Measure neutral functional differential equations as generalized ODEs
The authors consider integral equations of the form \[ x(t)=x(t_0)+\int_{t_0}^t f(x_s,s)\,\mathrm{d}g(s)+\int_{-r}^0 \mathrm{d}_\theta[\mu(t,\theta)]x(t+\theta)-\int_{-r}^0 \mathrm{d}_\theta[\mu(t_0,\theta)]\varphi(\theta), \] for which they coin the term ``measure neutral functional differential equations''. The solution $x$ and the function $f$ take values in $\mathbb{R}^n$, $g$ is a real nondecreasing function, $\mu$ takes values in $\mathbb{R}^{n\times n}$ and is left-continuous with bounded variation with respect to the second variable, and $\varphi$ is the initial condition on $[-r,0]$. \par It is shown that under certain assumptions, the integral equations of the above-mentioned type represent a special case of Kurzweil's generalized ordinary differential equations. Hence, one can easily obtain some basic results on the local existence and uniqueness of solutions, as well as continuous dependence on parameters.
generalized ordinary differential equations
measure neutral functional differential equations
scientific article; zbMATH DE number 7047664
scientific article