General Wiener-Hopf Research Articles

On the symmetrization of general Wiener-Hopf operators

On the symmetrization of general Wiener-Hopf operators

Sensitivity Analysis for General Nonlinear Nonconvex Variational Inequalities

In this communication, we proved that the parametric general nonlinear nonconvex variational inequalities are equivalent to the parametric general Wiener-Hopf equations. We use this alternative equivalence formulation to studied the sensitivity analysis for general nonlinear nonconvex variational inequalities without assuming the differentiability of the given data.

EXTENDED GENERAL NONLINEAR QUASI-VARIATIONAL INEQUALITIES AND PROJECTION DYNAMICAL SYSTEMS

The aim of this paper is to introduce and study a new class of the extended general nonlinear quasi-variational inequalities and a new class of the extended general Wiener-Hopf equations. The equivalence between the extended general nonlinear quasi-variational inequalities and the fixed point problems, and as well as the extended general Wiener-Hopf equations is established. Then by using these equivalences, we discuss the existence and uniqueness of a solution of the extended general nonlinear quasi-variational inequalities. Applying the equivalent alternative formulation and a nearly uniformly Lipschitzian mapping $S$, we define some new $p$-step projection iterative algorithms with mixed errors for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping $S$ which is also a unique solution of the extended general nonlinear quasi-variational inequalities. The convergence analysis of the suggested iterative schemes under some suitable conditions is studied. We also suggest and analyze a class of extended general projection dynamical systems associated with the extended general nonlinear quasi-variational inequalities. We show that the trajectory of the solution of the extended general projection dynamical system converges globally exponential to a unique solution of the extended general nonlinear quasi-variational inequalities. Results obtained in this paper may be viewed as an refinement and improvement of the previously known results.

The Wiener-Hopf Equation Technique for Solving General Nonlinear Regularized Nonconvex Variational Inequalities

AbstractIn this paper, we introduce and study some new classes of extended general nonlinear regularized non-convex variational inequalities and the extended general nonconvex Wiener-Hopf equations, and by the projection operator technique, we establish the equivalence between the extended general nonlinear regularized nonconvex variational inequalities and the fixed point problems as well as the extended general nonconvex Wiener-Hopf equations. Then by using this equivalent formulation, we discuss the existence and uniqueness of solution of the problem of extended general nonlinear regularized nonconvex variational inequalities. We apply the equivalent alternative formulation and a nearly uniformly Lipschitzian mapping S for constructing some new p-step projection iterative algorithms with mixed errors for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping S which is unique solution of the problem of extended general nonlinear regularized nonconvex variational inequalities. We also consider the convergence analysis of the suggested iterative schemes under some suitable conditions.Mathematics Subject Classification (2010)Primary 47H05; Secondary 47J20, 49J40

Projection iterative methods for extended general variational inequalities

In this paper, we introduce and study a new class of variational inequalities involving three operators, which is called the extended general variational inequality. Using the projection technique, we show that the extended general variational inequalities are equivalent to the fixed point and the extended general Wiener-Hopf equations. This equivalent formulation is used to suggest and analyze a number of projection iterative methods for solving the extended general variational inequalities. We also consider the convergence of these new methods under some suitable conditions. Since the extended general variational inequalities include general variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems.

General Wiener-Hopf equations and nonexpansive mappings

In this paper, we show that the general variational inequalities are equivalent to a new class of general Wiener-Hopf equations involving the nonexpansive mappings. Using this equivalence, we suggest and analyze an iterative method for finding the common elements of the solution set of the general variational inequalities and the solution set of the fixed-point of the nonexpansive mapping. We also consider the convergence criteria of the proposed method under some mild conditions. Since the general variational inequalities and the Wiener-Hopf equations include several classes of variational inequalities and Wiener-Hopf equations as special cases, our results continue to hold for these problems. Results obtained in this paper may be viewed as a refinement and improvement of the previously known results.

Wiener-Hopf Equations Technique for General Variational Inequalities Involving Relaxed Monotone Mappings and Nonexpansive Mappings

We show that the general variational inequalities are equivalent to the general Wiener-Hopf equations and use this alterative equivalence to suggest and analyze a new iterative method for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality involving multivalued relaxed monotone operators. Our results improve and extend recent ones announced by many others.

A contribution to the quarter-plane problem in diffraction theory

The three-dimensional analog of Sommerfeld's famous half-plane problem for a screen Σ 1: x 1 > 0, x 2 > 0, x 3 = 0 is treated by an operator theoretic approach. This paper shows that the Dirichlet problem is well-posed for any complex wave-number k ∉ R, if we look for a solution u ∈ H 1 ( R 3) of the Helmholtz equation ( Δ + k 2) u = 0 in e 7mid; ∑ 1=g ∈ H 1 2 (∑ 1) with u ¦ Σ 1 = g ϵ H 1 2 (σ 1) . An explicit representation formula for the solution is established by use of general Wiener-Hopf operator theory, Sobolev space properties, and Hilbert space methods. The corresponding Neumann problem can be solved by analogy.

