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Stability of a quartic functional equation

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Abstract

In this paper, the authors investigate the general solution and generalized Hyers–Ulam stability of the n-dimensional quartic functional equation of the form

$$\begin{aligned} f\left( \sum _{i=1}^{n}x_i\right)&= \sum _{1 \le i<j< k< l\le n} f\left( x_i+x_j+x_k+x_l\right) +\left( -n+4\right) \nonumber \\ {}&\sum _{1 \le i< j< k \le n} f\left( x_i+x_j+x_k\right) +\left( \frac{n^2-7n+12}{2}\right) \sum _{ \begin{array}{c} 1=i;\\ i\ne j \end{array}}^{n} f\left( x_i+x_j\right) \nonumber \\&- \sum _{i=1}^{n} f\left( 2x_i\right) + \left( \frac{-n^3+9n^2-26n+120}{6}\right) \ \ \sum _{i=1}^{n}\left( \frac{f(x_i)+f(-x_i)}{2}\right) \end{aligned}$$

where n is a positive integer with \({\mathbb {N}}- \{0,1,2,3,4\}\). The stability of this quartic functional equation is introduced in Banach space using direct and fixed point methods.

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Acknowledgements

This work was completed with the support of our TeX-pert. The publication was supported by the Ministry of Education and Science of the Russian Federation (the agreement number N.02.a03.21.0008).

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Authors and Affiliations

  1. RUDN University, 6 Miklukho-Maklaya Str., 117198, Moscow, Russia

    Sandra Pinelas

  2. Sri Vidya Mandir Arts and Science College, Katteri, Uthangarai, Tamilnadu, 636902, India

    V. Govindan & K. Tamilvanan

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  1. Sandra Pinelas
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  2. V. Govindan
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  3. K. Tamilvanan
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Correspondence to Sandra Pinelas.

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Pinelas, S., Govindan, V. & Tamilvanan, K. Stability of a quartic functional equation. J. Fixed Point Theory Appl. 20, 148 (2018). https://doi.org/10.1007/s11784-018-0629-z

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  • DOI: https://doi.org/10.1007/s11784-018-0629-z

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