Stability of a Mixed Type Additive, Quadratic, Cubic and Quartic Functional Equation

  • M. Eshaghi-Gordji
  • S. Kaboli-Gharetapeh
  • M.S. Moslehian
  • S. Zolfaghari
Part of the Springer Optimization and Its Applications book series (SOIA, volume 35)


We find the general solution of the functional equation
$$\begin{array}{l} D_f {\rm{(}}x,y{\rm{)}}\,\,{\rm{: = }}f{\rm{(}}x + {\rm{2}}y{\rm{)}} + f{\rm{(}}x - {\rm{2}}y{\rm{)}} - {\rm{4[}}f{\rm{(}}x + y{\rm{)}} - f{\rm{(}}x - y{\rm{)]}} - f{\rm{(4}}y{\rm{)}} + {\rm{4}}f{\rm{(3}}y{\rm{)}} \\ - {\rm{6}}f{\rm{(2}}y{\rm{)}} + {\rm{4}}f{\rm{(}}y{\rm{)}} + {\rm{6}}f{\rm{(}}x{\rm{) }} = {\rm{ 0}}{\rm{.}} \\ \end{array}$$
in the context of linear spaces. We prove that if a mapping f from a linear space X into a Banach space Y satisfies f(0)=0 and
$$\|D_f(x,y)\|\leq\epsilon \quad (x,y\in X),$$
where ε > 0, then there exist a unique additive mapping \(A:X\to Y,\) a unique quadratic mapping \(Q_1:X\to Y,\) a unique cubic mapping \(C:X\to Y\) and a unique quartic mapping \(Q_2:X\to Y\) such that
$$\|f(x)-A(x)-Q_1(x)-C(x)-Q_2(x)\|\leq\frac{1087 \epsilon}{140}\quad \forall x\in X.$$


Banach Space Functional Equation Positive Real Number Real Vector Space Ulam Stability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • M. Eshaghi-Gordji
    • 1
  • S. Kaboli-Gharetapeh
    • 2
  • M.S. Moslehian
    • 3
  • S. Zolfaghari
    • 1
  1. 1.Department of MathematicsSemnan UniversitySemnanIran
  2. 2.Department of MathematicsPayame Noor University of MashhadMashhadIran
  3. 3.Department of Pure Mathematics and Center of Excellence in Analysis on Algebraic Structures (CEAAS)Ferdowsi University of MashhadMashhadIran

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