The following problem is considered \[ \dot{u}(t)=(lu)(t)+f(t),\quad t\in J=[a,b],\;u(t)\in \mathbb{R}^{n},\quad u(\tau)=c, \] where \(f\in L(J,\mathbb{R}^n)\); \(l:C(J,\mathbb{R}^n)\rightarrow L(J,\mathbb{R}^n)\) is a \((\sigma_1,\dots,\sigma_n,\tau)\) positive linear operator \((\sigma_i \in\{-1,1\}\), \(\tau \in J)\), i.e., if \(\sigma_iu_i(t)\geqslant 0\), \(i=1,\dots,n\), then \(\sigma_i(lu)_i(t)(t-\tau)\geqslant 0\), \(i=1,\dots,n\). Two theorems and some corollaries are proved on the unique solvability of the problem above.