This paper studies one-parameter formal deformations of Hom-Lie-Yamaguti algebras. The first, second, and third cohomology groups on Hom-Lie-Yamaguti algebras extending ones on Lie-Yamaguti algebras are provided. It is proved that first and second cohomology groups are suitable to the deformation theory involving infinitesimals, equivalent deformations, and rigidity. However, the third cohomology group is not suitable for the obstructions.

We introduce the relevant concepts ofn-ary multiplicative Hom-Nambu-Lie superalgebras and construct three classes ofn-ary multiplicative Hom-Nambu-Lie superalgebras. As a generalization of the notion of derivations forn-ary multiplicative Hom-Nambu-Lie algebras, we discuss the derivations ofn-ary multiplicative Hom-Nambu-Lie superalgebras. In addition, the theory of one parameter formal deformation ofn-ary multiplicative Hom-Nambu-Lie superalgebras is developed by choosing a suitable cohomology.