两类泛函方程在几类空间中的稳定性
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摘要
泛函方程理论中一个典型的问题是稳定性问题.泛函方程的稳定性问题源自Ulam在1940年提出的关于群同态的稳定性问题：
给定一个群(G1,*)和一个度量群(G2,.,d),其中d((?))为一个度量.给定一个ε>0,存在一个δ>0使得如果h：G1→G2为一个映射且对所有的x,y∈G1均有d(h(x*y),h(x)·h(y))<δ是否存在一个同态H：G1→G2使得对所有的x∈G1, d(h(x),H(x))<ε?
1941年,D.H.Hyers解决了Banach空间上可加映射的稳定性问题.在接下来的几十年里,许多数学家对各种不同的泛函方程的稳定性进行了系统的研究,例如指数方程,二次泛函方程,三次泛函方程以及广义可加的泛函方程等.1978年Th.M.Rassias解决了线性映射在Banach空间中的稳定性问题；1999年Y.Lee和K.Jun研究了广义Jensen方程的稳定性.这些稳定性的成果在随机分析,金融数学和精算数学等领域中均有广泛的应用.
本文共分为两章.
在第一章中,我们研究了一个源自Jensen可加泛函方程和二次泛函方程的混合二次可加泛函方程在β-巴拿赫空间和拟巴拿赫空间中的稳定性问题.首先,我们讨论了上述方程在β-巴拿赫空间中的稳定性,接下来我们又考虑了这个方程在拟巴拿赫空间中的稳定性.
在第二章中,我们研究了一个源自四次泛函方程的n维四次方程在非阿基米德巴拿赫模和随机巴拿赫模中的稳定性问题.在这里V={I(?)M：1∈I},M={1,2,...,n},且M/I=Ic
The stability problem is a classical question in the theory of functional equations.The stability problem concerning the stability of group homomorphisms was firstly raisedby Ulam in1940:
Give a group (G1,) and a metric group (G2,·, d) with the metric d(·.·). Give>0,does there exists a δ>0such that if h: G1→G2satisfies d(h(x y), h(x)· h(y))<δ forall x, y∈G1, then is there a homomorphism H: G1→G2with d(h(x), H(x))<ε for allx∈G1?
In1941, D. H. Hyers solved the stability problem of additive mapping on Banachspaces. In the following decades, many mathematicians have studied the stability ofdiferent kinds of functional equations such as exponential equation, quadratic functionalequation, cubic functional equation, generalized additive equation and so on. In1978, Th.M. Rassias solved the stability problem of linear mapping in Banach space. In1999, Y.Lee and K. Jun studied the stability of generalized Jense equation. These stability resultshave applications in some related fields such as random analysis, financial mathematicsand actuarial mathematics.
This thesis consists of two chapters.
In chapter1, we consider the Hyers-Ulam stability of a mixed additive-quadraticfunctional equation deriving from the Jensen additive functional equationin β-Banach space and quasi-Banach space. We firstly discuss the Hyers-Ulam stability ofabove mixed additive-quadratic functional equation in β-Banach space. Then we considerthe Hyers-Ulam stability of this functional equation in quasi-Banach spaces.
In chapter2, we consider the following n-dimensional functional equation (here V={I M:1∈I}, M={1,2,..., n}, M/I=Ic)deriving from the quarticfunctional equationsf(2x+y)+f(2x y)=4f(x+y)+4f(x y)+24f(x)6f(y),and discussed its Hyers-Ulam stability in non-Archimedean Banach Modules and randomBanach Modules.
给定一个群(G1,*)和一个度量群(G2,.,d),其中d((?))为一个度量.给定一个ε>0,存在一个δ>0使得如果h：G1→G2为一个映射且对所有的x,y∈G1均有d(h(x*y),h(x)·h(y))<δ是否存在一个同态H：G1→G2使得对所有的x∈G1, d(h(x),H(x))<ε?
1941年,D.H.Hyers解决了Banach空间上可加映射的稳定性问题.在接下来的几十年里,许多数学家对各种不同的泛函方程的稳定性进行了系统的研究,例如指数方程,二次泛函方程,三次泛函方程以及广义可加的泛函方程等.1978年Th.M.Rassias解决了线性映射在Banach空间中的稳定性问题；1999年Y.Lee和K.Jun研究了广义Jensen方程的稳定性.这些稳定性的成果在随机分析,金融数学和精算数学等领域中均有广泛的应用.
