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 article, we apply three numerical methods to study the L∞-convergence of the Newton-multigrid method for parabolic quasi-Variational inequalities with a nonlinear right-hand side. To discretize the problem, we utilize a finite element method for the operator and Euler scheme for the time. To obtain the system discretization of the problem, we reformulate the parabolic quasi-Variational inequality as a Hamilton–Jacobi–Bellman equation. For linearizing the problem on the coarse grid, we employ Newton’s method as an external interior iteration of the Jacobian system. On the smooth grid, we apply the multigrid method as an interior iteration on the Jacobian system. Finally, we provide a proof for the L∞-convergence of the Newton-multigrid method for parabolic quasi-Variational inequalities with a nonlinear right-hand, by giving a numerical example for this problem.

Based on the time-fractional  Black-Scholes pricing model, the evaluation of an American-style option problem  can be formulated as  a   free boundary problem.  It is equivalent  to a time-fractional parabolic Variational inequality. Due to the time-fractional derivative involved in  the problem, increasing the computational cost for large final times has been expected in  the numerical   solution for this problem. In this paper, we want to propose a new adaptive numerical method to solve   this problem   accurately, with low computational cost. The presented method is based on interpolating wavelets family. An adaptive scheme in time discretization with an adaptive wavelet collocation method for space discretization has been used for  the given problem. We show  that combination of interpolating wavelet basis and finite difference method, makes an accurate structure to design an optimal adaptive mesh  for this problem. The presented computational mesh  by this method can prevent growing of computational cost by time. The  performance of the proposed method has been tested by means of some numerical experiments. We show that,  in comparison with the full grid algorithms, the presented adaptive algorithm can capture the priori unknown free boundary and is able to find the value of American put option price with high accuracy and reasonable CPU time.

In this paper,  a new version of a higher-order strongly convex function is introduced which is named    $(\omega-\sigma)$-higher-order strongly convex function. Sufficient conditions for the existence of minimum for  $(\omega-\sigma)$-higher order strongly convex function is provided. The vector version of $(\omega-\sigma)$-higher order strongly convex function is given and by using  KKM theory an existence results for a solution of it is proved. Moreover, the compactness of the solution set of the vector version of $(\omega-\sigma)$-higher order strongly convex function is investigated. The results of this article improve and extend the corresponding results presented in this area.

In this paper, we introduce a new iterative algorithm for approximating a common solution of certain class of multiple{sets split Variational inequality problems. The sequence of the proposed iterative algorithm is proved to converge strongly in Hilbert spaces. As application, we obtain some strong convergence results for some classes of multiple{sets split convex minimization problems.

In this article, we introduced an iterative scheme for finding a common element of the set of fixed points of a multi-valued hemicontractive-type mapping, the set of common solutions of a finite family of split equilibrium problems and the set of common solutions of a finite family of Variational inequality problems in real Hilbert spaces. Moreover, the sequence generated by the proposed algorithm is proved to be strongly convergent to a common solution of these three problems under mild conditions on parameters. Our results improve and generalize many well-known recent results existing in the literature in this field of research.