  Many problems in nonlinear analysis and optimization, among them variational
inequalities and minimization of convex functions, can be reduced to finding
zeros (namely, roots) of set-valued operators. Hence numerous algorithms have
been devised in order to achieve this task. A lot of these algorithms are
inexact in the sense that they allow perturbations to appear during the
iterative process, and hence they enable one to better deal with noise and
computational errors, as well as superiorization. For many years a certain
fundamental question has remained open regarding many of these known inexact
algorithmic schemes in various finite and infinite dimensional settings, namely
whether there exist sequences satisfying these inexact schemes when errors
appear. We provide a positive answer to this question. Our results also show
that various theorems discussing the convergence of these inexact schemes have
a genuine merit beyond the exact case. As a by-product we solve the standard
and the strongly implicit inexact resolvent inclusion problems, introduce a
promising class of functions (fully Legendre functions), establish continuous
dependence (stability) properties of the solution of the inexact resolvent
inclusion problem and continuity properties of the protoresolvent, and
generalize the notion of strong monotonicity.
