关于Jensen泛函方程Hyers-Ulam稳定性的一个结论
|
摘要
对于Jensen泛函方程的Hyers-Ulam稳定性的证明,S.-M.Jung在1998年完成了对0≤p<1及p>1情形的证明,并给出反例说明当p=1时,Jensen泛函方程不具有Hyers-Ulam稳定性.利用Janusz Brzdek给出的一个不动点方法讨论Jensen泛函方程当p<0时的情形,并给出了结论,同时进一步讨论其在限定区域上的Hyers-Ulam稳定性.
S.M.Jung had Proved the Hyers-Ulam stability of Jensen functional equation in 1998 for the case 0≤P<1 and P>1,and gave an counterexamples to prove that while P=1 Jensen functional equation didn't have the character of Hyers-Ulam stability. In this article we u-sed a method of fixed point theorem that proved by Janusz Brzdek,and proved the Jensen functional equation for the case P<0,and gave a conclusion,while proved its Hyers-Ulam on a restrrict domain.
S.M.Jung had Proved the Hyers-Ulam stability of Jensen functional equation in 1998 for the case 0≤P<1 and P>1,and gave an counterexamples to prove that while P=1 Jensen functional equation didn't have the character of Hyers-Ulam stability. In this article we u-sed a method of fixed point theorem that proved by Janusz Brzdek,and proved the Jensen functional equation for the case P<0,and gave a conclusion,while proved its Hyers-Ulam on a restrrict domain.
引文
[1]Ulam S.M.Problem in Modern Mathematics[M].chapter6,John Wiey&Sons,New York,NY,USA,1940.
[2]Hyers D.H.On the stability of the linear functional equation[J].Proceedings of the American Mathematical society,1941,72(2):222-224.
[3]Rassias Th.M.On the stability of the linear mapping in Banach spaces[J].Proceedings of the American Mathematical society,1978,72(2):297-300.
[4]Jung S M.Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis[M].Springer,2011:155-158.
[5]Brzdk J,Chudziak J,Páles Z.A fixed point approach to stability of functional equations[J].Nonlin ear Analysis:Theory,Methods&Applications,2011,74(17):6728-6732.
[6]Forti G L.Comments on the core of the direct method for proving Hyers–Ulam stability of functional equations[J].Journal of Mathematical Analysis and Applications,2004,295(1):127-133.
[7]Forti G L.Elementary remarks on Ulam–Hyers stability of linear functional equations[J].Journal of Mathematical Analysis and Applications,2007,328(1):109-118.
[8]Sikorska J.On a direct method for proving the Hyers–Ulam stability of functional equations[J].Journal of Mathematical Analysis and Applications,2010,372(1):99-109.
[9]Brzdk J.Hyperstability of the Cauchy equa tion on restricted domains[J].Acta Mathematica Hungarica,1-10.
[10]Baker J A.The stability of certain functional equations[J].Proceedings of the American Mathematical Society,1991:729-732.
[11]Radu V.The fixed point alternative and the stability of functional equations[J].Fixed Point Theory,2003,4(1):91-96.
[2]Hyers D.H.On the stability of the linear functional equation[J].Proceedings of the American Mathematical society,1941,72(2):222-224.
[3]Rassias Th.M.On the stability of the linear mapping in Banach spaces[J].Proceedings of the American Mathematical society,1978,72(2):297-300.
[4]Jung S M.Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis[M].Springer,2011:155-158.
[5]Brzdk J,Chudziak J,Páles Z.A fixed point approach to stability of functional equations[J].Nonlin ear Analysis:Theory,Methods&Applications,2011,74(17):6728-6732.
[6]Forti G L.Comments on the core of the direct method for proving Hyers–Ulam stability of functional equations[J].Journal of Mathematical Analysis and Applications,2004,295(1):127-133.
[7]Forti G L.Elementary remarks on Ulam–Hyers stability of linear functional equations[J].Journal of Mathematical Analysis and Applications,2007,328(1):109-118.
[8]Sikorska J.On a direct method for proving the Hyers–Ulam stability of functional equations[J].Journal of Mathematical Analysis and Applications,2010,372(1):99-109.
[9]Brzdk J.Hyperstability of the Cauchy equa tion on restricted domains[J].Acta Mathematica Hungarica,1-10.
[10]Baker J A.The stability of certain functional equations[J].Proceedings of the American Mathematical Society,1991:729-732.
[11]Radu V.The fixed point alternative and the stability of functional equations[J].Fixed Point Theory,2003,4(1):91-96.