대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제39권3호
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  • Pages.673-692
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  • 2024
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  • 1225-1763(pISSN)
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  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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MULTI-JENSEN AND MULTI-EULER-LAGRANGE ADDITIVE MAPPINGS

  • Abasalt Bodaghi (Department of Mathematics West Tehran Branch Islamic Azad University) ;
  • Amir Sahami (Department of Mathematics Faculty of Basic Science Ilam University)
  • 투고 : 2023.07.07
  • 심사 : 2024.05.31
  • 발행 : 2024.07.31
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초록

In this work, an alternative fashion of the multi-Jensen is introduced. The structures of the multi-Jensen and the multi-Euler-Lagrange-Jensen mappings are described. In other words, the system of n equations defining each of the mentioned mappings is unified as a single equation. Furthermore, by applying a fixed point theorem, the Hyers-Ulam stability for the multi-Euler-Lagrange-Jensen mappings in the setting of Banach spaces is established. An appropriate counterexample is supplied to invalidate the results in the case of singularity for multiadditive mappings.

키워드

과제정보

The authors sincerely thank the anonymous reviewer for his/her careful reading, constructive comments that improved the manuscript substantially.

참고문헌

  1. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66. https://doi.org/10.2969/jmsj/00210064
  2. A. Bahyrycz and K. Cieplinski, On an equation characterizing multi-Jensen-quadratic mappings and its Hyers-Ulam stability via a fixed point method, J. Fixed Point Theory Appl. 18 (2016), no. 4, 737-751. https://doi.org/10.1007/s11784-016-0298-8
  3. A. Bahyrycz, K. Cieplinski, and J. Olko, On an equation characterizing multi-additive-quadratic mappings and its Hyers-Ulam stability, Appl. Math. Comput. 265 (2015), 448-455. https://doi.org/10.1016/j.amc.2015.05.037
  4. A. Bahyrycz, K. Cieplinski, and J. Olko, On an equation characterizing multi-Cauchy-Jensen mappings and its Hyers-Ulam stability, Acta Math. Sci. Ser. B (Engl. Ed.) 35 (2015), no. 6, 1349-1358. https://doi.org/10.1016/S0252-9602(15)30059-X
  5. A. Bahyrycz and J. Olko, On stability and hyperstability of an equation characterizing multi-Cauchy-Jensen mappings, Results Math. 73 (2018), no. 2, Paper No. 55, 18 pp. https://doi.org/10.1007/s00025-018-0815-8
  6. A. Bodaghi, Functional inequalities for generalized multi-quadratic mappings, J. Inequal. Appl. 2021, Paper No. 145, 13 pp. https://doi.org/10.1186/s13660-021-02682-z
  7. A. Bodaghi, Approximation of the multi-m-Jensen-quadratic mappings and a fixed point approach, Math. Slovaca 71 (2021), no. 1, 117-128. https://doi.org/10.1515/ms-2017-0456
  8. J. Brzdek, J. Chudziak, and Z. Pales, A fixed point approach to stability of functional equations, Nonlinear Anal. 74 (2011), no. 17, 6728-6732. https://doi.org/10.1016/j.na.2011.06.052
  9. K. Cieplinski, On multi-Jensen functions and Jensen difference, Bull. Korean Math. Soc. 45 (2008), no. 4, 729-737. https://doi.org/10.4134/BKMS.2008.45.4.729
  10. K. Cieplinski, Stability of the multi-Jensen equation, J. Math. Anal. Appl. 363 (2010), no. 1, 249-254. https://doi.org/10.1016/j.jmaa.2009.08.021
  11. 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
  12. K. Cieplinski, On Ulam stability of a functional equation, Results Math. 75 (2020), no. 4, Paper No. 151, 11 pp. https://doi.org/10.1007/s00025-020-01275-4
  13. M. Eshaghi Gordji, M. B. Ghaemi, S. Kaboli Gharetapeh, S. Shams, and A. Ebadian, On the stability of J*-derivations, J. Geom. Phys. 60 (2010), no. 3, 454-459. https://doi.org/10.1016/j.geomphys.2009.11.004
  14. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431-434 https://doi.org/10.1155/S016117129100056X
  15. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436. https://doi.org/10.1006/jmaa.1994.1211
  16. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  17. S.-M. Jung, Hyers-Ulam-Rassias stability of Jensen's equation and its application, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3137-3143. https://doi.org/10.1090/S0002-9939-98-04680-2
  18. E. Keyhani, M. Hassani, and A. Bodaghi, On set-valued multiadditive functional equations, Filomat 38 (2024), no. 5, 1847-1857.
  19. Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math. 22 (1989), no. 2, 499-507.
  20. 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
  21. Y.-H. Lee and K.-W. Jun, A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation, J. Math. Anal. Appl. 238 (1999), no. 1, 305-315. https://doi.org/10.1006/jmaa.1999.6546
  22. M. Maghsoudi and A. Bodaghi, On the stability of multi m-Jensen mappings, Casp. J. Math. Sci. 9 (2020), no. 2, 199-209.
  23. C. Park and A. Bodaghi, On the stability of *-derivations on Banach *-algebras, Adv. Difference Equ. 2012, 2012:138, 10 pp. https://doi.org/10.1186/1687-1847-2012-138
  24. C. Park and J. M. Rassias, Stability of the Jensen-type functional equation in C*-algebras: a fixed point approach, Abstr. Appl. Anal. 2009, Art. ID 360432, 17
  25. W. Prager and J. Schwaiger, Multi-affine and multi-Jensen functions and their connection with generalized polynomials, Aequationes Math. 69 (2005), no. 1-2, 41-57. https://doi.org/10.1007/s00010-004-2756-4
  26. W. Prager and J. Schwaiger, Stability of the multi-Jensen equation, Bull. Korean Math. Soc. 45 (2008), no. 1, 133-142. https://doi.org/10.4134/BKMS.2008.45.1.133
  27. T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300. https://doi.org/10.2307/2042795
  28. J. M. Rassias, On the stability of the Euler-Lagrange functional equation, Chinese J. Math. 20 (1992), no. 2, 185-190.
  29. J. M. Rassias, On the stability of the non-linear Euler-Lagrange functional equation in real normed linear spaces, J. Math. Phys. Sci. 28 (1994), no. 5, 231-235.
  30. P. K. Sahoo and P. Kannappan, Introduction to Functional Equations, CRC, Boca Raton, FL, 2011.
  31. S. Salimi and A. Bodaghi, A fixed point application for the stability and hyperstability of multi-Jensen-quadratic mappings, J. Fixed Point Theory Appl. 22 (2020), no. 1, Paper No. 9, 15 pp. https://doi.org/10.1007/s11784-019-0738-3
  32. J. Tipyan, C. Srisawat, P. Udomkavanich, and P. Nakmahachalasint, The generalized stability of an n-dimensional Jensen type functional equation, Thai J. Math. 12 (2014), no. 2, 265-274.
  33. S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, Inc., New York, 1964.
  34. T. Z. Xu, Stability of multi-Jensen mappings in non-Archimedean normed spaces, J. Math. Phys. 53 (2012), no. 2, 023507, 9 pp. https://doi.org/10.1063/1.3684746
  35. T. Z. Xu, Approximate multi-Jensen, multi-Euler-Lagrange additive and quadratic mappings in n-Banach spaces, Abstr. Appl. Anal. 2013, Art. ID 648709, 12 pp. https://doi.org/10.1155/2013/648709