Generalized Wiener-Hopf Equations Research Articles

Homogeneous conservative Wiener-Hopf equation

The existence of a -solution of the homogeneous generalized Wiener-Hopf equation is proved, where is a probability distribution of recurrent type in . Asymptotic properties of this solution are established. Bibliography: 10 titles.

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.

The Wiener--Hopf Technique for Impenetrable Wedges Having Arbitrary Aperture Angle

Diffraction by impenetrable wedges having arbitrary aperture angle is studied by means of the Wiener--Hopf (W-H) technique. A system of functional equations called generalized Wiener--Hopf equations (GWHE) is obtained. Only for certain values of the aperture angle are these equations recognizable as standard or classical Wiener--Hopf equations (CWHE). However, in all cases a mapping is found that reduces the GWHE to CWHE. It means that the diffraction by an impenetrable wedge always reduces to a standard W-H factorization. The solution for the diffraction by a wedge with given face impedances (the Malyuzhinets problem) is obtained in closed form by an explicit factorization of the kernel.

Iterative methods for generalized nonlinear variational inequalities

In this paper, we establish the equivalence between the generalized nonlinear variational inequalities and the generalized Wiener-Hopf equations. This equivalence is used to suggest and analyze a number of iterative algorithms for solving generalized variational inequalities. We also discuss the convergence analysis of the proposed algorithms. As a special case, we obtain various known results from our results.

Equivalence of Variational Inequalities with Wiener-Hopf Equations

We show that a variational inequality is equivalent to a generalized Wiener-Hopf equation in the sense that, if one of them has a solution so does the other one. Moreover, their solutions can be transformed to each other by a simple formula. Applications are considered.

Equivalence of variational inequalities with Wiener-Hopf equations

We show that a variational inequality is equivalent to a generalized Wiener-Hopf equation in the sense that, if one of them has a solution so does the other one. Moreover, their solutions can be transformed to each other by a simple formula. Applications are considered.

THE WIENER-HOPF EQUATION IN NEVANLINNA AND SMIRNOV ALGEBRAS

A generalized Wiener-Hopf equation on the semiaxis is considered in the class of analytic functionals which are the Fourier transform of Nevanlinna algebras or Smirnov algebras . The problem, connected with this equation, of factoring measurable functions on the axis in the algebras which satisfy the condition and also the problem of linear junction in the algebras and are also considered. Bibliography: 24 titles.

On the Solution of a Generalized Wiener-Hopf Equation

This paper deals with the generalized Wiener-Hopf equation G(α)X+(α)+H(α)X+(−α)−Y−(α)+Ψ(i)(α)=0, τ1<τ<τ2,where α (= σ + iτ) is a complete variable, G(α), H(α), and Ψ(i)(α) are known functions, and X+(α) and Y−(α) are unknowns, analytic in upper and lower half-planes, respectively, as indicated by their respective subscripts. This type of equation arises in a class of boundary-value problems in electromagnetic theory, the geometries of which may be described as modified Wiener-Hopf type. The method of approach, which is fundamentally different than those currently available in the literature, is based on a pairing of singularities in the complex α plane. This leads to a functional equation which is exactly solvable in its asymptotic form. The knowledge of this solution permits one to employ one of several rapidly converging numerical procedures available in the literature for a more accurate solution. Two examples illustrating the application of the procedure are included in the paper.

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 %,