On the hyperstability of a Cauchy-Jensen type functional equation in Banach spaces
DOI:
https://doi.org/10.4067/S0716-09172015000400005Keywords:
Hyperstability, Cauchy-Jensen, Fixed point theorem.Abstract
In this paper, we establish some hyperstability results of the following Cauchy-Jensen functional equation
in Banach spaces.
References
[1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2, pp. 64-66, (1950).
[2] C. Baak, Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces, Acta Mathematica Sinica, English Series, Vol.22, No. 6, pp. 1789-1796, (2006).
[3] A. Bahyrycz, M. Piszczek, Hyperstability of the Jensen functional equation, Acta Math. Hungar. 142 (2), pp. 353-365, (2014).
[4] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16, pp. 385-397, (1949).
[5] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57, pp. 223-237, (1951).
[6] J. Brzdek, J. Chudziak, Z. Páles, A fixed point approach to stability of functional equations, Nonlinear Anal., vol. 74, no. 17, pp. 6728-6732, (2011).
[7] J. Brzdek, Remarks on hyperstability of the the Cauchy equation, Aequations Mathematicae, 86, pp. 255-267, (2013).
[8] J. Brzdek, Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungarica, 141 (1-2), pp. 58-67, (2013).
[9] J. Brzdek, A hyperstability result for the Cauchy equation. Bulletin of the Australian Mathematical Society 89, pp. 33-40, (2014).
[10] L. Cadariu, V. Radu, Fixed points and the stability of Jensens functional equation, Journal of Inequalities in Pure and Applied Mathematics, 4(1), (2003).
[11] Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. 14, pp. 431-434, (1991).
[12] M. E. Gordji, H. Khodaei, M. Kamyar, Stability of Cauchy-Jensen type functional equation in generalized fuzzy normed spaces, Computers and Mathematics with Applications, 62, pp. 2950-2960, (2011).
[13] E. Gselmann, Hyperstability of a functional equation, Acta Mathematica Hungarica, vol. 124, no. 1-2, pp. 179-188, (2009).
[14] D. H. Hyers, On the stability of the linear functional equation, Proc.
Nat. Acad. Sci., U.S.A., 27, pp. 222-224, (1941).
[15] K. W. Jun, H. M. Kim, J. M. Rassias, Extended Hyers-Ulam stability for Cauchy-Jensen mappings, J. Diference Equ. Appl., 13(12), pp. 1139-1153, (2007).
[16] S. M. Jung, M. S. Moslehian, P. K. Sahoo, Stability of a generalized Jensen equation on restricted domains, J. Math. Ineq., 4, pp. 191-206, (2010).
[17] Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math. 22, pp. 499-507, (1989).
[18] Y. H. Lee, K. W. Jun, A generalization of the Hyers-Ulam-Rassias stability of Jensens equation, J. Math. Anal. Appl., 238, pp. 305-315, (1999).
[19] G. Maksa, Z. P´ales, Hyperstability of a class of linear functional equations, Acta Math., vol. 17, no. 2, pp. 107-112, (2001).
[20] C. Park, Fixed points and Hyers-Ulam-Rassias stability of CauchyJensen functional equations in Banach algebras, Fixed. Point. Theory. Appl., 15, (2007) : Article ID 50175.
[21] C. Park, J. M Rassias, Stability of the Jensen-type functional equation in C*-algebras: A fixed point approach. Abstract and Applied Analysis, Volume 2009 (2009), Article ID 360432, 17 pages.
[22] M. Piszczek, Remark on hyperstability of the general linear equation, Aequations Mathematicae, (2013).
[23] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, pp. 297-300, (1978).
[24] Th. M. Rassias, On a modified HyersUlam sequence, J. Math. Anal. Appl. 158, pp. 106-113, (1991).
[25] Th. M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114, pp. 989-993, (1992).
[26] J. M. Rassias, On the Ulam stability of Jensen and Jensen type mappings on restricted domains, J. Math. Anal. Appl. 281, pp. 516-524, (2003).
[27] S. M. Ulam, Problems in Modern Mathematics, Chapter IV, Science Editions, Wiley, New York, (1960).
