2-Banach空间上三次-四次混合型函数方程的Hyers-Ulam-Rassias稳定性

Abstract

Abstract:

In this paper, we investigate the general solution and the Hyers-Ulam-Rassias stability of the mixed type cubic-quartic functional equation

$\begin{eqnarray*} &&f(kx+y)+f(kx-y)\\ & = & \frac{k^2+k}{2}[f(x+y)+f(x-y)]+\frac{k^2 -k}{2}[f(-x-y)+f(y-x)]\\ && +(k^4+k^3-k^2 -k)f(x)+(k^4-k^3-k^2+k)f(-x)-(k^2 -1)f(y)-(k^2 -1)f(-y) \end{eqnarray*}$

in 2-Banach spaces, where k >1. The results improve and extend some recent results.

Key words: Hyers-Ulam-Rassias stability, General solution, Cubic function, Quartic function, 2-Banach space

CLC Number:

O177.9

Chun Wang,Tianzhou Xu. Hyers-Ulam-Rassias Stability of a Mixed Type Cubic-Quartic Functional Equation in 2-Banach Spaces[J].Acta mathematica scientia,Series A, 2020, 40(2): 352-368.

Trendmd

References 37

1	Aczél J, Dhombres J. Functional Equations in Several Variables. Cambridge: Cambridge Univ Press, 1989
2	Agarwal R P , Xu B , Zhang W . Stability of functional equations in single variable. J Math Anal Appl, 2003, 288, 852- 869 doi: 10.1016/j.jmaa.2003.09.032
3	Alotaibi A , Mohiuddine S A . On the stability of a cubic functional equation in random 2-normed spaces. Adv Difference Equ, 2012, 39, 1- 10
4	Aoki T . On the stability of the linear transformation in Banach spaces. J Math Soc Japan, 1950, 2, 64- 66 doi: 10.2969/jmsj/00210064
5	Brzdȩk J, Jung S M. A note on stability of an operator linear equation of the second order. Abstr Appl Anal, 2011, 2011: Article ID 602713
6	Brzdȩk J , Popa D , Xu B . On approximate solutions of the linear functional equation of higher order. J Math Anal Appl, 2011, 373, 680- 689 doi: 10.1016/j.jmaa.2010.08.028
7	Cholewa P W . Remark on the stability of functional equations. Aequationes Math, 1984, 27, 76- 86 doi: 10.1007/BF02192660
8	Ciepliński K . Generalized stability of multi-additive mappings. Appl Math Lett, 2010, 23, 1291- 1294 doi: 10.1016/j.aml.2010.06.015
9	Ciepliński K . Applications of fixed point theorems to the Hyers-Ulam stability of functional equations-a survey. Ann Funct Anal, 2012, 3, 151- 164 doi: 10.15352/afa/1399900032
10	Eskandani G Z , Rassias J M . Approximation of general α-cubic functional equations in 2-Banach spaces. Ukrainian Math J, 2017, 68, 1651- 1658 doi: 10.1007/s11253-017-1317-7
11	Fassi I E . Approximate solution of radical quartic functional equation related to additive mapping in 2-Banach spaces. J Math Anal Appl, 2017, 455, 2001- 2013 doi: 10.1016/j.jmaa.2017.06.078
12	Fassi I E . A new type of approximation for the radical quintic functional equation in non-Archimedean (2, β)-Banach spaces. J Math Anal Appl, 2018, 475, 322- 335
13	Găvruţa P . A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl, 1994, 184, 431- 436 doi: 10.1006/jmaa.1994.1211
14	Hatori O , Kobayasi K , Miura T , et al. On the best constant of Hyers-Ulam stability. J Nonlinear Convex Anal, 2004, 5, 387- 393
15	Hirasawa G , Miura T . Hyers-Ulam stability of a closed operator in a Hilbert space. Bull Korean Math Soc, 2006, 43, 107- 117 doi: 10.4134/BKMS.2006.43.1.107
16	Hyers D H . On the stability of the linear functional equation. Proc Natl Acad Sci USA, 1941, 27, 222- 224 doi: 10.1073/pnas.27.4.222
17	Jung S M. Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. New York: Springer, 2011
18	Kannappan P L . Quadratic functional equation and inner product spaces. Results Math, 1995, 27, 368- 372 doi: 10.1007/BF03322841
19	Mohiuddine S A , Rassias J M , Alotaibi A . Solution of the Ulam stability problem for Euler-Lagrange-Jensen k-quintic mappings. Math Methods Appl Sci, 2017, 40, 3017- 3025 doi: 10.1002/mma.4217