General Wiener-Hopf operators and the numerical range of an operator

Let H H be a separable Hilbert space and A A a bounded operator on H H . For a selfadjoint projection P P on H H we consider the general Wiener-Hopf operator T P ( A ) = P A R ( P ) {T_P}(A) = P{A_{R(P)}} where R ( P ) R(P) denotes the range of P P . In this paper we study the relation between T P ( A ) {T_P}(A) and W ( A ) W(A) , the numerical range of A A . In particular we characterize those operators A A such that T P ( A ) {T_P}(A) is invertible for every P P .

On the Invertibility of General Wiener-Hopf Operators

On the Invertibility of General Wiener-Hopf Operators

On the invertibility of general Wiener-Hopf operators

Let H \mathfrak {H} be a separable Hilbert space, B \mathfrak {B} the set of bounded linear operators on H \mathfrak {H} , and P P an orthogonal projection on H \mathfrak {H} . Denote the range of P P by R ( P ) R(P) . Let A A belong to B \mathfrak {B} . The general Wiener-Hopf operator associated with A A and P P is defined by T P ( A ) = P A | R ( P ) {T_P}(A) = PA|R(P) , the vertical bar denoting restriction. Let Q = I − P Q = I - P . The purpose of this paper is to disprove the general conjecture that if A A is an invertible element of B \mathfrak {B} , then the invertibility of T P ( A ) {T_P}(A) implies the invertibility of T Q ( A ) {T_Q}(A) . We also disprove the conjecture in an interesting special case.

Wiener-Hopf Equations Technique ForGeneral Variational Inequalities

In this paper, we show that the general variational inequality problem is equivalent to solving the Wiener-Hopf equations. We use this equivalence to suggest and analyze a number of iterative algorithms for solving general variational inequalities. INTRODUCTION Variational inequality theory, which was introduced by Stampacchia l] and Fichera [2] in the early sixties, describes a broad spectrum of very interesting and important developments involving a link among various fields of pure and applied sciences. It has been shown by many research workers that a wide class of contact, obstacle, unilateral, free, moving and general equilibrium problems arising in elasticity, fluid flow through porous media, economics, transportation, regional and engineering sciences can be studied in the general and unified framework of variational inequalities, see [3,4,5,6,7,8] for more details. Recently considerable efforts are being made to develop an efficient and implementable algorithm for computing the approximate solution of variational inequalities. There is a substantial number of iterative algorithms including the projection and auxiliary principle techniques, see [3,4,5,6,7,8,9,10] and the references cited therein. Transactions on Engineering Sciences vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3533 362 Contact Mechanics In this paper, using essentially a variant form of the projection techniques, we show that the general variational inequality problem is equivalent to solving the equations, which are called the Wiener-Hopf equations. This technique is mainly due to Shi [11] and Noor [12,6 . This alternate formulation is quite general and flexible. We use this equivalence to suggest and analyze a number of new iterative algorithms for variational inequalities and related complementarity problems. We also consider the convergence criteria of these algorithms. FORMULATION AND BASIC RESULTS Let H be a real Hilbert space, whose inner product and norm are denoted by and . respectively. Let K be a nonempty closed convex set in H. Given T, g : H —> H nonlinear operators, we consider the problem of finding ueH such that g(u}eK and >0, forallvcJf. (1) The inequality (l) is known as the general nonlinear variational inequality, which was introduced by Noor [9]. REMARK 1. I. We note that if g = /, then problem (l) is equivalent to finding ueK such that >0, for all veK, (2) a problem originally introduced and studied by Stampacchia 1]. II. If K* — {veH : > 0, for all ueK} is a polar cone of the convex cone K in H and K C g(K), then the problem (l) is equivalent to finding ueH such that 2(w)f#, TucJC and = 0, (3) which is known as the general nonlinear complementarity problem. For suitable and appropriate choices of the operators T, g and the convex set K, we can obtain many known classes of linear and nonlinear complementarity problems as special cases from the problem (3). III. If K = H, then the problem (l) is equivalent to finding ueH such that g(u)eH and Transactions on Engineering Sciences vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3533 Contact Mechanics 363 >= 0, for all gĤ ?, (4 which is known as the weak formulation of the odd order and nonsymmetric boundary value problems. From the above observations, it is clear that the problem (l) is more general and includes many known variational inequality and optimization problems as special cases. LEMMA 1 [7]. Given zeH, ueK satisfies the inequality > 0 for all veK, if and only if, u = PKZ, where PK is the projection of H into K. Furthermore, PK is a nonexpansive operator. Related to the general variational inequality (l), we now introduce a new class of the Wiener-Hopf equations. Let QK = / PK, where I is the identity operator. If g~* exists, then we consider the problem of finding zeH such that The equation of the type (7) are called the general Wiener-Hopf equations, see Speck [13] for the general treatment and physical formulations. DEFINITION 1. An operator T : H -> H is said to be: (a) Strongly monotone, if there exists a constant a > 0 such that > a u v || for all u.veH. (b) Lipschitz continuous, if there exists a constant /3 > 0 such that T% Tt, ||< /? % % ||, for all %,

An inversion formula for certain general Wiener-Hopf operators

An inversion formula for certain general Wiener-Hopf operators