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Nonlinear iterative methods for linear ill-posed problems in Banach spaces

We introduce and discuss nonlinear iterative methods to recover the minimum-norm solution of the operator equation Ax = y in Banach spaces X, Y, where A is a continuous linear operator from X to Y. The methods are nonlinear due to the use of duality mappings which reflect the geometrical aspects of the underlying spaces. The space X is required to be smooth and uniformly convex, whereas Y can be an arbitrary Banach space. The case of exact as well as approximate and disturbed data and operator are taken into consideration and we prove the strong convergence of the sequence of the iterates.