In this paper, we study non-abelian extensions of 3-Leibniz algebras through Maurer-Cartan elements. We construct a differential graded Lie algebra and prove that there is a one-to-one correspondence between the isomorphism classes of non-abelian extensions in 3-Leibniz algebras and the equivalence classes of Maurer-Cartan elements in this differential graded Lie algebra. And also the Leibniz algebra structure on the space of fundamental elements of 3-Leibniz algebras is analyzed. It is proved that the non-abelian extension of 3-Leibniz algebras induce the non-abelian extensions of Leibniz algebras.

In this paper, we prove an infinite dimensional KAM theorem and apply it to study 2-dimensional nonlinear Schrödinger equations with different large forcing terms and (2p + 1)-nonlinearities

　　　　　 \mathrm{i}{u}_t-\mathrm{\Delta }u+{\phi }_1\left({\overline{\omega }}_{1}t\right)u+{\phi }_2\left({\overline{\omega }}_{2}t\right)|u{|}^{2p}u=0,t\in \mathbb{R},x\in {\mathbb{T}}^2

under periodic boundary conditions. As a result, the existence of a Whitney smooth family of small-amplitude reducible quasi-periodic solutions is obtained.

Given a list assignment of L to graph G, assign a list L(v) of colors to each v\in V\left(G\right). An (L, d)*-coloring is a mapping π that assigns a color π(v) \inL(v) to each vertex v\inV(G) such that at most d neighbors of v receive the color v. If there exists an (L, d)*-coloring for every list assignment L with |L\left(v\right)|⩾k for all v\in V\left(G\right), then G is called to be (k, d)^*-choosable. In this paper, we prove every planar graph G without adjacent k-cycles is (3, 1)*-choosable, where k\in {3, 4, 5}.