Luděk Nechvátal

ON (q, h)-ANALOGUE OF FRACTIONAL CALCULUS

The paper discusses fractional integrals and derivatives appearing in the so-called (q, h)-calculus which is reduced for h = 0 to quantum calculus and for q = h = 1 to difference calculus. We introduce delta as well as nabla version of these notions and present their basic properties. Furthermore, we give comparisons with the known results and discuss possible extensions to more general settings.

Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations

This paper investigates some initial value problems in discrete fractional calculus. We introduce a linear difference equation of fractional order along with suitable initial conditions of fractional type and prove the existence and uniqueness of the solution. Then the structure of the solutions space is discussed, and, in a particular case, an explicit form of the general solution involving discrete analogues of Mittag-Leffler functions is presented. All our observations are performed on a special time scale which unifies and generalizes ordinary difference calculus and 𝑞-difference calculus. Some of our results are new also in these particular discrete settings.

Stability and asymptotic properties of a linear fractional difference equation

AbstractThis paper discusses qualitative properties of the two-term linear fractional difference equation ∇α0y(n)=λy(n), where α,λ∈R, 0<α<1, λ≠1 and ∇α0 is the α th order Riemann-Liouville difference operator. For this purpose, we show that this fractional equation is the Volterra equation of convolution type. This enables us to analyse its qualitative properties by use of tools standardly employed in the qualitative investigation of Volterra difference equations. As the main result, we derive a sharp condition for the asymptotic stability of the studied equation and, moreover, give a precise asymptotic description of its solutions.MSC:39A30, 39A12, 26A33.

On two-scale convergence

Two-scale convergence is a special weak convergence used in homogenization theory. Besides the original definitions by Nguetseng and Allaire, the paper introduces an alternative approach based on Arbogasts idea of two-scale transform which changes a sequence of one-variable functions into a sequence of two-variable functions. Although it leads to the same convergence, it simplifies some proofs.

Homogenization of monotone type problems with uncertain data

Abstract The paper deals with homogenization of nonlinear differential operators with monotone behaviour. We consider a situation, when the coefficients of the operator are not known exactly, but in certain bounds only due to errors caused by measurements. We use the deterministic approach to the problem- -worst scenario method introduced by I. Hlaváček.

On a solution of monotone type problems with uncertain inputs

Abstract The paper deals with a nonlinear weak monotone type problem and its solution with respect to uncertain coefficients in the equation. The so- -called worst scenario method is adopted. The formulation of suitable conditions and a proof of the existence of a solution of the worst scenario problem is presented.