This result is very deep and its proof rather long and intricate. In the present paper we show that if one restricts attention to C ~ functions on R n which are of exponential growth, it is possible to derive the formula by classical methods. Moreover, it is possible to obtain the measures featuring in the representation explicitly in terms of the represented solution. A's a fringe benefit we a...

Even though the two term recurrence relation satisfied by the incomplete gamma function is asymptotically stable in at least one direction, for an imaginary second argument there can be a considerable loss of correct digits before stability sets in. We present an approach to compute the recurrence relation to full precision, also for small values of the arguments, when the first argument is neg...

Oscillation criteria are established for a second-order forced dynamic equation on time scales containing both delay and advance arguments. Moreover, the potentials are allowed to change sign. Several nontrivial examples from difference equations are provided to illustrate the easy application of the results. The theory can be applied to second-order dynamic equations regardless of the choice o...

In this paper we shall discuss the almost sure exponential stability for a neutral differential difference equation with damped stochastic perturbations of the form d[x(t) −G(x(t − τ ))] = f(t, x(t), x(t − τ ))dt + σ(t)dw(t). Several interesting examples are also given for illustration. It should be pointed out that our results are even new in the case when σ(t) ≡ 0, i.e. for deterministic neut...

We mainly investigate the global asymptotic stability and exponential convergence of positive solutions to two families of higher-order difference equations, one of which was recently studied in Stević’s paper 2010 . A new concise proof is given to a quite recent result by Stević and analogous parallel result of the other inverse equation, which extend related results of Aloqeili 2009 , Berenha...

By means of the averaging technique and the generalized Riccati transformation technique, we establish some oscillation criteria for the second-order quasilinear neutral delay dynamic equations r t |xΔ t |γ−1xΔ t Δ q1 t |y δ1 t |α−1y δ1 t q2 t |y δ2 t |β−1y δ2 t 0, t ∈ t0,∞ T, where x t y t p t y τ t , and the time scale interval is t0,∞ T : t0,∞ ∩ T. Our results in this paper not only extend t...

In this paper we establish some sufficient conditions for oscillation of second order delay dynamic equations on time scales. Our results not only unify the oscillation of second order delay differential and difference equations but also are new for q-difference equations and can be applied on any time scale. We illustrate our results with many examples.

This paper describes a modification for the practical relevance of unit root tests for time series generated by linear stochastic difference equations with an explosive root.