Continuous dependence of fixed points of condensing maps

Many problems in analysis are concerned with the dependence upon parameters of fixed points of maps. For contraction mappings, criteria are relatively easy to obtain and have been known for some time. In the study of solutions of functional differential equations, more general results were needed. It is the purpose of this paper to give a rather general fixed-point theorem for condensing maps depending on a parameter, to prove continuous dependence and to indicate how many of the previous results are special cases

Existence of Mild Solutions for Semilinear Impulsive Functional Mixed Integro-differential Equations with Nonlocal Conditions

In this paper, we prove the existence, uniqueness and continuous dependence of initial data on mild solutions of first order semilinear functional impulsive mixed integro-differential equations with nonlocal condition in general Banach spaces. The results are obtained by using the semigroup theory and Banach contraction theorem

Optimal control of continuous-time Markov chains with noise-free observation

We consider an infinite horizon optimal control problem for a continuous-time Markov chain $X$ in a finite set $I$ with noise-free partial observation. The observation process is defined as $Y_t = h(X_t)$, $t \geq 0$, where $h$ is a given map defined on $I$. The observation is noise-free in the sense that the only source of randomness is the process $X$ itself. The aim is to minimize a discounted cost functional and study the associated value function $V$. After transforming the control problem with partial observation into one with complete observation (the separated problem) using filtering equations, we provide a link between the value function $v$ associated to the latter control problem and the original value function $V$. Then, we present two different characterizations of $v$ (and indirectly of $V$): on one hand as the unique fixed point of a suitably defined contraction mapping and on the other hand as the unique constrained viscosity solution (in the sense of Soner) of a HJB integro-differential equation. Under suitable assumptions, we finally prove the existence of an optimal control

Dissipativity of the delay semigroup

Under mild conditions a delay semigroup can be transformed into a (generalized) contraction semigroup by modifying the inner product on the (Hilbert) state space into an equivalent inner product. Applications to stability of differential equations with delay and stochastic differential equations with delay are given as examples