A Fixed Point Approach To The Stability Of A Mixed Quadratic-Quartic (QQ) Functional Equation In Quasi−Β−Normed Spaces

K. Balamurugan; M. Arunkumar; P. Ravindiran

doi:https://doi.org/10.14445/22315373/IJMTT-V20P504

International Journal of Mathematics Trends and Technology

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Volume 20 | Number 1 | Year 2015 | Article Id. IJMTT-V20P504 | DOI : https://doi.org/10.14445/22315373/IJMTT-V20P504

A Fixed Point Approach To The Stability Of A Mixed Quadratic-Quartic (QQ) Functional Equation In Quasi−Β−Normed Spaces

K. Balamurugan, M. Arunkumar, P. Ravindiran

Abstract

In this paper we prove the generalized Hyers-Ulam stability of the following mixed quadratic -quartic functional equation f(3 x + 2y + z) + f(3 x + 2y − z) + f(3 x − 2y + z) + f(3 x − 2y − z) + 240 f(x) + 160 f(y) + 48 f(z) = 72[ f(x + y) + f(x − y)] + 18[ f(x + z) + f(x − z)] + 8[ f(y + z) + f(y − z)] + 24 f(2 x) + 4f(2 y) in the quasiquasi quasi− β− normed spaces via fixed point method. Counterexamples for non-stability stability cases are also discussed.

Keywords

Hyers-Ulam stability, Quadratic mapping, Quartic mapping, Mixed type functional equation, Quasi - Banach space, Fixed point

References

[1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ, Press, 1989.

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

[3] K. Balamurugan, M. Arunkumar, P. Ravindiran, Generalized Hyers-Ulam stability for a mixed additive-cubic(AC) Functional Equation in Quasi-Banach Spaces , Proceedings of the International Conference on Mathematics And its Applications-2014(ICMAA-2014), India, Vol.1(2014), pp. 234-261, ISBN-978-81-923752-6-7.

[4] K. Balamurugan, M. Arunkumar, P. Ravindiran, A Fixed Point Approach to the stability of a mixed additive-cubic(AC) Functional Equation in Quasi--normed Spaces , Special issue of the International Conference On Mathematical Methods and Computation, Jamal Academic Research Journal: an Interdisciplinary (January 2015), pp. 58-73.

[5] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002. ISSN: 2231-5373 http://www.ijmttjournal.orgPage 39 International Journal of Mathematics Trends and Technology - Volume 20 Number 1 - April 2015

[6] M. Eshaghi Gordji, S. Abbaszadeh, C.Park, On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces, Journal of Inequalities and Applications, vol. 2009, Article ID 153084, 26 pages.

[7] M. Eshaghi Gordji, M. Bavand Savadkouhi, C.Park, Quadratic-Quartic functional equations in RN-spaces, Journal of Inequalities and Applications, vol. 2009, Article ID 868423, 14 pages.

[8] P. G˘avrut¸a, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436.

[9] D.H. Hyers, On the stability of the linear functional equation, Proc.Nat. Acad.Sci.,U.S.A.,27 (1941) 222-224.

[10] Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer Monographs in Mathematics, 2009.

[11] H.M. Kim, On the stability problem for a mixed type of quartic and quadratic functional equation, Journal of Mathematical Analysis and Applications, vol. 324, no. 1(2006), pp. 358-372.

[12] J.M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA, 46, (1982) 126-130.

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

[14] M. Arunkumar and P. Agilan , Additive Quadratic functional equations are stable in Banach space: A Fixed Point Approach, International Journal of Pure and Applied Mathematics, Vol. 86, No. 6,2013, 951-963.

[15] K. Jun and H. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274(2002) 867-878.

[16] A. Nataji and G. Z. Eskandani, Stability of mixed additive and cubic functional equation in quasi- Banach spaces, J. Math. Anal. Appl. 342(2008) 1318-1331.

[17] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1, Colloq. Publi., vol. 48, Amer. Math. Soc., Providence, RI, 2000.

[18] S. Rolewicz, Metric Linear Spaces, PWN-Polish Sci. Publ./Reidel, Warszawa/Dordrecht, 1984.

[19] S.M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, NewYork, 1964.

[20] A. Nataji and M. B. Moghimi, Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces, J. Math. Anal. Appl. 337(2008) 399-415.

[21] J. Tober, Stability of cauchy functional equation in quasi-Banach spaces, Ann.Polon. Math. 83 (2004), 243-255.

[22] K. Zhou Xu, J.Michael Rassias, Matina J. Rassias, andW.Xin Xu, A Fixed Point Approach to the Stability of Quintic and Sextic functional equation in quasi- normed spaces, Journal of Inequalities and Applications, Volume 2010, Article ID 423231, 23 pages, 2010. doi:10.1155/2010/423231

[23] B.Margoils, J.B.Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull.Amer. Math. Soc. 74(1968), 305 309.

[24] 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, 24 pages, 2010.

[25] D. Mihet¸ and V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, Journal of Mathematical Analysis and Applications, Vol. 343 , no. 1, pp. 567-572, 2008.

[26] C. Park, Fixed Points and the Stability of an AQCQ functional equation in non-archimedean normed spaces, Abstract and Applied Analysis,, Volume 2010, Article ID 849543, 15 pages, 2010. ISSN: 2231-5373 http://www.ijmttjournal.org Page 40

Citation :

K. Balamurugan, M. Arunkumar, P. Ravindiran, "A Fixed Point Approach To The Stability Of A Mixed Quadratic-Quartic (QQ) Functional Equation In Quasi−Β−Normed Spaces," International Journal of Mathematics Trends and Technology (IJMTT), vol. 20, no. 1, pp. 25-40, 2015. Crossref, https://doi.org/10.14445/22315373/IJMTT-V20P504