On iterative methods of gradient type for solving nonlinear ill-posed equations

Iterative methods of gradient type for the approximate solution of noisy nonlinear equations without the property of regularity are proposed and investigated. We prove the convergence of the approximations generated by the methods to a neighborhood of the solution with a diameter proportional to the magnitude of errors in the input data and in a sourcewise representation of the initial residual. This is ensured by a suitable combination of the method of gradient descent for a residual functional with approximate projecting onto special finite-dimensional subspaces.