Abstract
In this paper, we unify the system of functional equations defining a multi-Jensen-quadratic mapping to a single equation. Using a fixed point theorem, we study the generalized Hyers–Ulam stability of such equation and thus generalizing some known results. As a result, we show that the multi-Jensen-quadratic functional equation is hyperstable.
Similar content being viewed by others
References
Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)
Bahyrycz, A.: On stability and hyperstability of an equation characterizing multi-additive mappings. Fixed Point Theory 18(2), 445–456 (2017)
Bahyrycz, A., Olko, J.: On stability and hyperstability of an equation characterizing multi-Cauchy–Jensen mappings. Results Math. 73, 55 (2018). https://doi.org/10.1007/s00025-018-0815-8
Bahyrycz, A., Ciepliński, K., Olko, J.: On an equation characterizing multi Cauchy-Jensen mappings and its Hyers-Ulam stability. Acta Math. Sci. Ser. B Engl. Ed. 35, 1349–1358 (2015)
Bahyrycz, A., Ciepliński, K., Olko, J.: On an equation characterizing multi-additive-quadratic mappings and its Hyers-Ulam stability. Appl. Math. Comput. 265, 448–455 (2015)
Bodaghi, A., Park, C., Mewomo, O.T.: Multiquartic functional equations. Adv. Differ. Equations 2019, 312 (2019). https://doi.org/10.1186/s13662-019-2255-5
Bodaghi, A., Shojaee, B.: On an equation characterizing multi-cubic mappings and its stability and hyperstability. Fixed Point Theory (2019). arXiv:1907.09378v2(to appear)
Brzdȩk, J.: A hyperstability result for the Cauchy equation. Bull. Aust. Math. Soc. 89, 33–40 (2014)
Brzdȩk, J.: Hyperstability of the Cauchy equation on restricted domains. Acta Math. Hung. 141, 58–67 (2013)
Brzdȩk, J.: Stability of the equation of the p-Wright affine functions. Aequat. Math. 85, 497–503 (2013)
Brzdȩk, J., Ciepliński, K.: Hyperstability and superstability. Abstr. Appl. Anal., 13 (2013) (Article ID 401756). https://doi.org/10.1155/2013/401756
Brzdȩk, J., Ciepliński, K.: Remarks on the Hyers-Ulam stability of some systems of functional equations. Appl. Math. Comput. 219, 4096–4105 (2012)
Brzdȩk, J., Chudziak, J., Palés, Zs.: A fixed point approach to the stability of functional equations. Nonlinear Anal. 74, 6728–6732 (2011)
Brzdȩk, J., Ciepliński, K.: On a fixed point theorem in \(2\)-Banach spaces and some of its applications. Acta Math. Sci. Ser. B Engl. Ed. 38, 377–390 (2018)
Ciepliński, K.: On the generalized Hyers-Ulam stability of multi-quadratic mappings. Comput. Math. Appl. 62, 3418–3426 (2011)
Ciepliński, K.: Generalized stability of multi-additive mappings. Appl. Math. Lett. 23, 1291–1294 (2010)
Ciepliński, K.: Stability of the multi-Jensen equation. J. Math. Anal. Appl. 363, 249–254 (2010)
Ciepliński, K.: On multi-Jensen functions and Jensen difference. Bull. Korean Math. Soc. 45(4), 729–737 (2008)
Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci., USA 27, 222–224 (1941)
Jun, K.-W., Lee, Y.-H.: A generalization of the Hyers–Ulam–Rassias stability of Jensen’s equation. J. Math. Anal. Appl. 238(1), 305–315 (1999)
Jung, S.-M.: Hyers-Ulam-Rassias stability of Jensen’s equation and its application. Proc. Am. Math. Soc. 126(11), 3137–3143 (1998)
Kominek, Z.: On a local stability of the Jensen functional equation. Demonstr. Math. 22(2), 499–507 (1989)
Prager, W., Schwaiger, J.: Stability of the multi-Jensen equation. Bull. Korean Math. Soc. 45(1), 133–142 (2008)
Prager, W., Schwaiger, J.: Multi-affne and multi-Jensen functions and their connection with generalized polynomials. Aequ. Math. 69(1–2), 41–57 (2005)
Rassias, J.M.: Solution of the Ulam stability problem for cubic mappings. Glasnik Mat. Serija III. 36(1), 63–72 (2001)
Rassias, T.M.: On the stability of the linear mapping in Banach Space. Proc. Am. Math. Soc. 72(2), 297–300 (1978)
Ulam, S.M.: Problems in Modern Mathematics. Science Editions, Wiley, New York (1964)
Zhao, X., Yang, X., Pang, C.-T.: Solution and stability of the multiquadratic functional equation. Abstr. Appl. Anal., 8 (2013) (Art. ID 415053). https://doi.org/10.1155/2013/415053
Xu, T.Z.: Stability of multi-Jensen mappings in non-Archimedean normed spaces. J. Math. Phys. 53, 023507 (2012). https://doi.org/10.1063/1.368474
Acknowledgements
The authors sincerely thank the anonymous reviewer for his/her careful reading, constructive comments and suggesting some related references to improve the quality of the first draft. They also would like to thank Dr. Sang Og Kim for pointing out the result in Lemma 3.1 is not correct for \(k=n\).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Salimi, S., Bodaghi, A. A fixed point application for the stability and hyperstability of multi-Jensen-quadratic mappings. J. Fixed Point Theory Appl. 22, 9 (2020). https://doi.org/10.1007/s11784-019-0738-3
Published:
DOI: https://doi.org/10.1007/s11784-019-0738-3