Aequationes mathematicae

, Volume 73, Issue 3, pp 249–259 | Cite as

Orthogonal stability of additive type equations

  • Mohammad Sal Moslehian
  • Themistocles M. Rassias


Suppose that (\(\mathcal{X}, \bot\)) is a symmetric orthogonality module and \({\mathcal{Y}}\) a Banach module over a unital Banach algebra \({\mathcal{A}}\) and \(f : \mathcal{X} \rightarrow {\mathcal{Y}}\) is a mapping satisfying
$$\|f(ax_{1} + ax_{2}) + (-1)^{k+1}f(ax_{1} - ax_{2}) - 2af(x_{k})\| \leq \epsilon$$
, for k = 1 or 2, for some ε ≥ 0, for all a in the unit sphere \({\mathcal{A}}_{1}\) of \({\mathcal{A}}\) and all \(x_{1}, x_{2} \in \mathcal{X}\) with \(x_{1} \bot x_{2}\). Assume that the mapping \(t \mapsto f(tx)\) is continuous for each fixed \(x \in \mathcal{X}\) . Then there exists a unique \({\mathcal{A}}\) -linear mapping \(T : \mathcal{X} \rightarrow {\mathcal{Y}}\) satisfying \(T(ax) = aT(x), a \in \mathcal{A}, x \in \mathcal{X}\) such that
$$\|f(x) - f(0) - T(x)\|\leq\frac{5}{2}\epsilon$$
, for all \(x \in \mathcal{X}\).

Mathematics Subject Classification (2000).

Primary 39B55 secondary 46L05, 39B52, 46H25, 39B82 


Hyers-Ulam stability orthogonality orthogonally additive mapping orthogonally quadratic mapping orthogonally additive type equation orthogonality space Banach module C*-algebra 


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Copyright information

© Birkhäuser Verlag, Basel 2007

Authors and Affiliations

  • Mohammad Sal Moslehian
    • 1
    • 2
  • Themistocles M. Rassias
    • 3
  1. 1.Department of MathematicsFerdowsi UniversityMashhadIran
  2. 2.Centre of Excellence in Analysis on Algebraic Structures (CEAAS)Ferdowsi UniversityFerdowsiIran
  3. 3.Department of MathematicsNational Technical University of AthensAthensGreece

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