Stability of a generalization of the Fréchet functional equation

Renata Malejki

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 the function spaces. We end the paper with some new inequalities characterizing the inner product spaces.

Keywords


stability; hyperstability; characterization of inner product spaces; fixed point theorem; Fréchet equation

Mathematics Subject Classification (2010)


39B52; 39B82; 47H10

References


Alsina, C. , J. Sikorska and M.S. Tomás. "Norm derivatives and characterizations of inner product spaces." Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2010.

Bahyrycz, A. et al. "Hyperstability of the Fréchet equation and a characterization of inner product spaces." J. Funct. Spaces Appl. 2013: Art. ID 496361, 6 pp.

Brillouët-Belluot, N. , J. Brzdęk and K. Ciepliński. "On some recent developments in Ulam’s type stability." Abstr. Appl. Anal. 2012: Art. ID 716936, 41 pp.

Brzdęk, J. "Remarks on hyperstability of the Cauchy functional equation." Aequationes Math. 86.3 (2013): 255-267.

---. "Hyperstability of the Cauchy equation on restricted domains." Acta Math. Hungar. 141.1-2 (2013): 58-67.

Brzdęk, J., J. Chudziak and Z. Páles. "A fixed point approach to stability of functional equations." Nonlinear Anal. 74.17 (2011): 6728-6732.

Brzdęk, J. and K. Cieplinski. "A fixed point approach to the stability of functional equations in non-Archimedean metric spaces." Nonlinear Anal. 74.18 (2011): 6861-6867.

Chudziak, M. "On solutions and stability of functional equations connected to the Popoviciu inequality" Ph.D. Thesis (in Polish). Cracow: Pedagogical University of Cracow (Poland), 2012.

Cadariu, L., L. Gavruta and P. Gavruta. "Fixed points and generalized Hyers-Ulam stability." Abstr. Appl. Anal. 2012: Art. ID 712743, 10 pp.

Dragomir, S.S. "Some characterizations of inner product spaces and applications." Studia Univ. Babes-Bolyai Math. 34.1 (1989): 50-55.

Fechner, W. "On the Hyers-Ulam stability of functional equations connected with additive and quadratic mappings." J. Math. Anal. Appl. 322.2 (2006): 774-786.

Fréchet, M. "Sur la définition axiomatique d’une classe d’espaces vectoriels distanciés applicables vectoriellement sur l’espace de Hilbert." Ann. of Math. (2) 36.3 (1935): 705-718.

Gselmann, E. "Hyperstability of a functional equation, Acta Math. Hungar. 124.1-2 (2009): 179-188.

Hyers, D.H., G. Isac and Th.M. Rassias. "Stability of functional equations in several variables." Progress in Nonlinear Differential Equations and their Applications, 34. Boston, MA: Birkhäuser Boston, Inc., 1998.

Jordan, P. and J. Von Neumann. "On inner products in linear, metric spaces." Ann. of Math. (2) 36.3 (1935): 719-723.

Jung, S.-M. "On the Hyers-Ulam stability of the functional equation that have the quadratic property." J. Math. Anal. Appl. 222.1 (1998): 126-137.

---. "Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis." Springer Optimization and Its Applications, 48. Springer, New York: 2011.

Kannappan, P. "Functional equations and inequalities with applications." Springer Monographs in Mathematics. New York: Springer, 2009.

Lee, Y.-H. "On the Hyers-Ulam-Rassias stability of the generalized polynomial function of degree 2." Journal of the Chungcheong Mathematical Society 22.2 (2009): 201-209.

Maksa, G. and Z. Páles. "Hyperstability of a class of linear functional equations." Acta Math. Acad. Paedagog. Nyházi. (N.S.) 17.2 (2001): 107-112.

Moslehian, M.S. and J.M. Rassias. "A characterization of inner product spaces concerning an Euler-Lagrange identity." Commun. Math. Anal. 8.2 (2010): 16-21.

Nikodem, K. and Z. Páles. "Characterizations of inner product spaces by strongly convex functions." Banach J. Math. Anal. 5.1 (2011): 83-87.

Piszczek, M. "Remark on hyperstability of the general linear equation." Aequationes Math. 88.1-2 (2014): 163-168.

Rassias, Th.M. "New characterizations of inner product spaces." Bull. Sci. Math. (2) 108.1 (1984): 95-99.

Sikorska, J. "On a direct method for proving the Hyers-Ulam stability of functional equations." J. Math. Anal. Appl. 372.1 (2010): 99-109.


Full Text: PDF

Download statistics: 1910

Licencja Creative Commons
This article by Renata Malejki is governed by the Creative Commons Attribution-ShareAlike 4.0 International(CC BY-SA 4.0) licence.


e-ISSN: 2300-133X, ISSN: 2081-545X

Since 2017 Open Access in De Gruyter and CrossCheck access cofinanced by The Ministry of Science and Higher Education - Republic of Poland - DUN 775/P-DUN/2017 see more

The Journal is indexed in:
and others see Abstracting and Indexing list

AUPC SM is on the List of the Ministry’s scored journals with 20 points for 2019

Deklaracja dostępności cyfrowej