠、蕭勝彥
(Chin-cheng Lin、Sen-Yen Shaw)

審核日期

2008-4-14

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