Stability of a generalization of the Fréchet functional equation

Malejki, Renata

doi:10.1515/aupcsm-2015-0006

Abstract

We prove some stability and hyperstability results for a generalization of the well known Fréchet functional equation, stemming from one of the characterizations of the inner product spaces. As the main tool we use a fixed point theorem for some function spaces. We end the paper with some new inequalities characterizing the inner product spaces.

References

[1] C. Alsina, J. Sikorska, M.S. Tomás, Norm derivatives and characterizations of inner product spaces, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2010. Cited on 70.
Search in Google Scholar
[2] A. Bahyrycz, J. Brzdęk, M. Piszczek, J. Sikorska, Hyperstability of the Fréchet equation and a characterization of inner product spaces, J. Funct. Spaces Appl. 2013, Art. ID 496361, 6 pp. Cited on 69, 70 and 71.
Search in Google Scholar
[3] N. Brillouët-Belluot, J. Brzdęk, K. Ciepliński, On some recent developments in Ulam’s type stability, Abstr. Appl. Anal. 2012, Art. ID 716936, 41 pp. Cited on 71 and 76.
Search in Google Scholar
[4] J. Brzdęk, Remarks on hyperstability of the Cauchy functional equation, Aequationes Math. 86 (2013), no. 3, 255-267. Cited on 71.
Search in Google Scholar
[5] J. Brzdęk, Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungar. 141 (2013), no. 1-2, 58-67. Cited on 71 and 76.
Search in Google Scholar
[6] J. Brzdęk, J. Chudziak, Z. Páles, A fixed point approach to stability of functional equations, Nonlinear Anal. 74 (2011), no. 17, 6728-6732. Cited on 71.
Search in Google Scholar
[7] J. Brzdęk, K. Ciepliński, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal. 74 (2011), no. 18, 6861-6867. Cited on 71.
Search in Google Scholar
[8] M. Chudziak, On solutions and stability of functional equations connected to the Popoviciu inequality, Ph.D. Thesis (in Polish), Pedagogical University of Cracow (Poland), Cracow 2012. Cited on 71.
Search in Google Scholar
[9] L. Cădariu, L. Găvruta, P. Găvruta, Fixed points and generalized Hyers-Ulam stability, Abstr. Appl. Anal. 2012, Art. ID 712743, 10 pp. Cited on 71.
Search in Google Scholar
[10] S.S. Dragomir, Some characterizations of inner product spaces and applications, Studia Univ. Babes-Bolyai Math. 34 (1989), no. 1, 50-55. Cited on 70.
Search in Google Scholar
[11] W. Fechner, On the Hyers-Ulam stability of functional equations connected with additive and quadratic mappings, J. Math. Anal. Appl. 322 (2006), no. 2, 774-786. Cited on 71.
Search in Google Scholar
[12] M. Fréchet, Sur la définition axiomatique d’une classe d’espaces vectoriels distanciés applicables vectoriellement sur l’espace de Hilbert, Ann. of Math. (2) 36 (1935), no. 3, 705-718. Cited on 70.
Search in Google Scholar
[13] E. Gselmann, Hyperstability of a functional equation, Acta Math. Hungar. 124 (2009), no. 1-2, 179-188. Cited on 71.
Search in Google Scholar
[14] D.H. Hyers, G. Isac, Th.M. Rassias, Stability of functional equations in several variables. Progress in Nonlinear Differential Equations and their Applications, 34. Birkhäuser Boston, Inc., Boston, MA, 1998. Cited on 71 and 76.
Search in Google Scholar
[15] P. Jordan, J. Von Neumann, On inner products in linear, metric spaces, Ann. of Math. (2) 36 (1935), no. 3, 719-723. Cited on 70.
Search in Google Scholar
[16] S.-M. Jung, On the Hyers-Ulam stability of the functional equation that have the quadratic property, J. Math. Anal. Appl. 222 (1998), no. 1, 126-137. Cited on 71.
Search in Google Scholar
[17] S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, Springer Optimization and Its Applications, 48, Springer, New York, 2011. Cited on 71 and 76.
Search in Google Scholar
[18] P. Kannappan, Functional equations and inequalities with applications, Springer Monographs in Mathematics, Springer, New York, 2009. Cited on 71.
Search in Google Scholar
[19] Y.-H. Lee, On the Hyers-Ulam-Rassias stability of the generalized polynomial function of degree 2, Journal of the Chungcheong Mathematical Society 22 (2009), no. 2, 201-209. Cited on 71.
Search in Google Scholar
[20] G. Maksa, Z. Páles, Hyperstability of a class of linear functional equations, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 17 (2001), no. 2, 107-112. Cited on 71.
Search in Google Scholar
[21] M.S. Moslehian, J.M. Rassias, A characterization of inner product spaces concerning an Euler-Lagrange identity, Commun. Math. Anal. 8 (2010), no. 2, 16-21. Cited on 70.
Search in Google Scholar
[22] K. Nikodem, Z. Pales, Characterizations of inner product spaces by strongly convex functions, Banach J. Math. Anal. 5 (2011), no. 1, 83-87. Cited on 70.
Search in Google Scholar
[23] Th. M. Rassias, New characterizations of inner product spaces, Bull. Sci. Math. (2) 108 (1984), no. 1, 95-99. Cited on 70.
Search in Google Scholar
[24] M. Piszczek, Remark on hyperstability of the general linear equation, Aequationes Math. 88 (2014), no. 1-2, 163-168. Cited on 71 and 76.
Search in Google Scholar
[25] J. Sikorska, On a direct method for proving the Hyers-Ulam stability of functional equations, J. Math. Anal. Appl. 372 (2010), no. 1, 99-109. Cited on 71.
Search in Google Scholar

Stability of a generalization of the Fréchet functional equation

Abstract

Paradigm

My account