The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler–Lagrange equation for the energy functional is Darboux integrable. The time evolution of the Cauchy data is reduced to an ordinary differential equation of Lie type associated t...

We present a recently developed complete version of the Pontryagin maximum principle for a class of infinite-horizon optimal control problems arising in economics. The peculiarity of the result is that the adjoint variable is explicitly specified by a formula which resembles the Cauchy formula for solutions of linear differential systems. In certain situations this formula implies the “standard...

The relation showing that the Grünwald-Letnikov and generalised Cauchy derivatives are equal is presented. This establishes a bridge between two different formulations and simultaneously between the classic integer order derivatives and the fractional ones. Starting from the generalised Cauchy derivative formula, new relations are obtained, namely a regularised version that makes the concept of...

Recent results concerning the existence and uniqueness of mild and classical solutions for non-local Cauchy problems are extended to the following non-autonomous semilinear integrodifferential equation u′(t) = A(t) [ u(t) + ∫ t 0 F (t, s)u(s) ds ] + f(t, u(t)), 0 ≤ t ≤ T, u(0) + g(t1, . . . , tp, u(t1), . . . , u(tp)) = u0, in a Banach space X, with A(·) the generators of strongly continuous se...