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

In: Nonlinear Analysis and Variational Problems

Author

Listed:
  • M. Eshaghi-Gordji

    (Semnan University)

  • S. Kaboli-Gharetapeh

    (Payame Noor University of Mashhad)

  • M.S. Moslehian

    (Ferdowsi University of Mashhad)

  • S. Zolfaghari

    (Semnan University)

Abstract

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 $$\|f(x)-A(x)-Q_1(x)-C(x)-Q_2(x)\|\leq\frac{1087 \epsilon}{140}\quad \forall x\in X.$$

Suggested Citation

  • M. Eshaghi-Gordji & S. Kaboli-Gharetapeh & M.S. Moslehian & S. Zolfaghari, 2010. "Stability of a Mixed Type Additive, Quadratic, Cubic and Quartic Functional Equation," Springer Optimization and Its Applications, in: Panos M. Pardalos & Themistocles M. Rassias & Akhtar A. Khan (ed.), Nonlinear Analysis and Variational Problems, chapter 0, pages 65-80, Springer.
    • Handle: RePEc:spr:spochp:978-1-4419-0158-3_6
    DOI: 10.1007/978-1-4419-0158-3_6
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