International Journal of Nonlinear Analysis and Applications
Hyers-Ulam-Rassias stability of orthogonality equation on restricted domains
Document Type : Research Paper
Authors
1 Department of Electronic and Electrical Engineering, College of Science and Technology, Hongik University, 30016 Sejong, Republic of Korea
2 Mathematics Section, College of Science and Technology, Hongik University, 30016 Sejong, Republic of Korea
Abstract
In this paper, we prove some theorems about the Hyers-Ulam-Rassias stability of linear isometries. In particular, this paper will address the stability of the orthogonality equation, $\langle f(x), f(y) \rangle = \langle x, y \rangle$, on the restricted domains.
Keywords
s
[1] J.A. Baker, Isometries in normed spaces, Amer. Math. Month. 78 (1971), no. 6, 655–658.
[2] D.G. Bourgin, Approximate isometries, Bull. Amer. Math. Soc. 52 (1946), 704–714.
[3] R. Bhatia and P. Semrl, ˇ Approximate isometries on Euclidean spaces, Amer. Math. Monthly 104 (1997), 497–504.
[4] D.G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke
Math. J. 16 (1949), 385–397.
[5] R.D. Bourgin, Approximate isometries on finite dimensional Banach spaces, Trans. Amer. Math. Soc. 207 (1975),
309–328.
[6] J. Chmieli´nski, On the superstability of the generalized orthogonality equation in Euclidean spaces, Ann. Math.
Sil. 8 (1994), 127–140.
[7] J. Chmieli´nski, On the Hyers-Ulam stability of the generalized orthogonality equation in real Hilbert spaces, Stability of Mappings of Hyers-Ulam Type (Th. M. Rassias and J. Tabor, ed.), Hadronic Press, Palm Harbor, 1994,
pp. 31–41.
[8] J. Chmieli´nski, The stability of the Wigner equation in complex Hilbert spaces, Wy˙z. Szko la Ped. Krak´ow Rocznik
Nauk.-Dydakt. Prace Matematyczne 15 (1998), 49–55.
[9] J. Chmieli´nski, On a singular case in the Hyers-Ulam-Rassias stability of the Wigner equation, J. Math. Anal.
Appl. 289 (2004), 571–583.
[10] J. Chmieli´nski, Stability of the orthogonality preserving property in finite-dimensional inner product spaces, J.
Math. Anal. Appl. 318 (2006), 433–4[11] J. Chmielinski and S.-M. Jung, The stability of the Wigner equation on a restricted domain, J. Math. Anal. Appl.
254 (2001), 309–320.
[12] G. Choi and S.-M. Jung, The stability of isometries on restricted domains, Symmetry-Basel 13 (2021), Issue 2,
282, 11 pages.
[13] G. Dolinar, Generalized stability of isometries, J. Math. Anal. Appl. 242 (2000), 39–56.
[14] J.W. Fickett, Approximate isometries on bounded sets with an application to measure theory, Studia Math. 72
(1981), 37–46.
[15] J. Gevirtz, Stability of isometries on Banach spaces, Proc. Amer. Math. Soc. 89 (1983), 633–636.
[16] P.M. Gruber, Stability of isometries, Trans. Amer. Math. Soc. 245 (1978), 263–277.
[17] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224.
[18] D.H. Hyers and S.M. Ulam, On approximate isometries, Bull. Amer. Math. Soc. 51 (1945), 288–292.
[19] D.H. Hyers and S.M. Ulam, Approximate isometries of the space of continuous functions, Ann. Math. 48 (1947),
285–289.
[20] S.-M. Jung, Hyers-Ulam-Rassias stability of isometries on restricted domains, Nonlinear Stud. 8 (2001), 125–134.
[21] S.-M. Jung, Asymptotic properties of isometries, J. Math. Anal. Appl. 276 (2002), 642–653.
[22] S.-M. Jung and B. Kim, Stability of isometries on restricted domains, J. Korean Math. Soc. 37 (2000), no. 1,
125–137.
[23] J. Lindenstrauss and A. Szankowski, Non linear perturbations of isometries, Ast´erisque 131 (1985), 357–371.
[24] M. Omladiˇc and P. Semrl, ˇ On non linear perturbations of isometries, Math. Ann. 303 (1995), 617–628.
[25] J.M. Rassias, On a new approximation of approximately linear mappings by linear mappings, Discuss. Math. 7
(1985), 193–196.
[26] Th.M. Rassias, Properties of isometric mappings, J. Math. Anal. Appl. 235 (1999), 108–121.
[27] Th.M. Rassias, Isometries and approximate isometries, Internat. J. Math. Math. Sci. 25 (2001), 73–91.
[28] Th.M. Rassias and C. S. Sharma, Properties of isometries, J. Natural Geom. 3 (1993), 1–38.
[29] P. Semrl, ˇ Hyers-Ulam stability of isometries on Banach spaces, Aequationes Math. 58 (1999), 157–162.
[30] F. Skof, Sulle δ-isometrie negli spazi normati, Rendiconti Di Mat. Ser. VII, Roma 10 (1990), 853–866.
[31] F. Skof, On asymptotically isometric operators in normed spaces, Ist. Lomb. Sc. Lett. A 131 (1997), 117–129.
[32] R.L. Swain, Approximate isometries in bounded spaces, Proc. Amer. Math. Soc. 2 (1951), 727–729.
