An iterative scheme for solving equations with locally $\sigma$-inverse monotone operators

TL;DR

This paper introduces an iterative method for solving ill-posed nonlinear equations involving locally σ-inverse monotone operators, including a stopping rule, and proves convergence to the minimal-norm solution.

Contribution

It presents a new iterative scheme with a discrepancy-based stopping rule for locally σ-inverse monotone operators, with proven convergence to the minimal-norm solution.

Findings

Existence of a solution satisfying the stopping rule is established.

The iterative scheme converges to the minimal-norm solution.

The stopping rule ensures reliable termination of the iteration.

Abstract

An iterative scheme for solving ill-posed nonlinear equations with locally $\sigma$-inverse monotone operators is studied in this paper. A stopping rule of discrepancy type is proposed. The existence of $u_{n_\delta}$ satisfying the proposed stopping rule is proved. The convergence of this element to the minimal-norm solution is justified mathematically.