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This paper aims to study the oscillatory nature of solutions for fourth-order neutral differential equations with mixed deviating arguments and improved oscillation conditions obtained in various cases. We reduced the problem to first-order differential inequality by using suitable substitutions that enabled us to use comparison theorems. Further, we discuss the asymptotic nature of solutions, and in the end, an example is given to validate the results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the permanence of a class of discrete single-species population model with time delays. In most of the previous literature, the population models have only one positive equilibrium.While the system we considered has more than one positive equilibrium point and its dynamic behaviors will be more complex. By using the theory of oscillation of difference equation, we have established a new and rather weaker condition to ensure the permanence of the system. [ABSTRACT FROM AUTHOR]

This paper is concerned with the third order functional delay difference equation of the form... We obtain some new oscillation criteria by using Riccati transformation technique. Examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type (r(t)(z'(t))γ)' + Σmi=1 qi(t)xαi(σi(t)) = 0, t > t0, where z(t) = x(t)+p(t)x(τ(t)). Under the assumption ∫∞(r(η))-1/γ dη = ∞, we consider two cases when γ > αi and γ < αi. Our main tool is Lebesgue's dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the oscillatory character of a generalized di-fferential equation of order ff with ff 2 (0; 1]. Generalized criteria are obtained, of type Kamenev, which are extensions of several known in the literature both integer and fractional. [ABSTRACT FROM AUTHOR]

The qualitative study of mathematical models is an important area in applied mathematics. In this paper, a version of the food-limited population model with piecewise constant argument under impulse effect is investigated. Differential equations with piecewise constant arguments are related to difference equations. First, a representation for the solutions of the food-limited population model is stated in terms of the solutions of corresponding difference equation. Then using linearized oscillation theory for difference equations, a sufficient condition for the oscillation of the solutions about positive equilibrium point is obtained. Moreover, asymptotic behavior of the non-oscillatory solutions are investigated. Later, applying the same theory, non-impulsive model is also studied. Numerical examples are given to compare the results of impulsive model with the results of nonimpulsive case. The results show that when the solutions of impulsive differential equation model are oscillatory about positive equilibrium under suitable conditions the solutions of the corresponding non-impulsive model is not oscillatory. This situation indicates to the importance of impulse effects on the asymptotic behavior of the solutions. [ABSTRACT FROM AUTHOR]

Consider the first-order linear delay (advanced) differential equation x'(t) + p(t)x(Τ (t)) = 0 (x '(t) -- q(t)x(σ(t)) = 0), t ≥ t0, where p (q) is a continuous function of nonnegative real numbers and the argument Τ (t) (σ(t)) is not necessarily monotone. Based on an iterative technique, a new oscillation criterion is established when the well-known conditions ... and ... are not satisfied. An example, numerically solved in MATLAB, is also given to illustrate the applicability and strength of the obtained condition over known ones. [ABSTRACT FROM AUTHOR]

We have established sufficient conditions for oscillation of a class of first order neutral impulsive difference equations with deviating arguments and fixed moments of impulsive effect. [ABSTRACT FROM AUTHOR]

Abstract The purpose of this paper is to study the second-order nonlinear noncanonical differential equation r (t) y ′ (t) ′ + p (t) f (y (τ (t))) = 0 under the condition ∫ ∞ 1 r (t) d t < ∞. Contrary to most existing results, oscillation of the studied equation is attained via only one condition. We consider both delay and advanced differential equations. A particular example of Euler type equation is provided in order to illustrate the significance of our main results. [ABSTRACT FROM AUTHOR]

Abstract We consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. In our consideration we do not impose any controllability and strict normality assumptions and omit the Legendre condition for the Hamiltonian. We adapt the theory developed by A.A. Abramov for the Hamiltonian spectral problems based on piecewise constant transformations of their conjoined bases using the notion of the comparative index. We introduce the concept of oscillation numbers which generalize the notion of proper focal points and prove the oscillation theorem relating the number of finite eigenvalues in the given interval with the values of the oscillation numbers at the end points of this interval. [ABSTRACT FROM AUTHOR]

