halanay type inequalities on time scales with applications a b s t r a c t this paper aims to introduce halanay type inequalities on time scales by means of these inequalities we derive new global stability conditions for nonlinear dynamic equations on time scales giving several examples we show that besides generalization and extension to q difference case our results also provide improvements for the existing theory regarding differential and difference inequalities which are the most important particular cases of dynamic inequalities on time scales

periodic solutions for a neutral nonlinear dynamical equation on a time scale abstract let t be a periodic time scale we use a fixed point theorem due to krasnosel ski to show that the nonlinear neutral dynamic system with delay x t a t x t c t x t k q t x t x t k t t has a periodic solution we assume that k is a fixed constant if t r and is a multiple of the period of t if t r under a slightly more stringent inequality we show that the periodic solution is unique using the contraction mapping principle

existence and exponential stability of piecewise mean square almost periodic solutions for impulsive stochastic nicholson s blowflies model on time scales the concept of piecewise mean square almost periodic stochastic process is introduced a new impulsive stochastic nicholsons blowflies model is introduced and studied on time scales the first results of the piecewise mean square almost periodic solutions for the model are obtained in this paper a class of impulsive stochastic nicholson s blowflies model with patch structure and nonlinear harvesting terms is introduced and studied on time scales by using contraction mapping principal and gronwall bellman inequality technique some sufficient conditions for the existence and exponential stability of piecewise mean square almost periodic solutions for the model with infinite delays are established on time scales finally an example is given to demonstrate the validity of the conditions of the main theorem

estimation of stochastic volatility models via monte carlo maximum likelihood this paper discusses the monte carlo maximum likelihood method of estimating stochastic volatility sv models the basic sv model can be expressed as a linear state space model with log chi square disturbances the likelihood function can be approximated arbitrarily accurately by decomposing it into a gaussian part constructed by the kalman filter and a remainder function whose expectation is evaluated by simulation no modifications of this estimation procedure are required when the basic sv model is extended in a number of directions likely to arise in applied empirical research this compares favorably with alternative approaches the finite sample performance of the new estimator is shown to be comparable to the monte carlo markov chain mcmc method

nonparametric specification tests for stochastic volatility models based on volatility density this paper develops a specification test for stochastic volatility models by comparing the nonparametric kernel deconvolution density estimator of an integrated volatility density with its parametric counterpart l2 distance is used to measure the discrepancy the asymptotic null distributions of the test statistics are established and the asymptotic power functions are computed through monte carlo simulations the size and power properties of the test statistics are studied the tests are applied to an empirical example

mcmc maximum likelihood for latent state models abstract this paper develops a pure simulation based approach for computing maximum likelihood estimates in latent state variable models using markov chain monte carlo methods mcmc our mcmc algorithm simultaneously evaluates and optimizes the likelihood function without resorting to gradient methods the approach relies on data augmentation with insights similar to simulated annealing and evolutionary monte carlo algorithms we prove a limit theorem in the degree of data augmentation and use this to provide standard errors and convergence diagnostics the resulting estimator inherits the sampling asymptotic properties of maximum likelihood we demonstrate the approach on two latent state models central to financial econometrics a stochastic volatility and a multivariate jump diffusion models we find that convergence to the mle is fast requiring only a small degree of augmentation

likelihood analysis of non gaussian measurement time series summary in this paper we provide methods for estimating non gaussian time series models these techniques rely on markov chain monte carlo to carry out simulation smoothing and bayesian posterior analysis of parameters and on importance sampling to estimate the likelihood function for classical inference the time series structure of the models is used to ensure that our simulation algorithms are efficient

existence of periodic solutions in shifts δ for neutral nonlinear dynamic systems abstract this paper focuses on the existence of a periodic solution of the delay neutral nonlinear dynamic systems x t a t x t q t x s t g t x t x s t in our analysis we utilize a new periodicity concept in terms of shifts operators which allows us to extend the concept of periodicity to time scales where the additivity requirement t t t for all t t and for a fixed t 0 may not hold more importantly the new concept will easily handle time scales that are not periodic in the conventional way such as q z and k 1 3 k 2 3 k 0 hence we will develop the tool that enables us to investigate the existence of periodic solutions of q difference systems since we are dealing with systems in order to convert our equation to an integral systems we resort to the transition matrix of the homogeneous floquet system y t a t y t and then make use of krasnoselskii s fixed point theorem to obtain a fixed point

monte carlo methods for estimating smoothing and filtering one and two factor stochastic volatility models one and two factor stochastic volatility models are assessed over three sets of stock returns data sp the problem is to get the shape of the conditional returns distribution right none of the models come close to matching the tails of this distribution including a second factor provides only a relatively small improvement over the single factor models fitting this aspect of the data is important for option pricing and risk management

