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

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1 Answer

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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