Deformations and extensions of modified $\lambda$-differential $3$-Lie Algebras

In this paper, we introduce the representation of modified $\lambda$-differential $3$-Lie algebras and define the cohomology of modified $\lambda$-differential $3$-Lie algebras with coefficients in a representation. As applications of the proposed cohomology theory, we study linear deformations, abelian extensions and $T^{\ast }$-extensions of modified $\lambda$-differential $3$-Lie algebras.