Solution and Stability of a Cubic Type Functional Equation: Using Direct and Fixed Point Methods


Download PDF

Authors: V. GOVINDAN, S. MURTHY AND M. SARAVANAN

DOI: 10.46793/KgJMat2001.007G

Abstract:

In this concept, we investigate the generalized Ulam-Hyers-Rassias stability for the new type of cubic functional equation of the form

g(ax   + bx   + 2cx  )
    1      2       3 + g(ax  +  bx  −  2cx  )
    1      2       3 + 8 a3g(x 1) + 8 b3g(x 2)
=2g(ax1 + bx2) + 4(g(ax1  +  cx3 ) + g(ax1  −  cx3) +  g(bx2 +  cx3 ) + g (bx2 −  cx3) )
by using direct and fixed point alternative.

Keywords:

Cubic functional equation, generalized Hyers-Ulam stability, fixed point.



References:

[1]   J. Aczel and J. Dhomberes, Functional Equations in Several Variables, Cambridge University Press, New York, New Rochelle, Melbourne, Sidney, 1989.

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

[3]   M. Arunkumar, S. Murthy and G. Ganapathy, Solution and Stability of n-dimensional quadratic functional equation, ICCMSC 2012, Computer and Information Science (CIS), Springer Verlag-Germany, 238, 2012, 384–394.

[4]   M. Arunkumar, S. Murthy and V. Govindan, General solution and generalized Ulam-Hyers stability of a generalized n-type additive quadratic functional equation in Banach spaces and Banach algebra: using direct and fixed point method, International Journal of Advanced Mathematical Sciences 3(1) (2015), 25–64.

[5]   M. Arunkumar, S. Murthy, V. Govindan and T. Namachivayam, General solution and four types of Ulam-Hyers stability of n-dimensional additive functional equation in Banach and fuzzy Banach spaces: Hyers direct and fixed point methods, International Journal of Applied Engineering and Research 11(1) (2016), 324–338.

[6]   M. Arunkumar, K. Ravi and M. J. Rassias, Stability of a quartic and orthogonally quartic functional equation, Bull. Math. Anal. Appl. 3(3) (2011), 13–24.

[7]   M. Arunkumar, S. Murthy and G. Ganapathy, Stability of a functional equation having nth order solution in generalized 2-normed spaces, International Journal of Mathematical Sciences and Engineering Applications 5(4) (2011), 361–369.

[8]   T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, The Journal of Fuzzy Mathematics 11(3) (2003), 687–705.

[9]   D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223–237.

[10]   S. C. Chang and J. N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86 (1994), 429–436.

[11]   P. W. Cholewa, On the stability of quadratic functional mappings in normed spaces, Abh. Math. Semin. Univ. Hambg. 62 (1992), 59–64.

[12]   S. Czerwik, Fundamental Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002.

[13]   C. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Semin. Univ. Hambg. 62 (1992), 59–64.

[14]   P. Guvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436.

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

[16]   D. H. Hyers, G. Issac and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.

[17]   K. W. Jun and H. M. Kim, The generalized Hyers-Ulam-Rassias stability of cubic functional equation, J. Math. Anal. Appl. 274(2) (2002), 267–278.

[18]   S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations, Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.

[19]   S. S. Jin and Y. H. Lee, Fuzzy stability of a general quadratic functional equation deriving from quadratic and additive mappings, Abstr. Appl. Anal. 2011 (2011), Article ID 534120, 15 pages.

[20]   S. S. Jin and Y. H. Lee, Fuzzy stability of a general quadratic functional equation, Adv. Fuzzy Syst. 2011 (2011), Article ID 791695, 9 pages.

[21]   S. S. Jin and Y. H. Lee, Fuzzy stability of the Cauchy additive and quadratic type functional equation, Commun. Korean Math. Soc. 27 (2012), 523–535.

[22]   W. Liguang, L. Bo and B. Ran, Stability of a mixed type functional equation on multi-Banach spaces: a fixed point approach, J. Fixed Point Theory Appl. (2010), Article ID 283827, 9 pages.

[23]   W. Liguang, The fixed point method for intuitionistic fuzzy stability of a quadratic functional equation, Fixed Point Theory Appl. (2010), Article ID 107182, 7 Pages.

[24]   W. Liguang, and L. Bo, The Hyers-Ulam stability of a functional equation deriving from quadratic and cubic functions, in quasi-β-normed spaces, Acta Math. Sin. (Engl. Ser.) 26(12) (2010), 2335–2348.

[25]   W. Liguang and L. Jing, On the stability of a functional equation deriving from additive and quadratic functions, Adv. Differential Equations 2012(98) (2012), 12 pages.

[26]   W. Liguang, K. Xu and Q. Liu, On the stability a mixed functional equation deriving from additive, quadratic and cubic mappings, Acta Math. Sin. (Engl. Ser.) 30(6) (2014), 1033–1049.

[27]   S. H. Lee, S. M. Im and I. S. Hwang, Quadratic functional equations, J. Math. Anal. Appl. 307 (2005), 387–394.

[28]   S. S. Kim, Y. J. Cho and A. White, Linear operators on linear 2-normed spaces, Glas. Math. 27(42) (1992), 63–70.

[29]   S. Murthy, M. Arunkumar and G. Ganapathy, Solution and stability of n-dimensional cubic functional equation in F-spaces: direct and fixed point methods, Proceedings of International Conference on Mathematical Sciences, Chennai, Tamil Nadu, July 17-19, 2014, 81–88.

[30]   Z. Lewandowska, Linear operators on a generalized 2-normed spaces, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 42(4) (1999), 353–368.

[31]   A. Najati, The generalized Hyers-Ulam stability of a cubic functional equation, Turkish J. Math. 31 (2007), 1–14.

[32]   J. H. Park, S. B. Lee and W. G. Park, Stability results in non Archimedean L-fuzzy normed spaces for a cubic functional equation, J. Inequal. Appl. (2012), 193–201.

[33]   J. M. Rassias, On approximately of approximately linear mappings of linear mappings, J. Funct. Anal. 46 (1982), 126–130.

[34]   Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72(2) (1978), 297–300.

[35]   K. Ravi and M. Arunkumar, Hyers-Ulam-Rassias stability of a quartic functional equation, International Journal of Pure and Applied Mathematics 34(2) (2007), 247–260.

[36]   K. Ravi and M. Arunkumar, On a General solution of a quartic functional equation, J. Comb. Inf. Syst. Sci. 33(1) (2008), 373–386.

[37]   F. Skof, Local properties and approximations of operators, Seminario Matematico e Fisico di Milano 53 (1983), 113–129.

[38]   S. M. Ulam, A Collection of Mathematical Problems, Interscience Publisher, New York, 1960.

[39]   S. M. Ulam, Problem in Modern Mathematics, Sciences Editions, John Wiley and Sons Inc. New York, 1969.