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**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].

“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].