Gibali [J. Nonlinear Anal. Optim., 2015, 6(1): 41‒51] presented a self-adaptive subgradient extragradient projection method for solving variational inequalities without Lipschitz continuity, where its next iterative point was obtained by projecting a vector onto a specific half-space. In this paper, we present new kinds of self-adaptive subgradient extragradient projection methods by using a new descent direction. With the help of the techniques in the method of He and Liao [J. Optim. Theory Appl, 2002, 112(1): 111‒128], we get a longer step-size for these kinds of algorithms, which proves the global convergence of the generated sequence. Numerical results show that these kinds of extragradient subgradient projection methods are less dependent on the choice of the initial point, the dimension of the variational inequalities, and the tolerance of accuracy than the known methods. Moreover, the new methods proposed in this paper outperform (with respect to the number of iterations and cpu-time) the method presented by Gibali.

Let G=(V,E,F) be a connected loopless plane graph, with vertex set V, edge set E, and face set F. For any adjacent faces e₁ and e₂, if they are incident to the same face and appear consecutively on the edge of that face, then it is said that e₁ and e₂ are facially adjacent. A plane graph G is called weakly edge-face k-colorable indicating that there is a mapping \pi :E\cup F\to\{1,2,\dots ,k\} such that any two incident edges and faces, adjacent faces, and facially adjacent edges receive distinct colors. The weakly edge-face chromatic number of G, denoted by {\overline{\chi }}_{ef}\left(G\right), is defined to be the smallest integer k such that G has a weakly edge-face k-coloring. In 2016, Fabrici conjectured that every connected, loopless, and bridgeless plane graph was weakly edge-face 5-colorable. In this paper, a sufficient condition is provided for the foregoing conjecture to prove that Halin graphs are weakly edge-face 5-colorable in which the upper bound 5 is the best possible.

In this paper, an iterative algorithm is introduced to find a common solution for the split variational inclusion problem and the fixed-point problem of a countable family of demicontractive mappings in Hilbert spaces. As a result, strong convergence theorem concerning the common element along with its applications and numerical examples is obtained.

We introduce some fundamental principles and classic theorems in the research of complex geometry, with emphasis on some major research fields and significant progresses in Kähler geometry in the past decades.