Analogue of the Cauchy matrix for system of quasi-integral equations with constant coefficients

V. I. Rodionov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 2 (2012), pp. 44-62

Résumé

In previous article we defined the concept of quasi-integral for two regulated functions on the interval and the special parameter, called “defect”. If there is the Riemann–Stieltjes integral, then for any defect there is a quasi-integral, and they are all equal. The Perron–Stieltjes integral, if it exists, coincides with one of quasi-integrals where the defect is defined in a special way. In the present article the theorem of existence and uniqueness of solution for a quasi-integral equation with a constant matrix is proved. System's kernel is a scalar piecewise continuous function of bounded variation. Components of the equation are regulated functions, spectral parameter is a regular number. Under certain conditions a quasi-integral equation can be interpreted as an impulse system. An explicit representation for the solution of a quasi-integral homogeneous equation is given. For an absolutely regular spectral parameter, the analogue of the Cauchy matrix is defined, its properties are investigated and the explicit representation for the solution of the nonhomogeneous quasi-integral equation in the Cauchy form is given. Similar results are obtained for the adjoint and associated equations. We discussed the possibility of restoration of the approximating defect of quasi-integral, which is defect generating approximated solutions of the impulse system.

Détail

Keywords: impulse system, regulated function, quasi-integral.