The stability problem of
functional equations originated from a question of
Stanisław Ulam, posed in 1940, concerning the stability of
group homomorphisms. In the next year, Donald H. Hyers gave a partial affirmative answer to the question of Ulam in the context of
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s in the case of ''additive'' mappings, 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 Hyers's theorem for mappings between Banach spaces by considering an unbounded Cauchy difference subject to a continuity condition upon the mapping. He was the first to prove the stability of the ''linear mapping''. This 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 equations. For an extensive presentation of the stability of functional equations in the context of Ulam's problem, the interested reader is referred to the books by S.-M. Jung, S. Czerwik, Y.J. Cho, C. Park, Th.M. Rassias and R. Saadati, Y.J. Cho, Th.M. Rassias and R. Saadati, and Pl. Kannappan, as well as to the following papers. In 1950, T. Aoki considered an unbounded Cauchy difference which was generalised later by Rassias to the linear case. This result is known as Hyers–Ulam–Aoki stability of the additive mapping.
[L. Maligranda, A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functions–a question of priority, Aequationes Mathematicae 75 (2008), 289-296.] Aoki (1950) had not considered continuity upon the mapping, whereas Rassias (1978) imposed extra continuity hypothesis which yielded a formally stronger conclusion.
References
See also
* Th. M. Rassias
''On the stability of functional equations and a problem of Ulam'' Acta Applicandae Mathematicae, 62(1)(2000), 23-130.
* 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.
* J. Chung
''Hyers-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.
* A. Najati and C. Park
''Hyers–Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation'' J. Math. Anal. Appl. 335(2007), 763–778.
* Th. M. Rassias and J. Brzdek (eds.), ''Functional Equations in Mathematical Analysis'', Springer, New York, 2012, .
* 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, New York, 2009, .
* P. K. Sahoo and Pl. Kannappan, ''Introduction to Functional Equations'', CRC Press, Chapman & Hall Book, Florida, 2011, .
* W. W. Breckner and T. Trif, ''Convex Functions and Related Functional Equations. Selected Topics'', Cluj University Press, Cluj, 2008.
{{DEFAULTSORT:Hyers-Ulam-Rassias Stability
Mathematical analysis
Functional equations
Functional analysis