×
Govindan, Vediyappan; Park, Choonkil; Pinelas, Sandra; Rassias, Themistocles M.

Hyers-Ulam stability of an additive-quadratic functional equation. (English) Zbl 1455.39007

Cubo 22, No. 2, 233-255 (2020).
Summary: In this paper, we introduce the following \((a, b, c)\)-mixed type functional equation of the form \[ \begin{split} g(ax_1+bx_2+cx_3)-g(-ax_1+bx_2+cx_3) +g(ax_1-bx_2+cx_3)-g(ax_1+bx_2-cx_3) +\\ 2a^2[g(x_1) +g(-x_1)] + 2b^2[g(x_2) +g(-x_2)] + 2c^2[g(x_3) +g(-x_3)] +a[g(x_1)-g(-x_1)] +\\ b[g(x_2)-g(-x_2)] +c[g(x_3)-g(-x_3)] = 4g(ax_1+cx_3) + 2g(-bx_2) + 2g(bx_2) \end{split} \] where \(a\), \(b\), \(c\) are positive integers with \(a >1\), and investigate the solution and the Hyers-Ulam stability of the above functional equation in Banach spaces by using two different methods.

Cited in 2 Documents

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges

Keywords:

Hyers-Ulam stability; mixed-type functional equation; Banach space; fixed point
Cite Review PDF
Full Text: DOI

References:

[1] J. Aczel and J. Dhombres, Functional equations in several variables, Encyclopedia of Mathematics and its Applications, 31, Cambridge University Press, Cambridge, 1989. · Zbl 0685.39006
[2] L. Aiemsomboon and W. Sintunavarat, Stability of the generalized logarithmic functional equations arising from fixed point theory, Rev. R. Acad. Cienc. Exactas Fıs. Nat. Ser. A Mat. RACSAM 112 (2018), no. 1, 229-238. · Zbl 06836246
[3] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66. · Zbl 0040.35501
[4] I. Chang, E. Lee and H. Kim, On the Hyers-Ulam-Rassias stability of a quadratic functional equations, Math. Inequal. Appl. 6 (2003), 87-95. · Zbl 1024.39008
[5] K. Cieplinski, Applications of fixed point theorems to the Hyers-Ulam stabiltiy of functional equations - a survey, Ann. Funct. Anal. 3 (2012), 151-164. · Zbl 1252.39032
[6] St. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64. · Zbl 0779.39003
[7] S. Czerwik, Functional equations and inequalities in several variables, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. · Zbl 1011.39019
[8] A. Ebadian, N. Ghobadipour and M. Eshaghi Gordji, A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in C∗-ternary algebras, J. Math. Phys. 51 (2010), no. 10, 103508, 10 pp. · Zbl 1314.46063
[9] V. Govindan, S. Murthy and M. Saravanan, Solution and stability of a cubic type functional equation: Using direct and fixed point methods, Kraguj. J. Math. 44 (2020), 7-26. · Zbl 1488.39067
[10] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436. · Zbl 0818.46043
[11] O. Hadzic and E. Pap, Fixed point theory in probabilistic metric spaces, Mathematics and its Applications, 536, Kluwer Academic Publishers, Dordrecht, 2001. · Zbl 0994.47077
[12] D. H. Hyers, G. Isac and T. M. Rassias, Stability of functional equations in several variables, Progress in Nonlinear Differential Equations and their Applications, 34, Birkhauser Boston, Inc., Boston, MA, 1998. · Zbl 0907.39025
[13] K. Jun and H. Kim , On the Hyers-Ulam-Rassias stability of a generalized quadratic and additive type functional equation, Bull. Korean Math. Soc. 42 (2005), 133-148. · Zbl 1071.39030
[14] S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, Hadronic Press, Inc., Palm Harbor, FL, 2001. · Zbl 0980.39024
[15] Y. Jung, The Ulam-Gavruta-Rassias stability of module left derivations, J. Math. Anal. Appl. 339 (2008), 108-114. · Zbl 1134.39023
[16] Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995), no. 3-4, 368-372. · Zbl 0836.39006
[17] Y. Lee, S.-M. Jung and M. T. Rassias. Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation. J. Math. Inequal. 12 (2018), no. 1, 43-61. · Zbl 1412.39028
[18] Y. Lee and G. Kim, Generalized Hyers-Ulam stability of the pexider functional equation, Math. 7 (2019), no. 3, Article No. 280.
[19] M. Mirzavaziri and M. S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc. (N.S.) 37 (2006), no. 3, 361-376. · Zbl 1118.39015
[20] M. S. Moslehian and Th. M. Rassias, Stabiltiy of functional equations in non-Archimedean spaces, Appl. Anal. Discrete Math. 1 (2007), 325-334. · Zbl 1257.39019
[21] A. Najati and M. B. Moghini, On the stabiltiy of a quadratic and additive functional equation, J. Math. Anal. Appl. 337 (2008), 399-415. · Zbl 1127.39055
[22] C. Park, Additive s-functional inequalities and partial multipliers in Banach algebras, J. Math. Inequal. 13 (2019), no. 3, 867-877. · Zbl 1426.39025
[23] C. Park, J. R. Lee and X. Zhang, Additive s-functional inequality and hom-derivations in Banach algebras, J. Fixed Point Theory Appl. 21 (2019), no. 1, Paper No. 18, 14 pp. · Zbl 1404.39028
[24] C. Park and T. M. Rassias, Fixed points and stability of the Cauchy functional equation, Aust. J. Math. Anal. Appl. 6 (2009), no. 1, Art. 14, 9 pp. · Zbl 1175.39027
[25] F. Skof, Proprieta locali e approssimazione di operatori, Rend. Semin. Mat. Fis. Milano 53 (1983), 113-129. · Zbl 0599.39007
[26] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300. · Zbl 0398.47040
[27] N. Sirovnik, On certain functional equation in semiprime rings and standard operator algebras, Cubo 16 (2014), no. 1, 73-80. · Zbl 1308.47048
[28] S. M. Ulam, Problems in modern mathematics, Science Editions John Wiley & Sons, Inc., New York, 1964 · Zbl 0137.24201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.