Perturbation Of Quasi-linear Generalized Inverse And Constrained Extremal Solution Of Linear Inclusions In Banach Space

The research on the generalized inverse theory is used to solve the linear ill-posed equations(including linear algebraic equations,differential equations,partial differential equations,integral equations,etc.).The content of the research for the generalized inverse theory is rich,and the most prominent one is the theoretical research on a variety of projection generalized inverse and its application.Since the 30’s of last century,the research results about this area are numerous,among which the most notable is the perturbation problem of the projection generalized inverse,including perturbation analysis of linear generalized inverse between Hilbert spaces,perturbation analysis of Moore-Penrose generalized inverse,perturbation analysis of generalized inverses of bounded linear operators in Banach spaces and perturbation analysis of metric generalized inverse of bounded linear operators in Banach spaces.The result of perturbation analysis on generalized inverse of operator has been widely used in many mathematical fields,which has attracted the attention of the academic circles,especially in computational mathematics,bifurcation theory in nonlinear analysis,generalized transversality research of Banach manifold etc..Because the linear generalized inverse of linear operator in Banach space is not suitable for ill-posed linear operator equations to establish extremal solution,minimum norm solution and the optimal approximate solution,the study of metric generalized inverse and its perturbation analysis in Banach spaces is particularly important.The primary purpose of this dissertation is to build a unified theory for the perturbation analysis of projection generalized inverse of closed linear operator.It not only includes the Moore-Penrose generalized inverse in Hilbert spaces,and the linear oblique projection generalized inverse of closed linear operator in Banach spaces,but also including the perturbation theory of metric generalized inverse of closed linear operator in Banach spaces.In short,this dissertation is to study the perturbation analysis of the Moore-Penrose quasi-linear projection generalized inverse of the closed linear operator in the Banach space and the research objective is to establish a unified theory of perturbation analysis.In this paper,first we study the perturbation of the Moore-Penrose quasi-linear projection generalized inverse from two aspects:On one hand,for a closed linear operator T with domain is dense under the conditions which the perturbation operator δT is T-bounded and specific conditions combined with the generalized Banach lemma for a homogeneous operator,we establish a unique perturbation theorem about the Moore-Penrose quasi-linear projection generalized inverse of closed linear operator and the characterizations of perturbation bounds.Such perturbation results make a large number of well-known results into a special case.On the other hand,also for a closed linear operator T with domain is dense using the generalized Neumann lemma and the perturbation operator δT satisfying a specific inequality,we give a new perturbation theorem and three inequalities for error estimates to Moore-Penrose metric generalized inverse of bounded linear operator and Moore-Penrose quasi-linear projection generalized inverse of closed linear operator.The result of this perturbation has also extended the corresponding known results in the reference.By the above two results,we construct a theoretical framework for the perturbation analysis of the Moore-Penrose bounded quasi-linear generalized inverse of the closed linear operator.At last,in this dissertation,we establish the representation form of constrained extremal solution for linear inclusions in Banach spaces by using the algebraic operator parts,the metric generalized inverse of multi-valued linear operator,and the dual mapping in Banach spaces.It follows from the main results in this dissertation that we may get the constrained extremal solution of multi-valued linear inclusions,by using the extremal solution of some interrelated multi-valued linear inclusions in the same spaces.The result is including the least-squares solution of linear inclusions in Hilbert spaces(unconstrained and constrained),and contains the results of the extremal solutions(unconstrained)of the linear inclusions in Banach spaces.As a result,it is seen that the application of the metric generalized inverse of multi-valued linear operator in Banach spaces.