Sufficient conditions are obtained for the third order nonlinear delay difference equation of the form Δ(an(Δbn(Δyn)α))) + qnf(yσ(n))=0 to have property (A) or to be oscillatory. These conditions improve and complement many known results reported in the literature. Examples are provided to illustrate the importance of the main results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain the some new oscillatory condi- tions for all solutions of higher order di¤erence equation with general argument (*) Δmx(n) + p(n)x (τ(n)) = 0; n = 0; 1; ... ; where (p(n)) is a sequence of nonnegative real numbers and (τ(n)) is a sequence of integers and non-monotone. [ABSTRACT FROM AUTHOR]

In this paper, we present some new sufficient conditions for oscillation of even order nonlinear neutral difference equation of the form Δm(xn + axn - τ1 + bxn + τ2 ) + pnxn - σ1α + qnxn + σ2β = 0, n ≥ n0 > 0, where m ≥ 2 is an even integer, using arithmetic-geometric mean inequality. Examples are provided to illustrate the main results. [ABSTRACT FROM AUTHOR]

Pogo phenomenon is considered as low-frequency instability at system level in heavy satellite launchers. This instability results from the frequency interactions between structure and propulsion subsystems. To compensate for or prevent the occurrence of this instability in a satellite launcher, an element that reduces the frequency interaction effects of these two systems is used. This element, which is known as 'accumulator', absorbs the pogo vibrations in satellites. In this paper, by dynamically modeling the engine elements (tank, supply line, accumulator, pump, and thrust chamber), and by considering the connection of structure to engine at two points of tank and pump, the effects of the accumulator element and parameters on the natural frequency of the propulsion system have been investigated and the considerations related to operating conditions and the placement of this element along the supply line of the propulsion system have been determined. The findings indicate that placing the accumulator near the pump assembly and along the suction path reduces the natural frequency of the propulsion system; which contributes to a greater stability of the satellite launcher. With the increase in the compliance and inertance of accumulator, accumulator's influence on the natural frequency of the propulsion system becomes more significant. [ABSTRACT FROM AUTHOR]

In this paper, we aim to establish the sharp maximal pointwise estimates for the multilinear commutators generated by multilinear strongly singular Calderón–Zygmund operators and BMO functions or Lipschitz functions, respectively. As applications, the boundedness of these multilinear commutators on product of weighted Lebesgue spaces are obtained. It is interesting to note that there is no size condition assumption for the kernel of the multilinear strongly singular Calderón–Zygmund operator. Due to the stronger singularity for the kernel of the multilinear strongly singular Calderón–Zygmund operator, we need to be more careful in estimating the mean oscillation over the small balls to get the sharp maximal function estimates. [ABSTRACT FROM AUTHOR]

In this paper, we derive new singular Sturmian separation theorems for nonoscillatory symplectic difference systems on unbounded intervals. The novelty of the presented theory resides in two aspects. We introduce the multiplicity of a focal point at infinity for conjoined bases, which we incorporate into our new singular Sturmian separation theorems. At the same time we do not impose any controllability assumption on the symplectic system. The presented results naturally extend and complete the known Sturmian separation theorems on bounded intervals by J. V. Elyseeva [Comparative index for solutions of symplectic difference systems, Differential Equations 45(3) (2009), pp. 445-459, translated from Differencial'nyje Uravnenija 45 (2009), no. 3, 431-444], as well as the singular Sturmian separation theorems for eventually controllable symplectic systems on unbounded intervals by O. Došlý and J. Elyseeva [Singular comparison theorems for discrete symplectic systems, J. Difference Equ. Appl. 20(8) (2014), pp. 1268-1288]. Our approach is based on developing the theory of comparative index on unbounded intervals and on the recent theory of recessive and dominant solutions at infinity for possibly uncontrollable symplectic systems by the authors [P. Šepitka and R. Šimon Hilscher, Recessive solutions for nonoscillatory discrete symplectic systems, Linear Algebra Appl. 469 (2015), pp. 243-275; P. Šepitka and R. Šimon Hilscher, Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems, J. Difference Equ. Appl. 23(4) (2017), pp. 657-698]. Some of our results, including the notion of the multiplicity of a focal point at infinity, are new even for an eventually controllable symplectic difference system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the oscillation of the following nth-order nonlinear dynamic equation (r(t)Φγ(an-1(t)( . . . (a1(t)xΔ(t))Δ. . .)Δ)Δ)Δ+∑κi=0qi(t)Φαi(x(δi(t)))= 0 on a time scale T with n ≥ 2: We obtain some new oscillation criteria of the above equation. [ABSTRACT FROM AUTHOR]

