Some properties of solutions to a family of integral equations arising in the models of living systems (Q2400743)

The author considers the integral equation \[ x(t)=e^{-\int_0^tg\left(s,x_s\right)\;ds}\psi(t)+\int_0^tP(a)e^{-\int_{t-a}^{t}g\left(s,x_s\right)\;ds}f\big(t-a,x_{t-a}\big)\;da, \] for \(t\geq0\), subject to the condition that \[ x(t)=\psi(t), \] whenever \(t\in[-\omega,0]\). We note that in the above integral equation, the notation \[ y_t(x):=y(t+x) \] is utilized. Some assumptions are made about the functions \(\psi\), \(P\), \(f\), and \(g\). For example, it is assumed that the map \(\psi\) has the form \[ \psi(t)=\int_0^{\infty}P_0(a)\varphi(t-a)\;da, \] for \(t\in[-\omega,0]\), and \[ \psi(t)=\int_{t}^{\infty}P_0(a)\varphi(t-a)\;da, \] for \(t\in[0,+\infty)\). Here the function \(P_0\) is a decreasing map on \([0,+\infty)\) satisfying \(0\leq P_0(a)\leq 1\) and \(P_0(0)=1\). Moreover, the map \(\varphi\) is a nonnegative, continuous map on \((-\infty,0]\) and then satisfies \(\varphi(s)\equiv0\) for all \(s\in\big(-\infty,\tau_{\varphi}\big]\) for some constant \(\tau_{\varphi}<0\). The author then studies the existence of a unique solution to this problem. Moreover, the continuous dependence of the solution on the map \(\varphi\) is also investigated. The paper seems to be well written and relatively easy to follow. In addition, the author has some interesting commentary on the relation of the problem to models in mathematical biology. Therefore, it seems that this article might be of interest to researchers with an interest either in the theory of integral equations or mathematical biology.