## Tbilisi Mathematical Journal

### A fixed point approach to Ulam-Hyers stability of duodecic functional equation in quasi-$β$-normed spaces

#### Abstract

In this study, we achieve the general solution and investigate Ulam-Hyers stabilities involving a general control function, sum of powers of norms, product of powers of norms and mixed product-sum of powers of norms of the duodecic functional equation in quasi-$β$-normed spaces via fixed point method. We also illustrate a counter-example for non-stability of the duodecic functional equation in singular case.

#### Article information

Source
Tbilisi Math. J., Volume 10, Issue 4 (2017), 83-101.

Dates
Accepted: 25 September 2017
First available in Project Euclid: 21 April 2018

https://projecteuclid.org/euclid.tbilisi/1524276060

Digital Object Identifier
doi:10.1515/tmj-2017-0048

Mathematical Reviews number (MathSciNet)
MR3719329

Zentralblatt MATH identifier
1377.39044

#### Citation

Rassias, J. M.; Ravi, K.; Kumar, B. V. Senthil. A fixed point approach to Ulam-Hyers stability of duodecic functional equation in quasi-$β$-normed spaces. Tbilisi Math. J. 10 (2017), no. 4, 83--101. doi:10.1515/tmj-2017-0048. https://projecteuclid.org/euclid.tbilisi/1524276060

