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Information Center for Mathematical Science

논문검색

Information Center for Mathematical Science

논문검색

Glasnik Matematicki Serja III.
( Vol. 34 NO.2 / (1999))
Solution of the Ulam Stability Problem for Quartic Mappings
John Michael Rassias,
Pages. 243-252
Abstract In 1940 S.M. Ulam proposed at the University of Wisconsin the problem: ""Give
conditions in oder for a linear mapping near an approximately linear mapping to
exist."" In 1968 S.M. Ulam proposed the general problem: ""When is it true that by
changing a little the hypotheses of a theorem one can still assert that the
thesis of the theorem remains true or approximately true?"" In 1978 P. M. Gruber
proposed the Ulam type problem: ""Suppose a mathematical object satisfies a
certain property approximately. Is it then possible to approximate this object
by objects, satisfying the property exactly?"" According to P. M> Gruber this
kind of stability problems is of particular interest in probability theory and
in th case of functional equations of different types. In 1982-1998 we solved
the above Ulam problem, or equivalently the Ulam type problem for linear
mappings and also established analogous stability problems for quadratic and
cubic mappings. In this paper we introduce the new quartic mappings $F: X
ightarrow Y$, satisfying the new quartic functional equation \ $$ F(x_1 + 2x_2) + F(x_1 - 2x_2) + 6F(x_1) = $$ $$ hspace{3cm} 4 [ F(x_1 + x_2) + F(x_1 - x_2) + 6F(x_2)]$$ \ for all 2-dimensional vectors $(x_1, x_2) in X^2$, with $X$ a linear space $(Y: = $ a real complete linear space), and then solve the Ulam stability problem for the above mappings $F$.
Contents 1. Quartic functional equation 2. Quartic functional inequality
Key words Ulam problem, Ulam type problem, quartic mappings, quartic functional equation, quartic functional inequality, approximately quartic, stability problem
Mathmatical Subject Classification 39B