In this paper, some new results are obtained for the even order neutral delay difference equation ΔanΔm-1xn+pnxn-kα+qnxn-lβ=0, where m≥2 is an even integer, which ensure that all solutions of the studied equation are oscillatory. Our results extend, include, and correct some of the existing results. Examples are provided to illustrate the importance of the main results. [ABSTRACT FROM AUTHOR]

The paper presents a low-Mach number (LM) treatment technique for high-order, Finite-Volume (FV) schemes for the Euler and the compressible Navier–Stokes equations. We concentrate our efforts on the implementation of the LM treatment for the unstructured mesh framework, both in two and three spatial dimensions, and highlight the key differences compared with the method for structured grids. The main scope of the LM technique is to at least maintain the accuracy of low speed regions without introducing artefacts and hampering the global solution and stability of the numerical scheme. Two families of spatial schemes are considered within the k-exact FV framework: the Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and the Weighted Essentially Non-Oscillatory (WENO). The simulations are advanced in time with an explicit third-order Strong Stability Preserving (SSP) Runge–Kutta method. Several flow problems are considered for inviscid and turbulent flows where the obtained solutions are compared with referenced data. The associated benefits of the method are analysed in terms of overall accuracy, dissipation characteristics, order of scheme, spatial resolution and grid composition. [ABSTRACT FROM AUTHOR]

New oscillation criteria for third order noncanonical differential equations of the form (r2(t) (r1(t)y'(t)')' + p(t)y(τ(t)) = 0 are established. Our technique employs an equivalent canonical representation of the studied equation, which essentially simplifies the examination of noncanonical equations. The results obtained are supported by several illustrative examples. [ABSTRACT FROM AUTHOR]

Presents a study which examined the effects of the nonlinear power dissipation on the strength and the position of the cold-work internal friction peaks associated with the oscillation of geometric kinks in the atmosphere of paraelastic interstitials. Description of a nonlinear macroscopic mathematical model of dislocation damping due to the presence of dragging interstitial clouds; Results; Discussion.

We report the electrical transport properties of topological Dirac semimetal Cd3As2. Unlike the ordinary Hall resistance with linear slope and the parabolic magnetoresistance (MR) in conventional semiconductors, the linear-band dispersion in this system give rise to a high electron mobility of 3.4 × 104 cm/V s and a giant linear MR exceeding 5 × 103% at 2.1 K. More importantly, the large Hall angle-induced phenomena can be captured by applying a generic two-band magneto-transport model on both the Hall and MR data, and the fitting results are consistent over the entire temperature range from 2.1 K to 350 K. Furthermore, the large-Hall-angle condition also enables the observation of distinct quantum oscillations with non-trivial band topology of the Cd3As2 sample at low temperatures. Our work provides a general approach to investigate the high-mobility topological Dirac semimetals and their potential applications. [ABSTRACT FROM AUTHOR]

We report the electrical transport properties of topological Dirac semimetal Cd3As2. Unlike the ordinary Hall resistance with linear slope and the parabolic magnetoresistance (MR) in conventional semiconductors, the linear-band dispersion in this system give rise to a high electron mobility of 3.4 × 104 cm/V s and a giant linear MR exceeding 5 × 103% at 2.1 K. More importantly, the large Hall angle-induced phenomena can be captured by applying a generic two-band magneto-transport model on both the Hall and MR data, and the fitting results are consistent over the entire temperature range from 2.1 K to 350 K. Furthermore, the large-Hall-angle condition also enables the observation of distinct quantum oscillations with non-trivial band topology of the Cd3As2 sample at low temperatures. Our work provides a general approach to investigate the high-mobility topological Dirac semimetals and their potential applications. [ABSTRACT FROM AUTHOR]

