# why we are finding the stability for functional equations? [closed]

We know why we are finding stability of differential equation. but i need the answer for the question "why we are finding the stability for functional equations?" if possible explain with some sutable examples.

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## closed as unclear what you're asking by David White, Daniel Moskovich, Ramiro de la Vega, Andres Caicedo, Ryan BudneyJul 22 '13 at 1:00

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question.If this question can be reworded to fit the rules in the help center, please edit the question.

Maybe reasoning by analogy with the answer you do know might provide hints... This does not look like a good site to ask this question. –  Mariano Suárez-Alvarez Feb 5 '13 at 6:23
Who is "we"? And what functional equations? I think this question needs more thought or explanation from the OP? –  Yemon Choi Feb 5 '13 at 6:23

The starting point of the stability theory of functional equations was the problem formulated by S. M. Ulam in 1940 (see \cite{swe}), during a conference at Wisconsin University:

\emph{Let $(G,.)$ be a group $(B,.,d)$ be a metric group. Does for every $\varepsilon>0$, there exists a $\delta>0$ such that if a function $f:G\rightarrow B$ satisfies the inequality $$d(f(xy),f(x)f(y))\leq \delta,\ \ x,y\in G,$$ there exists a homomorphism $g:G\rightarrow B$ such that $$d(f(x),g(x))\leq\varepsilon,\ \ x\in G?$$} In 1941, D.H. Hyers \cite{hyer} gave an affirmative partial answer to this problem. This is the reason for which today this type of stability is called Hyers-Ulam stability of functional equation. In 1950, Aoki \cite{ghy} generalized Hyers' theorem for approximately additive functions. In 1978, Th. M. Rassias \cite{tem} generalized the theorem of Hyers by considering the stability problem with unbounded Cauchy differences. Taking this fact into account, the additive functional equation $f(x+y)=f(x)+f(y)$ is said to have the Hyers-Ulam-Rassias stability on $(X,Y)$. This terminology is also applied to the case of other functional equations. For more detailed definitions of such terminology one can refer to \cite{ter1} and \cite{ter2}. Thereafter, the stability problem of functional equations has been extended in various directions and studied by several mathematicians \cite{17,b1,izm,hgl,kaz,tem2,f3,sav,isac}.

The Hyers-Ulam stability of mappings is in development and several authors have remarked interesting applications of this theory to various mathematical problems. In fact the Hyers-Ulam stability has been mainly used to study problems concerning approximate isometries or quasi-isometries, the stability of Lorentz and conformal mappings, the stability of stationary points, the stability of convex mappings, or of homogeneous mappings, etc \cite{hyer2,hyer3,Cze,nik,Tab}.

\bibitem{ghy} {T. Aoki, On the stability of the linear transformation in Banach spaces, \em J. Math. Soc Japan.,} {\bf 2}(1950), 64–66.

\bibitem{b1} {J. A. Baker, A general functional equation and its stability, \em Proc. Amer. Math. Soc.} {\bf 133}(2005), 1657-1664.

\bibitem{b2} {J. A. Baker, The stability of the cosine equation, \em Proc. Amer. Math. Soc.} {\bf 80}(1980), 411-416.

\bibitem{b3} {J. A. Baker, J. Lawrence, and F. Zorzitto, The stability of the equation f(x + y) = f(x)f(y), \em Proc. Amer. Math. Soc.} {\bf 74}(1979), 242-246.

\bibitem{Cze} {S. Czerwik, On the stability of the homogeneous mapping, \em G. R. Math. Rep. Acad. Sci. Canada XIV} {\bf 6}(1992), 268-272.

\bibitem{asd} {S.Czerwik and M. Przybyla, A general Baker superstability criterion for the D'Alembert functional equation, \em Banach Spaces and their Applications in Analysis,Walter de Gruyter Gmbh @ Co.KG,Berlin} (2007), 303-306.

\bibitem{f4} {J. Diaz and B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, \em Bull. Amer. Math. Soc.} {\bf 74}(1968), 305-309.

\bibitem{ter1} {G. L. Forti, Hyers-Ulam stability of functional equations in several variables, \em Aeq. Math.} {\bf 50}(1995), 143-190.

\bibitem{Forti} {G. L. Forti, Remark 11 in: Report of the 22nd Internat. Symposium on Functional Equations, \em Aeq. Math.} {\b 29} (1980), (1985), 90-91.

\bibitem{izm} {R. Ger and P. $\check{S}$emrl, The stability of the exponential equation, \em Proc. Amer. Math. Soc.} {\bf 124}(1996), 779-787.

\bibitem{hyer} {D. H. Hyers, On the stability of the linear functional equation, \em Proc. Nat. Acad. Sci. U.S.A.} {\bf 27}(1941), 222-224.

\bibitem{ter2} {D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, \em Aeq. Math.} {\bf 44}(1992), 125-153.

\bibitem{hyer2} {D. H. Hyers, The Stability of homomorphisms and related topics, \em in Global Analysis-Analysis on manifolds (ed. Th. M Rassias), Teubner-Texte zur Math., Leipzig,} {\bf 57}(1983), 140-153.

\bibitem{hyer3} {D. H. Hyers and S. M. Ulam, Approximately convex functions,\em Proc. Amer. Math. Soc.} {\bf 3} (1952), 821-828.

\bibitem{isac} {G. Isac and Th. M. Rassias, Stability of $\Psi$-additive mappings: Applications to nonlinear analysis , \em Internat. J. Math. $\&$ Math. Sci.} {\bf 19(2)}(1996), 219-228.

\bibitem{hgl} {K. Jarosz, Almost multiplicative functionals, \em Studia Math.} {\bf 124}(1997), 37-58.

\bibitem{nik} {K. Nikodem, Approximately quasiconvex functions,\em C. R. Math. Rep. Acad. Sci. Canada,} {\bf 10} (1988), 291-294.

\bibitem{kaz} {B. E. Johnson, Approximately multiplicative functionals, \em J. London Math. Soc.} {\bf 34(2)}(1986), 489-510.

\bibitem{f3} {V. Radu, The fixed point alternative and the stability of functional equations, \em Fixed Point Theory.,} {\bf 4}(2003), 91-96.

\bibitem{tem2} {Th. M. Rassias, On the stability of functional equations and a problem of Ulam, \em Acta Applicandae Mathematicae.} {\bf 62}(2000), 23-130.

\bibitem{Ra9} {Th. M. Rassias, Problem $18$, In: Report on the $31$st ISFE, \em Aeq. Math.} {\bf 47}, 1994, 312-13.

\bibitem{tem} {Th. M. Rassias, On the stability of the linear mapping in Banach spaces, \em Proc. Amer. Math. Soc.} {\bf 72}(1978), 297-300.

\bibitem{sav} {Th. M. Rassias, The problem of S. M. Ulam for approximately multiplicative mappings, \em J. Math. Anal. Appl.} {\bf 246}(2000), 352-378.

\bibitem{Tab} {J. Tabor and J. Tabor, Homogenity is superstable, \em Publ. Math. Debrecen} {\bf 45}(1994), 123-130. \bibitem{swe} {S. M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, New York, 1960.}

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