Abstract
The “oldest quartic” functional equation
was introduced and solved by the second author of this paper (see J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) 1999, 2, 243–252). Similarly, an interesting “quintic” functional equation was introduced and investigated by I. G. Cho, D. Kang and H. Koh, Stability problems of quintic mappings in quasi-β-normed spaces, J. Inequal. Appl. 2010 2010, Article ID 368981, in the following form:
In this paper, we generalize this “Cho–Kang–Koh equation” by introducing pertinent Euler–Lagrange k-quintic functional equations, and investigate the “Ulam stability” of these new k-quintic functional mappings.
Funding statement: The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
References
[1]
A. Alotaibi, J. M. Rassias and S. A. Mohiuddine,
Solution of the Ulam stability problem for quartic
[2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. 10.2969/jmsj/00210064Search in Google Scholar
[3] I. G. Cho, D. Kang and H. Koh, Stability problems of quintic mappings in quasi-β-normed spaces, J. Inequal. Appl. 2010 (2010), Article ID 368981. 10.1155/2010/368981Search in Google Scholar
[4] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431–434. 10.1155/S016117129100056XSearch in Google Scholar
[5] P. Gǎvruţa, A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431–436. 10.1006/jmaa.1994.1211Search in Google Scholar
[6] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. 10.1073/pnas.27.4.222Search in Google Scholar
[7] K.-W. Jun and H.-M. Kim, The generalized Hyers–Ulam–Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), no. 2, 267–278. 10.1016/S0022-247X(02)00415-8Search in Google Scholar
[8] S.-M. Jung, On the Hyers–Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), no. 1, 126–137. 10.1006/jmaa.1998.5916Search in Google Scholar
[9] S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001. Search in Google Scholar
[10] S. A. Mohiuddine, Stability of Jensen functional equation in intuitionistic fuzzy normed space, Chaos Solitons Fractals 42 (2009), no. 5, 2989–2996. 10.1016/j.chaos.2009.04.040Search in Google Scholar
[11]
S. A. Mohiuddine, J. M. Rassias and A. Alotaibi,
Solution of the Ulam stability problem for Euler–Lagrange
[12] S. A. Mohiuddine, J. M. Rassias and A. Alotaibi, Solution of the Ulam stability problem for Euler–Lagrange–Jensen k-cubic mappings, Filomat 30 (2016), no. 2, 305–312. 10.2298/FIL1602305MSearch in Google Scholar
[13] S. A. Mohiuddine and H. Şevli, Stability of Pexiderized quadratic functional equation in intuitionistic fuzzy normed space, J. Comput. Appl. Math. 235 (2011), no. 8, 2137–2146. 10.1016/j.cam.2010.10.010Search in Google Scholar
[14] M. Mursaleen and K. J. Ansari, Stability results in intuitionistic fuzzy normed spaces for a cubic functional equation, Appl. Math. Inf. Sci. 7 (2013), no. 5, 1677–1684. 10.12785/amis/070504Search in Google Scholar
[15] A. Najati and C. Park, On the stability of an n-dimensional functional equation originating from quadratic forms, Taiwanese J. Math. 12 (2008), no. 7, 1609–1624. 10.11650/twjm/1500405074Search in Google Scholar
[16] J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), no. 1, 126–130. 10.1016/0022-1236(82)90048-9Search in Google Scholar
[17] J. M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989), no. 3, 268–273. 10.1016/0021-9045(89)90041-5Search in Google Scholar
[18] J. M. Rassias, On the stability of the Euler–Lagrange functional equation, Chinese J. Math. 20 (1992), no. 2, 185–190. Search in Google Scholar
[19] J. M. Rassias, Solution of the Ulam stability problem for Euler–Lagrange quadratic mappings, J. Math. Anal. Appl. 220 (1998), no. 2, 613–639. 10.1006/jmaa.1997.5856Search in Google Scholar
[20] J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) (1999), no. 2, 243–252. Search in Google Scholar
[21] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297–300. 10.1090/S0002-9939-1978-0507327-1Search in Google Scholar
[22] J. Rätz, On orthogonally additive mappings, Aequationes Math. 28 (1985), no. 1–2, 35–49. 10.1007/BF02189390Search in Google Scholar
[23] S. M. Ulam, A collection of Mathematical Problems, Interscience Tracts Pure Appl. Math. 8, Interscience Publishers, New York, 1960. Search in Google Scholar
[24] T. Z. Xu, J. M. Rassias and W. X. Xu, A generalized mixed quadratic-quartic functional equation, Bull. Malays. Math. Sci. Soc. (2) 35 (2012), no. 3, 633–649. Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
