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http://dx.doi.org/10.4134/BKMS.2010.47.4.777

STABILITY OF THE MONOMIAL FUNCTIONAL EQUATION IN QUASI NORMED SPACES  

Mirmostafaee, Alireza Kamel (DEPARTMENT OF PURE MATHEMATICS CENTER OF EXCELLENCE IN ANALYSIS ON ALGEBRAIC STRUCTURES FERDOWSI UNIVERSITY OF MASHHAD)
Publication Information
Bulletin of the Korean Mathematical Society / v.47, no.4, 2010 , pp. 777-785 More about this Journal
Abstract
Let X be a linear space and Y be a complete quasi p-norm space. We will show that for each function f : X $\rightarrow$ Y, which satisfies the inequality ${\parallel}{\Delta}_x^nf(y)\;-\;n!f(x){\parallel}\;{\leq}\;\varphi(x,y)$ for suitable control function $\varphi$, there is a unique monomial function M of degree n which is a good approximation for f in such a way that the continuity of $t\;{\mapsto}\;f(tx)$ and $t\;{\mapsto}\;\varphi(tx,\;ty)$ imply the continuity of $t\;{\mapsto}\;M(tx)$.
Keywords
quasi p-norm; monomial functional equation; fixed point alternative; Hyers-Ulam-Rassias stability;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
Times Cited By Web Of Science : 0  (Related Records In Web of Science)
Times Cited By SCOPUS : 1
연도 인용수 순위
1 D. Wolna, The stability of monomial functions on a restricted domain, Aequationes Math. 72 (2006), no. 1-2, 100–109.   DOI
2 A. Gilanyi, On the stability of monomial functional equations, Publ. Math. Debrecen 56 (2000), no. 1-2, 201–212.
3 D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224.   DOI   ScienceOn
4 D. H. Hyers, Transformations with bounded mth differences, Pacific J. Math. 11 (1961), 591–602.   DOI
5 S.-M. Jung, T.-S. Kim, and K.-S. Lee, A fixed point approach to the stability of quadratic functional equation, Bull. Korean Math. Soc. 43 (2006), no. 3, 531–541.   과학기술학회마을   DOI   ScienceOn
6 Z. Kaiser, On stability of the monomial functional equation in normed spaces over fields with valuation, J. Math. Anal. Appl. 322 (2006), no. 2, 1188–1198.   DOI   ScienceOn
7 Y.-H. Lee, On the stability of the monomial functional equation, Bull. Korean Math. Soc. 45 (2008), no. 2, 397–403.   과학기술학회마을   DOI   ScienceOn
8 V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), no. 1, 91–96.
9 J. M. Rassias, Alternative contraction principle and Ulam stability problem, Math. Sci. Res. J. 9 (2005), no. 7, 190–199.
10 Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297–300.   DOI   ScienceOn
11 S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, Inc., New York 1964.
12 M. Albert and J. A. Baker, Functions with bounded nth differences, Ann. Polon. Math. 43 (1983), no. 1, 93–103.   DOI
13 T. Aoki, Locally bounded linear topological spaces, Proc. Imp. Acad. Tokyo 18 (1942), 588–594.   DOI
14 T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66.   DOI
15 Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI, 2000.
16 D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223–237.   DOI
17 A. Gilanyi, Hyers-Ulam stability of monomial functional equations on a general domain, Proc. Natl. Acad. Sci. USA 96 (1999), no. 19, 10588–10590.   DOI   ScienceOn
18 L. Cadariu and V. Radu, Remarks on the stability of monomial functional equations, Fixed Point Theory 8 (2007), no. 2, 201–218.
19 J. B. Diaz and B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305–309.   DOI