Abstract
In this paper, using a purely fixed point approach, we produce a new proof of the Ulam-Hyers stability and hyperstability of the general functional equation:
considered in Bahyrycz and Olko (Aequationes Math 89:1461, 2015. https://doi.org/10.1007/s00010-014-0317-z), and in Bahyrycz and Olko (Aequationes Math 90:527, 2016. https://doi.org/10.1007/s00010-016-0418-y). Here m and n are positive integers, f is a mapping from a vector space X into a Banach space (Y, ∥ ∥), A ∈ Y and, for every i ∈{1, 2, …, m} and j ∈{1, …, n}, A i and a ij are scalars.
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References
Bahyrycz, A., Olko, J.: On stability of the general linear equation. Aequationes Math. 89, 1461–1474 (2015)
Bahyrycz, A., Olko, J.: Hyperstability of general linear functional equation. Aequationes Math. 90, 527–540 (2016)
Brzdek, J., Chudziak, J., Páles, Z.: A fixed point approach to stability of functional equations. Nonlinear Anal. (74), 6728–6732 (2011)
Cădariu, L., Radu, V.: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 4, Article 4 (2003)
Chu, H.Y., Kim, A., Yu, S.K.: On the stability of the generalized cubic set-valued functional equation. Appl. Math. Lett. 37, 7–14 (2014)
Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin. Univ. Hambg. 62, 59–64 (1992)
Dong, Z.: On Hyperstability of generalised linear functional equations in several variables. Bull. Aust. Math. Soc. 92, 259–267 (2015)
Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. U. S. A. 27, 222–224 (1941)
Jun, K.W., Kim, H.M., Rassais, T.M.: Extended Hyers-Ulam stability for Cauchy-Jensen mappings. J. Differ. Equ. Appl. 13, 1139–1153 (2007)
Moslehian, M.S.: Hyers-Ulam-Rassias stability of generalized derivations. Int. J. Math. Math. Sci. 2006, Article ID 93942, 8 (2006)
Oubbi, L.: Ulam-Hyers-Rassias stability problem for several kinds of mappings. Afr. Mat. 24, 525–542 (2013)
Patel, B.M., Patel, A.B.: Stability of Quartic functional equations in 2-Banach space. Int. J. Math. Anal. 7(23), 1097–1107 (2013)
Radu, V.: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4(1), 91–96 (2003)
Rassias, T.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)
Ulam, S.M.: Problems in Modern Mathematics. Chapter VI, Science Editions. Wiley, New York (1960)
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Benzarouala, C., Oubbi, L. (2019). A Purely Fixed Point Approach to the Ulam-Hyers Stability and Hyperstability of a General Functional Equation. In: Brzdęk, J., Popa, D., Rassias, T. (eds) Ulam Type Stability . Springer, Cham. https://doi.org/10.1007/978-3-030-28972-0_2
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DOI: https://doi.org/10.1007/978-3-030-28972-0_2
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