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

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Book cover Nonlinear Analysis and Variational Problems

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 35))

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

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

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Author information

Authors and Affiliations

  1. Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran

    M. Eshaghi-Gordji & S. Zolfaghari

  2. Department of Mathematics, Payame Noor University of Mashhad, Mashhad, Iran

    S. Kaboli-Gharetapeh

  3. Department of Pure Mathematics and Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P.O. Box 1159, 91775, Mashhad, Iran

    M.S. Moslehian

Authors
  1. M. Eshaghi-Gordji
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  2. S. Kaboli-Gharetapeh
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  3. M.S. Moslehian
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  4. S. Zolfaghari
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Corresponding author

Correspondence to M. Eshaghi-Gordji .

Editor information

Editors and Affiliations

  1. Dept. Industrial & Systems, Engineering, University of Florida, Weil Hall 303, Gainesville, 32611-6595, Florida, USA

    Panos M. Pardalos

  2. Zagoras Street 4, Athens, 151 25, Greece

    Themistocles M. Rassias

  3. School of Mathematical Sciences, Rochester Institute of Technology, Lomb Memorial Drive 85, Rochester, 14623-5602, USA

    Akhtar A. Khan

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Dedicated to the memory of Professor George Isac

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Eshaghi-Gordji, M., Kaboli-Gharetapeh, S., Moslehian, M., Zolfaghari, S. (2010). Stability of a Mixed Type Additive, Quadratic, Cubic and Quartic Functional Equation. In: Pardalos, P., Rassias, T., Khan, A. (eds) Nonlinear Analysis and Variational Problems. Springer Optimization and Its Applications, vol 35. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0158-3_6

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