[33] J. V¨ais¨al¨a, Isometric approximation property of unbounded sets, Result. Math. 43 (2003), 359–372.
[34] E.P. Wigner, Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren, Friedr. Vieweg
und Sohn, Braunschweig, 1931.
[1] J.A. Baker, Isometries in normed spaces, Amer. Math. Month. 78 (1971), no. 6, 655–658.
[2] D.G. Bourgin, Approximate isometries, Bull. Amer. Math. Soc. 52 (1946), 704–714.
[3] R. Bhatia and P. Semrl, ˇ Approximate isometries on Euclidean spaces, Amer. Math. Monthly 104 (1997), 497–504.
[4] D.G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke
Math. J. 16 (1949), 385–397.
[5] R.D. Bourgin, Approximate isometries on finite dimensional Banach spaces, Trans. Amer. Math. Soc. 207 (1975),
309–328.
[6] J. Chmieli´nski, On the superstability of the generalized orthogonality equation in Euclidean spaces, Ann. Math.
Sil. 8 (1994), 127–140.
[7] J. Chmieli´nski, On the Hyers-Ulam stability of the generalized orthogonality equation in real Hilbert spaces, Stability of Mappings of Hyers-Ulam Type (Th. M. Rassias and J. Tabor, ed.), Hadronic Press, Palm Harbor, 1994,
pp. 31–41.
[8] J. Chmieli´nski, The stability of the Wigner equation in complex Hilbert spaces, Wy˙z. Szko la Ped. Krak´ow Rocznik
Nauk.-Dydakt. Prace Matematyczne 15 (1998), 49–55.
[9] J. Chmieli´nski, On a singular case in the Hyers-Ulam-Rassias stability of the Wigner equation, J. Math. Anal.
Appl. 289 (2004), 571–583.
[10] J. Chmieli´nski, Stability of the orthogonality preserving property in finite-dimensional inner product spaces, J.
Math. Anal. Appl. 318 (2006), 433–4[11] J. Chmielinski and S.-M. Jung, The stability of the Wigner equation on a restricted domain, J. Math. Anal. Appl.
254 (2001), 309–320.
[12] G. Choi and S.-M. Jung, The stability of isometries on restricted domains, Symmetry-Basel 13 (2021), Issue 2,
282, 11 pages.
[13] G. Dolinar, Generalized stability of isometries, J. Math. Anal. Appl. 242 (2000), 39–56.
[14] J.W. Fickett, Approximate isometries on bounded sets with an application to measure theory, Studia Math. 72
(1981), 37–46.
[15] J. Gevirtz, Stability of isometries on Banach spaces, Proc. Amer. Math. Soc. 89 (1983), 633–636.
[16] P.M. Gruber, Stability of isometries, Trans. Amer. Math. Soc. 245 (1978), 263–277.
[17] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27 (1941), 222–224.
[18] D.H. Hyers and S.M. Ulam, On approximate isometries, Bull. Amer. Math. Soc. 51 (1945), 288–292.
[19] D.H. Hyers and S.M. Ulam, Approximate isometries of the space of continuous functions, Ann. Math. 48 (1947),
285–289.
[20] S.-M. Jung, Hyers-Ulam-Rassias stability of isometries on restricted domains, Nonlinear Stud. 8 (2001), 125–134.
[21] S.-M. Jung, Asymptotic properties of isometries, J. Math. Anal. Appl. 276 (2002), 642–653.
[22] S.-M. Jung and B. Kim, Stability of isometries on restricted domains, J. Korean Math. Soc. 37 (2000), no. 1,
125–137.
[23] J. Lindenstrauss and A. Szankowski, Non linear perturbations of isometries, Ast´erisque 131 (1985), 357–371.
[24] M. Omladiˇc and P. Semrl, ˇ On non linear perturbations of isometries, Math. Ann. 303 (1995), 617–628.
[25] J.M. Rassias, On a new approximation of approximately linear mappings by linear mappings, Discuss. Math. 7
(1985), 193–196.
[26] Th.M. Rassias, Properties of isometric mappings, J. Math. Anal. Appl. 235 (1999), 108–121.
[27] Th.M. Rassias, Isometries and approximate isometries, Internat. J. Math. Math. Sci. 25 (2001), 73–91.
[28] Th.M. Rassias and C. S. Sharma, Properties of isometries, J. Natural Geom. 3 (1993), 1–38.
[29] P. Semrl, ˇ Hyers-Ulam stability of isometries on Banach spaces, Aequationes Math. 58 (1999), 157–162.
[30] F. Skof, Sulle δ-isometrie negli spazi normati, Rendiconti Di Mat. Ser. VII, Roma 10 (1990), 853–866.
[31] F. Skof, On asymptotically isometric operators in normed spaces, Ist. Lomb. Sc. Lett. A 131 (1997), 117–129.
[32] R.L. Swain, Approximate isometries in bounded spaces, Proc. Amer. Math. Soc. 2 (1951), 727–729.
[33] J. V¨ais¨al¨a, Isometric approximation property of unbounded sets, Result. Math. 43 (2003), 359–372.
[34] E.P. Wigner, Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren, Friedr. Vieweg
und Sohn, Braunschweig, 1931.
)
Volume 13, Issue 2
July 2022Pages 3213-3222
- Receive Date: 08 March 2021
- Revise Date: 17 April 2021
- Accept Date: 11 May 2021