Some problems in the fixed point theory

Abstract

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 ?

Keywords

• ordered set,L-space,metric space,Banach space,Picard operator,weakly Picard operator,fixed point,fixed point structure,iterative algorithm,retraction-displacement condition,well-posedness of fixed point problem,Ostrowski property,global asymptotic stability,open problem,conjecture

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