×

Banach limit in the stability problem of a linear functional equation. (English) Zbl 1467.39019

Summary: We present some applications of the Banach limit in the study of the stability of the linear functional equation in a single variable.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges

References:

[1] Agarwal, RP; Xu, B.; Zhang, W., Stability of functional equations in single variable, J. Math. Anal. Appl., 288, 852-869 (2003) · Zbl 1053.39042 · doi:10.1016/j.jmaa.2003.09.032
[2] Badora, R., Report of meeting, Ann. Math. Sil., 20, 88 (2006)
[3] Badora, R., Report of Meeting, Aequ. Math., 79, 175 (2010)
[4] Baker, JA, The stability of certain functional equations, Proc. Am. Math. Soc., 112, 729-732 (1991) · Zbl 0735.39004 · doi:10.1090/S0002-9939-1991-1052568-7
[5] Brillouët-Belluot, N.; Brzdęk, J.; Ciepliñski, K., On some recent developments in Ulam’s type stability, Abstr. Appl. Anal., 2012, 41 (2012) · Zbl 1259.39019 · doi:10.1155/2012/716936
[6] Brydak, D., On the stability of the functional equation \(\varphi [f(x)]=g(x)\varphi (x)+F(x)\), Proc. Am. Math. Soc., 26, 455-460 (1970) · Zbl 0209.15802
[7] Brzdęk, J.; Ciepliński, K.; Leśniak, Z., On Ulam’s type stability of the linear equation and related issues, Discrete Dyn. Nat. Soc., 2014, 14 (2014) · Zbl 1419.39053 · doi:10.1155/2014/536791
[8] Brzdęk, J.; Popa, D.; Raşa, I.; Xu, B., Ulam Stability of Operators (2018), Oxford: Academic Press, Oxford · Zbl 1393.39001
[9] Brzdęk, J.; Popa, D.; Xu, B., Remarks on stability and nonstability of the linear functional equation of the first order, Appl. Math. Comput., 238, 141-148 (2014) · Zbl 1334.39057
[10] Forti, GL, Hyers-Ulam stability of functional equations in several variables, Aequ. Math., 50, 143-190 (1995) · Zbl 0836.39007 · doi:10.1007/BF01831117
[11] Hyers, DH; Isac, G.; Rassias, ThM, Stability of Functional Equations in Several Variables (1998), Basel: Birkhäuser, Basel · Zbl 0907.39025 · doi:10.1007/978-1-4612-1790-9
[12] Jung, S-M, Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, Springer Optimization and Its Applications (2011), New York: Springer, New York · Zbl 1221.39038
[13] Kim, GH, On the stability of generalized gamma functional equation, Int. J. Math. Math. Sci., 23, 513-520 (2000) · Zbl 0955.39010 · doi:10.1155/S0161171200003598
[14] Kuczma, M., Functional Equations in a Single Variable (1968), Warszawa: PWN-Polish Scientific Publishers, Warszawa · Zbl 0196.16403
[15] Kuczma, M.; Choczewski, B.; Ger, R., Iterative Functional Equations, Encyclopedia of Mathematics and its Applications (1990), Cambridge: Cambridge University Press, Cambridge · Zbl 0703.39005
[16] Sofi, MA, Banach limits: some new thoughts and perspectives, J. Anal. (2019) · Zbl 1482.46022 · doi:10.1007/s41478-019-00184-2
[17] Trif, T., On the stability of a general gamma-type functional equation, Publ. Math. Debrecen, 60, 47-61 (2002) · Zbl 1004.39023
[18] Turdza, E., On the stability of the functional equation \(\varphi [f(x)]=g(x)\varphi (x)+F(x)\), Proc. Am. Math. Soc., 30, 484-486 (1971) · Zbl 0224.39005
[19] Ulam, S.M.: A collection of mathematical problems. In: Interscience Tract 8 (= Problems in Modern Mathematics, Science Edition), Interscience, New York (1960) (1964) · Zbl 0086.24101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.