Korean J. Math. Vol. 27 No. 2 (2019) pp.343-355
DOI: https://doi.org/10.11568/kjm.2019.27.2.343

A fixed point approach to the stability of a quadratic-cubic functional equation

Article Sidebar

PDF
Published: Jun 30, 2019
Keywords:
fixed point method, quadratic-cubic functional equation.
Mathematics Subject Classification:
39B52

Main Article Content

Yang-Hi Lee
Department of Mathematics Education Gongju National University of Education Gongju 32553, Republic of Korea

Abstract

In this paper, we investigate the stability of the functional equation
f(x+ky)−kf(x+y)+kf(x−y)−f(x−ky)−f(ky)+k3+k2−2k2f(−y)−k3−k2−2k2f(y)=0
by using the fixed point theory in the sense of L. C\u adariu and V. Radu.



Article Details

References

[1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. Google Scholar

[2] J. Baker, A general functional equation and its stability, Proc. Natl. Acad. Sci. 133 (6) (2005), 1657–1664. Google Scholar

[3] L. C ̆adariu and V. Radu, Fixed points and the stability of quadratic functional equations, An. Univ. Timisoara Ser. Mat.-Inform. 41 (2003), 25–48. Google Scholar

[4] L. C ̆adariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach in Iteration Theory, Grazer Mathematische Berichte, Karl- Franzens-Universit ̈aet, Graz, Graz, Austria 346 (2004), 43–52. Google Scholar

[5] I.-S. Chang and Y.-S. Jung, Stability of a functional equation deriving from cubic and quadratic functions, J. Math. Anal. Appl. 283 (2003), 491–500. Google Scholar

[6] Y.-J. Cho, M. Eshaghi Gordji, and S. Zolfaghari, Solutions and Stability of Generalized Mixed Type QC Functional Equations in Random Normed Spaces, J. Inequal. Appl. 2010 (2010), Art. ID 403101. Google Scholar

[7] S. Czerwik, On the stability of the quadratic mapping in the normed space, Abh. Math. Sem. Hamburg, 62 (1992), 59–64. Google Scholar

[8] 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 (1968), 305–309. Google Scholar

[9] P. G ̆avruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. and Appl. 184 (1994), 431–436. Google Scholar

[10] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27 (1941), 222–224. Google Scholar

[11] C.-J. Lee and Y.-H. Lee, On the stability of a mixed type quadratic and cubic functional equation, J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 19 (2012), 383–396. Google Scholar

[12] Y.-H. Lee and S.-M. Jung, Fuzzy stability of the cubic and quadratic functional equation, Appl. Math. Sci. (Ruse) 10 (2016), 2671–2686. Google Scholar

[13] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. Google Scholar

[14] W. Towanlong and P. Nakmahachalasint, A Mixed-Type Quadratic and Cubic Functional Equation and Its Stability, Thai J. Math. 8 (4) (2012), 61–71. Google Scholar

[15] S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1964. Google Scholar

[16] Z. Wang and W. X. Zhang, Fuzzy stability of quadratic-cubic functional equations, Acta Math. Sin. (Engl. Ser.) 27 (2011), 2191–2204. Google Scholar