On the Application of New Convergence Criteria for Kantorovich Method to Nonlinear Singular Integral Equation with Shift

Abstract

Abstract-The paper is concerned with the applicability of some new conditions for the convergence of Newton-kantorovich approximations to solution of nonlinear singular integral equation with shift of Uryson type. The results are illustrated in generalized Holder space. Keywords: Newton-Kantorovich approximations, Nonlinear singular integral equations of Uryson type, Noether operator, Carleman shift. 1-I TRODUCTIOThe theory of approximation methods and its applications for the solution of linear and nonlinear singular integral equations (LSIE) and (NSIE) has been developed by many authors [5,11,12,17,21]. There is a literature on the successful development of the nonlinear singular integral equations with shift (NSIES) [1,3,4,15,18,20]. The Noether theory of singular integral operators with shift (SIOS) is developed for a closed and open contour ([2,10,13,14,16,18] and others). The theory of singular integral equations with shift (SIES) is an important part of integral equations because of its recent applications in many fields of physics and engineering, [6,14,16]. It is known [6,7], that Weiner-Hope equations are a natural apparatus for the solution of problems of synthesis of signais for linear systems with continuous time and stationary parameters. If the problem of synthesis is not stationary, then the Weiner-Hope method is not applicable and the problem is reduced to singular integral equation. In this paper, some new conditions for the convergence of Newton-Kantorovich approximations have been applied to solution of the following NSIES of Uryson type: