Some problems in the fixed point theory

In this paper we present some of my favorite problems, all the time open, in the fixed point theory. These problems are in connection with the following two: Which properties have the fixed point equations for which an iterative algorithm is convergent? Let us have a fixed point theorem, T, and an operator f (single or multivalued) which does not satisfy the conditions in T. In which conditions the operator f has an invariant subset Y such that the restriction of f to Y, fY, satisfies the conditions of T