On Ulam stability of generalized Hosszú functional equation

Salman Alsaeed; El-sayed El-hady; Janusz Brzdęk; Salman Alsaeed; El-sayed El-hady; Janusz Brzdęk

doi:10.3934/math.2026466

AIMS Mathematics

2026, Volume 11, Issue 4: 11332-11346. doi: 10.3934/math.2026466

Previous Article Next Article

Research article Special Issues

On Ulam stability of generalized Hosszú functional equation

1.
Department of Mathematics, College of Science, Jouf University, Sakaka, P.O. Box 2014, Saudi Arabia
2.
Faculty of Applied Mathematics, AGH University of Kraków, Mickiewicza 30, 30-059 Kraków, Poland

Received: 26 January 2026 Revised: 25 March 2026 Accepted: 07 April 2026 Published: 22 April 2026
MSC : 39B52, 39B82, 47A30

We investigated Hyers-Ulam stability (H-US) of the following generalization of the Hosszú equation (HFE): $ \Upsilon(x_1 + x_2 - \alpha x_1x_2) + \Upsilon(\alpha x_1x_2) = \Upsilon(x_1) +\Upsilon(x_2) $, in the class of maps $ \Upsilon $ from a quadratically closed field $ \mathbb{K} $ into a linear space, where $ \alpha\in \mathbb{K} $ is fixed. We considered this stability in cases where the linear space is equipped with either the $ m $-norm or the classical norm. In this way, we extended some earlier stability outcomes obtained for maps from the set of reals $ \mathbb{R} $ into a Banach space. We also proved some auxiliary stability results for the Cauchy additive equation $ \psi(x+y) = \psi(x)+\psi(y) $ in $ m $-Banach spaces. Finally, we discussed the symmetry issues that can be observed in these results.
- functional equations,
- Ulam stability,
- Hosszúequation,
- $ m $-norm
Citation: Salman Alsaeed, El-sayed El-hady, Janusz Brzdęk. On Ulam stability of generalized Hosszú functional equation[J]. AIMS Mathematics, 2026, 11(4): 11332-11346. doi: 10.3934/math.2026466

Related Papers:

Abstract

We investigated Hyers-Ulam stability (H-US) of the following generalization of the Hosszú equation (HFE): $ \Upsilon(x_1 + x_2 - \alpha x_1x_2) + \Upsilon(\alpha x_1x_2) = \Upsilon(x_1) +\Upsilon(x_2) $, in the class of maps $ \Upsilon $ from a quadratically closed field $ \mathbb{K} $ into a linear space, where $ \alpha\in \mathbb{K} $ is fixed. We considered this stability in cases where the linear space is equipped with either the $ m $-norm or the classical norm. In this way, we extended some earlier stability outcomes obtained for maps from the set of reals $ \mathbb{R} $ into a Banach space. We also proved some auxiliary stability results for the Cauchy additive equation $ \psi(x+y) = \psi(x)+\psi(y) $ in $ m $-Banach spaces. Finally, we discussed the symmetry issues that can be observed in these results.