[2] C. Baak, Cauchy-Rassias stability of Cauchy-Jensen additive mappings in Banach spaces, Acta Mathematica Sinica, English Series, Vol.22, No. 6, pp. 1789-1796, (2006).
[3] A. Bahyrycz, M. Piszczek, Hyperstability of the Jensen functional equation, Acta Math. Hungar. 142 (2), pp. 353-365, (2014).
[4] D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J., 16, pp. 385-397, (1949).
[5] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57, pp. 223-237, (1951).
[6] J. Brzdek, J. Chudziak, Z. Páles, A fixed point approach to stability of functional equations, Nonlinear Anal., vol. 74, no. 17, pp. 6728-6732, (2011).
[7] J. Brzdek, Remarks on hyperstability of the the Cauchy equation, Aequations Mathematicae, 86, pp. 255-267, (2013).
[8] J. Brzdek, Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungarica, 141 (1-2), pp. 58-67, (2013).
[9] J. Brzdek, A hyperstability result for the Cauchy equation. Bulletin of the Australian Mathematical Society 89, pp. 33-40, (2014).
[10] L. Cadariu, V. Radu, Fixed points and the stability of Jensens functional equation, Journal of Inequalities in Pure and Applied Mathematics, 4(1), (2003).
[11] Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. 14, pp. 431-434, (1991).
[12] M. E. Gordji, H. Khodaei, M. Kamyar, Stability of Cauchy-Jensen type functional equation in generalized fuzzy normed spaces, Computers and Mathematics with Applications, 62, pp. 2950-2960, (2011).
[13] E. Gselmann, Hyperstability of a functional equation, Acta Mathematica Hungarica, vol. 124, no. 1-2, pp. 179-188, (2009).
[14] D. H. Hyers, On the stability of the linear functional equation, Proc.
Nat. Acad. Sci., U.S.A., 27, pp. 222-224, (1941).
[15] K. W. Jun, H. M. Kim, J. M. Rassias, Extended Hyers-Ulam stability for Cauchy-Jensen mappings, J. Diference Equ. Appl., 13(12), pp. 1139-1153, (2007).
[16] S. M. Jung, M. S. Moslehian, P. K. Sahoo, Stability of a generalized Jensen equation on restricted domains, J. Math. Ineq., 4, pp. 191-206, (2010).
[17] Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math. 22, pp. 499-507, (1989).
[18] Y. H. Lee, K. W. Jun, A generalization of the Hyers-Ulam-Rassias stability of Jensens equation, J. Math. Anal. Appl., 238, pp. 305-315, (1999).
[19] G. Maksa, Z. P´ales, Hyperstability of a class of linear functional equations, Acta Math., vol. 17, no. 2, pp. 107-112, (2001).
[20] C. Park, Fixed points and Hyers-Ulam-Rassias stability of CauchyJensen functional equations in Banach algebras, Fixed. Point. Theory. Appl., 15, (2007) : Article ID 50175.
[21] C. Park, J. M Rassias, Stability of the Jensen-type functional equation in C*-algebras: A fixed point approach. Abstract and Applied Analysis, Volume 2009 (2009), Article ID 360432, 17 pages.
[22] M. Piszczek, Remark on hyperstability of the general linear equation, Aequations Mathematicae, (2013).
[23] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, pp. 297-300, (1978).
[24] Th. M. Rassias, On a modified HyersUlam sequence, J. Math. Anal. Appl. 158, pp. 106-113, (1991).
[25] Th. M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114, pp. 989-993, (1992).
[26] J. M. Rassias, On the Ulam stability of Jensen and Jensen type mappings on restricted domains, J. Math. Anal. Appl. 281, pp. 516-524, (2003).
[27] S. M. Ulam, Problems in Modern Mathematics, Chapter IV, Science Editions, Wiley, New York, (1960).
How to Cite
[1]
I.-I. El-Fassi and S. Kabbaj, “On the hyperstability of a Cauchy-Jensen type functional equation in Banach spaces”, Proyecciones (Antofagasta, On line), vol. 34, no. 4, pp. 359-375, 1.
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