This paper is concerned with double families of evolution operators employed in the study of dynamical systems in which cause and effect are represented in different Banach spaces. The main tool is the Laplace transform of vector-valued functions. It is used to define the generator of the double family which is a pair of unbounded linear operators and relates to implicit evolution equations in a direct manner. The characterization of generators for a special class of evolutions is presented.

The existence of mean periodic functions in the sense of L. Schwartz, generated, in various ways, by an equicontinuous group $U$ or an equicontinuous cosine function $C$ forces the spectral structure of the infinitesimal generator of $U$ or $C$. In particular, it is proved under fairly general hypotheses that the spectrum has no accumulation point and that the continuous spectrum is empty.

Let $\left\{T\right(t):t>0\}$ be a strongly continuous semigroup of linear contractions in $L_p$, $1<p<\infty$, of a $\sigma$-finite measure space. In this paper we prove that if there corresponds to each $t>0$ a positive linear contraction $P\left(t\right)$ in $L_p$ such that $\left|T\right(t\left)f\right|\le P\left(t\right)\left|f\right|$ for all $f\in L_p$, then there exists a strongly continuous semigroup $\left\{S\right(t):t>0\}$ of positive linear contractions in $L_p$ such that $\left|T\right(t\left)f\right|\le S\left(t\right)\left|f\right|$ for all $t>0$ and $f\in L_p$. Using this and Akcoglu’s dominated ergodic theorem for positive linear contractions in $L_p$, we also prove multiparameter pointwise ergodic and local ergodic theorems...

We provide sufficient conditions for sums of two unbounded operators on a Banach space to be (pre-)generators of contraction semigroups. Necessary conditions and applications to positive emigroups on Banach lattices are also presented.

We consider a class of perturbations of the degenerate Ornstein-Uhlenbeck operator in ${ℝ}^N$. Using a revised version of Bernstein’s method we provide several uniform estimates for the semigroup ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ associated with the realization of the operator $𝒜$ in the space of all the bounded and continuous functions in ${ℝ}^N$

We prove $L^2$-maximal regularity of the linear non-autonomous evolutionary Cauchy problem$$\dot{u}\left(t\right)+A\left(t\right)u\left(t\right)=f\left(t\right)\text{for}\text{a.e.}t\in [0,T],u\left(0\right)=u_0,$$ where the operator $A\left(t\right)$ arises from a time depending sesquilinear form $𝔞(t,·,·)$ on a Hilbert space $H$ with constant domain $V.$ We prove the maximal regularity in $H$ when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed by El-Mennaoui, Keyantuo, Laasri (2011), El-Mennaoui, Laasri (2013), and Laasri (2012). As a consequence, we obtain an invariance...

Similarly to quasidifferential equations of Panasyuk, the so-called mutational equations of Aubin provide a generalization of ordinary differential equations to locally compact metric spaces. Here we present their extension to a nonempty set with a possibly nonsymmetric distance. In spite of lacking any linear structures, a distribution-like approach leads to so-called right-hand forward solutions. These extensions are mainly motivated by compact subsets of the Euclidean space...

We give sufficient conditions for the existence of the fundamental solution of a second order evolution equation. The proof is based on stable approximations of an operator A(t) by a sequence $A_n\left(t\right)$ of bounded operators.