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AN EXAMPLE FOR THE NON-STABILITY OF MULTI-ADDITIVE-QUADRATIC-CUBIC MAPPINGS

  • Abasalt Bodaghi (Department of Mathematics West Tehran Branch Islamic Azad University)
  • Received : 2023.03.18
  • Accepted : 2023.06.16
  • Published : 2024.01.31

Abstract

In this paper, we improve Corollary 1 of [4] and then present an example to show that the assertion in the mentioned corollary can not be valid in the singularity case.

Keywords

Acknowledgement

The author sincerely thanks the anonymous reviewer for her/his careful reading, constructive comments and suggesting some related references to improve the quality of the first draft of the paper.

References

  1. A. Bodaghi, An example for the nonstability of multicubic mappings, Int. J. Nonlinear Anal. Appl. 14 (2023), no. 3, 273-277. https://doi.org/10.22075/IJNAA.2022.25878.3150 
  2. A. Bodaghi, H. Moshtagh, and H. Dutta, Characterization and stability analysis of advanced multi-quadratic functional equations, Adv. Difference Equ. 2021 (2021), Paper No. 380, 15 pp. https://doi.org/10.1186/s13662-021-03541-3 
  3. A. Bodaghi, H. Moshtagh, and A. Mousivand, Characterization and stability of multi-Euler-Lagrange quadratic functional equations, J. Funct. Spaces 2022 (2022), Art. ID 3021457, 9 pp. https://doi.org/10.1155/2022/3021457 
  4. A. Bodaghi and Th. M. Rassias, Functional inequalities for multi-additive-quadratic-cubic mappings, in Approximation and computation in science and engineering, 103-126, Springer Optim. Appl., 180, Springer, Cham., 2022. https://doi.org/10.1007/978-3-030-84122-5_7 
  5. A. Bodaghi and B. Shojaee, On an equation characterizing multi-cubic mappings and its stability and hyperstability, Fixed Point Theory 22 (2021), no. 1, 83-92. https://doi.org/10.24193/fpt-ro.2021.1.06 
  6. K. Cieplinski, Generalized stability of multi-additive mappings, Appl. Math. Lett. 23 (2010), no. 10, 1291-1294. https://doi.org/10.1016/j.aml.2010.06.015 
  7. K. Cieplinski, On the generalized Hyers-Ulam stability of multi-quadratic mappings, Comput. Math. Appl. 62 (2011), no. 9, 3418-3426. https://doi.org/10.1016/j.camwa.2011.08.057 
  8. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64. https://doi.org/10.1007/BF02941618 
  9. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431-434. https://doi.org/10.1155/S016117129100056X 
  10. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of functional equations in several variables, Progress in Nonlinear Differential Equations and their Applications, 34, Birkhauser Boston, Inc., Boston, MA, 1998. https://doi.org/10.1007/978-1-4612-1790-9 
  11. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, second edition, Birkhauser Verlag, Basel, 2009. https://doi.org/10.1007/978-3-7643-8749-5 
  12. Y.-H. Lee, S.-M. Jung, and M. Th. Rassias, On an n-dimensional mixed type additive and quadratic functional equation, Appl. Math. Comput. 228 (2014), 13-16. https://doi.org/10.1016/j.amc.2013.11.091 
  13. Y.-H. Lee, S.-M. Jung, and M. Th. Rassias, Uniqueness theorems on functional inequalities concerning cubic-quadratic-additive equation, J. Math. Inequal. 12 (2018), no. 1, 43-61. https://doi.org/10.7153/jmi-2018-12-04 
  14. C. Park, Multi-quadratic mappings in Banach spaces, Proc. Amer. Math. Soc. 131 (2003), no. 8, 2501-2504. https://doi.org/10.1090/S0002-9939-02-06886-7 
  15. C. Park and A. Bodaghi, Two multi-cubic functional equations and some results on the stability in modular spaces, J. Inequal. Appl. 2020 (2020), Paper No. 6, 16 pp. https://doi.org/10.1186/s13660-019-2274-5 
  16. C. Park and M. Th. Rassias, Additive functional equations and partial multipliers in C∗-algebras, Rev. R. Acad. Cienc. Exactas F'is. Nat. Ser. A Mat. RACSAM 113 (2019), no. 3, 2261-2275. https://doi.org/10.1007/s13398-018-0612-y 
  17. Th. M. Rassias, Functional equations and inequalities, Mathematics and its Applications, 518, Kluwer Acad. Publ., Dordrecht, 2000. https://doi.org/10.1007/978-94-011-4341-7 
  18. S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, Inc., New York, 1964. 
  19. X. Zhao, X. Yang, and C.-T. Pang, Solution and stability of the multiquadratic functional equation, Abstr. Appl. Anal. 2013 (2013), Art. ID 415053, 8 pp. https://doi.org/10.1155/2013/415053