This paper is devoted to the study of the approximation problem for the abstract hyperbolic differential equation u'(t) = A(t)u(t) for t ∈ [0,T], where A(t):t ∈ [0,T] is a family of closed linear operators, without assuming the density of their domains.

We show that the critical nonlinear elliptic Neumann problem $\Delta u-\mu u+u^{7/3}=0$ in $\Omega$, $u>0$ in $\Omega$, $\frac{\partial u}{\partial \nu }=0$ on $\partial \Omega$, where $\Omega$ is a bounded and smooth domain in ${ℝ}^5$, has arbitrarily many solutions, provided that $\mu >0$ is small enough. More precisely, for any positive integer $K$, there exists ${\mu }_K>0$ such that for $0<\mu <{\mu }_K$, the above problem has a nontrivial solution which blows up at $K$ interior points in $\Omega$, as $\mu \to 0$. The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional...

We show that the result of Kato on the existence of a semigroup solving the Kolmogorov system of equations in l₁ can be generalized to a larger class of the so-called Kantorovich-Banach spaces. We also present a number of related generation results that can be proved using positivity methods, as well as some examples.

Let ϰ be a positive, continuous, submultiplicative function on ${ℝ}^{+}$ such that $li{m}_{t\to \infty }{e}^{-\omega t}{t}^{-\alpha }\varkappa \left(t\right)=a$ for some ω ∈ ℝ, α ∈ $\overline{{ℝ}^{+}}$ and $a\in {ℝ}^{+}$. For every λ ∈ (ω,∞) let ${\varphi }_{\lambda }\left(t\right)=e^{-\lambda t}$ for $t\in {ℝ}^{+}$. Let $L_{\varkappa }^1\left({ℝ}^{+}\right)$ be the space of functions Lebesgue integrable on ${ℝ}^{+}$ with weight $\varkappa$, and let E be a Banach space. Consider the map ${\varphi }_{•}:(\omega ,\infty )\ni \lambda \to {\varphi }_{\lambda }\in L_{\varkappa }^1\left({ℝ}^{+}\right)$. Theorem 5.1 of the present paper characterizes the range of the linear map $T\to T{\varphi }_{•}$ defined on $L(L_{\varkappa }^1\left({ℝ}^{+}\right);E)$, generalizing a result established by B. Hennig and F. Neubrander for $\varkappa \left(t\right)=e^{\omega t}$. If ϰ ≡ 1 and E =ℝ then Theorem 5.1 reduces to D. V. Widder’s characterization...