We consider a perturbed linear boundary-value problem for a weakly singular integral equation. Assume that the generating boundary-value problem is unsolvable for arbitrary inhomogeneities. Efficient conditions for the coefficients guaranteeing the appearance of the family of solutions of the perturbed linear boundary-value problem in the form of Laurent series in powers of a small parameter ε with singularity at the point ε = 0 are established.
We establish consistency conditions for systems of linear differential equations with constant delay of neutral type and restrictions. The applicability of the projection-iterative method to these problems is justified.Functional-differential equations and their systems are mathematical models of various physical processes. Their broad applications stimulated the extensive development of the theory of these equations in the last 50 years (see [1][2][3][4]). The investigation of consistency conditions and justification of approximate methods for the solution of functional differential equations under additional restrictions imposed on their solutions form a separate branch. In particular, the approach proposed for the first time in [7] and the method developed in [8] were used in [5,6] for the investigation of systems of differential equations with delay and restrictions. In the present paper, we apply this approach to the study of analogous problems for systems of linear differential equations with delay of neutral type and additional conditions.
Object of InvestigationConsider the problemwhere ∆ > 0 is a constant delay, N (t), L(t), M(t), and S(t) are, respectively, m × m, m × m, m × m, and l × m matrices whose elements are square summable on the segment [a, b],(1), condition (2), and restriction (3) almost everywhere. If this vector function does not exist, then problem (1)-(3) is inconsistent. Under conditions (3) (l > 0), the latter case is, generally speaking, typical.The aim of the present paper is to establish consistency conditions for problem (1)-(3) and to justify the applicability of a version of the projection-iterative method to it.
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