  In this paper, we introduce compatible ternary Leibniz algebras,
(dual)Nijenhuis pairs from the second-order deformation of ternary Leibniz
algebras with a representarion and study the invariance of certains operators
(generalized derivations, Rota-Baxter operators, Reynolds operators, element of
centroid, averaging operators, Nijenhuis operators) whenever we go from Leibniz
algebras to ternary Leibniz algebras. We also give contructions of (compatible)
ternary Leibniz algebras either from averaging operators, modules or
O-operator.
