Iteration method for the solution of integro-differential equations with constraints

Abstract

517.968.7 We establish conditions for the existence of solutions of boundary-value problems for integro-differential equations with constraints and substantiate the application of the iteration method to the solution of these equations. Statement of the ProblemIn the present paper, we consider the integro-differential equationand study the problem of finding a function y ∈ W m 2 [a; b] and a parameter λ ∈ R n that satisfy Eq. (1) almost everywhere and satisfy the boundary conditionsIf such a pair y(x), λ exists, then the problem is regarded as consistent. In Eq. (1) and relations (2) and (3), we assume thatThe 1 × n matrix ξ(x) and the n × 1 matrix S(x) whose elements are linearly independent functions square summable on the segment [a, b], the constant m × 1 matrix U whose elements have the form(α νs y (s−1) (a) + β νs y (s−1) (b)),

On the uniqueness of the solution of a boundary-value problem for weakly nonlinear integro-differential equations with parameters

“…Using the method developed in [5][6][7][8][9], we will show that problem (1.1), (1.2) is equivalent to an integral equation without limitations. Let us consider the generating problem…”

“…B. Polishchuk [12], and V. A. Feruk [2,3,7], the investigation of problems with limitations were carried out with the help of problems with parameters. In [9], the author has established the conditions of existence of solutions of a boundary-value problem for the linear integro-differential equations with limitations and has substantiated the application of the iteration method to them. The present work considers the questions of existence and uniqueness of the solution of a boundary-value problem for weakly nonlinear integro-differential equations with parameters.…”

“…The present work considers the questions of existence and uniqueness of the solution of a boundary-value problem for weakly nonlinear integro-differential equations with parameters. While solving these questions, the method developed in [5][6][7][8][9] is used.…”

Projection method for the solution of integro-differential equations with restrictions and control

“…In [5], the problem of the existence and uniqueness of a solution of problem (1)-(3) was considered and the possibility of the application of an iterative method to it was substantiated. The iterative method has the bounded range of applicability and may be slowly convergent.…”

“…These problems were studied in numerous works (see, e.g., [1][2][3][4]). The application of an iterative method to integro-differential equations with restrictions and control was studied in [5]. In the present paper, which is a continuation of the work indicated, we investigate the application of a projection method to a boundary-value problem for integro-differential equations.…”