distributional solutions of burgers equation and intrinsic regular graphs in heisenberg groups abstract in the present paper we will characterize the continuous distributional solutions of burgers equation as those which induce intrinsic regular graphs in the first heisenberg group h 1 r 3 endowed with a left invariant metric d equivalent to its carnot caratheodory metric we will also extend the characterization to higher heisenberg groups h n r 2 n 1

intersections of intrinsic submanifolds in the heisenberg group in the first heisenberg group we show that the intersection of two intrinsic submanifolds with linearly independent horizontal normals locally coincides with the image of an injective continuous curve the key tool is a chain rule that relies on a recent result by dafermos

massera s theorems for various types of equations with discontinuous solutions abstract we present new massera type theorems for various types of equations with periodic right hand sides we deal with generalized ordinary differential equations measure differential equations impulsive equations all of which might have discontinuous solutions as well as dynamic equations on time scales for scalar nonlinear equations we find sufficient conditions guaranteeing that each bounded solution is asymptotic to a periodic solution for linear systems we show that the existence of a bounded solution implies the existence of a periodic solution we include some examples to illustrate our results

estimation of affine asset pricing models using the empirical characteristic function abstract the known functional form of the conditional characteristic function ccf of discretely sampled observations from an affine diffusion is used to develop computationally tractable and asymptotically efficient estimators of the parameters of affine diffusions and of asset pricing models in which the state vectors follow affine diffusions both time domain estimators based on fourier inversion of the ccf and frequency domain estimators based directly on the ccf are constructed a method of moments estimator based on the ccf is shown to approximate the efficiency of maximum likelihood for affine diffusion and asset pricing models

uniform exponential stability of first order dynamic equations with several delays abstract this paper is concerned with the uniform exponential stability of ordinary and delay dynamic equations after revealing the equivalence between various types of uniform exponential stability definitions on time scales with bounded graininess and demonstrating their relation when the graininess is arbitrary we confine our attention to the uniform exponential stability of ordinary dynamic equations we introduce and prove the bohl perron criterion for delay dynamic equations if for any bounded right hand side the solution of the delay dynamic equation with bounded coefficients and delays is bounded then the trivial solution of the equation is uniformly exponentially stable we also obtain some corollaries of this criterion based on these results explicit exponential stability tests are derived for delay dynamic equations with nonnegative coefficients which are illustrated with an example on a nonstandard time scale

time series analysis of non gaussian observations based on state space models from both classical and bayesian perspectives the analysis of non gaussian time series by using state space models is considered from both classical and bayesian perspectives the treatment in both cases is based on simulation using importance sampling and antithetic variables markov chain monte carlo methods are not employed non gaussian disturbances for the state equation as well as for the observation equation are considered methods for estimating conditional and posterior means of functions of the state vector given the observations and the mean square errors of their estimates are developed these methods are extended to cover the estimation of conditional and posterior densities and distribution functions the choice of importance sampling densities and antithetic variables is discussed the techniques work well in practice and are computationally efficient their use is illustrated by applying them to a univariate discrete time series a series with outliers and a volatility series

a powerful portmanteau test of lack of fit for time series a new portmanteau test for time series more powerful than the tests of ljung and box and monti is proposed the test is based on the mth root of the determinant of the mth autocorrelation matrix it is shown that the proposed statistic is a function of all of the squared multiple correlation coefficients of the regressions of the residuals on their lags when the number of lags goes from 1 to m it can also be written as a function of the first m partial autocorrelation coefficients the asymptotic distribution of the test statistic is a linear combination of chi squared distributions and can be approximated by a gamma distribution it is shown depending on the model and sample size that this test can be up to 50 more powerful than the ljung and box and monti tests the test is applied to the detection of several types of nonlinearity by using the autocorrelation matrix of the squared residuals and it is shown that in general the new test is more powerful than the test of mcleod and li an example

regular submanifolds graphs and area formula in heisenberg groups we describe intrinsically regular submanifolds in heisenberg groups h low dimensional and low codimensional submanifolds turn out to be of a very different nature the first ones are legendrian surfaces while low codimensional ones are more general objects possibly non euclidean rectifiable nevertheless we prove that they are graphs in a natural group way as well as that an area formula holds for the intrinsic hausdorff measure finally they can be seen as federer fleming currents given a natural complex of differential forms on h

periodicity and attractivity of a nonlinear higher order difference equation in this paper a nonlinear higher order difference equation is considered the existence and nonexistence of two period positive solution and the attractivity are investigated

