Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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HYPERSTABILITY CRITERION FOR A NEW TYPE OF 2-VARIABLE RADICAL FUNCTIONAL EQUATIONS

  • EL-Fassi, Iz-iddine (Department of Mathematics Faculty of Sciences and Techniques Sidi Mohamed ben abdellah University)
  • Received : 2020.07.09
  • Accepted : 2020.07.31
  • Published : 2021.04.30
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Abstract

The aim of this paper is to obtain the general solution of the 2-variable radical functional equations $f({\sqrt[k]{x^k+z^k}},\;{\sqrt[{\ell}]{y^{\ell}+w^{\ell}}})=f(x,y)+f(z,w)$, x, y, z, w ∈ ℝ, for f a mapping from the set of all real numbers ℝ into a vector space, where k and ℓ are fixed positive integers. Also using the fixed point result of Brzdęk and Ciepliński [11, Theorem 1] in (2, 𝛽)-Banach spaces, we prove the generalized hyperstability results of the 2-variable radical functional equations. In the end of this paper we derive some consequences from our main results.

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References

  1. M. R. Abdollahpour, R. Aghayari, and M. Th. Rassias, Hyers-Ulam stability of associated Laguerre differential equations in a subclass of analytic functions, J. Math. Anal. Appl. 437 (2016), no. 1, 605-612. https://doi.org/10.1016/j.jmaa.2016.01.024
  2. M. R. Abdollahpour and M. Th. Rassias, Hyers-Ulam stability of hypergeometric differential equations, Aequationes Math. 93 (2019), no. 4, 691-698. https://doi.org/10.1007/s00010-018-0602-3
  3. J. Aczel and J. Dhombres, Functional equations in several variables, Encyclopedia of Mathematics and its Applications, 31, Cambridge University Press, Cambridge, 1989. https://doi.org/10.1017/CBO9781139086578
  4. 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
  5. D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223-237. https://doi.org/10.1090/S0002-9904-1951-09511-7
  6. J. Brzdek, Remarks on hyperstability of the Cauchy functional equation, Aequationes Math. 86 (2013), no. 3, 255-267. https://doi.org/10.1007/s00010-012-0168-4
  7. J. Brzdek, Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungar. 141 (2013), no. 1-2, 58-67. https://doi.org/10.1007/s10474-013-0302-3
  8. J. Brzdek, A hyperstability result for the Cauchy equation, Bull. Aust. Math. Soc. 89 (2014), no. 1, 33-40. https://doi.org/10.1017/S0004972713000683
  9. J. Brzdek, Remarks on stability of some inhomogeneous functional equations, Aequationes Math. 89 (2015), no. 1, 83-96. https://doi.org/10.1007/s00010-014-0274-6
  10. J. Brzdek and K. Cieplinski, Hyperstability and superstability, Abstr. Appl. Anal. 2013 (2013), Art. ID 401756, 13 pp. https://doi.org/10.1155/2013/401756
  11. J. Brzdek and K. Cieplinski, On a fixed point theorem in 2-Banach spaces and some of its applications, Acta Math. Sci. Ser. B (Engl. Ed.) 38 (2018), no. 2, 377-390. https://doi.org/10.1016/S0252-9602(18)30755-0
  12. K. Cieplinski and T. Z. Xu, Approximate multi-Jensen and multi-quadratic mappings in 2-Banach spaces, Carpathian J. Math. 29 (2013), no. 2, 159-166. https://doi.org/10.37193/CJM.2013.02.14
  13. Y. J. Cho, P. C. S. Lin, S. S. Kim, and A. Misiak, Theory of 2-Inner Product Spaces, Nova Science Publishers, Inc., Huntington, NY, 2001.
  14. St. 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
  15. St. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. https://doi.org/10.1142/9789812778116
  16. Iz. EL-Fassi, Approximate solution of radical quartic functional equation related to additive mapping in 2-Banach spaces, J. Math. Anal. Appl. 455 (2017), no. 2, 2001-2013. https://doi.org/10.1016/j.jmaa.2017.06.078
  17. Iz. EL-Fassi, A new type of approximation for the radical quintic functional equation in non-Archimedean (2, β)-Banach spaces, J. Math. Anal. Appl. 457 (2018), no. 1, 322-335. https://doi.org/10.1016/j.jmaa.2017.08.015
  18. Iz. EL-Fassi and S. Kabbaj, On the hyperstability of a Cauchy-Jensen type functional equation in Banach spaces, Proyecciones 34 (2015), no. 4, 359-375. https://doi.org/10.4067/S0716-09172015000400005