本文共分为两章.
在第一章中,我们研究了一个源自Jensen可加泛函方程和二次泛函方程的混合二次可加泛函方程在β-巴拿赫空间和拟巴拿赫空间中的稳定性问题.首先,我们讨论了上述方程在β-巴拿赫空间中的稳定性,接下来我们又考虑了这个方程在拟巴拿赫空间中的稳定性.
在第二章中,我们研究了一个源自四次泛函方程的n维四次方程在非阿基米德巴拿赫模和随机巴拿赫模中的稳定性问题.在这里V={I(?)M：1∈I},M={1,2,...,n},且M/I=Ic
The stability problem is a classical question in the theory of functional equations.The stability problem concerning the stability of group homomorphisms was firstly raisedby Ulam in1940:
Give a group (G1,) and a metric group (G2,·, d) with the metric d(·.·). Give>0,does there exists a δ>0such that if h: G1→G2satisfies d(h(x y), h(x)· h(y))<δ forall x, y∈G1, then is there a homomorphism H: G1→G2with d(h(x), H(x))<ε for allx∈G1?
In1941, D. H. Hyers solved the stability problem of additive mapping on Banachspaces. In the following decades, many mathematicians have studied the stability ofdiferent kinds of functional equations such as exponential equation, quadratic functionalequation, cubic functional equation, generalized additive equation and so on. In1978, Th.M. Rassias solved the stability problem of linear mapping in Banach space. In1999, Y.Lee and K. Jun studied the stability of generalized Jense equation. These stability resultshave applications in some related fields such as random analysis, financial mathematicsand actuarial mathematics.
This thesis consists of two chapters.
In chapter1, we consider the Hyers-Ulam stability of a mixed additive-quadraticfunctional equation deriving from the Jensen additive functional equationin β-Banach space and quasi-Banach space. We firstly discuss the Hyers-Ulam stability ofabove mixed additive-quadratic functional equation in β-Banach space. Then we considerthe Hyers-Ulam stability of this functional equation in quasi-Banach spaces.
In chapter2, we consider the following n-dimensional functional equation (here V={I M:1∈I}, M={1,2,..., n}, M/I=Ic)deriving from the quarticfunctional equationsf(2x+y)+f(2x y)=4f(x+y)+4f(x y)+24f(x)6f(y),and discussed its Hyers-Ulam stability in non-Archimedean Banach Modules and randomBanach Modules.
引文
[1] Th. M. Rassias. On the stability of functional equations and a problem of Ulam. Acta AppplicandaeMathematicae62:23-130,2000.
[2] D. H. Hyers. On the stability of the linear functional equation, Pro. Natl. Acad. Sci. USA27(1941)222-224.
[3] D. Mihet. V. Radu. On the stability of the additive Cauchy functional equation in random normedspaces. J. Math. Anal. Appl.343(2008)567-572.
[4] Th. M. Rassias, M. S. Moslehian. Stability of functional equations in non-Archimedean spaces.Appl. Anal. Discrete Math.1(2007)325-334.
[5] A. Najati, M. Eshaghi Gordji. Fixed points and quartic functional equations in β-Banach Modules.Results Math. Dol10.1007/s00025-001-0135-8.
[6] M. E. Gordji, A. Ebadian, S. Zolfaghari. Stability of a function equation deriving from cubic andquartic functions [J]. Abstr. Appl. Anal.2008, Article ID801904,17pages.
[7] J. M. Rassias, H-M Kim. Geberalized Hyers-Ulam stability for additive functional equations inquasi-β-normed spaces [J]. J. Math. Anal. Appl.2009,356:302-309.
[8] A. Najati, G. Zamani Eskandani. Stability of a mixed additive and cubic functional equation inquasi-Banach spaces [J]. J. Math. Anal. Appl.2008,342:1318-1331.
[9] A. Najati, M. B. Moghimi. Stability of a functional equation deriving from quadratic and additivefunctions in quasi-Banach spaces [J]. J. Math. Anal. Appl.2008,337(1):399-415.
