Boundary-value problem with pulsed action. Critical case of the second order (Q6564109)

The authors investigate the impulsive boundary value problem\N\[\N\dot z = A(t)z + f(t) + \epsilon Z(z,t,\epsilon), \quad t \ne \tau_i \in [a,b],\N\]\N\[\N\triangle z_{|t = \tau_i} - S_iz(\tau_i-) = a_i + \epsilon J_i(z(\tau_i-,\epsilon),\epsilon), \quad i = 1,\ldots,k,\N\]\N\[\Nlz = \alpha + \epsilon J(z(\cdot,\epsilon),\epsilon),\N\]\Nwhere \(a < \tau_1 < \ldots < \tau_p < b\); \(a_i \in {\mathbb R}^n\), \(i=1,\ldots,k\); \(A : [a,b] \to {\mathbb R}^{n \times n}\), \(f : [a,b] \to {\mathbb R}^n\) have piecewise continuous components with possible discontinuities at \(\tau_i \in [a,b]\); \(l = (l_1,\ldots,l_m)\) is a linear bounded \(m\)-valued functional; \(\alpha \in {\mathbb R}^m\) and \(Z : {\mathbb R}^3 \to {\mathbb R}^n\) is in general nonlinear vector function; \(J_i(z(\tau_i-),\epsilon)\) and \(J(z(\cdot,\epsilon),\epsilon)\) are nonlinear \(n\)- and \(m\)-valued vector functionals with respect to \(z\), respectively; \(\epsilon \geq 0\).\N\NThe existence and uniqueness result for a solution of this IBVP for sufficiently small values of the parameter \(\epsilon\) is obtained. Also, iterative process converging to this solution is constructed.