Approximate mixed type quadratic-cubic functional equation

Zhihua Wang; Zhihua Wang

doi:10.3934/math.2021211

AIMS Mathematics

2021, Volume 6, Issue 4: 3546-3561. doi: 10.3934/math.2021211

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Research article

Approximate mixed type quadratic-cubic functional equation

Zhihua Wang ^,

School of Science, Hubei University of Technology, Wuhan, Hubei 430068, China

Received: 22 October 2020 Accepted: 11 January 2021 Published: 21 January 2021
MSC : 39B82, 39B72

In this paper, we investigate the generalized Hyers-Ulam stability of the following mixed type quadratic-cubic functional equation

$ \begin{align*} 2f(2x+y)+2f(2x-y) = 4f(x+y)+4f(x-y)+4f(2x)+f(2y)-8f(x)-8f(y) \end{align*} $

in non-Archimedean $ (n, \beta) $-normed spaces.
Citation: Zhihua Wang. Approximate mixed type quadratic-cubic functional equation[J]. AIMS Mathematics, 2021, 6(4): 3546-3561. doi: 10.3934/math.2021211

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Abstract

In this paper, we investigate the generalized Hyers-Ulam stability of the following mixed type quadratic-cubic functional equation

$ \begin{align*} 2f(2x+y)+2f(2x-y) = 4f(x+y)+4f(x-y)+4f(2x)+f(2y)-8f(x)-8f(y) \end{align*} $

in non-Archimedean $ (n, \beta) $-normed spaces.

