拟Banach空间上含参数的二次-可加混合型函数方程的解和Hyers-Ulam-Rassias稳定性
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摘要
该文讨论了带有参数s的二次-可加混合型函数方程2k[f(x+ky)+f(kx+y)]=k(1-s+k+ks+2k~2)f(x+y)+k(1-s-3k+ks+2k~2)f(x-y)+2kf(kx)+2k(s+k-ks-2k~2)f(x)+2(1-k-s)f(ky)+2ksf(y)的一般解,同时研究了该函数方程在拟Banach空间上的Hyers-Ulam-Rassias稳定性,这里k>1,s≠1-2k.
This paper establishes the general solution of the mixed type quadratic-additive functional equation2 k[f(x+ky)+f(kx+y)]=k(1-s+k+ks+2 k~2)f(x+y)+k(1-s-3 k+ks+2 k~2)f(x-y)+2 kf(kx)+2 k(s+k-ks-2 k~2)f(x)+2(1-k-s)f(ky)+2 ksf(y)with a parameter s,and investigates the Hyers-Ulam-Rassias stability of this functional equation in quasi-Banach spaces, where k>1 and s≠1-2 k.
This paper establishes the general solution of the mixed type quadratic-additive functional equation2 k[f(x+ky)+f(kx+y)]=k(1-s+k+ks+2 k~2)f(x+y)+k(1-s-3 k+ks+2 k~2)f(x-y)+2 kf(kx)+2 k(s+k-ks-2 k~2)f(x)+2(1-k-s)f(ky)+2 ksf(y)with a parameter s,and investigates the Hyers-Ulam-Rassias stability of this functional equation in quasi-Banach spaces, where k>1 and s≠1-2 k.
引文
[1]Aczel J,Dhombres J.Functionsl Equations in Several Variables.Cambridge:Cambridge Univ Press,1989
[2]Benyamini Y,Lindenstrauss J.Geometric Nonlinear Functional Analysis.Providence:Amer Math Soc Colloq Publ,2000
[3]Brzdek J,Jung S M.A note on stability of an operator linear equation of the second order.Abstr Appl Anal,2011,2011:1-15
[4]Brzdek J,Popa D,Xu B.On approximate solutions of the linear functional equation of higher order.J Math Anal Appl,2011,373:680-689
[5]Cholewa P W.Remark on the stability of functional equations.Aequat Math,1984,27:76-86
[6]Cieplinski K.Generalized stability of multi-additive mappings.Appl Math Lett,2010,23:1291-1294
[7]Ciepliriski K.Applications of fixed point theorems to the Hyers-Ulam stability of functional equations-a survey.Ann Funct Anal,2012,3(1):151-164
[8]Czerwik S.On the stability of the quadratic mapping in normed spaces.Abh Math Sem Univ Hamburg,1992,62:59-64
[9]Gavruta P.A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings.J Math Anal Appl,1994,184:431-436
[10]Grabiec A.The generalized Hyers-Ulam stability of a class of functional equations.Publ Math Debrecen,1996,48:217-235
[11]Hatori O,Kobayasi K,Miura T,Takagi H,Takahasi S E.On the.best constant of Hyers-Ulam stability.J Nonlinear Convex Anal,2004,5:387-393
[12]Hirasawa G,Miura T.Hyers-Ulam stability of a closed operator in a Hilbert space.Bull Korean Math Soc,2006,43:107-117
[13]Hyers D H.On the stability of the linear functional equation.Proc Natl Acad Sci USA,1941,27:222-224
[14]Kannappan P L.Quadratic functional equation and inner product spaces.Results Math,1995,27:368-372
[15]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
[16]Rassias T M.On the stability of the linear mapping in Banach spaces.Proc Amer Math Soc,1978,72:297-300
[17]Rolewicz S.Metric Linear Spaces.Warszawa:Pwn-Polish Sci Publ,1984
[18]Skof F.Local properties an approximations of operators.Rend Sem Mat Fis Milano,1983,53:113-129
[19]Ulam S M.A Collection of Mathematical Problems.New York:Interscience,1960
[20]Wang C,Xu T Z.Hyers-Ulam stability of differentiation operator on Hilbert spaces of entire functions.J Funct Spaces,2014,2014:1-6
[21]Wang C,Xu T Z.Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives.Appl Math,2015,60:383-393
[22]Wang C,Xu T Z.Hyers-Ulam stability of differential operators on reproducing kernel function spaces.Complex Anal Oper Theory,2016,10:795-813
[23]Wang C,Xu T Z.Hyers-Ulam stability of a class of fractional linear differential equationsz.Kodai Math J,2015,38:510-520
