## General solution and stability of Quattuorvigintic functional equation in matrix paranormed spaces.(English)Zbl 1434.39022

The authors consider the following functional equation: \begin{multline*} Df(v,w) = f(v+12w)-24f(v+11w)+276f(v+10w)-2024f(v+9w)\\ + 10626f(v+8w)-42504f(v+7w)+134596f(v+6w)-346104f(v+5w)\\ + 735471f(v+4w)-1307504f(v+3w)+1961256f(v+2w)-2496144f(v+w)\\ + 2704156f(v)-2496144f(v-w)+1961256f(v-2w)-1307504f(v-3w)\\ + 735471f(v-4w)+134596f(v-6w)-42504f(v-7w)+10626f(v-8w)\\ - 2024f(v-9w)+276f(v-10w)-346104f(v-5w)-24f(v-11w)\\ + f(v-12w) -1.124000728-10^{21}f(w), \end{multline*} which is called quattuorvigintic functional equation since the function $$f(x)=cx^{24}$$ is its solution. They provide the general solution of this functional equation and prove its Hyers-Ulam stability in matrix paranormed spaces by using a fixed point approach and a direct method. The motivation behind the above functional equation is not clear from the text.

### MSC:

 39B52 Functional equations for functions with more general domains and/or ranges 39B82 Stability, separation, extension, and related topics for functional equations 47H10 Fixed-point theorems
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### References:

  T. Aoki, On the Stability of the Linear Transformation in Banach Spaces, J. Math. Soc. Japan, 2 (1950), 64-66. · Zbl 0040.35501  L. Cadariu and V. Radu, Fixed Points and the Stability of Jensen’s Functional Equation.,J. Inequal. Pure Appl. Math., 4(1) (2003), 1-7. · Zbl 1043.39010  P. Gavruta, A Generalization of the Hyers-Ulam Rassias Stability of Approximately Additive Mappings, J. Math. Anal. Appl., 184 (1994), 431-436. · Zbl 0818.46043  D. H. Hyers, On the Stability of the Linear Functional Equation, Proc. Natl. Acad. Sci. USA, 27 (1941), 222-224. · JFM 67.0424.01  G. Isac and Th. M. Rassias, Stability of $$\varphi$$ -Additive Mappings: Applications to Nonlinear Analysis, J. Funct. Anal., 19 (1996), 219-228. · Zbl 0843.47036  J. Lee, D. Shin and C. Park, Functional Equations and Inequalities in Matrix Paranormed Spaces, Journal of Inequalities and Applications,2013:547, (2013), 1-13. · Zbl 06313669  D. Mihet and V. Radu, On the Stability of the Additive Cauchy Functional Equation in Random Normed Spaces, J. Math. Anal. Appl., 343 (2008), 567-572. · Zbl 1139.39040  R. Murali, A. Antony Raj and M. Boobalan, Hyers-Ulam-Rassias Stability of Quartic Functional Equations in Matrix Paranormed Spaces, International Journal of Mathematics and Applications, 4(I-B) (2016), 81-86.  C. Park, Stability of an AQCQ-Functional Equation in Paranormed Spaces, Advances in Difference Equations., 2012:148, 1-20. · Zbl 1346.39035  Th. M. Rassias, On the Stability of the Linear Mapping in Banach Spaces, Proc. Am. Math. Soc., 72 (1978), 297-300. · Zbl 0398.47040  K. Ravi, J.M. Rassias and B.V. Senthil Kumar, Stability of Reciprocal Difference and Adjont Functional Equations in Paranormed Spaces: Direct and Fixed Point Methods, Tbilisi Mathematical Science, 5:1 (2013), 57-72. · Zbl 1352.39023  S. M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, NewYork (1964). · Zbl 0137.24201
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