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Stability and Superstability of Generalized (θ, ϕ)‐Derivations in Non‐Archimedean Algebras: Fixed Point Theorem via the Additive Cauchy Functional Equation

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  • M. Eshaghi Gordji
  • M. B. Ghaemi
  • G. H. Kim
  • Badrkhan Alizadeh

Abstract

Let A be an algebra, and let θ, ϕ be ring automorphisms of A. An additive mapping H : A → A is called a (θ, ϕ)‐derivation if H(xy) = H(x)θ(y) + ϕ(x)H(y) for all x, y ∈ A. Moreover, an additive mapping F : A → A is said to be a generalized (θ, ϕ)‐derivation if there exists a (θ, ϕ)‐derivation H : A → A such that F(xy) = F(x)θ(y) + ϕ(x)H(y) for all x, y ∈ A. In this paper, we investigate the superstability of generalized (θ, ϕ)‐derivations in non‐Archimedean algebras by using a version of fixed point theorem via Cauchy’s functional equation.

Suggested Citation

  • M. Eshaghi Gordji & M. B. Ghaemi & G. H. Kim & Badrkhan Alizadeh, 2011. "Stability and Superstability of Generalized (θ, ϕ)‐Derivations in Non‐Archimedean Algebras: Fixed Point Theorem via the Additive Cauchy Functional Equation," Journal of Applied Mathematics, John Wiley & Sons, vol. 2011(1).
    • Handle: RePEc:wly:jnljam:v:2011:y:2011:i:1:n:726020
    DOI: 10.1155/2011/726020
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    References listed on IDEAS

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    1. M. Eshaghi Gordji & A. Ebadian & S. Zolfaghari, 2008. "Stability of a Functional Equation Deriving from Cubic and Quartic Functions," Abstract and Applied Analysis, Hindawi, vol. 2008, pages 1-17, February.
    2. M. Eshaghi Gordji & M. B. Ghaemi & Badrkhan Alizadeh, 2011. "A Fixed Point Approach to Superstability of Generalized Derivations on Non-Archimedean Banach Algebras," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-9, September.
    3. Zbigniew Gajda, 1991. "On stability of additive mappings," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 14, pages 1-4, January.
    4. Paisan Nakmahachalasint, 2007. "Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias Stabilities of an Additive Functional Equation in Several Variables," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2007, pages 1-6, July.
    5. M. Eshaghi Gordji & M. B. Ghaemi & Badrkhan Alizadeh, 2011. "A Fixed Point Approach to Superstability of Generalized Derivations on Non‐Archimedean Banach Algebras," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    6. M. Eshaghi Gordji & S. Zolfaghari & J. M. Rassias & M. B. Savadkouhi, 2009. "Solution and Stability of a Mixed Type Cubic and Quartic Functional Equation in Quasi-Banach Spaces," Abstract and Applied Analysis, Hindawi, vol. 2009, pages 1-14, August.
    7. M. Eshaghi Gordji & S. Zolfaghari & J. M. Rassias & M. B. Savadkouhi, 2009. "Solution and Stability of a Mixed Type Cubic and Quartic Functional Equation in Quasi‐Banach Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2009(1).
    8. Paisan Nakmahachalasint, 2007. "On the Generalized Ulam-Gavruta-Rassias Stability of Mixed-Type Linear and Euler-Lagrange-Rassias Functional Equations," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2007, pages 1-10, May.
    9. M. Eshaghi Gordji & A. Ebadian & S. Zolfaghari, 2008. "Stability of a Functional Equation Deriving from Cubic and Quartic Functions," Abstract and Applied Analysis, John Wiley & Sons, vol. 2008(1).
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