Stability of functional equations on restricted domains in a group and their asymptotic behaviors. (English) Zbl 1205.39022

Summary: We consider Hyers-Ulam stability problems for the Pexider equation, the Cauchy equation, and the Jensen equation in general restricted domains in a group. The main purpose of this paper is to find restricted domains such that the functional inequality satisfied in those domains extends to the inequality for the whole domain and such that the Hyers-Ulam stability theorem holds for the inequalities as it does when the inequality holds globally. We also consider a distributional version of the Hyers-Ulam stability of the Pexider equation in restricted domains and its asymptotic behaviors.


39B52 Functional equations for functions with more general domains and/or ranges
39B82 Stability, separation, extension, and related topics for functional equations
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[1] Ulam, S. M., A Collection of Mathematical Problems (1960), Interscience Publ.: Interscience Publ. New York · Zbl 0086.24101
[2] Hyers, D. H., On the stability of the linear functional equations, Proc. Natl. Acad. Sci. USA, 27, 222-224 (1941) · Zbl 0061.26403
[3] Aoki, T., On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2, 64-66 (1950) · Zbl 0040.35501
[4] Bourgin, D. G., Class of transformations and bordering transformations, Bull. Amer. Math. Soc., 57, 223-237 (1951) · Zbl 0043.32902
[5] Bourgin, D. G., Multiplicative transformations, Proc. Natl. Acad. Sci. USA, 36, 564-570 (1950) · Zbl 0039.33403
[6] Rassias, Th. M., On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300 (1978) · Zbl 0398.47040
[7] Baker, J. A., On a functional equation of Aczél and Chung, Aequationes Math., 46, 99-111 (1993) · Zbl 0791.39008
[8] Baker, J. A., The stability of cosine functional equation, Proc. Amer. Math. Soc., 80, 411-416 (1980) · Zbl 0448.39003
[9] Chung, J., Distributional method for a class of functional equations and their stabilities, Acta Math. Sinica, 23, 2017-2026 (2007) · Zbl 1190.39015
[10] Chung, J., Stability of approximately quadratic Schwartz distributions, Nonlinear Anal., 67, 175-186 (2007) · Zbl 1116.39017
[11] Chung, J., A distributional version of functional equations and their stabilities, Nonlinear Anal., 62, 1037-1051 (2005) · Zbl 1076.39025
[12] Forti, G. L., Hyers-Ulam stability of functional equation in several variables, Aequationes Math., 50, 143-190 (1995) · Zbl 0836.39007
[13] Forti, G. L., The stability of homomorphisms and amenability with applications to functional equations, Abh. Math. Sem. Univ. Hamburg, 57, 215-226 (1987) · Zbl 0619.39012
[14] Hyers, D. H.; Isac, G.; Rassias, Th. M., Stability of Functional Equations in Several Variables (1998), Birkhauser · Zbl 0894.39012
[15] Jung, S. M., Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis (2001), Hadronic Press, Inc.: Hadronic Press, Inc. Palm Harbor, Florida · Zbl 0980.39024
[16] Jung, S. M., On the Hyers-Ulam stability of functional equations that have the quadratic property, J. Math. Anal. Appl., 222, 126-137 (1998) · Zbl 0928.39013
[17] Jun, K. W.; Kim, H.-M., Stability problem for Jensen-type functional equations of cubic mappings, Acta Math. Sin. (Engl. Ser.), 22, 6, 1781-1788 (2006) · Zbl 1118.39013
[18] Jun, K.; Kim, H.-M.; Rassias, J. M., An answer to a question of John M. Rassias concerning the stability of Cauchy equation, Hadronic Math. Ser., 67-71 (1999)
[19] Kim, H.-M.; Rassias, J.; Cho, Y.-S., Stability Problem of Ulam for Euler-Lagrange Quadratic Mappings 2007 (2007), pp. 1-15
[20] Kim, G. H., On the stability of Pexiderized trigonometric functional equation, Appl. Math. Comput., 203, 99-105 (2008) · Zbl 1159.39013
[21] Kim, G. H.; Lee, Y. H., Boundedness of approximate trigonometric functional equations, Appl. Math. Lett., 31, 439-443 (2009) · Zbl 1168.39009
[22] Park, C. G., Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras, Bull. Sci. Math., 132, 87-96 (2008) · Zbl 1140.39016
[23] Rassias, J. M.; Rassias, M. J., On the Ulam stability of Jensen and Jensen type mappings on restricted domains, J. Math. Anal. Appl., 281, 516-524 (2003) · Zbl 1028.39011
[24] Rassias, J. M., On the Ulam stability of mixed type mappings on restricted domains, J. Math. Anal. Appl., 276, 747-762 (2002) · Zbl 1021.39015
[25] Rassias, J. M., On approximation of approximately linear mappings by linear mappings, J. Funct. Anal., 46, 126-130 (1982) · Zbl 0482.47033
[26] Rassias, J. M., On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math., 108, 445-446 (1984) · Zbl 0599.47106
[27] Rassias, J. M., Solution of a problem of Ulam, J. Approx. Theory, 57, 268-273 (1989) · Zbl 0672.41027
[28] Rassias, J. M., On the stability of the Euler-Lagrange functional equation, Chinese J. Math., 20, 185-190 (1992) · Zbl 0753.39003
[29] Rassias, M. J.; Rassias, J. M., On the Ulam stability for Euler-Lagrange type quadratic functional equations, Austral. J. Math. Anal. Appl., 2, 1-10 (2005) · Zbl 1094.39027
[30] Ravi, K.; Arunkumar, M., On the Ulam-Gavruta-Rassias stability of the orthogonally Euler-Lagrange type functional equation, Int. J. Appl. Math. Stat., 7, 143-156 (2007)
[31] Rassias, Th. M., On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., 251, 264-284 (2000) · Zbl 0964.39026
[32] Skof, F., Sull’approssimazione delle applicazioni localmente \(\delta \)-additive, Atii Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 117, 377-389 (1983)
[33] Skof, F., Proprietá locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53, 113-129 (1983)
[34] Jung, S. M., Hyers-Ulam stability of Jensen’s equation and its application, Proc. Amer. Math. Soc., 126, 3137-3143 (1998) · Zbl 0909.39014
[35] Hörmander, L., The Analysis of Linear Partial Differential Operator I (1983), Springer-Verlag: Springer-Verlag Berlin, New York
[36] Schwartz, L., Théorie des Distributions (1966), Hermann: Hermann Paris
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