Perturbation Analysis For The Solutions Of Operator Equation With Relative Bounded Perturbation

In this paper we study the unbounded operator equation Tx=b and its perturbation with a T-bounded perturbation. In order to find the solutions of Tx=b we firstly define the generalized inverse of an operator and give some conditions that guarantee the existence of generalized inverse; Secondly we define new norms on the domain of T. By means of this,we can transform the unbounded problem into a bounded operator problem, so we need to research some Banach properties whether keep under new norm.In this paper we prove the property P still holds under new norm; then we study stable perturbation of unbounded operator with T-bounded perturbation, by mean of this we can build the generalized inverse of perturbed operator with T-bounded perturbation; Thirdly, we study the analysis of the equation of Tx=b and its perturbation. In this part, we mainly discuss the minimum norm solution perturbation of operator equation Tx=b, we give the upper bounded and lower bounded of difference of the minimum norm solution of the equation and perturbation equation.At last we research a special example,which gives us an application of the theory that we have discussed.