Zdeněk Šmarda

On Some Solvable Difference Equations and Systems of Difference Equations

Here, we give explicit formulae for solutions of some systems of difference equations, which extend some very particular recent results in the literature and give natural explanations for them, which were omitted in the previous literature.

On a solvable system of rational difference equations

We show that the following system of difference equationswhere , , , and sequences , , and are real, can be solved in closed form. For the case when the sequences , , and are constant and , we apply obtained formulas in the investigation of the asymptotic behaviour of well-defined solutions of the system. We also find domain of undefinable solutions of the system. Our results considerably extend and improve some recent results in the literature.

Existence of positive solutions of discrete linear equations with a single delay

In the paper, we use a classical comparison result to prove the existence of positive solutions to a particular and very frequently investigated class of linear difference equations with a positive coefficient and a single delay. The relevant result is given in the form of an inequality (with a suitable auxiliary comparison function) for the equation coefficient. A parallel to known results is included as well as a discussion of future directions.

On the Critical Case in Oscillation for Differential Equations with a Single Delay and with Several Delays

New nonoscillation and oscillation criteria are derived for scalar delay differential equations 𝑥(𝑡)

An efficient new perturbative Laplace method for space-time fractional telegraph equations

In this paper, we propose a new technique for solving space-time fractional telegraph equations. This method is based on perturbation theory and the Laplace transformation. Fractional Taylor series and fractional initial conditions have been introduced. However, all the previous works avoid the term of fractional initial conditions in the space-time telegraph equations. The results of introducing fractional order initial conditions and the Laplace transform for the studied cases show the high accuracy, simplicity and efficiency of the approach.

On a delay population model with quadratic nonlinearity

AbstractA nonlinear delay differential equation with quadratic nonlinearity, x˙(t)=r(t)[∑k=1mαkx(hk(t))−βx2(t)],t≥0, is considered, where αk and β are positive constants, hk:[0,∞)→R are continuous functions such that t−τ≤hk(t)≤t, τ=const, τ>0, for any t>0 the inequality hk(t)<t holds for at least one k, and r:[0,∞)→(0,∞) is a continuous function satisfying the inequality r(t)≥r0=const for an r0>0. It is proved that the positive equilibrium is globally asymptotically stable without any further limitations on the parameters of this equation.

Asymptotic Convergence of the Solutions of a Dynamic Equation on Discrete Time Scales

The paper investigates a dynamic equation Δ𝑦(𝑡𝑛)=𝛽(𝑡𝑛)[𝑦(𝑡𝑛−𝑗)−𝑦(𝑡𝑛−𝑘)] for 𝑛→∞, where 𝑘 and 𝑗 are integers such that 𝑘>𝑗≥0, on an arbitrary discrete time scale 𝕋∶={𝑡𝑛} with 𝑡𝑛∈ℝ, 𝑛∈ℤ∞𝑛0−𝑘={𝑛0−𝑘,𝑛0−𝑘

Solving certain classes of Lane-Emden type equations using the differential transformation method

In this paper, the differential transformation method (DTM) is applied to solve singular initial problems represented by certain classes of Lane-Emden type equations. Some new differential transformation formulas for certain exponential and logarithmic nonlinearities are derived. The approximate and exact solutions of these equations are calculated in the form of series with easily computable terms. The results obtained with the proposed methods are in good agreement with those obtained by other methods. The advantages of this technique are shown as well.

Oscillation of Solutions of a Linear Second-Order Discrete-Delayed Equation

A linear second-order discrete-delayed equation with a positive coefficient is considered for . This equation is known to have a positive solution if fulfils an inequality. The goal of the paper is to show that, in the case of the opposite inequality for , all solutions of the equation considered are oscillating for .

Construction of the General Solution of Planar Linear Discrete Systems with Constant Coefficients and Weak Delay

Planar linear discrete systems with constant coefficients and weak delay are considered. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, the space of solutions with a given starting dimension is pasted after several steps into a space with dimension less than the starting one. In a sense this situation copies an analogous one known from the theory of linear differential systems with constant coefficients and weak delay when the initially infinite dimensional space of solutions on the initial interval on a reduced interval, turns (after several steps) into a finite dimensional set of solutions. For every possible case, general solutions are constructed and, finally, results on the dimensionality of the space of solutions are deduced.