The direct Lyapunov method for an almost periodic difference system on a compactum.
The paper deals with a difference equation arising from the scalar pantograph equation via the backward Euler discretization. A case when the solution tends to zero but after reaching a certain index it loses this tendency is discussed. We analyse this problem and estimate the value of such an index. Furthermore, we show that the utilized proof technique enables us to investigate some other numerical formulae, too.
In this note we derive a type of a three critical point theorem which we further apply to investigate the multiplicity of solutions to discrete anisotropic problems with two parameters.
Sufficient conditions for the absence of absolutely continuous spectrum for unbounded Jacobi operators are given. A class of unbounded Jacobi operators with purely singular continuous spectrum is constructed as well.