#### References

• T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math.Soc. Japan, 2 (1950), 64–66.
• D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223–237.
• L. Cadariu and V. Radu, Fixed points and stability for functional equations in probabilistic metric and random normed spaces, Fixed Point Theory and Applications, Vol. 2009, Article ID 589143 (2009), 18 pages.
• I. S. Chang and H. M. Kim, On the Hyers-Ulam stability of quadratic functional equations, J. Ineq. Appl. Math. 33 (2002), 1–12.
• Y. J. Cho, M. Eshaghi Gordji and S. Zolfaghari, Solutions and stability of generalized mixed type $QC$ functional equations in random normed spaces, U. Inequal. Appl. Article ID 403101, doi:10.1155/2010/403101, 16 pages.
• P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86.
• J. K. Chung and P. K. Sahoo, On the general solution of a quartic functional equation, Bull. Korean Math. Soc. 40 (4) (2003), 565–576.
• S. Czerwik, Functional equations and inequalities in several variables, (World Scientific Publishing Company, New Jersey, London, Singapore and Hong Kong), 2002.
• D. $\check{Z}$. Djokovi$\acute{c}$, A representation theorem for $(X_1-1)(X_2-1)\dots(X_n-1)$ and its applications, Ann. Polon. Math. 22 (1969/1970), 189–198.
• A. Ebadian and S. Zolfaghari, Stability of a mixed additive and cubic functional equation in several variables in non-Archimedean spaces, Ann. Univ. Ferrara. 58 (2012), 291–306.
• M. Eshaghi Gordji, A. Ebadian and S. Zolfaghari, Stability of a functional equation deriving from cubic and quartic functions, Abs. Appl. Anal. Article ID 801904, (2008), 17 pages.
• Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. 14 (3) (1991), 431–434.
• P. G$\check{a}$vrut$\check{a}$, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436.
• D. H. Hyers, On the stability of the linear functional equation, Proc.Nat. Acad.Sci. U.S.A. 27 (1941), 222–224.
• D. H. Hyers, G. Isac and T. M. Rassias, Stability of functional equations in several variables, Birkhauser, Basel, 1998.
• G. Isac and T. M. Rassias, Stability of $\psi$-additive mappings: applications to nonlinear analysis, Int. J. Math. Math. Sci. 19 (2) (1996), 219–228.
• K. W. Jun and H. M. Kim, On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces, J. Math. Anal. Appl. 332 (2) (2007), 1335–1350.
• P. Malliavin, Stochastic Analysis, Springer, Berlin, 1997.
• C. Park, Fixed points and the stability of an $AQCQ$-functional equation in non-Archimedean normed spaces, Abs. Appl. Anal. Vol. 2010, Article ID 849543 (2010), 15 pages.
• J. M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA, 46 (1982), 126–130.
• J. M. Rassias, On approximately of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (4) (1984), 445–446.
• J. M. Rassias, Solution of problem of Ulam, J.Approx. Theory. USA, 57 (3) (1989), 268–273.
• J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glasnic Matematicki. Serija III, 34 (2) (1999), 243–252.
• J. M. Rassias, Solution of the Ulam stablility problem for cubic mappings, Glasnik Matematicki. Serija III, 36 (1) (2001), 63–72.
• J. M. Rassias and Mohammad Eslamian, Fixed points and stability of nonic functional equation in quasi-$\beta$-normed spaces, Contemporary Anal. Appl. Math. 3(2) (2015), 293–309.
• T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.
• K. Ravi, M. Arunkumar and J. M. Rassias, Ulam stability for the orthogonally general Euler-Lagrange type functional equation, Int. J. Math. Stat. 3 (A08) (2008), 36–46.
• K. Ravi, J. M. Rassias and R. Kodandan, Generalized Ulam-Hyers stability of an AQ-functional equation in quasi-$\beta$-normed spaces, Math. AEterna, 1 (3-4) (2011), 217–236.
• K. Ravi, J. M. Rassias and B. V. Senthil Kumar, Ulam-Hyers stability of undecic functional equation in quasi-$\beta$-normed spaces: Fixed point method, Tbilisi J. Math., 9(2) (2016), 83–103.
• P. K. Sahoo, On a functional equation characterizing polynomials of degree three, Bull. Inst. Math. Acad. Sinica 32(1) (2004), 35–44.
• P. K. Sahoo, A generalized cubic functional equation, Acta Math. Sin. (Engl. Ser.) 21(5) (2005), 1159–1166.
• S. M. Ulam, Problems in Modern Mathematics, Rend. Chap. VI, Wiley, New York, 1960.
• T. Z. Xu, J. M. Rassias and W. X. Xu, A fixed point approach to the stability of a general mixed $AQCQ$-functional equation in non-Archimedean normed spaces, Discrete Dynamics in Nature and Society, Vol. 2010, Article ID 812545, (2010), 24 pages.
• T. Z. Xu, J. M. Rassias, M. J. Rassias and W. X. Xu, A fixed point approach to the stability of quintic and sextic functional equations in quasi-$\beta$-normed spaces, J. Inequal. Appl. Vol. 2010, Article ID 423231, (2010), 1–23.
• T. Z. Xu, J. M. Rassias and W. X. Xu, A fixed point approach to the stability of a general mixed additive-cubic functional equation in quasi fuzzy normed spaces, Int. J. Physical Sci. 6 (2) (2011), 12 pages.
• T. Z. Xu, J. M. Rassias and W. X. Xu, A generalized mixed additive-cubic functional equation, J. Comput. Anal. Appl. 13(7) (2011), 1273–1282.
• T. Z. Xu, J. M. Rassias and W. X. Xu, A generalized mixed quadratic-quartic functional equation, Bull. Malays. Math. Sci. Soc. 35(3) (2012), 633–649.
• T. Z. Xu and J. M. Rassias, Approximate septic and octic mappings in quasi-$\beta$-normed spaces, J. Comp. Anal. Appl. 15(6) (2013), 1110–1119.
• S. Zolfaghari, Stability of generalized $QCA$-functional equation in $p$-Banach spaces, Int. J. Nonlinear Anal. Appl. 1 (2010), 84–99.
• S. Zolfaghari, A. Ebadian, S. Ostadbashi, M. De La Se and M. Eshaghi Gordji, A fixed point approach to the Hyers-Ulam stability of an $AQ$ functional equation in probabilistic modular spaces, Int. J. Nonlinear Anal. Appl. 4 (2) (2013), 1–14.