Asymptotic behavior of a generalized functional equation

Mohammad Amin Tareeghee; Abbas Najati; Batool Noori; Choonkil Park; Mohammad Amin Tareeghee; Abbas Najati; Batool Noori; Choonkil Park

doi:10.3934/math.2022389

AIMS Mathematics

2022, Volume 7, Issue 4: 7001-7011. doi: 10.3934/math.2022389

Previous Article Next Article

Research article

Asymptotic behavior of a generalized functional equation

1.
Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran
2.
Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

Received: 02 December 2021 Revised: 09 January 2022 Accepted: 17 January 2022 Published: 07 February 2022
MSC : 39B82, 39B52, 39B62

In this paper, we investigate the Hyers-Ulam stability problem of the following functional equation

$ f(x+y)+g(x-y) = h(x)+k(y), $

on an unbounded restricted domain, which generalizes some of the results already obtained by other authors (for example [9,Theorem 2], [11,Theorem 5] and [21,Theorem 2]). Particular cases of this functional equation are Cauchy, Jensen, quadratic and Drygas functional equations. As a consequence, we obtain asymptotic behaviors of this functional equation.
- Hyers-Ulam stability,
- quadratic functional equation,
- asymptotic behavior
Citation: Mohammad Amin Tareeghee, Abbas Najati, Batool Noori, Choonkil Park. Asymptotic behavior of a generalized functional equation[J]. AIMS Mathematics, 2022, 7(4): 7001-7011. doi: 10.3934/math.2022389

Related Papers:

Abstract

In this paper, we investigate the Hyers-Ulam stability problem of the following functional equation

$ f(x+y)+g(x-y) = h(x)+k(y), $

on an unbounded restricted domain, which generalizes some of the results already obtained by other authors (for example [9,Theorem 2], [11,Theorem 5] and [21,Theorem 2]). Particular cases of this functional equation are Cauchy, Jensen, quadratic and Drygas functional equations. As a consequence, we obtain asymptotic behaviors of this functional equation.

References

[1]	J. Aczél, J. Dhombres, Functional equations in several variables, Cambridge University Press, Cambridge, 1989.
[2]	S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Hamburg, 62 (1992), 59–64. http://dx.doi.org/10.1007/BF02941618 doi: 10.1007/BF02941618
[3]	S. Czerwik, Functional equations and inequalities in several variables, World Scientific Publishing Company, Singapore, 2002.
[4]	B. R. Ebanks, P. Kannappan, P. K. Sahoo, A common generalization of functional equations characterizing normed and quasi-inner product spaces, Can. Math. Bull., 35 (1992), 321–327. http://dx.doi.org/10.4153/CMB-1992-044-6 doi: 10.4153/CMB-1992-044-6
[5]	B. Fadli, D. Zeglami, S. Kabbaj, A variant of the quadratic functional equation on semigroups, Proyecciones, 37 (2018), 45–55. http://dx.doi.org/10.4067/S0716-09172018000100045 doi: 10.4067/S0716-09172018000100045
[6]	D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27 (1941), 222–224. http://dx.doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
[7]	D. H. Hyers, G. Isac, T. M. Rassias, Stability of functional equations in several variables, Birkhäuser, Basel, 1998.
[8]	P. Jordan, J. V. Neumann, On inner products in linear metric spaces, Ann. Math., 36 (1935), 719–723. http://dx.doi.org/10.2307/1968653 doi: 10.2307/1968653
[9]	S. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl., 222 (1998), 126–137. http://dx.doi.org/10.1006/jmaa.1998.5916 doi: 10.1006/jmaa.1998.5916
[10]	S. Jung, Quadratic functional equations of Pexider type, Int. J. Math. Math. Sci., 24 (2000), 351–359. http://dx.doi.org/10.1155/S0161171200004075 doi: 10.1155/S0161171200004075
[11]	S. Jung, P. K. Sahoo, Stability of a functional equation of Drygas, Aequationes Math., 64 (2002), 263–273. http://dx.doi.org/10.1007/PL00012407 doi: 10.1007/PL00012407
[12]	S. Jung, Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, Springer, New York, 2011.
[13]	P. Kannappan, Quadratic functional equation and inner product spaces, Results Math., 27 (1995), 368–372. http://dx.doi.org/10.1007/BF03322841 doi: 10.1007/BF03322841
[14]	P. Kannappan, Functional equations and inequalities with applications, Springer, New York, 2009.
[15]	A. Najati, Hyers-Ulam stability of an $n$-Apollonius type quadratic mapping, B. Belg. Math. Soc.-Sim., 14 (2007), 755–774. http://dx.doi.org/10.36045/bbms/1195157142 doi: 10.36045/bbms/1195157142
[16]	A. Najati, S. Jung, Approximately quadratic mappings on restricted domains, J. Inequal. Appl., 2010 (2010). http://dx.doi.org/10.1155/2010/503458
[17]	A. Najati, C. Park, Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the pexiderized Cauchy functional equation, J. Math. Anal. Appl., 335 (2007), 763–778. http://dx.doi.org/10.1016/j.jmaa.2007.02.009 doi: 10.1016/j.jmaa.2007.02.009
[18]	A. Najati, C. Park, The pexiderized Apollonius-Jensen type additive mapping and isomorphisms between $C^$-algebras, J. Differ. Equ. Appl.*, 14 (2008), 459–479. http://dx.doi.org/10.1080/10236190701466546 doi: 10.1080/10236190701466546
[19]	B. Noori, M. B. Moghimi, B. Khosravi, C. Park, Stability of some functional equations on bounded domains, J. Math. Inequal., 14 (2020), 455–472. http://dx.doi.org/10.7153/jmi-2020-14-29 doi: 10.7153/jmi-2020-14-29
[20]	C. Park, A. Najati, B. Noori, M. B. Moghimi, Additive and Fréchet functional equations on restricted domains with some characterizations of inner product spaces, AIMS Math., 7 (2021), 3379–3394. http://dx.doi.org/10.3934/math.2022188 doi: 10.3934/math.2022188
[21]	J. M. Rassias, On the Ulam stability of mixed type mappings on restricted domains, J. Math. Anal. Appl., 276 (2002), 747–762. http://dx.doi.org/10.1016/S0022-247X(02)00439-0 doi: 10.1016/S0022-247X(02)00439-0
[22]	M. Sarfraz, Y. Li, Minimum functional equation and some Pexider-type functional equation on any group, AIMS Math., 6 (2021), 11305–11317. http://dx.doi.org/10.3934/math.2021656 doi: 10.3934/math.2021656
[23]	F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113–129. http://dx.doi.org/10.1007/BF02924890 doi: 10.1007/BF02924890
[24]	M. A. Tareeghee, A. Najati, M. R. Abdollahpour, B. Noori, On restricted functional inequalities associated with quadratic functional equations, In press.
[25]	Z. Wang, Approximate mixed type quadratic-cubic functional equation, AIMS Math., 6 (2021), 3546–3561. http://dx.doi.org/10.3934/math.2021211 doi: 10.3934/math.2021211

Reader Comments

Your name:*

Email:*
© 2022 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)