臺灣博碩士論文加值系統

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。

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.

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

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[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.

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[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.
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