Conditions for well-posedness of integral models of some living systems (Q1686939)

The author considers the existence, uniqueness, and nonnegativity of solutions to delay integral equations in $\mathbb{R}^m$: $$x(t) = \exp \left\{-\int^t_0G(s, x_s) ds\right\}\psi(t) + \int^t_0P(a) \exp \left\{-\int^t_{t-a}G(s, x_s) ds\right\} f(t-a, x_{t-a}) da,\quad t \ge 0, $$ with initial conditions $x(t) = \psi (t)$ $(t\in [-\omega, 0]),$ where $G$ and $P$ are diagonal matrices. Under some assumptions, well-posedness results are given using the Banach contraction mapping principle.