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Generalized Additive Cauchy Equations

In: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis

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  • Soon-Mo Jung

    (Hongik University)

Abstract

It is very natural for one to try to transform the additive Cauchy equation into other forms. Some typically generalized additive Cauchy equations will be introduced. The functional equation $$f(ax+by)=af(x)+bf(y)$$ appears in Section 3.1. The Hyers–Ulam stability problem is discussed in connection with a question of Th. M. Rassias and J. Tabor. In Section 3.2, the functional equation (3.3) is introduced, and the Hyers–Ulam–Rassias stability for this equation is also studied. The stability result for this equation will be used to answer the question of Rassias and Tabor cited above. The last section deals with the functional equation $$f(x+y)^2=(f(x)+f(y))^2$$ . The continuous solutions and the Hyers–Ulam stability for this functional equation will be investigated.

Suggested Citation

  • Soon-Mo Jung, 2011. "Generalized Additive Cauchy Equations," Springer Optimization and Its Applications, in: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, chapter 0, pages 87-103, Springer.
    • Handle: RePEc:spr:spochp:978-1-4419-9637-4_3
    DOI: 10.1007/978-1-4419-9637-4_3
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