In this paper, we introduce the notion of Lie $n$-centralizers. We then give a description of Lie $n$-centralizers on a generalized matrix algebra and present the necessary and sufficient conditions for a Lie $n$-centralizer to be proper. As applications, we determine generalized Lie $n$-derivations on a generalized matrix algebra and Lie $n$-centralizers of some operator algebras.

This paper explores the relationship between $m$-derivations and Jordan $n$-derivations, introducing the concept of $m$-Jordan $n$-derivations on triangular rings. By employing the maximal left ring of quotients, we first conduct a detailed analysis of 3-Jordan $n$-derivations. Subsequently, using an inductive approach, we prove that under specific hypotheses, an $m$-Jordan $n$-derivation must be an extremal $m$-derivation. Finally, we apply the results we obtained to upper triangular matrix rings.

The aim of this paper was to provide a characterization of nonlinear generalized Lie $n$-higher derivations for a certain class of triangular algebras. It was shown that, under some mild conditions, each component $G_r$ of a nonlinear generalized Lie $n$-higher derivation $\{G_r{\}}_{r\in N}$ of the triangular algebra $\mathcal{U}$ could be expressed as the sum of an additive generalized higher derivation and a nonlinear mapping vanishing on all ( $n-1$)-th commutators on $\mathcal{U}$.