Stability of an additive-quartic functional equation in modular spaces

Volume 26, Issue 1, pp 22--40 http://dx.doi.org/10.22436/jmcs.026.01.04

Publication Date: September 17, 2021 Submission Date: July 10, 2021 Revision Date: July 21, 2021 Accteptance Date: August 10, 2021

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1108 Downloads
• 2088 Views

Authors

S. Karthikeyan - Department of Mathematics , R.M.K. Engineering College, Kavaraipettai - 601 206, Tamil Nadu, India. C. Park - Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea. P. Palani - Department of Mathematics, Sri Vidya Mandir Arts and Science College, Uthangarai- 636 902, Tamil Nadu, India. T. R. K. Kumar - Department of Mathematics, R.M.K. Engineering College, Kavaraipettai - 601 206, Tamil Nadu, India.

Abstract

In this paper, we prove the Ulam-Hyers stability of the following additive-quartic functional equation \[ f\left(\frac{u+v}{2}-w\right) +f\left(\frac{v+w}{2}-u\right)+f\left(\frac{w+u}{2}-v\right) =\frac{25}{32}\left(f(u-v)+f(v-w)+f(w-u)\right)-\frac{7}{32}\left(f(v-u)+f(w-v)+f(u-w)\right) \] in modular spaces by using the direct method.

Share and Cite

Keywords

• Ulam-Hyers stability
• additive functional equation
• quartic functional equation
• modular space

MSC

•  39B52
•  39B72
•  39B82

References

• [1] J. Aczél, J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge (1989)
  • View Article
  • MathSciNet


• [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64--66
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57 (1951), 223--237
  • View Article
  • MathSciNet
  • Google Scholar


• [4] Y. Cho, M. B. Ghaemi, M. Choubin, M. Eshaghi Gordji, On the Hyers-Ulam stability of sextic functional equations in $b$-homogeneous probabilistic modular spaces, Math. Inequal. Appl., 16 (2013), 1097--1114
  • View Article
  • Google Scholar


• [5] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62 (1992), 59--64
  • View Article
  • MathSciNet
  • Google Scholar


• [6] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Co., River Edge (2002)
  • View Article
  • MathSciNet
  • Google Scholar


• [7] E. Elqorachi, M. T. Rassias, Generalized Hyers-Ulam Stability of Trigonometric Functional Equations, Mathematics, 6 (2018), 11 pages
  • View Article
  • Google Scholar


• [8] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci., 14 (1991), 431--434
  • View Article
  • MathSciNet
  • Google Scholar


• [9] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431--436
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] M. Heydari, S. M. Hosseini, G. B. Loghmani, Numerical solution of singular IVPs of Lane-Emden type using integral operator and radial basis functions, Int. J. Ind. Math., 4 (2012), 135--146
  • View Article
  • Google Scholar


• [11] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222--224
  • View Article
  • MathSciNet
  • Google Scholar


• [12] D. H. Hyers, G. Isac, T. M. Rassias, Stability of Functional Equations in Several Variables, Birkauser, Boston (1998)
  • View Article
  • MathSciNet
  • Google Scholar


• [13] D. H. Hyers, T. M. Rassias, Approximate homomorphisms, Aequationes Math., 44 (1992), 125--153
  • View Article
  • MathSciNet
  • Google Scholar


• [14] Y. F. Jin, C. Park, M. T. Rassias, Hom-derivations in $C^*$-ternary algebras, Acta Math. Sin. (Engl. Ser.), 36 (2020), 1025--1038
  • View Article
  • MathSciNet
  • Google Scholar


• [15] S.-M. Jung, Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients, J. Math. Anal. Appl., 320 (2006), 549--561
  • View Article
  • MathSciNet
  • Google Scholar


• [16] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York (2011)
  • View Article
  • MathSciNet
  • Google Scholar


• [17] S.-M. Jung, K. S. Lee, M. T. Rassias, S. M. Yang, Approximation Properties of Solutions of a Mean Value-Type Functional Inequality, II, Mathematics, 8 (2020), 8 pages
  • View Article
  • Google Scholar


• [18] S.-M. Jung, D. Popa, M. T. Rassias, On the stability of the linear functional equation in a single variable on complete metric groups, J. Global Optim., 59 (2014), 165--171
  • View Article
  • MathSciNet
  • Google Scholar


• [19] S.-M. Jung, M. T. Rassias, A linear functional equation of third order associated to the Fibonacci numbers, Abstr. Appl. Anal., 2014 (2014), 8 pages
  • View Article
  • Google Scholar


• [20] S.-M. Jung, M. T. Rassias, C. Mortici, On a functional equation of trigonometric type, Appl. Math. Comput., 252 (2015), 294--303
  • View Article
  • MathSciNet
  • Google Scholar


