Two-sided estimates on solutions to an integro-differential equation which describes the process of hematogenic process
The paper contains an existence-uniqueness result for the following integro-differential equation describing the hematogenic process \[ \dot x(t)=f(x_t)-\lambda x(t)-(\rho x)(t),\;t\geq 0, \] \[ x(t)=\psi(t),\;-\omega_n\leq t\leq 0,\;\psi(0)=\int^r_0 R(a)\varphi (a)da, \] where \[ f(x_t)= \sum^n_{i=1} p_if\bigl(x(t-\omega_i) \bigr),\;p_i\in R^1,\;\sum^n_{i=1} p_i=1, \omega_i>0; \] \[ (\rho x)(t)= \begin{cases} e^{-\lambda t}\int^{\tau-t}_0\mu(a+t) \varphi(a) da+\int^t_0 \mu(a)e^{-\lambda a}f(x_{t-a})da,\;0\leq t\leq\tau,\\ \int^\tau_0\mu(a) e^{-\lambda a}f(x_{t-a})da,\;t\geq\tau;\end{cases} \] \(\lambda\) and \(\tau\) are parameters; \(R(0)=1\), \(R(a)=0\), \(a\geq\tau\), \(R(a)=\int^\pi_n \mu(s)d_s\) \(a\in[0,\tau]\).
scientific article; zbMATH DE number 1752672