Algorithms for General Monotone Mixed Variational Inequalities

AbstractIn this paper, we suggest and analyze some new iterative methods for solving general monotone mixed variational inequalities, which are being used to study odd-order and nonsymmetric boundary value problems arising in pure and applied sciences. These new methods can be viewed as generalizations and extensions of the methods of He, Solodov and Tseng, and Noor for solving monotone (mixed) variational inequalities

An iterative algorithm for variational inequalities

AbstractIn this paper, we introduce and study a new class of nonlinear variational inequalities. This new class enables us to apply variational techniques to the solution of differential equations of both odd and even orders. A projection method is used to suggest an iterative algorithm for finding the approximate solution of this class. We also discuss the convergence criteria of the proposed iterative algorithm. Several special cases are discussed, which can be obtained from the general result

Projection-proximal methods for general variational inequalities

AbstractIn this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems

Iterative methods for a class of complementarity problems

AbstractIn this paper, we propose and study an algorithm for a new class of complementarity problems of finding u ϵ Rn such that u ⩾ 0, Tu + A(u) ⩾ 0; (u, Tu + A(u)) = 0, where T is a continuous mapping and A is a nonlinear transformation from Rn into itself. It is proved that the approximate solution obtained from the iterative scheme converges to the exact solution. Several special cases are also discussed

Three-Step Iterative Algorithms for Multivalued Quasi Variational Inclusions

AbstractIn this paper, we suggest and analyze some new classes of three-step iterative algorithms for solving multivalued quasi variational inclusions by using the resolvent equations technique. New iterative algorithms include the Ishikawa, Mann, and Noor iterations for solving variational inclusions (inequalities) and optimization problems as special cases. The results obtained in this paper represent an improvement and a significant refinement of previously known results

Some iterative methods for solving a system of nonlinear equations

AbstractIn this paper, we suggest and analyze two new two-step iterative methods for solving the system of nonlinear equations using quadrature formulas. We prove that these new methods have cubic convergence. Several numerical examples are given to illustrate the efficiency and the performance of the new iterative methods. These new iterative methods may be viewed as an extension and generalizations of the existing methods for solving the system of nonlinear equations

Harmonic Variational Inequalities

In this paper, we consider a new class of variational inequalities, which is called the harmonic variational inequality. It is shown that that the minimum of a differentiable harmonic convex function on the harmonic convex set can be characterized by the harmonic variational inequality. We use the auxiliary principle technique to discuss the existence of a solution of the harmonic variational inequality. Results proved in this paper may stimulate further research in this field

On sensitivity analysis of general variational inequalities

It is well known that the Wiener-Hopf equations are equivalent to the general variational inequalities. We use this alternative equivalent formulation to study the sensitivity of the general variational inequalities without assuming the differentiability of the given data. Since the general variational inequalities include classical variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. In fact, our results can be considered as a significant extension of previously known results

Proximal Point Methods for Solving Mixed Variational Inequalities on the Hadamard Manifolds

We use the auxiliary principle technique to suggest and analyze a proximal point method for solving the mixed variational inequalities on the Hadamard manifold. It is shown that the convergence of this proximal point method needs only pseudomonotonicity, which is a weaker condition than monotonicity. Some special cases are also considered. Results can be viewed as refinement and improvement of previously known results

New classes of exponentially general nonconvex variational inequalities

In this paper, some new classes of exponentially general nonconvex variational inequalities are introduced and investigated. Several special cases are discussed as applications of these nonconvex variational inequalities. Projection technique is used to establish the equivalence between the non covex variational inequalities and fixed point problem. This equivalent formulation is used to discuss the existence of the solution. Several inertial type methods are suggested and analyzed for solving exponentially general nonconvex variational inequalities. using the technique of the projection operator and dynamical systems. Convergence analysis of the iterative methods is analyzed under suitable and appropriate weak conditions. In this sense, our result can be viewed as improvement and refinement of the previously known results. Our methods of proof are very simple as compared with other techniques