We introduce the notions of generalized and weighted generalized $ψ$-estimators as unique points of sign change of some appropriate functions, and we give necessary as well as sufficient conditions for their existence. We also derive a set of sufficient conditions under which the so-called $ψ$-expectation function has a unique point of sign change. We present several examples from statistical estimation theory, where our results are well-applicable. For example, we consider the cases of empirical quantiles, empirical expectiles, some $ψ$-estimators that are important in robust statistics, and some examples from maximum likelihood theory as well. Further, we introduce Bajraktarević-type (in particular, quasi-arithmetic-type) $ψ$-estimators. Our results specialized to $ψ$-estimators with a function $ψ$ being continuous in its second variable provide new results for (usual) $ψ$-estimators (also called Z-estimators).