We consider the zigzag half-nanotubes (tight-binding approximation) in a uniform magnetic field which is described by the magnetic Schrödinger operator with a periodic potential plus a finitely supported perturbation. We describe all eigenvalues and resonances of this operator, and theirs dependence on the magnetic field. The proof is reduced to the analysis of the periodic Jacobi operators on the half-line with finitely supported perturbations.

The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition $$\begin{array}{c}{u}^{\text{'}\text{'}}+g\left(t\right)f(t,u)=0,t\in (0,1),\\ u\left(0\right)=\alpha u\left(\xi \right)+\lambda ,u\left(1\right)=\beta u\left(\eta \right)+\mu .Criteriafortheexistenceofnontrivialsolutionsoftheproblemareestablished.Thenonlineartermf(t,x)maytakenegativevaluesandmaybeunboundedfrombelow.Conditionsaredeterminedbytherelationshipbetweenthebehavioroff(t,x)/xforxnear0and\pm \infty ,andthesmallestpositivecharacteristicvalueofanassociatedlinearintegraloperator.Theanalysismainlyreliesontopologicaldegreetheory.Thisworkcomplementssomerecentresultsintheliterature.Theresultsareillustratedwithexamples.\end{array}$$

The paper deals with the following second order Dirichlet boundary value problem with p ∈ ℕ state-dependent impulses: z″(t) = f (t,z(t)) for a.e. t ∈ [0, T], z(0) = z(T) = 0, z′(τ i+) − z′(τ i−) = I i(τ i, z(τ i)), τ i = γ i(z(τ i)), i = 1,..., p. Solvability of this problem is proved under the assumption that there exists a well-ordered couple of lower and upper functions to the corresponding Dirichlet problem without impulses.

In this study, we establish existence and uniqueness theorems for solutions of second order nonlinear differential equations on a finite interval subject to linear impulse conditions and periodic boundary conditions. The results obtained yield periodic solutions of the corresponding periodic impulsive nonlinear differential equation on the whole real axis.

Motivated by structured parasite populations in aquaculture we consider a class of size-structured population models, where individuals may be recruited into the population with distributed states at birth. The mathematical model which describes the evolution of such a population is a first-order nonlinear partial integro-differential equation of hyperbolic type. First, we use positive perturbation arguments and utilise results from the spectral...

Let be a locally compact Hausdorff space. Let $A_i$, i = 0,1,...,N, be generators of Feller semigroups in C₀() with related Feller processes $X_i=X_i\left(t\right),t\ge 0$ and let ${\alpha }_i$, i = 0,...,N, be non-negative continuous functions on with ${\sum }_{i=0}^N{\alpha }_i=1$. Assume that the closure A of ${\sum }_{k=0}^N{\alpha }_k{A}_k$ defined on ${\bigcap }_{i=0}^N\left(A_i\right)$ generates a Feller semigroup T(t), t ≥ 0 in C₀(). A natural interpretation of a related Feller process X = X(t), t ≥ 0 is that it evolves according to the following heuristic rules: conditional on being at a point p ∈ , with probability ${\alpha }_i\left(p\right)$, the process...

The paper is devoted to a careful analysis of the shape-preserving properties of the strongly continuous semigroup generated by a particular second-order differential operator, with particular emphasis on the preservation of higher order convexity and Lipschitz classes. In addition, the asymptotic behaviour of the semigroup is investigated as well. The operator considered is of interest, since it is a unidimensional Black-Scholes operator so that our results provide qualitative information on the...

Let $ℒ$ be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure $\gamma$ on ${ℝ}^d.$ We prove a sharp estimate of the operator norm of the imaginary powers of $ℒ$ on $L^p\left(\gamma \right),$$1\<p\<\infty ...$

Signals generated in circuits that include nano-structured elements typically have strongly distinct characteristics, particularly the hysteretic distortion. This is due to memristance, which is one of the key electronic properties of nanostructured materials. In this article, we consider signals generated from a memrsitive circuit model. We demonstrate numerically that such signals can be efficiently represented in certain custom-designed nonorthogonal bases. The proposed method ensures that the...