On a type of evolution of self-referred and hereditary phenomena (Q2494452)

The present paper is devoted to the analysis of the integral evolution and hereditary equations. In particular, the equation \[ u(x,t)=u_0(x)+\int_0^t u\left(\int_0^{\tau}u(x,s)ds,\tau\right)d\tau,\;t\geq 0,\;x\in \mathbb R \] is considered. Using the method of successive approximation, the authors establish several results on the local existence and uniqueness of such an equation, or equivalently of the initial value problem for first order partial functional-differential equations. Some open problems referring mainly to the initial datum \(u_0(x)\) are also posed.