scholarly journals Some Iterative Methods for Solving Nonconvex Bifunction Equilibrium Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Saira Zainab
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Iterative Methods ◽  
Equilibrium Problems ◽  
Implicit Method ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
New Class ◽  
Special Cases ◽  
Proof Of Convergence

We introduce and consider a new class of equilibrium problems and variational inequalities involving bifunction, which is called the nonconvex bifunction equilibrium variational inequality. We suggest and analyze some iterative methods for solving the nonconvex bifunction equilibrium variational inequalities using the auxiliary principle technique. We prove that the convergence of implicit method requires only monotonicity. Some special cases are also considered. Our proof of convergence is very simple. Results proved in this paper may stimulate further research in this dynamic field.

scholarly journals Iterative Schemes for a Class of Mixed Trifunction Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Eisa Al-Said
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Iterative Methods ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
Iterative Schemes ◽  
New Class ◽  
Special Cases

We use the auxiliary principle technique to suggest and analyze some iterative methods for solving a new class of variational inequalities, which is called the mixed trifunction variational inequality. The mixed trifunction variational inequality includes the trifunction variational inequalities and the classical variational inequalities as special cases. Convergence of these iterative methods is proved under very mild and suitable assumptions. Several special cases are also considered. Results proved in this paper continue to hold for these known and new classes of variational inequalities and its variant forms.

scholarly journals Mixed quasi invex equilibrium problems

2004 ◽  
Vol 2004 (57) ◽  
pp. 3057-3067 ◽  
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Variational Inequalities ◽  
Equilibrium Problems ◽  
Strong Monotonicity ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
Iterative Schemes ◽  
New Class ◽  
Special Cases ◽  
Weaker Conditions

We introduce a new class of equilibrium problems, known asmixed quasi invex equilibrium(orequilibrium-like) problems. This class of invex equilibrium problems includes equilibrium problems, variational inequalities, and variational-like inequalities as special cases. Several iterative schemes for solving invex equilibrium problems are suggested and analyzed using the auxiliary principle technique. It is shown that the convergence of these iterative schemes requires either pseudomonotonicity or partially relaxed strong monotonicity, which are weaker conditions than the previous ones. As special cases, we also obtained the correct forms of the algorithms for solving variational-like inequalities, which have been considered in the setting of convexity. In fact, our results represent significant and important refinements of the previously known results.

scholarly journals Some New Classes of Extended General Mixed Quasi-Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Auxiliary Principle ◽  
Existence Of A Solution ◽  
Auxiliary Principle Technique ◽  
New Class ◽  
Special Cases ◽  
Quasi Variational Inequality

We consider and study a new class of variational inequality, which is called the extended general mixed quas-variational inequality. We use the auxiliary principle technique to study the existence of a solution of the extended general mixed quasi-variational inequality. Several special cases are also discussed. Results proved in this paper may stimulate further research in this area.

scholarly journals On a class of multivalued variational inequalities

1998 ◽  
Vol 11 (1) ◽  
pp. 79-93 ◽  
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Variational Inequalities ◽  
Optimization Problems ◽  
Iterative Algorithms ◽  
Auxiliary Principle ◽  
Existence Of A Solution ◽  
Auxiliary Principle Technique ◽  
New Class ◽  
Special Cases ◽  
Projection Techniques ◽  
Multivalued Variational Inequalities

In this paper, we introduce and study a new class of variational inequalities, which are called multivalued variational inequalities. These variational inequalities include as special cases, the previously known classes of variational inequalities. Using projection techniques, we show that multivalued variational inequalities are equivalent to fixed point problems and Wiener-Hopf equations. These alternate formulations are used to suggest a number of iterative algorithms for solving multivalued variational inequalities. We also consider the auxiliary principle technique to study the existence of a solution of multivalued variational inequalities and suggest a novel iterative algorithm. In addition, we have shown that the auxiliary principle technique can be used to find the equivalent differentiable optimization problems for multivalued variational inequalities. Convergence analysis is also discussed.

scholarly journals Trifunction Bihemivariational Inequalities

2021 ◽  
pp. 287-313
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Hemivariational Inequalities ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
New Class ◽  
Mild Conditions ◽  
Special Cases

