A Gerschgorin theorem for difference equations—II

It is shown that the bounded solutions of the difference equation x(m + 1) = A(m + 1)x(m) arise as fixed points of a contraction mapping when A(m) satisfies a diagonal dominance condition on {0, 1, 2,…}.     

A link between the Perron-Frobenius theorem and Perron’s theorem for difference equations

Let A_n,n∈N, be a sequence of k×k matrices which converge to a matrix A as n→∞. It is shown that if x_n,n∈N, is a sequence of nonnegative nonzero vectors such that

     

Second bogolyubov theorem for systems of difference equations

We establish an analog of the second Bogolyubov theorem for a system of difference equations.     

A Hopf bifurcation theorem for difference equations approximating a differential equation

If an ordinary differential equation is discretizised near an asymptotically stable stationary solution with a pair of imaginary eigenvalues by Euler's method with constant step lengthh, small invariant attracting cycles of radiusO(h ^1/2) will appear. This Hopf bifurcation theorem is applied to prove the existence of limit cycles in certain difference equations occurring in biomathematics (hypercycle, two loci-two alleles) and is also extended to general Runge—Kutta methods.     

A Gerschgorin theorem for linear difference equations and eigenvalues of matrix products

The Gerschgorin circle theorem is used here to give sufficient conditions for the solution space of the difference equation x(m+1) = A (m+1)x(m) to admit a type of exponential dichotomy. The result obtained is then used to establish a result on regions of eigenvalue inclusion for the product of finitely many square matrices. An application to differential equations is also given.     

Chevalley restriction theorem for the cyclic quiver

We prove a Chevalley restriction theorem and its double analogue for the cyclic quiver.     

Asymptotic expansions for time-varying scalar differential equations using the residue theorem for their discretized difference equations

A fictitious time-dependent time-varying finite sampling period is defined for each time instant at which the asymptotic expansion of the solution of a continuous-time differential equation is investigated. Such a time-dependent sampling period is defined as the quotient of each time instant and a positive integer which tends to infinity as time tends to infinity. The asymptotic expansion formulas are extendable to the case of stable Lyapunov’s equations and to the use of a constant sampling period with minor modifications in the required mathematical proofs. Additional stability results are also discussed.     

An extension of Sharkovsky's theorem to periodic difference equations

We present an extension of Sharkovsky's theorem and its converse to periodic difference equations. In addition, we provide a simple method for constructing a p-periodic difference equation having an r-periodic geometric cycle with or without stability properties.     

A residual existence theorem for linear equations

A residual existence theorem for linear equations is proved: if ${A \in \mathbb{R}^{m\times n}}$ , ${b \in \mathbb{R}^{m}}$ and if X is a finite subset of ${\mathbb{R}^{n}}$ satisfying ${{\rm max}_{x \in X}p^T(Ax-b) \geq 0}$ for each ${p \in \mathbb{R}^{m}}$ , then the system of linear equations Ax = b has a solution in the convex hull of X. An application of this result to unique solvability of the absolute value equation Ax + B|x| = b is given.     

A local compactness theorem for Maxwell's equations

The paper gives a proof, valid for a large class of bounded domains, of the following compactness statements: Let G be a bounded domain, β be a tensor-valued function on G satisfying certain restrictions, and let {_n} be a sequence of vector-valued functions on G where the L₂-norms of {_n}, {curl _n}, and {div(β _n)} are bounded, and where all _n either satisfy x _n = 0 or (β F_n) = 0 at the boundary ?G of G ( = normal to ?G): then {_n} has a L₂-convergent subsequence. The first boundary condition is satisfied by electric fields, the second one by magnetic fields at a perfectly conducting boundary ?G if β is interpreted as electric dielectricity ? or as magnetic permeability μ, respectively. These compactness statements are essential for the application of abstract scattering theory to the boundary value problem for Maxwell's equations.