Generalized Ulam—Hyers—Rassias stability of a Cauchy type functional equation
DOI:
https://doi.org/10.4067/S0716-09172013000100002Keywords:
Alternative fixed point, Generalized Hyers—Ulam—Rassias stability, Cauchy type functional equation, Additive mappings, β-normed spaces.Abstract
Using the alternative fixed point theorem, we establish the generalized Hyers—Ulam—Rassias stability of a Cauchy type functional equationfor functions takin values in arbitrary complete (real or complex)
β-normed spaces.
References
[1] 1. M. Akkouchi, Stability of certain functional equations via a fixed point of Ciric, Filomat, 25(2), pp. 121-127, (2011).
[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2, pp. 64-66, (1950).
[3] J. A. Baker, The stability of certain functional equations, Proc. Amer. Math. Soc., 112 (3), pp. 729-732, (1991).
[4] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16, pp. 385—397, (1949).
[5] L. Cadariu and V. Radu, Fixed points and the stability of quadratic functional equations, Analele Universitatii de Vest din Timisoara, 41, pp. 25—48, (2003).
[6] L. Cadariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math., 4 (2003), Art. ID 4.
[7] L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Mathematische Berichte, 346, pp. 43—52, (2004).
[8] L. Cadariu and V. Radu, The fixed points method for the stability of some functional equations, Carpathian Journal of Mathematics, 23, pp. 63—72, (2007).
[9] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27 (1984) 76-86. MR0758860 (86d:39016).
[10] S. Czerwik, Stability of functional equations of Ulam-Hyers-Rassias type, Hadronic Press, (2003).
[11] 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, pp. 305-309, (1968).
[12] [3] G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math., 50, pp. 143-190, (1995).
[13] Z. Gajda, On stability of additive mappings, Internat. J. Math.
Math. Sci., 14, pp. 431-434, (1991).
[14] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27, pp. 222-224, (1941).
[15] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of functional Equations in Several Variables, Birkhauser, Boston, Basel, Berlin, (1998).
[16] G. Isac and Th. M. Rassias, On the Hyers-Ulam stability of -additive mappings, J. Approx. Theory, 72, pp. 131-137, (1993).
[17] 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(3), pp. 531-541, (2006).
[18] [14] Zs. Páles, Generalized stability of the Cauchy functional equation, Aequationes Math., 56(3), pp. 222-232, (1998).
[19] C. Park and J. M. Rassias, Stability of the Jensen-type functional equation in C*−algebras: a fixed point approach, Abstract and Applied Analysis Volume 2009, Article ID 360432, 17 pages.
[20] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4, pp. 91—96, (2003).
[21] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, pp. 297-300, (1978).
[22] J. M. Rassias, Solution of a problem of Ulam, J. Approx. Theory, 57, No. 3, pp. 268-273, (1989).
[23] Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Math. Appl., 62 (2000), pp. 23-130. MR1778016 (2001j:39042).
[24] Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., 251 (2000), pp. 264-284. MR1790409 (2003b:39036).
[25] Th. M. Rassias, The problem of S.M.Ulam for approximately multiplicative mappings, J. Math. Anal. Appl., 246 (2), pp. 352-378, (2000).
[26] I. A. Rus, Principles and Applications of Fixed Point Theory, Ed. Dacia, Cluj-Napoca, (1979) (in Romanian).
[27] [22] L. Székelyhidi, On a stability theorem, C. R. Math. Rep. Acad. Sci. Canada, 3(5), pp. 253-255, (1981).
[28] [23] L. Székelyhidi, The stability of linear functional equations, C. R. Math. Rep. Acad. Sci. Canada, 3(2), pp. 63-67, (1981).
[29] [24] L. Székelyhidi, Ulam’s problem, Hyers’s solution and to where they led, in Functional Equations and Inequalities, Th. M. Rassias (Ed.), Vol. 518 of Mathematics and Its Applications, Kluwer Acad. Publ., Dordrecht, pp. 259-285, (2000).