References

[1]	D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
[2]	D. H. Hyers, G. Isac, T. M. Rassias, Stability of functional equations in several variables, Boston, MA: Birkhäuser, 1998. https://doi.org/10.1007/978-1-4612-1790-9
[3]	S. M. Jung, Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, New York: Springer, 2011. https://doi.org/10.1007/978-1-4419-9637-4
[4]	T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64–66.
[5]	T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. https://doi.org/10.1090/S0002-9939-1978-0507327-1 doi: 10.1090/S0002-9939-1978-0507327-1
[6]	Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci., 14 (1991), 431–434.
[7]	J. Brzdęk, D. Popa, I. Raşa, B. Xu, Ulam stability of operators, Academic Press, 2018.
[8]	J. Aczél, J. Dhombres, Functional equations in several variables, Cambridge University Press, 1989.
[9]	M. Kuczma, An introduction to the theory of functional equations and inequalities, Basel: Birkhäuser, 2009. https://doi.org/10.1007/978-3-7643-8749-5
[10]	L. Losonczi, On the stability of Hosszú's functional equation, Results Math., 29 (1996), 305–310.
[11]	P. Găvruţă, Hyers-Ulam stability of Hosszú's equation, In: Functional equations and inequalities, Dordrecht: Springer, 2000,105–110. https://doi.org/10.1007/978-94-011-4341-7_9
[12]	H. Vu, J. M. Rassias, N. V. Hoa, Hyers-Ulam stability for boundary value problem of fractional differential equations with $\kappa$-Caputo fractional derivative, Math. Methods Appl. Sci., 46 (2023), 438–460. https://doi.org/10.1002/mma.8520 doi: 10.1002/mma.8520
[13]	B. Shiri, Y. G. Shi, D. Baleanu, Ulam-Hyers stability of incommensurate systems for weakly singular integral equations, J. Comput. Appl. Math., 474 (2026), 116920. https://doi.org/10.1016/j.cam.2025.116920 doi: 10.1016/j.cam.2025.116920
[14]	T. Z. Yuan, B. Xu, J. Brzdęk, On the best Ulam constant and solutions of the higher order linear difference equation with constant coefficients, Math. Ann., 393 (2025), 2057–2093. https://doi.org/10.1007/s00208-025-03280-w doi: 10.1007/s00208-025-03280-w
[15]	B. Choczewski, The first international conference on functional equations in Poland: Zakopane 1967, Rocznik Nauk. Dydakt. Prace Mat., 16 (1999), 5–13.
[16]	M. Hosszú, A remark on the dependence of functions, Zeszyty Nauk. Uniw. Jagiell. Prace Mat., 14 (1970), 127–129.
[17]	D. Blanuša, The functional equation $f(x+ y-xy) +f(xy) = f(x) + f(y)$, Aequationes Math., 5 (1970), 63–67. https://doi.org/10.1007/BF01819272 doi: 10.1007/BF01819272
[18]	Z. Daróczy, On the general solution of the functional equation $f(x + y - xy) + f(xy) = f(x) + f(y)$, Aequationes Math., 6 (1971), 130–132. https://doi.org/10.1007/BF01819743 doi: 10.1007/BF01819743
[19]	H. Światak, A proof of the equivalence of the equation $^(x +y-xy)+ ^(xy) = ^(x)+^(y)$ and Jensen's functional equation, Aequationes Math., 6 (1971), 24–29. https://doi.org/10.1007/BF01833233 doi: 10.1007/BF01833233
[20]	I. Fenyö, On the general solution of a functional equation in the domain of distributions, Aequationes Math., 3 (1969), 236–246. https://doi.org/10.1007/BF01817444 doi: 10.1007/BF01817444
[21]	E. L. Koh, On Hosszú's functional equation in distributions, Proc. Amer. Math. Soc., 120 (1994), 1123–1129. https://doi.org/10.1090/S0002-9939-1994-1186990-8 doi: 10.1090/S0002-9939-1994-1186990-8
[22]	H. Światak, Remarks on the functional equation $f(x + y - xy) + f(xy) = f(x) + f(y)$, Aequationes Math., 1 (1968), 239–241. https://doi.org/10.1007/BF01817420 doi: 10.1007/BF01817420
[23]	Z. Daróczy, Über die Funktionalgleichung $f(xy) + f(x + y - xy) = f(x) + f(y)$, Publ. Math. Debrecen, 16 (1969), 129–132.
[24]	T. M. K. Davison, The complete solution of Hosszú's functional equation over a field, Aequationes Math., 11 (1974), 273–276. https://doi.org/10.1007/BF01834926 doi: 10.1007/BF01834926
[25]	A. Najati, J. Brzdęk, E. Mohammadi, Set-valued solutions of the Hosszú functional equation, Aequationes Math., 100 (2026), 15. https://doi.org/10.1007/s00010-026-01265-x doi: 10.1007/s00010-026-01265-x