  19. Iz. EL-Fassi, S. Kabbaj, and A. Charifi, Hyperstability of Cauchy-Jensen functional equations, Indag. Math. (N.S.) 27 (2016), no. 3, 855-867. https://doi.org/10.1016/j.indag.2016.04.001
  20. R. W. Freese and Y. J. Cho, Geometry of Linear 2-Normed Spaces, Nova Science Publishers, Inc., Hauppauge, NY, 2001.
  21. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431-434. https://doi.org/10.1155/S016117129100056X
  22. S. Gahler, Lineare 2-normierte Raume, Math. Nachr. 28 (1964), 1-43. https://doi.org/10.1002/mana.19640280102
  23. S. Gahler, Uber 2-Banach-Raume, Math. Nachr. 42 (1969), 335-347. https://doi.org/10.1002/mana.19690420414
  24. E. Gselmann, Hyperstability of a functional equation, Acta Math. Hungar. 124 (2009), no. 1-2, 179-188. https://doi.org/10.1007/s10474-009-8174-2
  25. 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
  26. 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. https://doi.org/10.1007/978-1-4612-1790-9
  27. D. H. Hyers and T. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), no. 2-3, 125-153. https://doi.org/10.1007/BF01830975
  28. S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, Springer Optimization and Its Applications, 48, Springer, New York, 2011. https://doi.org/10.1007/978-1-4419-9637-4
  29. S.-M. Jung, D. Popa, and M. Th. Rassias, On the stability of the linear functional equation in a single variable on complete metric groups, J. Global Optim. 59 (2014), no. 1, 165-171. https://doi.org/10.1007/s10898-013-0083-9
  30. S.-M. Jung and M. Th. Rassias, A linear functional equation of third order associated with the Fibonacci numbers, Abstr. Appl. Anal. 2014 (2014), Art. ID 137468, 7 pp. https://doi.org/10.1155/2014/137468
  31. Pl. Kannappan, Functional equations and inequalities with applications, Springer Monographs in Mathematics, Springer, New York, 2009. https://doi.org/10.1007/978-0-387-89492-8
  32. H. Khodaei, M. Eshaghi Gordji, S. S. Kim, and Y. J. Cho, Approximation of radical functional equations related to quadratic and quartic mappings, J. Math. Anal. Appl. 395 (2012), no. 1, 284-297. https://doi.org/10.1016/j.jmaa.2012.04.086
  33. 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
  34. 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
  35. G. Maksa and Z. Pales, Hyperstability of a class of linear functional equations, Acta Math. Acad. Paedagog. Nyh'azi. (N.S.) 17 (2001), no. 2, 107-112.
  36. C. Mortici, M. Th. Rassias, and S.-M. Jung, On the stability of a functional equation associated with the Fibonacci numbers, Abstr. Appl. Anal. 2014 (2014), Art. ID 546046, 6 pp. https://doi.org/10.1155/2014/546046
  37. W.-G. Park, Approximate additive mappings in 2-Banach spaces and related topics, J. Math. Anal. Appl. 376 (2011), no. 1, 193-202. https://doi.org/10.1016/j.jmaa.2010.10.004
  38. C. Park and M. Th. Rassias, Additive functional equations and partial multipliers in C*-algebras, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113 (2019), no. 3, 2261-2275. https://doi.org/10.1007/s13398-018-0612-y
  39. 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
  40. T. M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1991), no. 1, 106-113. https://doi.org/10.1016/0022-247X(91)90270-A
  41. T. M. Rassias, Functional equations and inequalities, Mathematics and its Applications, 518, Kluwer Academic Publishers, Dordrecht, 2000. https://doi.org/10.1007/978-94-011-4341-7
  42. P. K. Sahoo and P. Kannappan, Introduction to Functional Equations, CRC Press, Boca Raton, FL, 2011.
  43. W. Smajdor, Note on a Jensen type functional equation, Publ. Math. Debrecen 63 (2003), no. 4, 703-714.
  44. T. Trif, On the stability of a functional equation deriving from an inequality of Popoviciu for convex functions, J. Math. Anal. Appl. 272 (2002), no. 2, 604-616. https://doi.org/10.1016/S0022-247X(02)00181-6
  45. S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York, 1960.
  46. J. Wang, Some further generalizations of the Hyers-Ulam-Rassias stability of functional equations, J. Math. Anal. Appl. 263 (2001), no. 2, 406-423. https://doi.org/10.1006/jmaa.2001.7626
  47. A. G. White, Jr., 2-Banach spaces, Math. Nachr. 42 (1969), 43-60. https://doi.org/10.1002/mana.19690420104