Quadratic $\rho$-functional inequalities in Banach spaces: a fixed point approach

Choonkil Park; Jeong Pil Seo

doi:10.11568/kjm.2015.23.2.231

Korean J. Math. Vol. 23 No. 2 (2015) pp.231-248
DOI: https://doi.org/10.11568/kjm.2015.23.2.231

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Published: Jun 30, 2016

Keywords:

Hyers-Ulam stability, quadratic -functional equation, fixed point, complex Banach space.

Mathematics Subject Classification:

quadratic Ï-functional equation, 39B62, 39B52, 47H10.

Choonkil Park

Research Institute for Natural Sciences Hanyang University Seoul 133-791, Korea

Jeong Pil Seo

Ohsang High School Gumi 730-842, Korea

Abstract

In this paper, we solve the following quadratic $ρ$ -functional inequalities
$\begin{array}{rcl} ‖ f (\frac{x + y + z}{2}) + f (\frac{x - y - z}{2}) + f (\frac{y - x - z}{2}) \\ + f (\frac{z - x - y}{2}) - f (x) - f (y) - f (z) ‖ \\ \leq ‖ ρ (f (x + y + z) + f (x - y - z) + f (y - x - z) \\ + f (z - x - y) - 4 f (x) - 4 f (y) - 4 f (z)) ‖, \end{array}$
where $ρ$ is a fixed complex number with $| ρ | < \frac{1}{8}$ ,
and
$\begin{array}{rcl} ‖ f (x + y + z) + f (x - y - z) + f (y - x - z) \\ + f (z - x - y) - 4 f (x) - 4 f (y) - 4 f (z) ‖ \\ \leq ‖ ρ (f (\frac{x + y + z}{2}) + f (\frac{x - y - z}{2}) + f (\frac{y - x - z}{2}) \\ + f (\frac{z - x - y}{2}) - f (x) - f (y) - f (z)) ‖, \end{array}$
where $ρ$ is a fixed complex number with $| ρ | < 4$ .

Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic $ρ$ -functional inequalities (0.1) and (0.2) in complex Banach spaces.

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] L. C ̆adariu, V. Radu, Fixed points and the stability of Jensen’s functional equa- tion, J. Inequal. Pure Appl. Math. 4, no. 1, Art. ID 4 (2003). Google Scholar

[3] L. C ̆adariu, V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (2004), 43–52. Google Scholar

[4] L. C ̆adariu, V. Radu, Fixed point methods for the generalized stability of func- tional equations in a single variable, Fixed Point Theory and Applications 2008, Art. ID 749392 (2008). Google Scholar

[5] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86. Google Scholar

[6] J. Diaz, 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

[7] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149–161. Google Scholar

[8] P. Gˇavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–43. Google Scholar

[9] A. Gil anyi, Eine zur Parallelogrammgleichung aquivalente Ungleichung, Aequationes Math. 62 (2001), 303-309. Google Scholar

[10] A. Gil anyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707-710. Google Scholar

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

[12] G. Isac, Th. M. Rassias, Stability of ψ-additive mappings: Appications to non-linear analysis, Internat. J. Math. Math. Sci. 19 (1996), 219–228. Google Scholar

[13] D. Mihe ̧t, V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567–572. Google Scholar

[14] C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175 (2007). Google Scholar

[15] C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equa- tions: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751 (2008). Google Scholar

[16] C. Park, Y. Cho, M. Han, Functional inequalities associated with Jordan-von Neumann-type additive functional equations, J. Inequal. Appl. 2007 (2007), Ar- ticle ID 41820, 13 pages. Google Scholar

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

[18] V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory 4 (2003), 91–96. Google Scholar

[19] J. R ̈atz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191–200. Google Scholar

[20] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. Google Scholar

[21] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ., New York, 1960. Google Scholar

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