[30] S. M. Ulam, Problems in Modern Mathematics, Chapter VI, science ed. Wiley, New York, (1940).
[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2, pp. 64-66, (1950).
[3] J. A. Baker, The stability of certain functional equations, Proc. Amer. Math. Soc., 112 (3), pp. 729-732, (1991).
[4] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16, pp. 385—397, (1949).
[5] L. Cadariu and V. Radu, Fixed points and the stability of quadratic functional equations, Analele Universitatii de Vest din Timisoara, 41, pp. 25—48, (2003).
[6] L. Cadariu and V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math., 4 (2003), Art. ID 4.
[7] L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Mathematische Berichte, 346, pp. 43—52, (2004).
[8] L. Cadariu and V. Radu, The fixed points method for the stability of some functional equations, Carpathian Journal of Mathematics, 23, pp. 63—72, (2007).
[9] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27 (1984) 76-86. MR0758860 (86d:39016).
[10] S. Czerwik, Stability of functional equations of Ulam-Hyers-Rassias type, Hadronic Press, (2003).
[11] 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, pp. 305-309, (1968).
[12] [3] G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math., 50, pp. 143-190, (1995).
[13] Z. Gajda, On stability of additive mappings, Internat. J. Math.
Math. Sci., 14, pp. 431-434, (1991).
[14] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27, pp. 222-224, (1941).
[15] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of functional Equations in Several Variables, Birkhauser, Boston, Basel, Berlin, (1998).
[16] G. Isac and Th. M. Rassias, On the Hyers-Ulam stability of -additive mappings, J. Approx. Theory, 72, pp. 131-137, (1993).
[17] 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(3), pp. 531-541, (2006).
[18] [14] Zs. Páles, Generalized stability of the Cauchy functional equation, Aequationes Math., 56(3), pp. 222-232, (1998).
[19] C. Park and J. M. Rassias, Stability of the Jensen-type functional equation in C*−algebras: a fixed point approach, Abstract and Applied Analysis Volume 2009, Article ID 360432, 17 pages.
[20] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4, pp. 91—96, (2003).
[21] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, pp. 297-300, (1978).
[22] J. M. Rassias, Solution of a problem of Ulam, J. Approx. Theory, 57, No. 3, pp. 268-273, (1989).
[23] Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Math. Appl., 62 (2000), pp. 23-130. MR1778016 (2001j:39042).
[24] Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., 251 (2000), pp. 264-284. MR1790409 (2003b:39036).
[25] Th. M. Rassias, The problem of S.M.Ulam for approximately multiplicative mappings, J. Math. Anal. Appl., 246 (2), pp. 352-378, (2000).
[26] I. A. Rus, Principles and Applications of Fixed Point Theory, Ed. Dacia, Cluj-Napoca, (1979) (in Romanian).
[27] [22] L. Székelyhidi, On a stability theorem, C. R. Math. Rep. Acad. Sci. Canada, 3(5), pp. 253-255, (1981).
[28] [23] L. Székelyhidi, The stability of linear functional equations, C. R. Math. Rep. Acad. Sci. Canada, 3(2), pp. 63-67, (1981).
[29] [24] L. Székelyhidi, Ulam’s problem, Hyers’s solution and to where they led, in Functional Equations and Inequalities, Th. M. Rassias (Ed.), Vol. 518 of Mathematics and Its Applications, Kluwer Acad. Publ., Dordrecht, pp. 259-285, (2000).
[30] S. M. Ulam, Problems in Modern Mathematics, Chapter VI, science ed. Wiley, New York, (1940).
Published
2013-06-23
How to Cite
[1]
M. Akkouchi, “Generalized Ulam—Hyers—Rassias stability of a Cauchy type functional equation”, Proyecciones (Antofagasta, On line), vol. 32, no. 1, pp. 15-29, Jun. 2013.
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