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# Hyers–Ulam–Rassias stability

The stability problem of functional equations originated from a question of Stanislaw 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 spaces, 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.[1][2] 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[3][4] of functional equations.

References

Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72(1978), 297–300.

D. H. Hyers, G. Isac and Th. M. Rassias,Stability of Functional Equations in Several Variables, Birkhäuser Verlag, Boston, Basel, Berlin, 1998.

S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York (2011) ISBN 978-1-4419-9636-7.

Hyers-Ulam-Rassias stability, in: Encyclopaedia of Mathematics, Supplement III, M. Hazewinkel (ed.), Kluwer Academic Publishers, Dordrecht, 2001, pp.194-196.

See also

P. M. Pardalos, P. G. Georgiev and H. M. Srivastava (eds.), Nonlinear Analysis. Stability, Approximation, and Inequalities. In honor of Themistocles M. Rassias on the occasion of his 60th birthday, Springer, New York, 2012.

Soon-Mo Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Inc., Florida, 2001.

Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Applicandae Mathematicae, 62(1)(2000), 23-130.

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.

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, 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.

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, ISBN 978-0-387-89491-1.

P. K. Sahoo and Pl. Kannappan, Introduction to Functional Equations, CRC Press, Chapman & Hall Book, Florida, 2011, ISBN 978-1-4398-4111-2.

Th. M. Rassias and J. Brzdek (eds.), Functional Equations in Mathematical Analysis, Springer, New York, 2012, ISBN 978-1-4614-0054-7.

W. W. Breckner and T. Trif, Convex Functions and Related Functional Equations. Selected Topics, Cluj University Press, Cluj, 2008.

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