In this paper, we study the time asymptotic behavior of the solution to an abstract Cauchy problem on Banach spaces without restriction on the initial data. The abstract results are then applied to the study of the time asymptotic behavior of solutions of an one-dimensional transport equation with boundary conditions in $L_1$-space arising in growing cell populations and originally introduced by M. Rotenberg, J. Theoret. Biol. 103 (1983), 181–199.

A condition on a family $B\left(t\right):t\in [0,T]$ of linear operators is given under which the inhomogeneous Cauchy problem for u"(t)=(A+ B(t))u(t) + f(t) for t ∈ [0,T] has a unique solution, where A is a linear operator satisfying the conditions characterizing infinitesimal generators of cosine families except the density of their domains. The result obtained is applied to the partial differential equation $$u_{tt}=u_{xx}+b(t,x)u_x(t,x)+c(t,x)u(t,x)+f(t,x)for(t,x)\in [0,T]\times [0,1],u(t,0)=u(t,1)=0fort\in [0,T],u(0,x)=u_0\left(x\right),u_t(0,x)=v_0\left(x\right)forx\in [0,1]$$ in the space of continuous functions on [0,1].

We give here a survey of some recent results on applications of topological quasi *-algebras to the analysis of the time evolution of quantum systems with infinitely many degrees of freedom.

2000 Mathematics Subject Classification: Primary 47A48, Secondary 60G12.In this paper classes of K^r -operators are considered – the classes of bounded and unbounded operators A with equal domains of A and A*, finite dimensional imaginary parts and presented as a coupling of a dissipative operator and an antidissipative one with real absolutely continuous spectra and the class of unbounded dissipative K^r -operators A with different domains of A and A* and with real absolutely continuous spectra....

In the first note we show for a strongly continuous family of operators ${\left(T\left(t\right)\right)}_{t\ge 0}$ that if every orbit $t\mapsto T\left(t\right)x$ is differentiable for $t>t_x$, then all orbits are differentiable for $t>t_0$ with $t_0$ independent of $x$. In the second note we give an example of an eventually differentiable semigroup which is not differentiable on the same interval in the operator norm topology.

Suppose A is a (possibly unbounded) linear operator on a Banach space. We show that the following are equivalent. (1) A is well-bounded on [0,∞). (2) -A generates a strongly continuous semigroup ${e^{-sA}}_{s\le 0}$ such that ${(1/s^2)e^{-sA}}_{s>0}$ is the Laplace transform of a Lipschitz continuous family of operators that vanishes at 0. (3) -A generates a strongly continuous differentiable semigroup ${e^{-sA}}_{s\ge 0}$ and ∃ M < ∞ such that $\parallel H_n\left(s\right)\parallel \equiv \parallel ({\sum }_{k=0}^n\left(s^k{A}^k\right)/k!)e^{-sA}\parallel \le M$, ∀s > 0, n ∈ ℕ ∪ 0. (4) -A generates a strongly continuous holomorphic semigroup ${e^{-zA}}_{Re\left(z\right)>0}$ that is O(|z|) in all...

An increasing sequence ${\left(n_k\right)}_{k\ge 0}$ of positive integers is said to be a Jamison sequence if for every separable complex Banach space X and every T ∈ ℬ(X) which is partially power-bounded with respect to ${\left(n_k\right)}_{k\ge 0}$, the set ${\sigma }_p\left(T\right)\cap$ is at most countable. We prove that for every separable infinite-dimensional complex Banach space X which admits an unconditional Schauder decomposition, and for any sequence ${\left(n_k\right)}_{k\ge 0}$ which is not a Jamison sequence, there exists T ∈ ℬ(X) which is partially power-bounded with respect to ${\left(n_k\right)}_{k\ge 0}$ and has the...

Let $(X,d_X)$, $(\Omega ,d_{\Omega })$ be complete separable metric spaces. Denote by (X) the space of probability measures on X, by $W_p$ the p-Wasserstein metric with some p ∈ [1,∞), and by ${}_p\left(X\right)$ the space of probability measures on X with finite Wasserstein distance from any point measure. Let $f:\Omega {\to }_p\left(X\right)$, $\omega \mapsto f_{\omega }$, be a Borel map such that f is a contraction from $(\Omega ,d_{\Omega })$ into ${(}_p\left(X\right),W_p)$. Let ν₁,ν₂ be probability measures on Ω with $W_p(\nu ₁,\nu ₂)$ finite. On X we consider the subordinated measures ${\mu }_i={\int }_{\Omega }{f}_{\omega }d{\nu }_i\left(\omega \right)$. Then $W_p(\mu ₁,\mu ₂)\le W_p(\nu ₁,\nu ₂)$. As an application we show that the solution measures ${\varrho }_{\alpha }\left(t\right)$ to the partial...

Some criteria for weak compactness of set valued integrals are given. Also we show some applications to the study of multimeasures on Banach spaces with the Radon-Nikodym property.

We propose a concept of weighted pseudo almost automorphic functions on almost periodic time scales and study some important properties of weighted pseudo almost automorphic functions on almost periodic time scales. As applications, we obtain the conditions for the existence of weighted pseudo almost automorphic mild solutions to a class of semilinear dynamic equations on almost periodic time scales.

Using known results on operator-valued Fourier multipliers on vector-valued function spaces, we give necessary or sufficient conditions for the well-posedness of the second order degenerate equations (P₂): d/dt (Mu’)(t) = Au(t) + f(t) (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), (Mu’)(0) = (Mu’)(2π), in Lebesgue-Bochner spaces $L^p(,X)$, periodic Besov spaces $B_{p,q}^s(,X)$ and periodic Triebel-Lizorkin spaces $F_{p,q}^s(,X)$, where A and M are closed operators in a Banach space X satisfying D(A) ⊂ D(M). Our results...

Let L be a nonsymmetric second order uniformly elliptic operator with generalWentzell boundary conditions. We show that a suitable version of L generates a quasicontractive semigroup on an Lp space that incorporates both the underlying domain and its boundary. This extends the earlier work of the authors on the symmetric case.

We investigate the relations between local α-times integrated semigroups and (α + 1)-times integrated Cauchy problems, and then the relations between global α-times integrated semigroups and regularized semigroups.