stability and instability for dynamic equations on time scales in this paper we examine the stability and instability of the equilibrium solution x 0 to the first order system of dynamic equations x d f t x t t 0 x d r n where t is from a so called time scale t with t 0 tand d is a compact set our methods involve the existence of a positive definite liapunov function v such that its delta derivative v d satisfies certain integral definite or semidefinite sign properties finally we use liapunov functions to develop an invariance principle regarding solutions to the above dynamic equation

testing the assumptions behind importance sampling importance sampling is used in many areas of modern econometrics to approximate unsolvable integrals its reliable use requires the sampler to possess a variance for this guarantees a square root speed of convergence and asymptotic normality of the estimator of the integral however this assumption is seldom checked in this paper we use extreme value theory to empirically assess the appropriateness of this assumption our main application is the stochastic volatility model where importance sampling is commonly used for maximum likelihood estimation of the parameters of the model

the structure of dynamic correlations in multivariate stochastic volatility models this paper proposes two types of stochastic correlation structures for multivariate stochastic volatility msv models namely the constant correlation cc msv and dynamic correlation dc msv models from which the stochastic covariance structures can easily be obtained both structures can be used for purposes of determining optimal portfolio and risk management strategies through the use of correlation matrices and for calculating value at risk var forecasts and optimal capital charges under the basel accord through the use of covariance matrices a technique is developed to estimate the dc msv model using the markov chain monte carlo mcmc procedure and simulated data show that the estimation method works well various multivariate conditional volatility and msv models are compared via simulation including an evaluation of alternative var estimators the dc msv model is also estimated using three sets of empirical data namely nikkei 225 index hang seng index and straits times index returns and significant dynamic correlations are found the dynamic conditional correlation dcc model is also estimated and is found to be far less sensitive to the covariation in the shocks to the indexes the correlation process for the dcc model also appears to have a unit root and hence constant conditional correlations in the long run in contrast the estimates arising from the dc msv model indicate that the dynamic correlation process is stationary

positive periodic solutions of delay difference equations and applications in population dynamics with the help of continuation theorem based on mawhin s coincidence degree the existence of a positive periodic solution of the delay difference equation x k 1 x k exp f k x k 1 x k n is studied when these results are applied to some special delay population models some new results are obtained

monte carlo maximum likelihood estimation for non gaussian state space models state space models are considered for observations which have non gaussian distributions we obtain accurate approximations to the loglikelihood for such models by monte carlo simulation devices are introduced which improve the accuracy of the approximations and which increase computational efficiency the loglikelihood function is maximised numerically to obtain estimates of the unknown hyperparameters standard errors of the estimates due to simulation are calculated details are given for the important special cases where the observations come from an exponential family distribution and where the observation equation is linear but the observation errors are non gaussian the techniques are illustrated with a series for which the observations have a poisson distribution and a series for which the observation errors have a t distribution

efficient sequential monte carlo with multiple proposals and control variates sequential monte carlo is a useful simulation based method for online filtering of state space models for certain complex state space models a single proposal distribution is usually not satisfactory and using multiple proposal distributions is a general approach to address various aspects of the filtering problem this article proposes an efficient method of using multiple proposals in combination with control variates the likelihood approach of tan 2004 is used in both resampling and estimation the new algorithm is shown to be asymptotically more efficient than the direct use of multiple proposals and control variates the guidance for selecting multiple proposals and control variates is also given numerical examples are used to demonstrate that the new algorithm can significantly improve over the bootstrap filter and auxiliary particle filter

sobolev s spaces on time scales and its applications to a class of second order hamiltonian systems on time scales abstract in this paper we study the sobolev s spaces on time scales and their properties as applications we present a recent approach via variational methods and the critical point theory to obtain the existence of solutions for a class of second order hamiltonian systems on time scales by establishing a proper variational setting three existence results for systems under consideration are obtained finally three examples are presented to illustrate the feasibility and effectiveness of the existence results

almost automorphic solutions of discrete delayed neutral system abstract we study almost automorphic solutions of the discrete delayed neutral dynamic system x t 1 a t x t q t x t g t g t x t x t g t by means of a fixed point theorem due to krasnoselskii using discrete variant of exponential dichotomy and proving uniqueness of projector of discrete exponential dichotomy we invert the equation and obtain some limit results leading to sufficient conditions for the existence of almost automorphic solutions of the neutral system unlike the existing literature we prove our existence results without assuming boundedness of inverse matrix a t 1 hence we significantly improve the results in the existing literature we provide two examples to illustrate effectiveness of our results finally we also provide an existence result for almost periodic solutions of the system