In this paper, we consider a new class of hemivariational inequalities, which is called the trifunction bihemivariational inequality. We suggest and analyze some iterative methods for solving the trifunction bihemivariational inequality using the auxiliary principle technique. The convergence analysis of these iterative methods is also considered under some mild conditions. Several special cases are also considered. Results proved in this paper can be viewed as a refinement and improvement of the known results.

scholarly journals General biconvex functions and bivariational inequalities

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Variational Inequalities ◽  
Convergence Analysis ◽  
Optimality Conditions ◽  
Banach Spaces ◽  
Higher Order ◽  
Implicit Method ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
New Concepts ◽  
Special Cases

<p style='text-indent:20px;'>In this paper, we define and introduce some new concepts of the higher order strongly general biconvex functions involving the arbitrary bifunction and a function. Some new relationships among various concepts of higher order strongly general biconvex functions have been established. It is shown that the new parallelogram laws for Banach spaces can be obtained as applications of higher order strongly affine general biconvex functions, which is itself an novel application. It is proved that the optimality conditions of the higher order strongly general biconvex functions are characterized by a class of variational inequalities, which is called the higher order strongly general bivariational inequality. Auxiliary principle technique is used to suggest an implicit method for solving strongly general bivariational inequalities. Convergence analysis of the proposed method is investigated using the pseudo-monotonicity of the operator. Some special cases also discussed. Results obtained in this paper can be viewed as refinement and improvement of previously known results.</p>

scholarly journals Hemiequilibrium problems

2004 ◽  
Vol 2004 (3) ◽  
pp. 235-244 ◽  
Author(s):  
Muhammad Aslam Noor
Keyword(s):  
Equilibrium Problems ◽  
Iterative Algorithms ◽  
New Method ◽  
Hemivariational Inequalities ◽  
Strong Monotonicity ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
New Class ◽  
Special Cases ◽  
Special Case

We consider a new class of equilibrium problems, known as hemiequilibrium problems. Using the auxiliary principle technique, we suggest and analyze a class of iterative algorithms for solving hemiequilibrium problems, the convergence of which requires either pseudomonotonicity or partially relaxed strong monotonicity. As a special case, we obtain a new method for hemivariational inequalities. Since hemiequilibrium problems include hemivariational inequalities and equilibrium problems as special cases, the results proved in this paper still hold for these problems.

scholarly journals Implicit Methods for Equilibrium Problems on Hadamard Manifolds

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Saira Zainab ◽  
Yonghong Yao
Keyword(s):  
Equilibrium Problems ◽  
Implicit Method ◽  
Hadamard Manifolds ◽  
Implicit Methods ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
Special Cases

We use the auxiliary principle technique to suggest and analyze an implicit method for solving the equilibrium problems on Hadamard manifolds. The convergence of this new implicit method requires only the speudomonotonicity, which is a weaker condition than monotonicity. Some special cases are also considered.

scholarly journals Existence Results for General Mixed Quasivariational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Variational Inequality ◽  
Existence Results ◽  
Quasivariational Inequality ◽  
Quasivariational Inequalities ◽  
Auxiliary Principle ◽  
Existence Of A Solution ◽  
Auxiliary Principle Technique ◽  
New Class ◽  
Special Cases

We consider and study a new class of variational inequality, which is called the general mixed quasivariational inequality. We use the auxiliary principle technique to study the existence of a solution of the general mixed quasivariational inequality. Several special cases are also discussed. Results proved in this paper may stimulate further research in this area.

scholarly journals SOME NONLINEAR VARIATIONAL INEQUALITIES

1995 ◽  
Vol 26 (2) ◽  
pp. 97-105
Author(s):  
MUHAMMAD ASLAM NOOR
Keyword(s):  
Variational Inequalities ◽  
Auxiliary Principle ◽  
Existence Of A Solution ◽  
Auxiliary Principle Technique ◽  
New Class ◽  
Special Cases

In this paper, we introduce and study a new class of variational in- equalities. Using the auxiliary principle technique, we prove the existence of a solution of this class of variational inequalities and suggest a new and novel itera- tive algorithm. Several special cases, which can be obtained from the main results, are also discussed.

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