• [21] P. Kannappan, Functional Equations and Inequalities with Applications, Springer, New York (2009)
  • View Article
  • MathSciNet
  • Google Scholar


• [22] H.-M. Kim, H.-Y. Shin, Refined stability of additive and quadratic functional equations in modular spaces, J. Inequal. Appl., 2017 (2017), 13 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [23] Y.-H. Lee, S.-M. Jung, M. T. Rassias, On an $n$-dimensional mixed type additive and quadratic functional equation, Appl. Math. Comput., 228 (2014), 13--16
  • View Article
  • MathSciNet
  • Google Scholar


• [24] Y.-H. Lee, S.-M. Jung, M. T. Rassias, Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation, J. Math. Inequal., 12 (2018), 43--61
  • View Article
  • MathSciNet
  • Google Scholar


• [25] W. A. J. Luxemburg, Banach Function Spaces, Ph.D. Thesis (Delft University of Technology), Delft (1955)
  • View Article
  • MathSciNet
  • Google Scholar


• [26] S. Mazur, W. Orlicz, On some classes of linear spaces, Studia Math., 17 (1958), 97--119
  • View Article
  • MathSciNet
  • Google Scholar


• [27] T. Miura, S. Miyajima, S. E. Takahasi, A characterization of Hyers-Ulam stability of first order linear differential operators, J. Math. Anal. Appl., 286 (2003), 136--146
  • View Article
  • MathSciNet
  • Google Scholar


• [28] C. Mortici, M. T. Rassias, S.-M. Jung, On the stability of a functional equation associated with the Fibonacci numbers, Abstr. Appl. Anal., 2014 (2014), 6 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [29] J. Musielak, W. Orlicz, On modular spaces, Studia Math., 18 (1959), 591--597
  • View Article
  • Google Scholar


• [30] J. Musielak, W. Orlicz, Some remarks on modular spaces, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys., 7 (1959), 661--668
  • MathSciNet
  • Google Scholar


• [31] H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co., Tokyo (1950)
  • MathSciNet
  • Google Scholar


• [32] C. Park, A. Bodaghi, S. O. Kim, A fixed point approach to stability of additive mappings in modular spaces without $\Delta_2$-conditions, J. Comput. Anal. Appl., 24 (2018), 1038--1048
  • View Article
  • MathSciNet
  • Google Scholar


• [33] C. Park, M. T. Rassias, Additive functional equations and partial multipliers in $C^*$-algebras, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 113 (2019), 2261--2275
  • View Article
  • MathSciNet
  • Google Scholar


• [34] C. Park, J. M. Rassias, A. Bodaghi, S. O. Kim, Approximate homomorphisms from ternary semigroups to modular spaces, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 113 (2019), 2175--2188
  • View Article
  • MathSciNet
  • Google Scholar


• [35] M. Ramdossa, D. Pachaiyappana, H. Duttab, Functional equation and its modular stability with and without $\Delta_p$ condition, Filomat, 34 (2020), 919--930
  • View Article
  • MathSciNet
  • Google Scholar


• [36] T. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297--300
  • View Article
  • MathSciNet
  • Google Scholar


• [37] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. (2), 108 (1984), 445--446
  • View Article
  • MathSciNet
  • Google Scholar


• [38] T. M. Rassias, Functional Equations and Inequalities, Springer Science & Business Media, Berlin (2000)
  • View Article
  • Google Scholar


• [39] T. M. Rassias, Solution of a functional equation problem of Steven Butler, Octogon Math. Mag., 12 (2004), 152--153
  • Google Scholar


• [40] K. Ravi, M. Arunkumar, J. M. Rassias, Ulam stability for the orthogonally general Euler-Lagrange type functional equation, Int. J. Math. Stat., 3 (2008), 36--46
  • View Article
  • Google Scholar


• [41] P. K. Sahoo, P. Kannappan, Introduction to Functional Equations, CRC Press, Boca Raton (2011)
  • View Article
  • MathSciNet
  • Google Scholar


• [42] T. Trif, On the stability of a functional equation deriving from an inequality of Popoviciu for convex functions, J. Math. Anal. Appl., 272 (2002), 604--616
  • View Article
  • MathSciNet
  • Google Scholar


• [43] S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, New York (1964)
  • MathSciNet
  • Google Scholar


• [44] K. Wongkum, P. Chaipunya, P. Kumam, On the generalized Ulam-Hyers-Rassias stability of quadratic mappings in modular spaces without $\Delta_2$-conditions, J. Funct. Spaces, 2015 (2015), 6 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [45] K. Wongkum, P. Kumam, Y. J. Cho, P. Thounthong, P. Chaipunya, On the generalized Ulam-Hyers-Rassias stability for quartic functional equation in modular spaces, J. Nonlinear Sci. Appl., 10 (2017), 1399--1406
  • View Article
  • MathSciNet
  • Google Scholar