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 New classes of preinvex functions and variational-like inequalities

Filomat ◽  
2021 ◽  
Vol 35 (6) ◽  
pp. 2081-2097
Author(s):  
Muhammad Noor ◽  
Khalida Noor
Keyword(s):  
Variational Inequalities ◽  
Monotone Operators ◽  
Higher Order ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
New Concepts ◽  
Special Cases ◽  
Preinvex Functions ◽  
Strongly Monotone Operators ◽  
And Function

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

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 New Trends in General Variational Inequalities

2020 ◽  
Vol 170 (1) ◽  
pp. 981-1064
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Michael Th. Rassias
Keyword(s):  
Variational Inequalities ◽  
Convex Functions ◽  
Optimization Problems ◽  
Higher Order ◽  
Auxiliary Principle ◽  
Open Problems ◽  
Auxiliary Principle Technique ◽  
Special Cases ◽  
General Variational Inequalities ◽  
General Convex

Abstract It is well known that general variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of unrelated problems, which arise in pure and applied sciences. In this paper, we present a number of new and known numerical techniques for solving general variational inequalities and equilibrium problems using various techniques including projection, Wiener-Hopf equations, dynamical systems, the auxiliary principle and the penalty function. General variational-like inequalities are introduced and investigated. Properties of higher order strongly general convex functions have been discussed. The auxiliary principle technique is used to suggest and analyze some iterative methods for solving higher order general variational inequalities. Some new classes of strongly exponentially general convex functions are introduced and discussed. Our proofs of convergence are very simple as compared with other methods. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Since the general variational inequalities include (quasi) variational inequalities and (quasi) implicit complementarity problems as special cases, these results continue to hold for these problems. Some numerical results are included to illustrate the efficiency of the proposed methods. Several open problems have been suggested for further research in these areas.

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 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 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 Strongly Convex Functions of Higher Order Involving Bifunction

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1028 ◽  
Author(s):  
Bandar B. Mohsen ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Mihai Postolache
Keyword(s):  
Banach Spaces ◽  
Convex Functions ◽  
Higher Order ◽  
Strongly Convex Functions ◽  
Strongly Convex ◽  
New Concepts ◽  
Special Cases ◽  
Novel Applications

Some new concepts of the higher order strongly convex functions involving an arbitrary bifuction are considered in this paper. Some properties of the higher order strongly convex functions are investigated under suitable conditions. Some important special cases are discussed. The parallelogram laws for Banach spaces are obtained as applications of higher order strongly affine convex functions as novel applications. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

scholarly journals Regularized Mixed Variational-Like Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Saira Zainab ◽  
Eisa Al-Said
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Regularization Method ◽  
Auxiliary Principle ◽  
Iterative Regularization ◽  
Auxiliary Principle Technique ◽  
Iterative Schemes ◽  
Special Cases

We use auxiliary principle technique coupled with iterative regularization method to suggest and analyze some new iterative methods for solving mixed variational-like inequalities. The convergence analysis of these new iterative schemes is considered under some suitable conditions. Some special cases are also discussed. Our method of proofs is very simple as compared with other methods. Our results represent a significant refinement of the previously known results.

scholarly journals Auxiliary Principle for a System of Generalized Set-Valued Nonlinear Mixed Variational-Like Inequalities in Banach Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Yangqing Qiu ◽  
Xiaolin Zhan ◽  
Luchuan Ceng
Keyword(s):  
Variational Inequalities ◽  
Banach Spaces ◽  
Iterative Algorithm ◽  
Original Problem ◽  
Auxiliary Problem ◽  
Uniformly Convex ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
Auxiliary Variational Inequalities ◽  
Set Valued Mappings

In this paper, the auxiliary principle technique is extended to study a system of generalized nonlinear mixed variational-like inequalities problem for set-valued mappings without compact values in Banach spaces with p-uniformly convex bidual spaces. First, the existence of the solutions of the related auxiliary problem is proved. Then, a new iterative algorithm based on the system of auxiliary variational inequalities is constructed. Finally, both the existence of the solutions of the original problem and the convergence of the iterative sequences generated by the algorithm are proved. And we also present a numerical example to demonstrate the result. Our results improve and extend some known results.

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.

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