Generalized Inverse And Hyers-Ulam Stability

As we all know, it is difficult to obtain the exact solution for nonlinear problems. However, for the Hyers-Ulam stability system, this stability ensures the existence of approximate solutions, so it is unnecessary to get exact solutions of this system. It provides important theoretical basis for solving such nonlinear problems. The research of Hyers-Ulam stability originates from the stability for the general functional equations proposed by S. Ulam:When is it true that the solution of an equation differing slightly from a given one, must of necessity be close to the solution of the given equation?D. Hyers used the direct method and solved part of Ulam’s problem in the framework of Banach space. T. Rassias introduced the concept of unbounded Cauchy difference and extended the Hyers’s theorem to the case of approximate linear mappings. After that, such problems were called Hyers-Ulam (or Hyers-Ulam-Rassias) stability problem, which attracted many mathe-maticians’attention. Many important results are obtained. For instance, S. Jung studied the Hyers-Ulam stability of the Jensen functional equations and first order linear differential equations. T. Miura, S. Miyajima and S. Takahasi investigated the n-th order differential operator with constant coefficients and the first order linear differential operator with a continuous function as coefficients. Particularly, for the first order linear differential operator Th:C1R,X)â†'C(R,X), S. Takahasi, H. Takagi, T. Miura and S. Miyajima proved that the infimum KTh becomes the minimum of all Hyers-Ulam stability constants and proposed the following open question:what properties does the mapping hâ†'KT possesses? Moreover, in2003, H. Takagi, T. Miura and S. Takahasi investigated the Hyers-Ulam stability of bounded linear operators in Banach space. After then, G. Hirasawa and T. Miura gave some necessary and sufficient conditions under which a closed linear operators in a Hilbert space has the Hyers-Ulam stability.In this paper, we first establish a relationship between the Moore-Penrose inverse and the Hyers-Ulam stability in Hilbert space. Then we utilize the perturbation theory of the generalized inverse to determine the Hyers-Ulam stability constants in terms of the Moore-Penrose inverse and prove that the mapping Tâ†'KT is semi-continuous.Moreover,some necessary and sufficient conditions for the Hyers-Ulam stability constants to be continuous or bounded are obtained.Therefore,we give a partial solution to S.Takahasi and H.Takagi’s problem in this paper.Theorem Let X and Y be Hilbert spaces.Assume that T∈C(X,Y) has Hyers-Ulam stability and δT∈B(X,Y) satisfies that T=T+δT∈C(X,Y) has Hyers-Ulam stability.If||δT||·T(?)||≤(3+2(?)3),then the mapping Tâ†'KT is lower semi-continuous.Theorem Let X and Y be two Hilbert spaces.Assume that T∈C(X,Y)has Hyers-Ulam stability and T=T+δT∈C(X,Y)with δT∈L(X,Y) and D(T)(?)D(δT).If I+δTT(?):Yâ†'Y is bijective,then the following statements are equivalent:(1)B=T(?)(I+δTT(?))-1=(I+T(?)δT)-1T(?):Yâ†'X is a bounded generalized invere of Tï¼›(2)Y=R(T)⊕N(T(?))ï¼›(3)(I+T(?)δT)-1N(T)(?)N(T)ï¼›(4)X=R(T(?))⊕N(T).Moreover,if one of the above conditions is true,then R(T) is closed,so T has Hyers-Ulam stability and KT=||T(?)||,whereTheorem Let X and Y be two Hilbert spaces.Assume that T∈C(X,Y) has Hyers-Ulam stability and I+δTT(?):Yâ†'Y is bijective with δT∈B(X,Y),then the following statements are equivalent:(1)B=T(?)(I+δTT(?))-1=(I+T(?)δT)-1T(?):Yâ†'X is a bounded generalized inverse of Tï¼› (2)Y=R(T)⊕N(T(?))ï¼›(3)(I+T(?)δT)-1N(T)(?)N(T)ï¼›(4)X=R(T(?))⊕N(T)ï¼›(6) T has Hyers-Ulam stabilty and there exist M>0and ε>0,such that KT≤M for all||δT||≤ε；...