Note on nonstability of the linear functional equation of higher order
We provide a complete solution of the problem of Hyers–Ulam stability for a large class of higher order linear functional equations in single variable, with constant coefficients. We obtain this by showing that such an equation is nonstable in the case where at least one of the roots of the characte...
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| Vydáno v: | Computers & mathematics with applications (1987) Ročník 62; číslo 6; s. 2648 - 2657 |
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| Jazyk: | angličtina |
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Elsevier Ltd
01.09.2011
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| ISSN: | 0898-1221, 1873-7668 |
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| Abstract | We provide a complete solution of the problem of Hyers–Ulam stability for a large class of higher order linear functional equations in single variable, with constant coefficients. We obtain this by showing that such an equation is nonstable in the case where at least one of the roots of the characteristic equation is of module 1. Our results are related to the notions of shadowing (in dynamical systems and computer science) and controlled chaos. They also correspond to some earlier results on approximate solutions of functional equations in single variable. |
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| AbstractList | We provide a complete solution of the problem of Hyers-Ulam stability for a large class of higher order linear functional equations in single variable, with constant coefficients. We obtain this by showing that such an equation is nonstable in the case where at least one of the roots of the characteristic equation is of module 1. Our results are related to the notions of shadowing (in dynamical systems and computer science) and controlled chaos. They also correspond to some earlier results on approximate solutions of functional equations in single variable. |
| Author | Brzdȩk, Janusz Popa, Dorian Xu, Bing |
| Author_xml | – sequence: 1 givenname: Janusz surname: Brzdȩk fullname: Brzdȩk, Janusz email: jbrzdek@ap.krakow.pl organization: Department of Mathematics, Pedagogical University, Podchora̧żych 2, PL-30-084 Kraków, Poland – sequence: 2 givenname: Dorian surname: Popa fullname: Popa, Dorian email: Popa.Dorian@math.utcluj.ro organization: Department of Mathematics, Technical University, Str. C. Daicoviciu 15, Cluj-Napoca, 400020, Romania – sequence: 3 givenname: Bing surname: Xu fullname: Xu, Bing email: bxu@scu.edu.cn, xb0408@yahoo.com.cn organization: Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China |
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| Cites_doi | 10.4064/ap-43-3-305-316 10.1016/0021-9045(72)90016-0 10.1016/0021-9045(73)90101-9 10.1016/j.jmaa.2003.09.032 10.1007/BF02189389 10.1016/j.apnum.2004.08.011 10.1155/2009/181678 10.1016/j.jmaa.2004.10.013 10.1007/s000100050159 10.1007/s00013-003-0471-3 10.1007/BF02960864 10.1007/s10474-007-7069-3 10.4064/ap-49-3-247-252 10.5486/PMD.2002.2522 10.1007/s00010-008-2945-7 10.1016/j.chaos.2007.07.037 |
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| Keywords | Characteristic root Normed space Linear equation Nonstability Hyers–Ulam stability |
| Language | English |
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Math. doi: 10.4064/ap-49-3-247-252 – volume: 60 start-page: 47 year: 2002 ident: 10.1016/j.camwa.2011.08.007_br000080 article-title: On the stability of a general gamma-type functional equation publication-title: Publ. Math. Debrecen doi: 10.5486/PMD.2002.2522 – year: 1968 ident: 10.1016/j.camwa.2011.08.007_br000005 – volume: 77 start-page: 33 year: 2009 ident: 10.1016/j.camwa.2011.08.007_br000100 article-title: On the stability of functional equations publication-title: Aequationes Math. doi: 10.1007/s00010-008-2945-7 – volume: 35 start-page: 238 year: 2008 ident: 10.1016/j.camwa.2011.08.007_br000030 article-title: Bounded solutions of a class of difference equations in Banach spaces producing controlled Chaos publication-title: Chaos Solitons Fractals doi: 10.1016/j.chaos.2007.07.037 |
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| SubjectTerms | Approximation Chaos theory Characteristic root Dynamical systems Hyers–Ulam stability Linear equation Mathematical analysis Mathematical models Modules Nonstability Normed space Roots Stability |
| Title | Note on nonstability of the linear functional equation of higher order |
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