Biderivations and linear commuting maps on the restricted Cartan-type Lie algebras and

Let W and S denote the restricted Cartan-type Lie algebras W(n,1̲) and S(n,1̲) in a field of characteristic p>3, respectively. We prove that each anti-symmetric biderivation of W and S is inner. As applications of the anti-symmetric biderivations, it is shown that commuting maps on W and S are scalar multiplication maps. Moreover, we prove that the commuting automorphisms and dertivations of W and S are the identity mappings and zero mappings, respectively.