On Osgood theorem in Banach spaces

Let $X$ be a real Banach space, $\omega:[0,+\infty)\to\R$ be an increasing continuous function such that $\omega(0)=0$ and $\omega(t+s)\leq\omega(t)+\omega(s)$ for all $t,s\in[0,+\infty)$. By the Osgood theorem, if $\int_{0}^1\frac{dt}{\omega(t)}=\infty$, then for any $(t_0,x_0)\in R\times X$ and any continuous map $f: R\times X\to X$ and such that $\|f(t,x)-f(t,y)\|\leq\omega(\|x-y\|)$ for all $t\in R$, $x,y\in X$, the Cauchy problem $\dot x(t)=f(t,x(t))$, $(t_0)=x_0$ has a unique solution in a neighborhood of $t_0$ . We prove that if $X$ has a complemented subspace with an unconditional Schauder basis and $\int_{0}^1\frac{dt}{\omega(t)}

Counterexample to Peano's theorem in infinite-dimensional F '-spaces

Let $E$ be a nonnormable Frechet space, and let $E'$ be the space of all continuous linear functionals on $E$ in the strong topology. A continuous mapping $f : E' \to E'$ such that for any $t_0\in R$ and $x_0\in E'$, the Cauchy problem $\dot x= f(x)$, x(t_0) = x_0$ has no solutions is constructed.

A Conditional Approach for Modelling Patient Re-admissions in Hospital using a mixture of Coxian phase-type distributions incorporating Bayes Theorem

The number of elderly patients requiring hospitalisation in Europe is rising. With a greater proportion of elderly people in the population comes a greater demand for health services and, in particular, hospital care. Thus, with a growing number of elderly patients requiring hospitalisation competing with non-elderly patients for a fixed (and in some cases, decreasing) number of hospital beds, this results in much longer waiting times for patients, often with a less satisfactory hospital experience. However, if a better understanding of the recurring nature of elderly patient movements between the community and hospital can be developed, then it may be possible for alternative provisions of care in the community to be put in place and thus prevent readmission to hospital. The research in this paper aims to model the multiple patient transitions between hospital and community by utilising a mixture of conditional Coxian phase-type distributions that incorporates Bayes' theorem. For the purpose of demonstration, the results of a simulation study are presented and the model is applied to hospital readmission data from the Lombardy region of Italy.