References

[1]	T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64–66. doi: 10.2969/jmsj/00210064
[2]	Y. J. Cho, P. C. S. Lin, S. S. Kim, A. Misiak, Theory of 2-Inner Product Spaces, Nova Science Publishers, Inc., New York, 2001.
[3]	Y. J. Cho, Th. M. Rassias, R. Saadati, Stability of Functional Equations in Random Normed Spaces, Springer Science, New York, 2013.
[4]	Y. J. Cho, C. Park, Th. M. Rassias, R. Saadati, Stability of Functional Equations in Banach Algebras, Springer Science, New York, 2015.
[5]	P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76–86. doi: 10.1007/BF02192660
[6]	K. Ciepliński, T. Z. Xu, Approximate multi-Jensen and multi-quadratic mappings in $2$-Banach spaces, Carpathian J. Math., 29 (2013), 159–166.
[7]	S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62 (1992), 59–64. doi: 10.1007/BF02941618
[8]	R. W. Freese, Y. J. Cho, Geometry of Linear 2-Normed Spaces, Nova Science Publishers, Inc., New York, 2001.
[9]	S. Gähler, $2$-metrische Räume und ihere topologische struktur, Math. Nachr., 26 (1963), 115–148. doi: 10.1002/mana.19630260109
[10]	S. Gähler, Lineare $2$-normierte Räume, Math. Nachr., 28 (1964), 1–43. doi: 10.1002/mana.19640280102
[11]	S. Gähler, Über $2$-Banach Räume, Math. Nachr., 42 (1969), 335–347. doi: 10.1002/mana.19690420414
[12]	P. Găvruţă, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431–436. doi: 10.1006/jmaa.1994.1211
[13]	M. E. Gordji, M. B. Savadkouhi, Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces, Appl. Math. Lett., 23 (2010), 1198–1202. doi: 10.1016/j.aml.2010.05.011
[14]	M. E. Gordji, M. B. Savadkouhi, Stability of cubic and quartic functional equations in non-Archimedean spaces, Acta Appl. Math., 110 (2010), 1321–1329. doi: 10.1007/s10440-009-9512-7
[15]	H. Gunawan, M. Mashadi, On $n$-normed spaces, Int. J. Appl. Math. Sci., 27 (2001), 631–639.
[16]	K. Hensel, Über eine neue Begründung der Theorie der algebraischen Zahlen, Jahresber. Deutsch. Math. Verein, 6 (1897), 83–88.
[17]	D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27 (1941), 222–224. doi: 10.1073/pnas.27.4.222
[18]	D. H. Hyers, G. Isac, Th. M. Rassias, Stability of Functional Equations in Several variables, Birkh$\ddot{a}$user, Basel, 1998.
[19]	K. W. Jun, H. M. Kim, The generalized of the Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl., 274 (2002), 267–278.
[20]	S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer Science, New York, 2011.
[21]	Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer Science, New York, 2009.
[22]	A. K. Katsaras, A. Beoyiannis, Tensor products of non-Archimedean weighted spaces of continuous functions, Georgian Math. J., 6 (1999), 33–44. doi: 10.1023/A:1022926309318
[23]	A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic Publishers, Dordrecht, 1997.
[24]	C. I. Kim, S. W. Park, The generalized Hyers-Ulam stability of additive functional inequalities in non-Archimedean $2$-normed space, Korean J. Math., 22 (2014), 339–348. doi: 10.11568/kjm.2014.22.2.339
[25]	S. S. Kim, Y. J. Cho, Strict convexity in linear $n$-normed spaces, Demonstr. Math., 29 (1996), 739–744.
[26]	J. R. Lee, S. Y. Jang, C. Park, D. Y. Shin, Fuzzy stability of quadratic functional equations, Adv. Differ. Equ., 2010 (2010), 412160. doi: 10.1186/1687-1847-2010-412160
[27]	R. Malceski, Strong $n$-convex $n$-normed spaces, Mat. Bilt., 21 (1997), 81–102.
[28]	A. K. Mirmostafaee, Approximately additive mappings in non-Archimedean normed spaces, Bull. Korean Math. Soc., 46 (2009), 387–400. doi: 10.4134/BKMS.2009.46.2.387
[29]	A. Misiak, $N$-inner product spaces, Math. Nachr., 140 (1989), 299–319.
[30]	M. S. Moslehian, Th. M. Rassias, Stability of functional equations in non-Archimedean spaces, Appl. Anal. Discrete Math., 1 (2007), 325–334. doi: 10.2298/AADM0702325M
[31]	M. S. Moslehian, Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal., 69 (2008), 3405–3408. doi: 10.1016/j.na.2007.09.023
[32]	A. Najati, F. Moradlou, Hyers-Ulam-Rassias stability of the Apollonius type quadratic mapping in non-Archimedean spaces, Tamsui Oxf. J. Math. Sci., 24 (2008), 367–380.
[33]	P. J. Nyikos, On some non-Archimedean spaces of Alexandroff and Urysohn, Topol. Appl., 91 (1999), 1–23. doi: 10.1016/S0166-8641(97)00239-3
[34]	W. G. Park, Approximate additive mappings in $2$-Banach spaces and related topics, J. Math. Anal. Appl., 376 (2011), 193–202. doi: 10.1016/j.jmaa.2010.10.004
[35]	C. Park, M. E. Gordji, M. B. Ghaemi, H. Majani, Fixed points and approximately octic mappings in non-Archimedean $2$-normed spaces, J. Ineq. Appl., 2012 (2012), 289. doi: 10.1186/1029-242X-2012-289
[36]	C. Park, Additive functional inequalities in $2$-Banach spaces, J. Ineq. Appl., 2013 (2013), 447. doi: 10.1186/1029-242X-2013-447
[37]	Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. doi: 10.1090/S0002-9939-1978-0507327-1
[38]	Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic, Dordrecht, 2003.
[39]	K. Ravi, R. Murali, M. Arunkumar, The generalized Hyers-Ulam-Rassias stability of a quadratic function equation, J. Ineq. Pure Appl. Math., 9 (2008), 20.
[40]	P. K. Sahoo, A generalized cubic functional equation, Acta Math. Sinica (English Series), 21 (2005), 1159–1166. doi: 10.1007/s10114-005-0551-3
[41]	P. K. Sahoo, P. Kannappan, Introduction to Functional Equations, CRC Press, Boca Raton, 2011.
[42]	F. Skof, Local properties and approximations of operators, Rend. Sem. Mat. Fis. Milano, 53 (1983), 113–129. doi: 10.1007/BF02924890
[43]	S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York, 1964.
[44]	L. G. Wang, B. Liu, The Hyers-Ulam stability of a functional equation deriving from quadratic and cubic functions in quasi-$\beta$-normed spaces, Acta Math. Sin., 26 (2010), 2335–2348. doi: 10.1007/s10114-010-9330-x
[45]	A. White, $2$-Banach spaces, Doctorial Diss., St. Louis Univ., 1968.
[46]	A. White, $2$-Banach spaces, Math. Nachr., 42 (1969), 43–60.
[47]	T. Z. Xu, Approximate multi-Jensen, multi-Euler-Lagrange additive and quadratic mappings in $n$-Banach spaces, Abst. Appl. Anal., 2013 (2013), 648709.
[48]	T. Z. Xu, J. M. Rassias, On the Hyers-Ulam stability of a general mixed additive and cubic functional equation in $n$-Banach spaces, Abst. Appl. Anal., 2012 (2012), 926390.
[49]	X. Z. Yang, L. D. Chang, G. F. Liu, G. N. Shen, Stability of functional equations in $(n, \beta)$-normed spaces, J. Ineq. Appl., 2015 (2015), 112. doi: 10.1186/s13660-015-0628-1

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