Some hyperstability results of a p-radical functional equation related to quartic mappings in non-archimedean Banach spaces

Authors

DOI:

https://doi.org/10.22199/issn.0717-6279-3704

Keywords:

Hhyperstability, Non-archimedean Banach spaces, Radical functional equations, Quartic functional equations

Abstract

The aim of this paper is to introduce and solve the following p-radical functional equation related to quartic mappings

where f is a mapping from R into a vector space X and p ≥ 3 is an odd natural number. Using an analogue version of Brzd¸ek’s fixed point theorem [13], we establish some hyperstability results for the considered equation in non-Archimedean Banach spaces. Also, we give some hyperstability results for the inhomogeneous p-radical functional equation related to quartic mapping.

Author Biographies

Ahmed Nuino, Ibn Tofaïl University.

Dept. of Mathematics

Mustapha Esseghyr Hryrou, Ibn Tofaïl University.

Dept. of Mathematics

Samir Kabbaj, Ibn Tofaïl University.

Dept. of Mathematics

References

L. Aiemsomboon and W. Sintunavarat, “On a new type of stability of a radical quadratic functional equation using Brzdȩk’s fixed point theorem”, Acta Mathematica Hungarica, vol. 151, no. 1, pp. 35–46, 2016, https://doi.org/10.1007/s10474-016-0666-2

L. Aiemsomboon and W. Sintunavarat, “On generalized hyperstability of a general linear equation”, Acta Mathematica Hungarica, vol. 149, no.2, pp. 413- 422, 2016, https://doi.org/10.1007/s10474-016-0621-2

M. Almahalebi, “Stability of a generalization of Cauchy’s and the quadratic functional equations”, Journal of Fixed Point Theory and Applications, vol. 20, no. 12, 2018. https://doi.org/10.1007/s11784-018-0503-z

M. Almahalebi, A. Charifi, and S. Kabbaj, “Hyperstability of a Cauchy functional equation”, Journal of Nonlinear Analysis and Optimization: Theory Applications, vol. 6, no. 2, pp. 127-137, 2015.

M. Almahalebi and C. Park, “On the hyperstability of a functional equation in commutative groups”, Journal of Computational Analysis Applications, vol. 20, no. 1, pp. 826-833, 2016.

M. Almahalebi and A. Chahbi, “Hyperstability of the Jensen functional equation in ultrametric spaces”, Aequationes mathematicae, vol. 91, no. 4, pp. 647661, 2017.

M. Almahalebi and A. Chahbi, “Approximate solution of p-radical functional equation in 2-Banach spaces”, Acta Mathematica Scientia, vol. 39, no. 2, pp. 551-566, 2019.

Z. Alizadeh and A. G. Ghazanfari, “On the stability of a radical cubic functional equation in quasi-β-spaces”, Journal of Fixed Point Theory and Applications, vol. 18, Art ID. 843, 2016. https://doi.org/10.1007/s11784-016-0317-9

T. Aoki, “On the stability of the linear transformation in Banach spaces”, Journal of Mathematical Society of Japan, vol. 2, pp. 64-66, 1950.

D. G. Bourgin, “Classes of transformations and bordering transformations”, Bulletin of the American Mathematical Society, vol. 57, pp. 223-237, 1951.

N. Brillouët-Belluot, J. Brzdȩk, and K. Ciepliński, “On some recent developments in Ulam’s type stability”, Abstract and Applied Analysis, vol. 2012, Art. ID 716936, 2012, https://doi.org/10.1155/2012/716936

J. Brzdȩk and J. Tabor, “A note on stability of additive mappings”, in Stability of mappings of Hyers-Ulam type, T. M. Rassias, Ed. Palm Harbor, FL: Hadronic Press, 1994, pp. 19–22.

J. Brzdȩk, J. Chudziak and Z. Páles, “A fixed point approach to stability of functional equations”, Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no.17, pp. 6728-6732, 2011.

J. Brzdȩk and K. Ciepliński, “A fixed point approach to the stability of functional equations in non-Archimedean metric spaces”, Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 18, pp. 6861-6867, 2011.

