×

Hyers-Ulam stability for linear equations of higher orders. (English) Zbl 1174.39012

The authors deal with the Hyers–Ulam stability of the \(m\)th-order iterative functional equation \[ \varphi(f^m(x))=\sum_{j=0}^{m-1}a_j\varphi(f^j(x))+F(x)\qquad (x\in S). \] The main result of the paper states that if, for all roots \(r\) of the characteristic equation \[ r^m=\sum_{j=0}^{m-1}r^j, \] the first-order equation \[ \varphi(f(x))=r\varphi(x)+F_0(x)\qquad (x\in S) \] is stable in the sense of Hyers and Ulam, then the \(m\)th-order equation is also stable in a similar sense. The main cases when the first-order equation is stable are also discussed. As an application, the solvability of the inhomogeneous \(m\)th-order equation is proved and an error-bound estimate is obtained for the distance from the solution of the corresponding homogeneous equation.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
39B12 Iteration theory, iterative and composite equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R. P. Agarwal, B. Xu and W. Zhang, Stability of functional equations in single variable, J. Math. Anal. Appl., 288 (2003), 852–869. · Zbl 1053.39042 · doi:10.1016/j.jmaa.2003.09.032
[2] J. A. Baker, The stability of certain functional equations, Proc. Amer. Math. Soc., 112 (1991), 729–732. · Zbl 0735.39004 · doi:10.1090/S0002-9939-1991-1052568-7
[3] D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57 (1951), 223–237. · Zbl 0043.32902 · doi:10.1090/S0002-9904-1951-09511-7
[4] D. Brydak, On the stability of the functional equation [f(x)] = g(x)(x) + F(x), Proc. Amer. Math. Soc., 26 (1970), 455–460. · Zbl 0209.15802
[5] M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucia, J. Pelant and V. Zizler, Functional Analysis and Infinite-Dimensional Geometry, Springer-Verlag (New York, 2001).
[6] G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math., 50 (1995), 143–190. · Zbl 0836.39007 · doi:10.1007/BF01831117
[7] R. Ger, A survey of recent results on stability of functional equations, in: Proc. of the 4th International Conference on Functional Equations and Inequalities, Pedagogical University of Cracow (Cracow, Poland, 1994), pp. 5–36.
[8] A. Gilányi, Hyers-Ulam stability of monomial functional equations on a general domain, Proc. Nat. Acad. Sci. USA, 96 (1999), 10588–10590. · Zbl 0961.39012 · doi:10.1073/pnas.96.19.10588
[9] P. M. Gruber, Stability of isometries, Trans. Amer. Math. Soc., 245 (1978), 263–277. · Zbl 0393.41020 · doi:10.1090/S0002-9947-1978-0511409-2
[10] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27 (1941), 222–224. · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[11] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser (1998). · Zbl 0907.39025
[12] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. I: Elementary Theory. Reprint of the 1983 original. American Mathematical Society (Providence, RI, 1997). · Zbl 0888.46039
[13] I. Kocsis and Gy. Maksa, The stability of a sum form functional equation arising in information theory, Acta Math. Hungar., 79 (1998), 39–48. · Zbl 0906.39016 · doi:10.1023/A:1006553604493
[14] M. Kuczma, Functional Equations in a Single Variable, Polish Scientific Publishers (Warszawa, 1968). · Zbl 0196.16403
[15] M. Kuczma, B. Choczewski and R. Ger, Iterative Functional Equations, Encyclopedia of Mathematics and its Applications, Cambridge University Press (1990). · Zbl 0703.39005
[16] Zs. Páles, Hyers-Ulam stability of the Cauchy functional equation on square-symmetric groupoids, Publ. Math. Debrecen, 58 (2001), 651–666. · Zbl 0980.39022
[17] L. Székelyhidi, Stability properties of functional equations describing the scientific laws, J. Math. Anal. Appl., 150 (1990), 151–158. · Zbl 0708.39006 · doi:10.1016/0022-247X(90)90202-Q
[18] L. Székelyhidi, The stability of the sine and cosine functional equations, Proc. Amer. Math. Soc., 110 (1990), 109–115. · Zbl 0718.39004
[19] T. Trif, On the stability of a general gamma-type functional equation, Publ. Math. Debrecen, 60 (2002), 47–61. · Zbl 1004.39023
[20] E. Turdza, On the stability of the functional equation [f(x)] = g(x)(x) + F(x), Proc. Amer. Math. Soc., 30 (1971), 484–486. · Zbl 0224.39005
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.