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Stabilities of Cubic Mappings in Various Normed Spaces: Direct and Fixed Point Methods

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  • H. Azadi Kenary
  • H. Rezaei
  • S. Talebzadeh
  • S. Jin Lee

Abstract

In 1940 and 1964, Ulam proposed the general problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?”. In 1941, Hyers solved this stability problem for linear mappings. According to Gruber (1978) this kind of stability problems are of the particular interest in probability theory and in the case of functional equations of different types. In 1981, Skof was the first author to solve the Ulam problem for quadratic mappings. In 1982–2011, J. M. Rassias solved the above Ulam problem for linear and nonlinear mappings and established analogous stability problems even on restricted domains. The purpose of this paper is the generalized Hyers‐Ulam stability for the following cubic functional equation: f(mx + y) + f(mx − y) = mf(x + y) + mf(x − y) + 2(m 3 − m)f(x), m ≥ 2 in various normed spaces.

Suggested Citation

  • H. Azadi Kenary & H. Rezaei & S. Talebzadeh & S. Jin Lee, 2012. "Stabilities of Cubic Mappings in Various Normed Spaces: Direct and Fixed Point Methods," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnljam:v:2012:y:2012:i:1:n:546819
    DOI: 10.1155/2012/546819
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    References listed on IDEAS

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    1. 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.
    2. 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).
    3. 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.
    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.
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