On boundary value problems for linear differential systems with singularities (Q1768962)

The author considers boundary value problems for linear differential systems with singularities \[ \frac{dx_i}{dt}=P_{i1}(t)x_1+P_{i2}(t)x_2+q_i(t),\quad i=1, 2,\tag{1} \] \[ l_i(x_1, x_2)=c_i,\quad i=1, 2,\tag{2} \] where \(P_{ik}\not\in L([a, b]; \mathbb{R}^{n_i\times n_k})\) and \(q_i\not\in L([a, b]; \mathbb{R}^{n_i})\) for some \(i, k\in \{1, 2\}\). Conditions are established for problem (1), (2) to be Fredholm with index zero. On the basis of this conclusion, several criteria for the existence of a unique solution satisfying boundary condition (2) is proved for the differential systems \[ \begin{aligned} \frac{dx_1}{dt}&=P_{11}(t)x_1+P_{12}(t)x_2+q_1(t),\\ \frac{dx_2}{dt}&=\epsilon P_{21}(t)x_1+P_{22}(t)x_2+q_2(t),\end{aligned} \] or \[ \begin{aligned} \frac{dx_1}{dt}&=P_{11}(t)x_1+\epsilon P_{12}(t)x_2+q_1(t),\\ \frac{dx_2}{dt}&=\epsilon P_{21}(t)x_1+P_{22}(t)x_2+q_2(t).\end{aligned} \] Moreover, optimal criteria for the existence of a unique solution of system (1) with the boundary conditions \(x_1(a)=c_1\), \(x_i(b)=c_2\), \(i\in \{1, 2\}\), are found. These criteria generalize and supplement the results concerning similar problems for second-order linear differential equations and two-dimensional differential systems with singularities.