In this paper, we propose a Split nonconvex variational inequality problem which is a natural extension of Split convex variational inequality problem in two different Hilbert spaces. Relying on the prox-regularity notion, we introduce and establish the convergence of an iterative method for the new Split nonconvex variational inequality problem. Further, we also establish the convergence of an iterative method for the Split convex variational inequality problem. The results presented in this paper are new and different form the previously known results for nonconvex (convex) variational inequality problems. These results also generalize, unify, and improve the previously known results of this area.

In this paper, an efficient numerical model for solution of the two-dimensional unsteady dam-break problem is described. The model solves the shallow water equations through Characteristic-Based Split (CBS) finite element method. The formulation of the model is based upon the fractional time step technique primarily used in the finite difference method for the incompressible Navier-Stokes equations. In addition to well-known advantages of the finite element discretization in introducing complex geometries and making accurate results near the boundaries, the CBS utilizes interesting advantages. These include the ability of the method to simulate both compressible and incompressible flows using the same formulation. Improved stability of the CBS algorithm along with its capability to simulate both sub-and super-critical flows are other main advantages of the method. These useful advantages of the algorithm introduce the CBS as a unique procedure to solve fluid dynamics problems under various conditions. Since dam-break problem has principally a high non-linear nature, the model is verified firstly by modeling one-dimensional problems of dam-break and bore formation problems. Furthermore, application of the model to a two-dimensional hypothetical dam-break problem shows the robustness and efficiency of the procedure. Despite the high non-linearity nature of the solved problems, the computational results, compared with the analytical solutions and reported results of other numerical models, indicate the favorable performance of the used procedure in modeling the dam-break problems.

In this paper, we introduce a new iterative algorithm of inertial form for approximating the solution of Split Variational Inclusion problem (SVIP) involving accrective operators in Banach space. Mo-tivated by the inertial technique, we incorporate the inertial term to accelerate the convergence of the proposed method. Under standard and mild assumption of monotonicity of the SVIP associated mappings, we establish the weak convergence of the sequence generated by our algorithm. Some applications and numerical example are presented to illustrate the performance of our method as well as comparing it with the non-inertial version.

A new iterative scheme for approximating a solution of the Split equality monotone inclusion problem (SEMIP) of maximal monotone mappings in the setting of Banach spaces is introduced. Strong convergence of the sequence generated by the proposed scheme to a solution of the SEMIP is then derived without prior knowledge of operator norms of the linear operators involved. In addition, we provide applications of our method and provide numerical examples to illustrate the convergence of the proposed scheme. Our results generalize, improve and extend many results in the literature.

Purpose: In this paper, we introduce and study an iterative method to approximate a common solution of a Split generalized equilibrium problem and a fixed point problem for a nonexpansive semigroup in real Hilbert spaces.Methods: We prove a strong convergence theorem of the iterative algorithm in Hilbert spaces under certain mild conditions.Results: We obtain a strong convergence result for approximating a common solution of a Split generalized equilibrium problem and a fixed point problem for a nonexpansive semigroup in real Hilbert spaces, which is a unique solution of a variational inequality problem. Further, we obtain some consequences of our main result.Conclusions: The results presented in this paper are the supplement, extension, and generalization of results in the study of Plubtieng and Punpaeng and that of Cianciaruso et al. The approach of the proof given in this paper is also different.