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Stability Of The Homogeneous Equation

Stability Of The Homogeneous Equation DEMONSTRATIO M A T H E M A T I C AVol. X X X INo 41998Jacek ChudziakSTABILITY OF T H E H O M O G E N E O U S E Q U A T I O NA b s t r a c t . T h e problem of Hyers-Ulam stability of the homogeneous equation is considered. We show, that under suitable assumptions on p and q, every function satisfying(4) is homogeneous.Let K be a real or complex field, let X and Y be normed spaces overK. By R + we denote the set of all non-negative real numbers and by 0° wemean 1.Let a function / : X —• Y satisfy the condition(1)|| f ( a x ) -a f ( x ) \ \ < H{a,x)for a £ K ,x £ Xwhere H : K x X — R + is given. Jacek Tabor and Jozef Tabor proved in[4] that under suitable assumptions on H the mapping / is homogeneous.Let E\ and E2 be real normed spaces and let Ei be complete. It hasbeen proved by Th.M. Rassias [3] and Z. Gajda [2] that if the function/ : E\ —• E2 satisfies http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

Stability Of The Homogeneous Equation

Chudziak, Jacek
Demonstratio Mathematica , Volume 31 (4): 8 – Jan 1, 1998
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References (3)

Publisher
de Gruyter
Copyright
© by Jacek Chudziak
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1998-0405
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO M A T H E M A T I C AVol. X X X INo 41998Jacek ChudziakSTABILITY OF T H E H O M O G E N E O U S E Q U A T I O NA b s t r a c t . T h e problem of Hyers-Ulam stability of the homogeneous equation is considered. We show, that under suitable assumptions on p and q, every function satisfying(4) is homogeneous.Let K be a real or complex field, let X and Y be normed spaces overK. By R + we denote the set of all non-negative real numbers and by 0° wemean 1.Let a function / : X —• Y satisfy the condition(1)|| f ( a x ) -a f ( x ) \ \ < H{a,x)for a £ K ,x £ Xwhere H : K x X — R + is given. Jacek Tabor and Jozef Tabor proved in[4] that under suitable assumptions on H the mapping / is homogeneous.Let E\ and E2 be real normed spaces and let Ei be complete. It hasbeen proved by Th.M. Rassias [3] and Z. Gajda [2] that if the function/ : E\ —• E2 satisfies

Journal

Demonstratio Mathematica – de Gruyter

Published: Jan 1, 1998

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