Regularization of Ill-Posed Problems with Both Data and Operator are Perturbed

Abstract

In this note, ill-posed problems are studied in the case when mapping A (not necessarily linear) is given only approximately. We prove the existence of three sequences: a sequence of operators $$\left( A_{n}\right) _{n}$$ , a sequence of arguments $$\left( x_{n}\right) _{n}$$ , and a sequence of second members $$\left( u_{n}\right) _{n}$$ converging on the exact values of the operator equation $$Ax=u$$ .