Nonexistence of Solutions of Some Non-Linear Non-Local Evolution Systems on the Heisenberg Group

We present non-existence results for systems of non-local in space hyperbolic equations, for systems of non-local in space parabolic equations, and for systems of non-local in space hyperbolic equations with linear damping terms. Our method of proof is based on the test function method with a help of a convexity inequality recently proved in [2].

A maximin characterisation of the escape rate of non-expansive mappings in metrically convex spaces

We establish a maximin characterisation of the linear escape rate of the orbits of a non-expansive mapping on a complete (hemi-)metric space, under a mild form of Busemann's non-positive curvature condition (we require a distinguished family of geodesics with a common origin to satisfy a convexity inequality). This characterisation, which involves horofunctions, generalises the Collatz–Wielandt characterisation of the spectral radius of a non-negative matrix. It yields as corollaries a theorem of Kohlberg and Neyman (1981), concerning non-expansive maps in Banach spaces, a variant of a Denjoy–Wolff type theorem of Karlsson (2001), together with a refinement of a theorem of Gunawardena and Walsh (2003), concerning order-preserving positively homogeneous self-maps of symmetric cones. An application to zero-sum stochastic games is also given.

Convexity, boundedness, and almost periodicity for differential equations in Hillbert space

There are three kinds of results. First we extend and sharpen a convexity inequality of Agmon and Nirenberg for certain differential inequalities in Hilbert space. Next we characterize the bounded solutions of a differential equation in Hilbert space involving and arbitrary unbounded normal operator. Finally, we give a general sufficient condition for a bounded solution of a differential equation in Hilbert space to be almost periodic.