| 复合泛函方程的稳定性 |
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| 引用本文： | 李静,杨莹,曹怀信.复合泛函方程的稳定性[J].陕西师范大学学报,2012(3):16-19. |
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| 作者姓名： | 李静 杨莹 曹怀信 |
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| 作者单位： | 陕西师范大学数学与信息科学学院;陕西铁路工程职业技术学院 |
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| 基金项目： | 国家自然科学基金资助项目(10571113,10871224);陕西铁路工程职业技术学院科研项目(2011-41) |
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| 摘 要： | 研究了复合泛函方程T(T(x)-T(y))=T(x+y)+T(x-y)-T(x)-T(y)在泛函Φ(x,y)限制下的稳定性问题.证明了:若E为Banach空间,泛函Φ:E×E→0,∞)连续使得级数Φ(x)d=sum (2-j-1Φ(2jx,2jx)) from j=1 to ∞在E的任一有界子集上一致收敛,F:E→E是连续映射且满足‖F(F(x)-F(y))-F(x+y)-F(x-y)+F(x)+F(y)‖≤Φ(x,y)(■x、y∈E),则存在唯一的连续2-齐次映射T:E→E满足以上复合泛函方程且‖T(x)-F(x)‖≤Φ(x),■x∈E.
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| 关 键 词： | 稳定性 复合泛函方程 幂等映射 连续性 |
Stability of composition functional equation |
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| LI Jing,YANG Ying,CAO Huai-xin.Stability of composition functional equation[J].Journal of Shaanxi Normal University: Nat Sci Ed,2012(3):16-19. |
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| Authors: | LI Jing YANG Ying CAO Huai-xin |
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| Institution: | 1*(1 College of Mathematics and Information Science, Shaanxi Normal University,Xi′an 710062,Shaanxi,China; 2 Shaanxi Railway Institute,Weinan 714000,Shaanxi,China) |
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| Abstract: | The stability of the composition functional equation T(T(x)-T(y))=T(x+y)+T(x-y)-T(x)-T(y) with a controlling function Φ(x,y) is discussed in this paper.It is proved that if E is a Banach spaces,Φ:E×E→0,∞) is continuous and such that the seriesΦ(x)d=sum (2-j-1Φ(2jx,2jx)) from j=1 to ∞ is uniformly convergent on every bounded subset of E,and F:E→E is a continuous mapping satisfying ‖F(F(x)-F(y))-F(x+y)-F(x-y)+F(x)+F(y)‖≤Φ(x,y)(■x,y∈E).Then there exists a unique 2-homogeneous continuous mapping T:E→E such that ■x∈E,F(F(x)-f(y))-F(x+y)-F(x-y)+F(x)+F(y)=0,‖T(x)-F(x)‖≤Φ(x). |
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| Keywords: | stability composition functional equation idempotent mapping continuity |
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