We study an alternative single species logistic distributed delay differential equation (DDE) with decay-consistent delay in growth. Population oscillation is rarely observed in nature, in contrast to the outcomes of the classical logistic DDE. In the alternative discrete delay model proposed by Arino et al. (J Theor Biol 241(1):109-119, 2006), oscillatory behavior is excluded. This study adapts their idea of the decay-consistent delay and generalizes their model. We establish a threshold for survival and extinction: In the former case, it is confirmed using Lyapunov functionals that the population approaches the delay modified carrying capacity; in the later case the extinction is proved by the fluctuation lemma. We further use adaptive dynamics to conclude that the evolutionary trend is to make the mean delay in growth as short as possible. This confirms Hutchinson’s conjecture (Hutchinson in Ann N Y Acad Sci 50(4):221-246, 1948) and fits biological evidence. [ABSTRACT FROM AUTHOR]

Consider the first-order delay difference equation with a constant argument Δxn+pnxn-k=0,  n=0,1,2,…, and the delay difference equation with a variable argument Δxn+pnxτn=0,  n=0,1,2,…, where p(n) is a sequence of nonnegative real numbers, k is a positive integer, Δx(n)=x(n+1)-x(n), and τ(n) is a sequence of integers such that τ(n)≤n-1 for all n≥0 and limn→∞τ(n)=∞. A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of monotone bounded solutions and of oscillatory solutions to a second-order differential equation with asymptotic conditions. Such asymptotic conditions arise in the study of the ocean fl ow in arctic gyres. Our approach relies on functional-analytic techniques. [ABSTRACT FROM AUTHOR]

For autonomous functional-differential equations with delays, we obtain an oscillation criterion, which allows one to reduce the oscillation problem to the calculation of a unique root of a real-valued function determined by the coefficients of the original equation. The criterion is illustrated by examples of equations with concentrated and distributed aftereffect, for which convenient oscillation tests are obtained. [ABSTRACT FROM AUTHOR]

We study the oscillatory behavior of differential equations with nonmonotone deviating arguments and nonnegative coefficients. New oscillation criteria, involving lim sup and lim inf, are obtained based on an iterative method. Examples, numerically solved in MATLAB, are given to illustrate the applicability and strength of the obtained conditions over known ones. [ABSTRACT FROM AUTHOR]

We deal with a nonlinear impulsive differential equation system with piecewise constant argument. We prove the existence and uniqueness of a solution. Moreover, we obtain sufficient conditions for the oscillation of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some sufficient criteria for the oscillation of the solutions of linear impulsive partial difference equations with continuous variables. [ABSTRACT FROM AUTHOR]

Medium-amplitude oscillatory shear (MAOS) is asymptotically nonlinear and a valuable tool for inferring structure from rheology. However, a drawback of conventional MAOS is the time and material intensive nature of experiments. Many strain amplitude sweeps, and typically multiple sample loadings, are required to obtain frequency-dependent properties. Here, we propose a new MAOS methodology that is much faster (fewer data points) and cheaper (fewer material loadings): The frequency-sweep MAOS. Similar to conventional frequency-sweep SAOS (small-amplitude oscillatory shear), we use only a frequency-sweep. A key challenge is that measurable MAOS data lie within a narrow and frequency-dependent domain between the instrument resolution (at small strain amplitudes) and nonlinear behavior beyond asymptotic nonlinearity (at large strain amplitudes). This necessitates a nonconstant strain amplitude c0(x) for the frequency-sweep instead of a single constant strain amplitude. We provide guidelines for finding this c0(x) trajectory for frequency-sweep MAOS. Full characterization of all four MAOS measures requires two frequency-sweeps: One frequency-sweep for the third-harmonic measures and two frequency-sweeps (at different input strain amplitudes) for the first-harmonic measures. We propose criteria to validate this MAOS data taken at a single strain amplitude based on ratios of stress harmonics; a similar idea is extended to validate frequency-sweep SAOS data as well. The proposed method of frequency-sweep MAOS is demonstrated for a polyvinyl alcohol-borax hydrogel. This new, faster, and material economical MAOS approach will be particularly beneficial for precious samples with very limited availability such as model polymers with well-defined architectures. [ABSTRACT FROM AUTHOR]