[24]Xu T Z,Wang C,Rassias T M.On the stability of multi-additive mappings in non-Archimedean normed spaces.J Comput Anal Appl,2015,18:1102-1110
[25]Xu T Z.Approximate multi-Jensen,multi-Euler-Lagrange additive and quadratic mappings in n-Banach spaces.Abstr Appl Anal,2013,2013:1-12
[26]Xu T Z.On the stability of multi-Jensen mappings in/3-normed spaces.Appl Math Lett,2012,25:1866-1870
[27]成立花,连铁艳.Banach空间三次方程的稳定性问题.数学物理学报.2014,34A(2):266-273Cheng L H,Lian T Y.Stability of cubic functional equations in Banach spaces.Acta.Math Sci,2014,34A(2):266-273
[2]Benyamini Y,Lindenstrauss J.Geometric Nonlinear Functional Analysis.Providence:Amer Math Soc Colloq Publ,2000
[3]Brzdek J,Jung S M.A note on stability of an operator linear equation of the second order.Abstr Appl Anal,2011,2011:1-15
[4]Brzdek J,Popa D,Xu B.On approximate solutions of the linear functional equation of higher order.J Math Anal Appl,2011,373:680-689
[5]Cholewa P W.Remark on the stability of functional equations.Aequat Math,1984,27:76-86
[6]Cieplinski K.Generalized stability of multi-additive mappings.Appl Math Lett,2010,23:1291-1294
[7]Ciepliriski K.Applications of fixed point theorems to the Hyers-Ulam stability of functional equations-a survey.Ann Funct Anal,2012,3(1):151-164
[8]Czerwik S.On the stability of the quadratic mapping in normed spaces.Abh Math Sem Univ Hamburg,1992,62:59-64
[9]Gavruta P.A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings.J Math Anal Appl,1994,184:431-436
[10]Grabiec A.The generalized Hyers-Ulam stability of a class of functional equations.Publ Math Debrecen,1996,48:217-235
[11]Hatori O,Kobayasi K,Miura T,Takagi H,Takahasi S E.On the.best constant of Hyers-Ulam stability.J Nonlinear Convex Anal,2004,5:387-393
[12]Hirasawa G,Miura T.Hyers-Ulam stability of a closed operator in a Hilbert space.Bull Korean Math Soc,2006,43:107-117
[13]Hyers D H.On the stability of the linear functional equation.Proc Natl Acad Sci USA,1941,27:222-224
[14]Kannappan P L.Quadratic functional equation and inner product spaces.Results Math,1995,27:368-372
[15]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
[16]Rassias T M.On the stability of the linear mapping in Banach spaces.Proc Amer Math Soc,1978,72:297-300
[17]Rolewicz S.Metric Linear Spaces.Warszawa:Pwn-Polish Sci Publ,1984
[18]Skof F.Local properties an approximations of operators.Rend Sem Mat Fis Milano,1983,53:113-129
[19]Ulam S M.A Collection of Mathematical Problems.New York:Interscience,1960
[20]Wang C,Xu T Z.Hyers-Ulam stability of differentiation operator on Hilbert spaces of entire functions.J Funct Spaces,2014,2014:1-6
[21]Wang C,Xu T Z.Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives.Appl Math,2015,60:383-393
[22]Wang C,Xu T Z.Hyers-Ulam stability of differential operators on reproducing kernel function spaces.Complex Anal Oper Theory,2016,10:795-813
[23]Wang C,Xu T Z.Hyers-Ulam stability of a class of fractional linear differential equationsz.Kodai Math J,2015,38:510-520
[24]Xu T Z,Wang C,Rassias T M.On the stability of multi-additive mappings in non-Archimedean normed spaces.J Comput Anal Appl,2015,18:1102-1110
[25]Xu T Z.Approximate multi-Jensen,multi-Euler-Lagrange additive and quadratic mappings in n-Banach spaces.Abstr Appl Anal,2013,2013:1-12
[26]Xu T Z.On the stability of multi-Jensen mappings in/3-normed spaces.Appl Math Lett,2012,25:1866-1870
[27]成立花,连铁艳.Banach空间三次方程的稳定性问题.数学物理学报.2014,34A(2):266-273Cheng L H,Lian T Y.Stability of cubic functional equations in Banach spaces.Acta.Math Sci,2014,34A(2):266-273