In this work, we consider the singular Hahn difference equation of the Sturm-Liouville type. We prove the existence of the spectral function for this equation. We establish Parseval equality and an expansion formula for this equation on a semi-unbounded interval.

The paper deals with an optimal control problem with a scalar first-order state constraint and a scalar control. In presence of (nonessential) touch points, the arc structure of the trajectory is not stable. Under some reasonable assumptions, we show that boundary arcs are structurally stable, and that touch point can either remain so, vanish or be transformed into a single boundary arc. Assuming a weak second-order optimality condition (equivalent to uniform quadratic growth), stability and...

Nonlinear Schrödinger equations are usually investigated with the use of the variational methods that are limited to energy-subcritical dimensions. Here we present the approach based on the shooting method that can give the proof of existence of the ground states in critical and supercritical cases. We formulate the assumptions on the system that are sufficient for this method to work. As examples, we consider Schrödinger-Newton and Gross-Pitaevskii equations with harmonic potentials.

We are concerned with first order set-valued problems with very general boundary value conditions $$\left\{\begin{array}{c}{u}_g^{\text{'}}\left(t\right)\in F(t,u\left(t\right)),{\mu }_g\text{-a.e.}\in [0,T],\hfill \\ L\left(u\right(0),T))=0\hfill \end{array}\right.$$ involving the Stieltjes derivative with respect to a left-continuous nondecreasing function $g:[0,T]\to ℝ$, a Carathéodory multifunction $F:[0,T]\times ℝ\to 𝒫\left(ℝ\right)$ and a continuous $L:{ℝ}^2\to ℝ$. Using appropriate notions of lower and upper solutions, we prove the existence of solutions via a fixed point result for condensing mappings. In the periodic single-valued case, combining an existence theory for the linear case with a recent result involving...

Strong convergence estimates for pseudospectral methods applied to ordinary boundary value problems are derived. The results are also used for a convergence analysis of the Schwarz algorithm (a special domain decomposition technique). Different types of nodes (Chebyshev, Legendre nodes) are examined and compared.

We study singular boundary value problems with mixed boundary conditions of the form $${\left(p\left(t\right)u^{\text{'}}\right)}^{\text{'}}+p\left(t\right)f(t,u,p\left(t\right)u^{\text{'}})=0,\underset{t\to 0+}{lim}p\left(t\right)u^{\text{'}}\left(t\right)=0,u\left(T\right)=0,$$ where $[0,T]\subset ℝ.$ We assume that ${ℝ}^2,$$f$ satisfies the Carathéodory conditions on $(0,T)\times$$p\in C...$