## On the stability of a multi-dimensional Cauchy type functional equation.(English)Zbl 0842.39014

Rassias, John M. (ed.), Geometry, analysis and mechanics. Dedicated to Archimedes on his 2281st birthday. Singapore: World Scientific. 365-376 (1994).
Summary: In 1940 S. M. Ulam [cf. A collection of mathematical problems (1960; Zbl 0086.24101)]imposed before the Mathematics Club of the University of Wisconsin the following problem:
“Give conditions in order for a linear mapping near an approximately linear mapping to exist”.
Then D. H. Hyers [Proc. Natl. Acad. Sci. 27, 222-224 (1941; Zbl 0061.26403)]established this stability problem with a Cauchy inequality involving a non-negative constant. Then the author [J. Approximation Theory 57, No. 3, 268-273 (1989; Zbl 0672.41027)]solved the Ulam problem with a Cauchy functional inequality, involving a product of powers of norms. Recently, the author [Discuss. Math. 12, 95-103 (1992; Zbl 0779.47005)]established the general version of this problem with multi-dimensional Cauchy inequalities involving a non-negative real-valued function $$K$$: $$K(0)= 0$$.
In this paper the author introduces the 2-dimensional Cauchy type functional inequality: $|f(x_1+ x_2)+ f(x_1- x_2)- 2f(x_1) |\leq c,$ for all $$x_1, x_2\in X$$ (:= normed linear space), $$c$$ $$(:=\text{const.})\geq 0$$ with $$|f(0) |\leq 0$$, $$c_0 (:=\text{const.})\geq 0$$, and assumes mapping $$f: X\to Y$$ (:= complete normed linear space) with $$f(tx)$$ continued in $$t$$ for each fixed $$x$$. Then he established the stability problem for above inequality and for the corresponding $$p$$-dimensional inequality with $$p= 2, 3, 4, \dots$$.
For the entire collection see [Zbl 0835.00005].

### MSC:

 39B72 Systems of functional equations and inequalities 39B52 Functional equations for functions with more general domains and/or ranges

### Citations:

Zbl 0086.24101; Zbl 0061.26403; Zbl 0672.41027; Zbl 0779.47005