[10] M. E. Gordji, S. Zolfaghari, J. M. Rassias. and M. B. Savadkouhi. Solution and stability of a mixedtype cubic and quartic functional equation in quasi-Banach spaces. Abstrat and Applied Analysis2009. Art. ID.417473,(2009).
[11] A. Najati, F. Moradlou. Stability of quadratic functional equation in quasi-Banach space [J]. Bull.Korean Math. Soc.2008,45(3):587-600.
[12] C. Park. On the Hyers-Ulam-Rassias stability of generalized quadratic mapping in Banach modules[J]. J. Math. Anal. Appl.2004,291(1):214-223.
[13] K. Ravi, R. Murali, M. Arunkumar. The generalized Hyers-Ulam stability of a quadratic functionequation [J]. J. Inequal. Pure. Appl. Math.2008,9(1): Article20.
[14] Th. M. Rassias. On the stability of functional equations and a problem of Ulam. Acta Appli.62(2000):23-130.
[15] Z. Gajda. On the stability of additive mappings. Int. J. Math. Sci.14(1991):431-434.
[16] Th. M. Rassias. On the stability of the linear mapping in Banach spaces. Pro. Amer. Math. Soc.72(1978):297-300.
[17] P. W. Cholewa. Remarks on the stability of functional equations. Aequationes Math.27(1984):76-86.
[18] S. M. Jung. Hyers-Ulam-Stability of Tensen’s equation and its application. Pro. Amer. Math. Soc.126(1998):3137-3143.
[19] I. Chang and H. Kim. On the Hyers-Ulam-Stability of quadratic functional equations. J. Ineq.Pure Appl. Math.3(2002):1-12.
[20] C. Park. On the stability of the quadratic mapping in Banach modules. J. Math. Anal. Appl.276(2002):135-144.
[21] J. Bae. K. Jun and Y. Lee. On the Hyers-Ulam-Stability of an n-dimensional Pexiderized quadraticequation. Math. Ineq. Appl.7(1)(2004):63-77.
[22] H. Chu, S. Koo and J.Park. Partial stabilities and partial derivations of n-variabie functions.Noulinear Anal. TMA(to appear).
[23] Y. Benyamini and J. Lindenstrauss. Geometric nonlinear functional analysis. Vol.1. Colloq. publ.48. Amer. Math. Soc. Providence,2000.
[24] C. Borelli and G. L. Forti. On a genaral Hyers-Ulam-Stability result. Internal. J. Math. Sci.18(1995):229-236.
[25] H. Azadi. Kenary, Y. Je Cho. Computers and Mathematics with Applications.61(2001):2704-2724.
[26] M. Mirzavaziri, M. S. Moslehian. A fixed point approach to stability of a quadratic equation. Bull.Braz. Math. Soc.(NS)37(3)(2006):361-376.
[27] S. M. Jung, T. S. Kim. A fixed point approach to the stability of a cubic functional equations.Bol. Soc. Mat. Mexicana(3)12(2006)51-57.
[28] D. H. Hyers, G. Isac and Th. M. Rassias. Stability of functional equations in several variables.Birkhauser. Basel.1998.
[29] S. Jung and J. M. Rassias. A fixed point approach to the stability of a functional equation of thespiral of Theodorns. Fixed point theory and applications.2008. Art. ID.915010(2008).
[30] M. Eshaghi Gordji. Stability of a functional equation deriving from quartic and additive functions.Bull. Korean Math. Soc. Vol.47, No.3(2010):491-502.
[2] D. H. Hyers. On the stability of the linear functional equation, Pro. Natl. Acad. Sci. USA27(1941)222-224.
[3] D. Mihet. V. Radu. On the stability of the additive Cauchy functional equation in random normedspaces. J. Math. Anal. Appl.343(2008)567-572.
[4] Th. M. Rassias, M. S. Moslehian. Stability of functional equations in non-Archimedean spaces.Appl. Anal. Discrete Math.1(2007)325-334.
[5] A. Najati, M. Eshaghi Gordji. Fixed points and quartic functional equations in β-Banach Modules.Results Math. Dol10.1007/s00025-001-0135-8.
