Stable Iterative Processes

This chapter is devoted to construction and investigation of stable iterative processes for irregular nonlinear equations in a Hilbert space. Here the stability is understood with respect both to variations of a starting point in a neighborhood of a solution and to errors in input data and in sourcewise–like representations of the solution. The stability of the processes means that iterative points generated by them are attracted to a neighborhood of the solution, as an iteration number increases. Diameters of the attracting neighborhoods arise to be proportional to levels of the mentioned errors. Therefore there is no a necessity to equip such iterative processes with stopping criterions. Let us remember that in Sections 4.1, 4.3, and 4.5, the construction of approximations adequate to error levels was ensured just by the stopping of iterative or continuous processes at an appropriate point.