[26]	G. Maksa, Z. Páles, On Hosszú's functional inequality, Publ. Math. Debrecen, 36 (1989), 187–189.
[27]	C. Borelli, On Hyers-Ulam stability of Hosszú's functional equation, Results Math., 26 (1994), 221–224. https://doi.org/10.1007/BF03323041 doi: 10.1007/BF03323041
[28]	Z. Kominek, On the Hyers-Ulam stability of Pexider-type extension of the Jensen-Hosszú equation, Bull. Int. Math. Virtual Inst., 1 (2011), 53–57.
[29]	Z. Kominek, Stability aspects of the Jensen-Hosszú equation, In: Traditional and present-day topics in real analysis, Wydawnictwo Uniwersytetu Łódzkiego, 2013,307–324. https://doi.org/10.18778/7525-971-1.19
[30]	Z. Kominek, On pexiderized Jensen-Hosszú functional equation on the unit interval, J. Math. Anal. Appl., 409 (2014), 722–728. https://doi.org/10.1016/j.jmaa.2013.07.001 doi: 10.1016/j.jmaa.2013.07.001
[31]	K. Lajkó, Applications of extensions of additive functions, Aequationes Math., 11 (1974), 68–76. https://doi.org/10.1007/BF01837734 doi: 10.1007/BF01837734
[32]	J. Tabor, Hosszú's functional equation on the unit interval is not stable, Publ. Math. Debrecen, 49 (1996), 335–340.
[33]	J. Schwaiger, On the existence of $m$-norms in vector spaces over valued fields, Aequationes Math., 97 (2023), 1051–1058. https://doi.org/10.1007/s00010-023-00956-z doi: 10.1007/s00010-023-00956-z
[34]	A. Misiak, $n$-inner product spaces, Math. Nachr., 140 (1989), 299–319. https://doi.org/10.1002/mana.19891400121 doi: 10.1002/mana.19891400121
[35]	S. Gähler, 2-metrisch Räume und ihre topologische Struktur, Math. Nachr., 26 (1963), 115–148. https://doi.org/10.1002/mana.19630260109 doi: 10.1002/mana.19630260109
[36]	S. Gähler, Lineare 2-normierte Räumen, Math. Nachr., 28 (1964), 1–43. https://doi.org/10.1002/mana.19640280102 doi: 10.1002/mana.19640280102
[37]	H. Gunawan, M. Mashadi, On $n$-normed spaces, Int. J. Math. Math. Sci., 27 (2001), 631–639.
[38]	J. Brzdęk, E. S. El-hady, On hyperstability of the Cauchy functional equation in $n$-Banach spaces, Mathematics, 8 (2020), 1886. https://doi.org/10.3390/math8111886 doi: 10.3390/math8111886
[39]	T. Z. Xu, J. M. Rassias, On the Hyers-Ulam stability of a general mixed additive and cubic functional equation in $n$-Banach spaces, Abstr. Appl. Anal., 2012 (2012), 926390. https://doi.org/10.1155/2012/926390 doi: 10.1155/2012/926390
[40]	H. Y. Chu, A. Kim, J. Park, On the Hyers-Ulam stabilities of functional equations on $n$-Banach spaces, Math. Nachr., 289 (2016), 1177–1188. https://doi.org/10.1002/mana.201400345 doi: 10.1002/mana.201400345
[41]	J. Brzdęk, K. Ciepliński, A fixed point theorem in $n$-Banach spaces and Ulam stability, J. Math. Anal. Appl., 470 (2019), 632–646. https://doi.org/10.1016/j.jmaa.2018.10.028 doi: 10.1016/j.jmaa.2018.10.028
[42]	E. S. El-hady, A. Bahyrycz, J. Brzdęk, Survey on Ulam stability with respect to $n$-norms and $(n, \beta)$-norms, Symmetry, 18 (2026), 411. https://doi.org/10.3390/sym18030411 doi: 10.3390/sym18030411
[43]	X. Y. Chen, M. M. Song, Characterizations on isometries in linear $n$-normed spaces, Nonlinear Anal., 72 (2010), 1895–1901. https://doi.org/10.1016/j.na.2009.09.029 doi: 10.1016/j.na.2009.09.029
[44]	H. Y. Chu, K. Lee, C. G. Park, On the Aleksandrov problem in linear $n$-normed spaces, Nonlinear Anal., 59 (2004), 1001–1011. https://doi.org/10.1016/j.na.2004.07.046 doi: 10.1016/j.na.2004.07.046
[45]	T. Trif, On the stability of a general gamma-type functional equation, Publ. Math. Debrecen, 60 (2002), 47–61. https://publi.math.unideb.hu/paper/2010/download/
[46]	J. Brzdęk, A. Fo$\breve{{\rm{s}}} $ner, Remarks on the stability of Lie homomorphisms, J. Math. Anal. Appl., 400 (2013), 585–596. https://doi.org/10.1016/j.jmaa.2012.11.008 doi: 10.1016/j.jmaa.2012.11.008
[47]	J. Brzdęk, Some remarks on the best Ulam constant, Symmetry, 16 (2024), 1644. https://doi.org/10.3390/sym16121644 doi: 10.3390/sym16121644

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)