Khalida Inayat Noor

Some aspects of variational inequalities

Abstract In this paper we provide an account of some of the fundamental aspects of variational inequalities with major emphasis on the theory of existence, uniqueness, computational properties, various generalizations, sensitivity analysis and their applications. We also propose some open problems with sufficient information and references, so that someone may attempt solution(s) in his/her area of special interest. We also include some new results, which we have recently obtained.

Some Relatively New Techniques for Nonlinear Problems

This paper outlines a detailed study of some relatively new techniques which are originated by He for solving diversified nonlinear problems of physical nature. In particular, we will focus on the variational iteration method (VIM) and its modifications, the homotopy perturbation method (HPM), the parameter expansion method, and exp-function method. These relatively new but very reliable techniques proved useful for solving a wide class of nonlinear problems and are capable to cope with the versatility of the physical problems. Several examples are given to reconfirm the efficiency of these algorithms. Some open problems are also suggested for future research work.

Traveling Wave Solutions of Seventh-order Generalized KdV Equations Using He's Polynomials

In this paper, we apply Hes polynomials to investigate propagating traveling solitary wave solutions of seventh order generalized KdV (SOG-KdV) equations which play a very important role in mathematical physics and engineering as well. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The suggested iterative scheme leads to the needed solution without any discretization, linearization or restrictive assumptions. The fact that proposed scheme solves nonlinear problems without using the Adomians polynomials can be considered as a clear advantage of this algorithm over the decomposition method.

On subclasses of close-to-convex functions of higher order.

The classes Tk(O), 0 0 2, of analytic functions, using the class Vk(O) of functions of bounded boundary rotation, are defined and it is shown that the functions In these classes are close-to- convex of higher order. Covering theorem, arc-length result and some radii problems are solved. We also discuss some properties of the class Vk(P) including distortion and coefficient results.

On quasi-convex functions and related topics

> O. g’(z) , In this paper, an up-to-date complete study of the class C is given. Its basic properties, its relationship with other subclasses of S, coefficient problems, arc length problem and many other results are included in this study. Some related classes are also defined and studied in some detail.

Self-adaptive projection algorithms for general variational inequalities

In this paper, we consider and analyze a new class of self-adaptive projection algorithms for solving general variational inequalities by using the technique of updating the solution. We prove that the convergence of these new methods only requires the pseudomonotonicity, which is a weaker condition than monotonicity. These new methods differ from the previously known splitting methods for solving variational inequalities and related complementarity problems. Proof of convergence is very simple. As special cases, we can obtain a number of four-step forward-backward splitting methods of Noor for solving variational inequalities.

An iterative method with cubic convergence for nonlinear equations

In this paper, we suggest and analyze a new three-step iterative method for solving nonlinear equations. We show that this new iterative method has third-order convergence. Several numerical examples are given to illustrate the efficiency and performance of this new method. New method can be viewed as an improvement of the previously known iterative methods.

Multivalued variational inequalities and resolvent equations

In this paper, we introduce and study some new classes of multivalued variational inequalities. These classes are more general and include the previously known classes of variational inequalities and related optimization problems as special cases. We also introduce some new resolvent equation problems. We prove that multivalued variational inequalities are equivalent to the fixed point problems and the resolvent equations. This equivalence is used to discuss the existence of the solution of the multivalued variational inequalities and suggest iterative algorithms for computing the approximate solutions. The convergence analysis is also studied.

On Bounded Boundary and Bounded Radius Rotations

We establish a relation between the functions of bounded boundary and bounded radius rotations by using three different techniques. A well-known result is observed as a special case from our main result. An interesting application of our work is also being investigated.

Modified Householder iterative method for nonlinear equations

Abstract In this paper, we suggest and analyze a new two-step predictor–corrector type iterative method for solving nonlinear equations of the type f ( x ) = 0 . This new method includes the two-step Newton method as a special case. We show that this new two-step method is a sixth-order convergent method. Several examples are given to illustrate the efficiency of this new method and its comparison with other sixth-order methods. This method can be considered as a significant improvement of the Newton method and its variant forms.