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We discuss an alternative approach to Fréchet derivatives on Banach spaces inspired by a characterisation of derivatives due to Carathéodory. The approach allows many questions of differentiability to be reduced to questions of continuity. We demonstrate how that simplifies the theory of differentiation, including the rules of differentiation and the Schwarz lemma on the symmetry of second-order derivatives. We also provide a short proof of the differentiable dependence of fixed points in the Banach fixed point theorem.
There is a long tradition of using geometry to solve problems from mechanics. Unfortunately this tradition is not practised much in schools and university any more.
With this exposition we would like to demonstrate how elementary properties of ellipses can be used to solve a problem related to the conical pendulum. The problem of the conical pendulum is to consider a mass attached to one end of a light inextensible string of length with the other end attached at the top of a vertical rod. The mass is moving about the rod in uniform circular motion in a horizontal plane. Given the angular velocity of the mass, the question is to determine the angle the string makes with the rod.
We carry out an in-depth study of some domination and smoothing properties of linear operators and of their role within the theory of eventually positive operator semigroups. On the one hand, we prove that, on many important function spaces, they imply compactness properties. On the other hand, we show that these conditions can be omitted in a number of Perron–Frobenius type spectral theorems. We furthermore prove a Kreĭn–Rutman type theorem on the existence of positive eigenvectors and eigenfunctionals under certain eventual positivity conditions.
It is proved that elliptic boundary-value problems have a global smoothing property in Lebesgue spaces, provided the underlying space of weak solutions admits a Sobolev-type inequality. The results apply to all standard boundary conditions, and a wide range of non-smooth domains, even if the classical estimates fail. The dependence on the data is explicit. In particular, this provides good control over the domain dependence, which is important for applications involving varying domains.
We prove that a class of weighted semilinear reaction diffusion equations on RN generates gradient-like semiflows on the Banach space of bounded uniformly continuous functions on RN. If N = 1 we show convergence to a single equilibrium. The key for getting the result is to show the exponential decay of the stationary solutions, which is obtained by means of a decay estimate of the kernel of the underlying semigroup.
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