Hyers–Ulam–Rassias stability
Encyclopedia
The stability problem of functional equation
s originated from a question of Stanislaw Ulam, posed in 1940, concerning the stability of group homomorphism
s. In the next year, Donald H. Hyers gave a partial affirmative answer to the question of Ulam in the context of Banach space
s, that was the first significant breakthrough and a step toward more solutions in this area. Since then, a large number of papers have been published in connection with various generalizations of Ulam’s problem and Hyers’s theorem. In 1978, Themistocles M. Rassias
succeeded in extending the Hyers’s theorem by considering an unbounded Cauchy difference. This exciting result of Rassias attracted several mathematicians worldwide who began to be stimulated to investigate the stability problems of functional equations.
By regarding a large influence of S. M. Ulam, D. H. Hyers, and Th. M. Rassias on the study of stability problems of functional equations, the stability phenomenon proved by Th. M. Rassias led to the development of what is now known as Hyers–Ulam–Rassias stability of functional equation
s.
Functional equation
In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...
s originated from a question of Stanislaw Ulam, posed in 1940, concerning the stability of group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
s. In the next year, Donald H. Hyers gave a partial affirmative answer to the question of Ulam in the context of Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s, that was the first significant breakthrough and a step toward more solutions in this area. Since then, a large number of papers have been published in connection with various generalizations of Ulam’s problem and Hyers’s theorem. In 1978, Themistocles M. Rassias
Themistocles M. Rassias
Themistocles M. Rassias is a Greek mathematician, and a professor at the National Technical University of Athens , Greece. He has published more than 220 papers, 6 research books and 30 edited volumes in research Mathematics as well as 4 textbooks in Mathematics for university students...
succeeded in extending the Hyers’s theorem by considering an unbounded Cauchy difference. This exciting result of Rassias attracted several mathematicians worldwide who began to be stimulated to investigate the stability problems of functional equations.
By regarding a large influence of S. M. Ulam, D. H. Hyers, and Th. M. Rassias on the study of stability problems of functional equations, the stability phenomenon proved by Th. M. Rassias led to the development of what is now known as Hyers–Ulam–Rassias stability of functional equation
Functional equation
In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...
s.
See also
- Soon-Mo Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Inc., Florida, 2001.
- J. Chung, http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WK2-4DSPR9R-6&_user=83473&_coverDate=12%2F15%2F2004&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1633114773&_rerunOrigin=google&_acct=C000059671&_version=1&_urlVersion=0&_userid=83473&md5=32f25b33ce19cdf5b0fdf58031a327fc&searchtype=aHyers-Ulam-Rassias stability of Cauchy equation in the space of Schwartz distributions], J. Math. Anal. Appl. 300(2)(2004), 343 – 350.
- T. Miura, S.-E. Takahasi, and G. Hirasawa, Hyers-Ulam-Rassias stability of Jordan homomorphisms on Banach algebras, J. Inequal. Appl. 4(2005), 435–441.
- P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184(1994), 431–436.
- P. Gavruta and L. Gavruta, A new method for the generalized Hyers–Ulam–Rassias stability, Int. J. Nonlinear Anal. Appl. 1(2010), No. 2, 6 pp.
- A. Najati and C. Park, http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WK2-4N25W07-C&_user=83473&_coverDate=11%2F15%2F2007&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000059671&_version=1&_urlVersion=0&_userid=83473&md5=cbaa154e49b04219c17b1d34fb1d8a00&searchtype=aHyers–Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation], J. Math. Anal. Appl. 335(2007), 763–778.
- D. Zhang and J. Wang, On the Hyers-Ulam-Rassias stability of Jensen’s equation, Bull. Korean Math. Soc. 46(4)(2009), 645–656.
- T. Trif, Hyers-Ulam-Rassias stability of a Jensen type functional equation, J. Math. Anal. Appl. 250(2000), 579–588.
- Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer Monographs in Mathematics, Springer, New York, 2009.
- P. K. Sahoo and Pl. Kannappan, Introduction to Functional Equations, Chapman & Hall, 2011.
- W. W. Breckner and T. Trif, Convex Functions and Related Functional Equations. Selected Topics, Cluj University Press, Cluj, 2008.