J. Brzdȩk, “Stability of additivity and fixed point methods”, Fixed Point Theory and Applications volume, vol. 2013, Art. ID. 265, 2013, https://doi.org/10.1186/1687-1812-2013-285

J. Brzdȩk, “Hyperstability of the Cauchy equation on restricted domains”, Acta Mathematica Hungarica, vol. 141, pp. 58-67, 2013.

J. Brzdȩk, L. Cădariu, and K. Ciepliński, “Fixed point theory and the Ulam stability”, Journal of Function Spaces, Vol. 2014, Art. ID 829419, 2014. https://doi.org/10.1155/2014/829419

J. Brzdȩk, W. Fechner, M. S. Moslehian and J. Sikorska, “Recent developments of the conditional stability of the homomorphism equation”, Banach Journal of the Mathematical Analysis, vol. 9, no. 3, pp. 278-327, 2015.

J. Brzdȩk, “Remark 3”, Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica, vol. 14, p. 196, 2015.

J. Brzdȩk, “Remarks on solutions to the functional equations of the radical type”, Advances in the Theory of Nonlinear Analysis and its Application, vol. 1, pp. 125-135, 2017.

M. Eshaghi Gordji and M. Parviz, “On the HyersUlam stability of the functional equation f (2√x2 + y2) = f(x) + f(y)”, Nonlinear functional analysis and applications, vol. 14, pp. 413-420, 2009.

M. Eshaghi Gordji, H. Khodaei, A. Ebadian and G. H. Kim, “Nearly radical quadratic functional equations in p-2-normed spaces”, Abstract and Applied Analysis, vol 2012, Art. ID 896032, 2012, https://doi.org/10.1155/2012/896032

S. Gähler, “2-metrische Räume und ihre topologische Struktur”, Mathematische Nachrichten, vol. 26, pp. 115-148, 1963.

S. Gähler, “Linear 2-normiete Räumen”, Mathematische Nachrichten, vol. 28, pp. 1-43, 1964.

P. Gǎvruţa, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings”, Journal of Mathematical Analysis and Applications, vol. 184, pp. 431-436, 1994.

D. H. Hyers, “On the stability of the linear functional equation”, Proceedings of the National Academy of Sciences, vol. 27, no. 4, pp. 222-224, 1941.

H. Khodaei, M. Eshaghi Gordji, S. S. Kim, and Y. J. Cho, “Approximation of radical functional equations related to quadratic and quartic mappings”, Journal of Mathematical Analysis and Applications, vol. 395, pp. 284-297, 2012.

S. S. Kim, Y. J. Cho, and M. Eshaghi Gordji, “On the generalized HyersUlam-Rassias stability problem of radical functional equations”, Journal of Inequalities and Applications, vol. 2012, Art. ID. 186, 2012, https://doi.org/10.1186/1029-242X-2012-186

S. H. Lee, S. M. Im, and I. S. Hwang, “Quartic functional equations”, Journal of Mathematical Analysis and Applications, vol. 307, no. 2, 387-394, 2005.

G. Maksa and Z. Pales, “Hyperstability of a class of linear functional equations”, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, vol. 17, no. 2, pp. 107-112, 2001.

W. -G. Park, “Approximate additive mappings in 2-Banach spaces and related topics”, Journal of Mathematical Analysis and Applications, vol. 376, no. 1, pp. 193-202, 2011.

“16th International Conference on Functional Equations and Inequalities, Będlewo, Poland, May 17-23, 2015 Report of Meeting”, Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica, vol. 14, pp. 163–202, 2015.

T. M. Rassias, “On the stability of the linear mapping in Banach spaces”, Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978.

T. M. Rassias, “Problem 16; 2”, Aequationes mathematicae, vol. 39, pp. 292-293, 1990.

T. M. Rassias, “On a modified Hyers-Ulam sequence”, Journal of Mathematical Analysis and Applications, vol. 158, pp. 106-113, 1991.

S. M. Ulam, Problems in Modern Mathematics. New York (NY): John-Wiley & Sons Inc., 1964.

Published

2021-06-16

How to Cite

[1]
A. Nuino, M. Esseghyr Hryrou, and S. Kabbaj, “Some hyperstability results of a p-radical functional equation related to quartic mappings in non-archimedean Banach spaces”, Proyecciones (Antofagasta, On line), vol. 40, no. 5, pp. 1155-1177, Jun. 2021.

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