Using purely elementary methods, necessary and sufficient conditions are given for the existence of T-periodic and 2T-periodic solutions around the upper equilibrium of the mathematical pendulum when the suspension point is vibrating vertically with asymmetric high frequency. The equation of the motion is of the form ... Ah, Ae, Th, Te are positive constants (Th + Te = T); g and l denote the acceleration of gravity and the length of the pendulum, respectively. An extended Oscillation Theorem is given. The exact stability regions for the upper equilibrium are presented. [ABSTRACT FROM AUTHOR]

We consider difference equations with a single delay term and obtain sufficient conditions for both oscillation and nonoscillation. Moreover, our nonoscillation theorem improves the affirmative answer to Ladas’ corrected conjecture given by Tang and Yu in [Comput. Math. Appl. 38(11–12) (1999), pp. 229–237]. We also present a detailed discussion with examples to emphasize the significance and the sharpness of the new results. [ABSTRACT FROM AUTHOR]

Voltage–current (V–I) droop control has been proposed for real and reactive power sharing. Compared with the conventional droop control using real power/frequency and reactive power/voltage, it enhances stability since adding a V–I droop is equivalent to providing a resistor. However, for a system with parallel converters with V–I droops, oscillations are observed when small droop coefficients are adopted. In this paper, a multi-input multioutput (MIMO) model of the system is derived in the frequency domain. This model is based on the $dq$ -reference frame and consists of converter control and the power network model, which is represented by an admittance matrix. Linear system analysis is carried out to identify the root causes of oscillations in both grid-connected mode and autonomous mode. Analysis results are validated by simulation results of detailed model-based systems built in Matlab/Simpowersystems. [ABSTRACT FROM AUTHOR]

Strong coupling in nanostructures can bring intriguing optical phenomena such as ultrafast Rabi oscillation—periodical energy exchange phenomenon between two modes. Rabi splitting appears in the frequency-domain spectra for strong coupling system. However, in metallic nanosystems the time-domain Rabi oscillations are hard to be observed because the plasmon lifetime is limited by the heavy ohmic losses. Here we report a theoretical investigation of surface plasmon coupling behaviour of two gold nanorods with one being a core-shell rod filled with a gain material and find the periodic energy exchange phenomenon which recalls the concept of Rabi oscillation. The gain material-cored gold-shell structure dipolar mode hybridizes with the solid gold rod quadrupolar mode to form the Fano resonances. Energy exchange between the two rods happens through the near field coupling. Two approaches, to prolong plasmon lifetime by increasing the gain efficiency and to increase Rabi oscillation frequency by increasing the coupling strength, are suggested to increase the Rabi oscillation cycles. Our results offer a way to achieve unique control of light at the nanoscale and further to explore plasmonic Rabi oscillation phenomena in plasmonic nanosystems. [ABSTRACT FROM AUTHOR]

This work concerns the oscillation and asymptotic properties of solutions to the non-linear difference equation with advanced arguments xn+1 -- xn = Σi=1m fi, n (xn+hi, n) We establish sufficient conditions for the existence of positive, and negative solutions. Then we obtain conditions for solutions to be bounded, convergent to positive infinity and to negative infinity and to zero. Also we obtain conditions for all solutions to be oscillatory. [ABSTRACT FROM AUTHOR]

We discuss the concept of relative oscillation for linear Hamiltonian differential systems. We prove a statement which enables to study this oscillation using the criteria of the classical oscillation theory. We also discuss some other aspects of the problem. [ABSTRACT FROM AUTHOR]

For k ≥ 2 even, and α ≥ − ( 2 k + 1 ) ∕ 4 , we provide a uniform approximation of the ultraspherical polynomials P k ( α , α ) ( x ) in the oscillatory region with a very explicit error term. In fact, our result covers all α for which the expression “oscillatory region” makes sense. To that end, we construct the almost equioscillating function g ( x ) = c b ( x ) ( 1 − x 2 ) ( α + 1 ) ∕ 2 P k ( α , α ) ( x ) = cos B ( x ) + r ( x ) . Here the constant c = c ( k , α ) is defined by the normalization of P k ( α , α ) ( x ) , B ( x ) = ∫ 0 x b ( x ) d x , and the functions b ( x ) and B ( x ) , as well as bounds on the error term r ( x ) , are given by some rather simple elementary functions. [ABSTRACT FROM AUTHOR]

