On the uniform ergodic for α−times integrated semigroups

Let $A$ be a generator of an $\alpha-$times integrated semigroup$(S(t))_{t\geq 0}$. We study the uniform ergodicity of $(S(t))_{t\geq 0}$ and we show that the range of $A$ is closed if and only if $\lambda R(\lambda,A)$ is uniformly ergodic.Moreover, we obtain that $(S(t))_{t\geq 0}$ is uniformly ergodic if and only if $\alpha=0$. Finally, we get that $\frac{1}{t^{\alpha+1}}\int_{0}^{t}S(s)ds$ converge uniformly for all $\alpha\geq 0$.

A Parameter Property of Classical Solutions of Cauchy Problems

This work is concerned with the abstract Cauchy problems that depend on parameters. The goal is to study continuity in the parameters of the classical solutions of the Cauchy problems. The situation considered in this work is when the operator of the Cauchy problem is not densely defined. By applying integrated semigroup theory and the results on continuity in the parameters ofC0-semigroup and integrated semigroup, we obtain the results on the existence and continuity in parameters of the classical solutions of the Cauchy problems. The application of the obtained abstract results in a parabolic partial differential equation is discussed in the last section of the paper.

Existence Results of Nondensely Defined Fractional Evolution Differential Inclusions

This paper deals with the existence results of integral solutions for nondensely defined fractional evolution differential inclusions. Our approach is based on integrated semigroup theory and a fixed point theorem for condensing map due to Martelli. An example is also given to illustrate our results.

Quasilinear Stochastic Cauchy Problem in Abstract Colombeau Spaces

Generalized solutions to the abstract Cauchy problem for a quasilinear equation with the generator of an integrated semigroup and with terms reflecting nonlinear perturbations and white noise type perturbations are under consideration. An abstract stochastic Colombeau algebra is constructed, and solutions in the algebra are studied.

Integrated solutions of stochastic evolution equations with additive noise

We investigate the existence of a solution to the abstract stochastic evolution equation with additive noise: in the case when A is the generator of an n-times integrated semigroup.