On models of developing systems and their applications

Solvability of systems of volterra integral equations of the first kind with piecewise continuous kernels

“…The differentiation of VIE (3) leads to integral-functional equation and its solution in not unique in the general case [16]. That is why study of VIE (3) can not be performed using only the classic methods in the Volterra theory [1,6,15,14]. In this paper we continue our results on VIE studies [11], [8,9], [17,19].…”

“…x(t) ∈ C (0,T ) is equivalent to proving the existence of unique solution of equation (6) in C (0,T ) . Let us introduce the function…”

On the solvability of a class of Volterra operator equations of the first kind with piecewise continuous kernels

“…Clearly the inequality D(0) < 1 is fulfilled if |α (1) i (0)| are sufficiently small. Here and below an operator K(t, s) is defined with formula (2) in n 1 D i .…”

“…If condition (C1) is satisfied then an asymptotic approximationx of desired parametric family of solution of the equation (4) can be constructed. Indeed let operator B(0) is Fredholm operator, {φ (1) i } r i=1 is the basis in N(B(0)), {φ (l) i } i=1,r, l=1,p i is corresponding CJS satisfying equations (24) and condition (25) for j * = 0. Then the first coefficient x 0 (z) of desired approximationx, satisfy the difference equation (12) and can be constructed as polynomial…”

“…In Section 2 we continue our studies on the Volterra equations presented in papers [5,[11][12][13][14][15][16][17] and provide the sufficient conditions of existence and uniquness of continuous solution of the equation (3) with piece-wise continuous kernel (2). The special case of such equations are the Volterra equations which model the evolving dynamic systems [1][2][3]. To the best of our knowledge such equations have not been studied in the literature.…”