On a real Banach space, nonlinear equations with accretive operators are considered. For such equations, the notion of approximate \(K\)-controllability (or controllability with pre-assigned responses) is introduced and sufficient conditions of such controllabilities are validated. Various definitions of solutions are considered. Among them are so-called ``Evans-solutions'' and ``Kato-solutions'' for fully nonlinear problems and ``mild solutions'' for semilinear problems. The ``approximate \(K\)-controllability'' is also introduced with usage of Leray-Schauder theory. An application in the field of partial differential equations is represented.