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)),
We substantiate the application of the projection-iterative method to the solution of a boundary-value problem for integro-differential equations with restrictions and control.
Object of InvestigationConsider the problemwhere, the kernel H(t, s) is square summable in the collection of variables, C(t) and S(t) are 1 × l and l × 1 matrices whose elements are linearly independent functions square summable on the segment [a, b], U is a constant m × 1 matrix with elementsγ ∈ R m and α ∈ R l are given, and x ∈ W m 2 [a, b] and λ ∈ R l are to be determined. In [1,2], the problem of the existence and uniqueness of a solution of problem (1), (2) was considered and the application of iterative and projection methods to it was substantiated. However, these methods are not always applicable because, as is known, the iterative method has the bounded range of applicability and may be slowly convergent, whereas the projection method is often accompanied with the problem of instability. In the present paper, to reduce the impact of these flaws we propose to apply the projection-iterative method to problem (1), (2).
We consider an application of the projection method to a boundary-value problem for integro-differential equations with restrictions and control and propose a calculation scheme for the method.In the investigation of boundary-value problems for differential equations with restrictions and control, an important role is played by the determination of conditions for the existence of solutions and the development of efficient methods for their construction. 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.
Statement of the ProblemConsider the integro-differential equation
H(t, s)(Mx)(s)ds.(1)We seek its approximate solution (z(t), λ) that satisfies the boundary conditionsIt is assumed in Eq. (1) and conditions (2) and (3) that(Mx)(t) = q 0 (t)x r (t) + . . . + q r (t)x(t), r < m,
517.968.7 We establish conditions for the existence of solutions of boundary-value problems for weakly nonlinear integro-differential equations with parameters and restrictions. We also substantiate the applicability of iterative and projection-iterative methods for the solution of these problems.
Statement of the ProblemConsider the integro-differential equation(1)We pose the problem of finding a parameter λ ∈ R l and a function x(t, ε) continuous in ε on a segment [0, ε 0 ] and belonging to W m 2 [a, b] for every fixed ε ∈ [0, ε 0 ] that satisfy Eq. (1) almost everywhere, the boundary conditionsIf this pair (x(t), λ) exists, then problem (1)-(3) is regarded as consistent.In relations (1)- (3),t ∈ [a, b], ε is a sufficiently small nonnegative parameter, f ∈ L 2 [a, b], {p 1 , . . . , p m } ⊂ L 2 [a, b], the kernel H(t, s) is square summable in the collection of variables, C(t) and S(t) are, respectively, 1×l and l×1 matrices
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