[6] M. E. Gordji, A. Ebadian, S. Zolfaghari. Stability of a function equation deriving from cubic andquartic functions [J]. Abstr. Appl. Anal.2008, Article ID801904,17pages.
[7] J. M. Rassias, H-M Kim. Geberalized Hyers-Ulam stability for additive functional equations inquasi-β-normed spaces [J]. J. Math. Anal. Appl.2009,356:302-309.
[8] A. Najati, G. Zamani Eskandani. Stability of a mixed additive and cubic functional equation inquasi-Banach spaces [J]. J. Math. Anal. Appl.2008,342:1318-1331.
[9] A. Najati, M. B. Moghimi. Stability of a functional equation deriving from quadratic and additivefunctions in quasi-Banach spaces [J]. J. Math. Anal. Appl.2008,337(1):399-415.
[10] M. E. Gordji, S. Zolfaghari, J. M. Rassias. and M. B. Savadkouhi. Solution and stability of a mixedtype cubic and quartic functional equation in quasi-Banach spaces. Abstrat and Applied Analysis2009. Art. ID.417473,(2009).
[11] A. Najati, F. Moradlou. Stability of quadratic functional equation in quasi-Banach space [J]. Bull.Korean Math. Soc.2008,45(3):587-600.
[12] C. Park. On the Hyers-Ulam-Rassias stability of generalized quadratic mapping in Banach modules[J]. J. Math. Anal. Appl.2004,291(1):214-223.
[13] K. Ravi, R. Murali, M. Arunkumar. The generalized Hyers-Ulam stability of a quadratic functionequation [J]. J. Inequal. Pure. Appl. Math.2008,9(1): Article20.
[14] Th. M. Rassias. On the stability of functional equations and a problem of Ulam. Acta Appli.62(2000):23-130.
[15] Z. Gajda. On the stability of additive mappings. Int. J. Math. Sci.14(1991):431-434.
[16] Th. M. Rassias. On the stability of the linear mapping in Banach spaces. Pro. Amer. Math. Soc.72(1978):297-300.
[17] P. W. Cholewa. Remarks on the stability of functional equations. Aequationes Math.27(1984):76-86.
[18] S. M. Jung. Hyers-Ulam-Stability of Tensen’s equation and its application. Pro. Amer. Math. Soc.126(1998):3137-3143.
[19] I. Chang and H. Kim. On the Hyers-Ulam-Stability of quadratic functional equations. J. Ineq.Pure Appl. Math.3(2002):1-12.
[20] C. Park. On the stability of the quadratic mapping in Banach modules. J. Math. Anal. Appl.276(2002):135-144.
[21] J. Bae. K. Jun and Y. Lee. On the Hyers-Ulam-Stability of an n-dimensional Pexiderized quadraticequation. Math. Ineq. Appl.7(1)(2004):63-77.
[22] H. Chu, S. Koo and J.Park. Partial stabilities and partial derivations of n-variabie functions.Noulinear Anal. TMA(to appear).
[23] Y. Benyamini and J. Lindenstrauss. Geometric nonlinear functional analysis. Vol.1. Colloq. publ.48. Amer. Math. Soc. Providence,2000.
[24] C. Borelli and G. L. Forti. On a genaral Hyers-Ulam-Stability result. Internal. J. Math. Sci.18(1995):229-236.
[25] H. Azadi. Kenary, Y. Je Cho. Computers and Mathematics with Applications.61(2001):2704-2724.
[26] M. Mirzavaziri, M. S. Moslehian. A fixed point approach to stability of a quadratic equation. Bull.Braz. Math. Soc.(NS)37(3)(2006):361-376.
[27] S. M. Jung, T. S. Kim. A fixed point approach to the stability of a cubic functional equations.Bol. Soc. Mat. Mexicana(3)12(2006)51-57.
[28] D. H. Hyers, G. Isac and Th. M. Rassias. Stability of functional equations in several variables.Birkhauser. Basel.1998.
[29] S. Jung and J. M. Rassias. A fixed point approach to the stability of a functional equation of thespiral of Theodorns. Fixed point theory and applications.2008. Art. ID.915010(2008).
[30] M. Eshaghi Gordji. Stability of a functional equation deriving from quartic and additive functions.Bull. Korean Math. Soc. Vol.47, No.3(2010):491-502.