Investigation of the solvability of boundary value problems by applying vector a priori inequalities in presence of impulse disturbances (Q1897956)

There are used the following notations: \(R^n = \{\alpha = \text{col} (\alpha^1, \ldots, \alpha^n)\), \(\alpha^i \in R\), \(i = \overline {1,n}\}\), \(|\alpha |= \text{col} \{|\alpha^1 |, \cdots, |\alpha^n |\}\) \((\alpha = \text{col} (\alpha^1, \ldots, \alpha^n) \in R^n)\); \(L^n = L^n [a,b]\) is the space of the functions \(z : [a,b] \to R^n\), with summable components: \({\mathcal B} (x_0, \sigma, X) = \{x \in X, \rho (x, x_0) < \sigma\}\) \(((X, \rho)\) a metric space); \(DS^n (m)\) is the space of the functions \(y : [a,b] \to R\), with summable derivative, of the form \(y(t) = \int^t_a \dot y (s) ds + y(0) + \sum^m_{i = 1} \chi_{[t_i, b]} (t) \Delta y (t_i)\) \((a < t_1 < \cdots < t_m < b\) is a fixed system of points, \(\chi_{[t_i, b]}\) is the characteristic function of the segment \([t_i, b]\), \(\Delta y(t) = y(t) - y(t - 0))\). It is considered the equation (1) \(\dot y(t) = (Fy) (t)\), \(t \in [a,b]\) with the boundary condition \(\eta y = 0\), where \(F : DS^n (m) \to L^n\) is a completely continuous operator satisfying an a priori inequality \(|\dot y(t) |\leq {\mathcal M} (t, |\Delta y |)\) \(({\mathcal M} : [a,b] \times R_+^{nm + n} \to R^n\), is a function such that \({\mathcal M} (\cdot, s) \in L^n\) for every \(s \in {\mathcal B} (0, \tau, R_+^{nm + u})\), respectively for every \(s \in R^{nm + n}_+)\) almost everywhere on \([a,b]\), for every solution \(y\) such that \(|\Delta y |\in {\mathcal B} (0, \sigma, R_+^{nm + n})\), respectively for every solution of the equation (1) and, \(\eta : DS^n (m) \to R^{nm + n}\) is a continuous vector function. The existence of a solution \(y\) in \(DS^n (m)\) is established.