Proximal (sub-)differential inequalities (Q1817980)

The authors study the interesting question on a characterization of comparison properties of nonsmooth vector functions in terms of proximal subderivatives. For a function \(\phi: \mathbb{R}\to (-\infty,+\infty]\), the number \(\zeta\) is called a proximal subderivative of \(\phi\) at \(t\) if \(\exists\sigma>0\) such that for all \(\tau\) near \(t\), \[ \phi(\tau)\geq \phi(t)+ \zeta(\tau- t)- \sigma|\tau- t|^2. \] The set of all proximal subderivatives of \(\phi\) at \(t\) is denoted by \(\partial_p\phi(t)\). If \(\phi(t)= +\infty\), then by convention \(\partial_p\phi(t)= \emptyset\). Let \(\rho\) be a vector function with lower-semicontinuous components \(\rho^i: \mathbb{R}\to (-\infty,+\infty]\) \((i= 1,\dots, m)\), and \(f: (a,b)\times \mathbb{R}^m\to \mathbb{R}^m\) is a continuous function. Definition: The function \(\rho\) is said to satisfy the comparison principle with respect to solutions to the ODE \(\dot r= f(t,r)\) if for every \[ t_0\in \text{dom }\rho:= \bigcap^m_{i=1} \{t:\rho^i(t)< +\infty\}, \] and every \(r_0\in \mathbb{R}^m\) with \(r_0\geq \rho(t_0)\), there exists a solution \(r(\cdot)\) to this ODE satisfying \(\rho(t)\leq r(t)\) on the maximal existent interval of \(r(\cdot)\). The main result of the paper is embodied in the following theorem: Let the functions \(\rho\), \(f\) be as defined above. Then \(\rho(t)\) satisfies the comparison principle iff \[ \partial_p\rho^i(t)\leq f^i(t,r),\quad \forall i\in\{i: r^i= \rho^i(t)\},\quad \forall r\geq \rho(t),\quad \forall t\in \text{dom }\rho,\tag{1} \] and one of the following conditions holds: \(\forall t\in\text{dom }\rho\), \[ \partial_p\Biggl( \sum^m_{i=1} \rho^i\Biggr)(t)\leq \sum^m_{i= 1}f^i(t, \rho(t))\quad\text{or }\lim_{\tau\downarrow t} \rho(\tau)= \rho(t). \] Note, inequality (1) means that every element of the left set is less or equal to the right number. Related problems on invariance, weak and strong comparison principles are also discussed in detail.