Author(s)

Nguyen Van Kinh

Faculty of Information Technology, Gia Dinh University, Ho Chi Minh City, Vietnam.

Let $A:D\left(A\right)\subset X\to Y$ be a linear, closed, densely defined unbounded operator, where $X$ and $Y$ are Hilbert spaces. Assume that $A$ is not boundedly invertible. If equation (1) $Au=f$ is solvable, and $\Vert f_{\delta }-f\Vert \le \delta$ then the following results are provided: Problem $F_{\alpha ,\delta }\left(u\right):={\Vert Au-f_{\delta }\Vert }^2+\alpha {\Vert u\Vert }^2$ has a unique global minimizer $u_{\alpha ,\delta }$ for any $f_{\delta }\in Y$ , and $u_{\alpha ,\delta }=A^{*}{\left(A{A}^{*}+\alpha I_Y\right)}^{-1}{f}_{\delta }$ . Then there is a function $\alpha \left(\delta \right)$ , $\underset{\delta \to 0}{\text{lim}}\alpha \left(\delta \right)=0$ such that $\underset{\delta \to 0}{\text{lim}}\Vert u_{\alpha \left(\delta \right),\delta }-x_0\Vert =0$ , where $x_0$ is the unique minimal-norm solution to (1). In this paper we introduce the regularization method solving Equation (1) with $A$ being a linear, closed, densely defined unbounded operator. At the same time, an application is given to the weak derivative operator equation.

Ill-Posed Problem, Regularization Method, Unbounded Linear Operator

Kinh, N. (2025) On the Regularization Method to Stable Approximate Solution of Equations of the First Kind with Unbounded Operators. Open Journal of Optimization, 14, 1-9. doi: 10.4236/ojop.2025.141001.