The Structure of the Set of Extremal Solutions of Ill-Posed Operator Equation Tx = y with codim R(T) = 1

Abstract

In this paper, we study the problem of the ill-posed operator equation Tx = y with codim R(T) = 1 in normed linear spaces. The structure of the set of extremal solutions of the equation has been obtained by the maximal elements of the ones in N(T∗) and the generalized inverse T ⁺ of T. Furthermore, the representation of the set of the extremal solutions of the equation is given formally.

Keywords:

• Extremal solutions
• Ill-posed operator equation
• Normed linear spaces
• Structure of the solution set

Mathematics Subject Classification 2000:

• 35B32

ACKNOWLEDGMENT

This research was supported by the National Nature Science Foundation of China under grant 10671049.