How to solve the equation AuBu+Cu=f

摘要

In this paper, we first discuss the problem of how to solve a continuous linear operator equation in separable Hilbert space. We give the necessary and sufficient condition for the existence of solution, and the method to decide the existence for solutions of the equation is also given. If the solution exists, we give the representation and approxima of the minimal normal solution of the equation and formula are obtained. We conduct some research of how to solve the operator equation AuBu+Cu=f. We give the factorization method and characteristic value method to solve such an equation.