Abstract measure differential inequalities and applications (Q791736)

Let X denote a linear space and \(I=(-b,b)\) for some \(b>0\). For any \(x\in X\) define \(S_ x=\{rx: -\infty<r<1\}\) and \(\bar S_ x=\{rx: - \infty<r\leq 1\}\). Let \(x_ 0\in X\) be fixed. For any \(z\in X\) for which \(S_{x_ 0}\subset S_ z\) we denote \(\overline{x_ 0z}=S_ z- S_{x_ 0}\) and write \(z>x_ 0\). For any \(z>x_ 0\) let \(M_ z\) denote the smallest \(\sigma\)-algebra on \(S_ z\), containing \(\{x_ 0\}\) and sets \(\bar S_ x\), \(x\in \overline{x_ 0z}\). Let f be a real valued \(\mu\)-integrable function defined on \(S_ z\times I\), where \(\mu\) is a fixed finite positive measure. For any real number \(c\in I\) and real measure p on \(M_ z\) consider equation (1) \(dp/d\mu =f(x,p(\bar s_ x))\) with the initial condition (2) p(\(\bar S_{x_ 0})=c\), where dp/\(d\mu\) is the Radon-Nikodym derivative of p with respect to \(\mu\). Under some additional assumptions the authors establish a basic inequality between two approximate solutions of (1). Uniqueness of a solution of (1), (2) and its continuous dependence on initial condition follow as a consequence of this estimation.