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

曾冠逞(Guan-Cheng Zeng)

指導教授 ： 林欽誠 ； 蕭勝彥

國立中央大學/理學院/數學系/碩士(2008年)

摘要

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。

關鍵字

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

並列摘要

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.

並列關鍵字

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

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.

[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.