20	Mohiuddine S A , Rassias J M , Alotaibi A . Solution of the Ulam stability problem for Euler-Lagrange-Jensen k-cubic mappings. Filomat, 2016, 30, 305- 312 doi: 10.2298/FIL1602305M
21	Murali R , Boobalan M , Raj A A . Stability of a mixed type cubic and quartic functional equation in 2-Banach space. Asian J Math Comput Res, 2016, 12, 110- 119
22	Najati A , Moghimi M B . Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces. J Math Anal Appl, 2008, 337, 399- 415 doi: 10.1016/j.jmaa.2007.03.104
23	Park C, Gordji M E, Ghaemi M B, Majani H. Fixed points and approximately octic mappings in non-Archimedean 2-normed spaces. J Inequal Appl, 2012, 2012: Article ID 289
24	Park C. Additive functional inequalities in 2-Banach spaces. J Inequal Appl, 2013, 2013: Article ID 447
25	Park W G . Approximate additive mapping in 2-Banach spaces and related topics. J Math Anal Appl, 2011, 376, 193- 202 doi: 10.1016/j.jmaa.2010.10.004
26	Rassias Th M . On the stability of the linear mapping in Banach spaces. Proc Amer Math Soc, 1978, 72, 297- 300 doi: 10.1090/S0002-9939-1978-0507327-1
27	Ulam S M. A Collection of Mathematical Problems. New York: Interscience, 1960
28	Wang C, Xu T Z. Hyers-Ulam stability of differentiation operator on Hilbert spaces of entire functions. J Funct Spaces, 2014, 2014: Article ID 398673
29	Wang C , Xu T Z . Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives. Appl Math, 2015, 60, 383- 393 doi: 10.1007/s10492-015-0102-x
30	Wang C , Xu T Z . Hyers-Ulam stability of differential operators on reproducing kernel function spaces. Complex Anal Oper Theory, 2016, 10, 795- 813 doi: 10.1007/s11785-015-0486-3
31	Wang C , Xu T Z . Hyers-Ulam stability of a class of fractional linear differential equations. Kodai Math J, 2015, 38, 510- 520 doi: 10.2996/kmj/1446210592
32	Wang C , Xu T Z . Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete Contin Dyn Syst Ser S, 2017, 10, 505- 521
33	Wang J , Li X . A uniform method to Ulam-Hyers stability for some linear fractional equations. Mediterr J Math, 2016, 13, 625- 635 doi: 10.1007/s00009-015-0523-5
34	Xu B , Brzdȩk J , Zhang W . Fixed point results and the Hyers-Ulam stability of linear equations of higher orders. Pacific J Math, 2015, 273, 483- 498 doi: 10.2140/pjm.2015.273.483
35	Xu B , Zhang W . Hyers-Ulam stability for a nonlinear iterative equation. Colloq Math, 2002, 93, 1- 9 doi: 10.4064/cm93-1-1
36	Xu T Z , Wang C , Rassias Th M . On the stability of multi-additive mappings in non-Archimedean normed spaces. J Comput Anal Appl, 2015, 18, 1102- 1110
37	Xu T Z . On the stability of multi-Jensen mappings in β-normed spaces. Appl Math Lett, 2012, 25, 1866- 1870 doi: 10.1016/j.aml.2012.02.049

[1]	Chun Wang. Stability of Some Fractional Systems and Laplace Transform [J]. Acta mathematica scientia,Series A, 2019, 39(1): 49-58.
[2]	Wang Chun, Xu Tianzhou. Solution and Hyers-Ulam-Rassias Stability of a Mixed Type Quadratic-Additive Functional Equation with a Parameter in Quasi-Banach Spaces [J]. Acta mathematica scientia,Series A, 2017, 37(5): 846-859.
[3]	ZHANHua-Shui. On Solutions of Quasilinear Degenerate Parabolic Equations [J]. Acta mathematica scientia,Series A, 2005, 25(6): 832-838.
[4]	Yu Changan. A General Solution for Trinomial Linear Recurrence with two Indices [J]. Acta mathematica scientia,Series A, 1997, 17(3): 255-260.

Hyers-Ulam-Rassias Stability of a Mixed Type Cubic-Quartic Functional Equation in 2-Banach Spaces

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 37

Related Articles 4

Metrics

Comments

Recommended 10