On the asymptoticity aspect of Hyers-Ulam stability of mappings

Hyers, D.; Isac, G.; Rassias, Th.

doi:10.1090/S0002-9939-98-04060-X

On the asymptoticity aspect of Hyers-Ulam stability of mappings
HTML articles powered by AMS MathViewer

by D. H. Hyers, G. Isac and Th. M. Rassias

Proc. Amer. Math. Soc. 126 (1998), 425-430

DOI: https://doi.org/10.1090/S0002-9939-98-04060-X

PDF | Request permission

Abstract:

The object of the present paper is to prove an asymptotic analogue of Th. M. Rassias’ theorem obtained in 1978 for the Hyers-Ulam stability of mappings.

References

J. Ralph Alexander, Charles E. Blair, and Lee A. Rubel, Approximate versions of Cauchy’s functional equation, Illinois J. Math. 39 (1995), no. 2, 278–287. MR 1316538
Herbert Amann, Fixed points of asymptotically linear maps in ordered Banach spaces, J. Functional Analysis 14 (1973), 162–171. MR 0350527, DOI 10.1016/0022-1236(73)90048-7
Herbert Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620–709. MR 415432, DOI 10.1137/1018114
Nguyên Phuong Các and Juan A. Gatica, Fixed point theorems for mappings in ordered Banach spaces, J. Math. Anal. Appl. 71 (1979), no. 2, 547–557. MR 548782, DOI 10.1016/0022-247X(79)90208-7
Jacek Chmieliński, On the stability of the generalized orthogonality equation, Stability of mappings of Hyers-Ulam type, Hadronic Press Collect. Orig. Artic., Hadronic Press, Palm Harbor, FL, 1994, pp. 43–57. MR 1304268
P. D. T. A. Elliott, Cauchy’s functional equation in the mean, Adv. in Math. 51 (1984), no. 3, 253–257. MR 740585, DOI 10.1016/0001-8708(84)90010-0
Roman Ger, On functional inequalities stemming from stability questions, General inequalities, 6 (Oberwolfach, 1990) Internat. Ser. Numer. Math., vol. 103, Birkhäuser, Basel, 1992, pp. 227–240. MR 1213010, DOI 10.1007/978-3-0348-7565-3_{2}0
R. Ger and J. Sikorska, Stability of the orthogonal additivity, Bull. Polish Acad. Sci. Math. 43 (1995), 143–151.
Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
Donald H. Hyers and Themistocles M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), no. 2-3, 125–153. MR 1181264, DOI 10.1007/BF01830975
George Isac, Opérateurs asymptotiquement linéaires sur des espaces localement convexes, Colloq. Math. 46 (1982), no. 1, 67–72 (French). MR 672364, DOI 10.4064/cm-46-1-67-72
George Isac and Themistocles M. Rassias, On the Hyers-Ulam stability of $\psi$-additive mappings, J. Approx. Theory 72 (1993), no. 2, 131–137. MR 1204135, DOI 10.1006/jath.1993.1010
George Isac and Themistocles M. Rassias, Stability of $\Psi$-additive mappings: applications to nonlinear analysis, Internat. J. Math. Math. Sci. 19 (1996), no. 2, 219–228. MR 1375983, DOI 10.1155/S0161171296000324
M. A. Krasnosel′skiĭ, Positive solutions of operator equations, P. Noordhoff Ltd., Groningen, 1964. Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron. MR 0181881
Themistocles M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297–300. MR 507327, DOI 10.1090/S0002-9939-1978-0507327-1
Themistocles M. Rassias and Peter emrl, On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993), no. 2, 325–338. MR 1209322, DOI 10.1006/jmaa.1993.1070
Fulvia Skof, On the approximation of locally $\delta$-additive mappings, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 117 (1983), no. 4-6, 377–389 (1986) (Italian, with English summary). MR 929697
Fulvia Skof, On the stability of functional equations on a restricted domain and a related topic, Stability of mappings of Hyers-Ulam type, Hadronic Press Collect. Orig. Artic., Hadronic Press, Palm Harbor, FL, 1994, pp. 141–151. MR 1304278