Descent methods for convex optimization problems in Banach spaces

where f : E→ R is a convex function; see, for example, [1, 2, 8] and the references therein. It is well known that standard iterative methods for solving (1.2), which are designed for finite-dimensional optimization problems, cannot guarantee strong convergence of their iteration sequences to a solution of the initial problem if the cost function does not possess strengthened convexity properties such as strong convexity. Usually, these methods provide only weak convergence to a solution. However, such a convergence is not satisfactory for many real problems, which are ill-posed in general, since even small perturbations of the initial data may cause great changes in solutions. These questions are crucial for developing stable solution methods. Strong convergence ensuring stability and continuous dependence of the initial data can be obtained via the regularization approach