In less prescriptive environments where individuals are told ‘what to do’ but not ‘how to do’, synchronization can be a byproduct of strategic thinking, prediction, and local interactions. We prove this in the context of multi-population robust mean-field games. The model sheds light on a multi-scale phenomenon involving fast synchronization within the same population and slow inter-cluster oscillation between different populations. [ABSTRACT FROM AUTHOR]

In this article, we study oscillatory properties of the fourth-order difference equation with middle-term ... in case when the corresponding second-order difference equation Δ²hm - amhm+1 = 0 is nonoscillatory. [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions are established for the oscillation of solutions of q-fractional difference equations of the form where $m=\lceil\alpha\rceil$ , ${}_{q}\nabla_{0}^{\alpha}$ is the Riemann-Liouville q-differential operator and ${}_{q}I_{0}^{m-\alpha}$ is the q-fractional integral. The results are also obtained when the Riemann-Liouville q-differential operator is replaced by Caputo q-fractional difference. Examples are provided to demonstrate the effectiveness of the main result. [ABSTRACT FROM AUTHOR]

In this paper, a time-delay rumor propagation model with a saturated control function is established. Regarding time delay as a bifurcation parameter, Hopf bifurcation is studied. By means of the normal form and the center manifold theorem, a formula is put forward to determine both Hopf bifurcation direction and bifurcating periodic solution stability, together with some numerical simulations to illustrate the relevant theoretical results. Simulation results indicate that an appropriate government control could make periodic oscillation behaviors of the system become steady so as to improve the balance of a social system. [ABSTRACT FROM AUTHOR]

Measurements of the solar neutrino mass-squared difference from KamLAND and solar neutrino data are somewhat discrepant, perhaps due to nonstandard neutrino interactions in matter. We show that the zenith angle distribution of solar neutrinos at Hyper-Kamiokande and the energy spectrum of reactor antineutrinos at JUNO can conclusively confirm the discrepancy and detect new neutrino interactions. [ABSTRACT FROM AUTHOR]

We find very general classes of linear and half-linear difference equations which are conditionally oscillatory. We identify the critical oscillation constant whose value implies the oscillation or non-oscillation of studied equations. Our results are divided into oscillatory and non-oscillatory theorems which determine the critical oscillation constant for coefficients given by sequences having mean values. In addition, our approach enables to analyse also the oscillatory properties of equations whose coefficients are not given by sequences with mean values. We point out that the obtained results are new even for linear equations with periodic coefficients. Such consequences are formulated at the end of this paper. [ABSTRACT FROM AUTHOR]

We investigate oscillation about the positive equilibrium point of a population model with piecewise constant argument. By using linearized oscillation theory for difference equations a necessary and sufficient condition for the oscillation is obtained. Moreover it is showed that every nonoscillatory solution approaches to the equilibrium point as t tends to infinity. [ABSTRACT FROM AUTHOR]

In this study,we consider a stochastic optimal control problem whose state variables are described by the system of forward and backward stochastic differential equations with a general cost functional which relies on the global terminal condition. In the framework of Fréchet derivatives, we derive the corresponding maximum principle via constructing a series of adjoint equations which need to be solved step by step. [ABSTRACT FROM AUTHOR]

In this paper, oscillatory properties of solutions to a nonlinear impulsive parabolic differential system of neutral type are investigated. A series of sufficient conditions are established for problems with Robin and Dirichlet boundary conditions. Examples are provided to confirm the validity of the analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study the nth-order nonlinear dynamic equation with Laplacians and a deviating argument [ABSTRACT FROM AUTHOR]

This paper is concerned with the distributions of zeros of solutions to first order delay dynamic equations on time scales. The results are obtained using iterative sequences. [ABSTRACT FROM AUTHOR]