asymptotical properties of social network dynamics on time scales in this paper we develop conditions for various types of stability in social networks governed by imitation of success principle considering so called prisoner s dilemma as the base of node to node game in the network we obtain well known hopfield neural network model asymptotic behavior of the original model and dynamic hopfield model has a certain correspondence to obtain more general results we consider hopfield model dynamic system on time scales developed stability conditions combine main parameters of network structure such as network size and maximum relative nodes degree with the main characteristics of time scale nodes inertia and resistance rate of input output response

the generalized invariance principle for dynamic equations on time scales the study of long term behavior of a solution generated by dynamics on time scales always gains a great amount of attentions both from engineers and from mathematicians in the paper a generalized invariance principle which not only includes the case described by the conventional invariance principle but also involves the case where the sign of the delta derivative of the lyapunov function along with the solution is not determined is established this consequently allows us to verify the stability and boundedness of the solution generated by a much wider class of dynamic equations on time scales by utilizing this generalized principle

the simulation smoother for time series models summary recently suggested procedures for simulating from the posterior density of states given a gaussian state space time series are refined and extended we introduce and study the simulation smoother which draws from the multivariate posterior distribution of the disturbances of the model so avoiding the degeneracies inherent in state samplers the technique is important in gibbs sampling with non gaussian time series models and for performing bayesian analysis of gaussian time series

markov chain monte carlo methods for stochastic volatility models this paper is concerned with simulation based inference in generalized models of stochastic volatility defined by heavy tailed student t distributions with unknown degrees of freedom and exogenous variables in the observation and volatility equations and a jump component in the observation equation by building on the work of kim shephard and chib rev econom stud 65 1998 361 we develop efficient markov chain monte carlo algorithms for estimating these models the paper also discusses how the likelihood function of these models can be computed by appropriate particle filter methods computation of the marginal likelihood by the method of chib j amer statist assoc 90 1995 1313 is also considered the methodology is extensively tested and validated on simulated data and then applied in detail to daily returns data on the s p 500 index where several stochastic volatility models are formally compared under different priors on the parameters

growth of solutions for measure differential equations and dynamic equations on time scales abstract in this paper we investigate the asymptotic behavior for measure differential equations and dynamic equations on time scales via generalized ordinary differential equations generalized odes for short at first we establish new results that guarantee the existence of unbounded solutions for generalized odes and after that using the known correspondence between generalized odes and measure differential equations also between measure differential equations and dynamic equations on time scales we obtain similar results for these equations finally we present an application to illustrate our main results

stability and periodicity in dynamic delay equations let t be an arbitrary time scale that is unbounded above by means of a variation of lyapunov s method and contraction mapping principle this paper handles asymptotic stability of the zero solution of the completely delayed dynamic equations x d t a t x d t d d t moreover if t is a periodic time scale then necessary conditions are given for the existence of a unique periodic solution of the above mentioned equation

stability criteria for impulsive systems on time scales in this paper we study stability of impulsive system on time scales by using comparison method lyapunov function and analysis method the asymptotic stability criteria for system with impulses at fixed times and impulses at variable times on time scales are obtained respectively an example is presented to illustrate the efficiency of proposed result

uniform eventual lipschitz stability of impulsive systems on time scales in this paper we discuss the uniform eventual lipschitz stability of impulsive system on time scales by using comparison method lyapunov function and analysis technology some criteria of such stability for system with impulses on time scales are obtained an example is presented to illustrate the efficiency of proposed results

rectifiability and perimeter in the heisenberg group in this paper we fully extend to the heisenberg group endowed with its intrinsic carnot caratheodory metric and perimeter the classical de giorgi s rectifiability divergence theorems

on the asymptotic efficiency of approximate bayesian computation estimators many statistical applications involve models for which it is difficult to evaluate the likelihood but from which it is relatively easy to sample approximate bayesian computation is a likelihood free method for implementing bayesian inference in such cases we present results on the asymptotic variance of estimators obtained using approximate bayesian computation in a large data limit our key assumption is that the data is summarized by a fixed dimensional summary statistic that obeys a central limit theorem we prove asymptotic normality of the mean of the approximate bayesian computation posterior this result also shows that in terms of asymptotic variance we should use a summary statistic that is the same dimension as the parameter vector p and that any summary statistic of higher dimension can be reduced through a linear transformation to dimension p in a way that can only reduce the asymptotic variance of the posterior mean we look at how the monte carlo error of an importance sampling algorithm that samples from the approximate bayesian computation posterior affects the accuracy of estimators we give conditions on the importance sampling proposal distribution such that the variance of the estimator will be the same order as that of the maximum likelihood estimator based on the summary statistics used this suggests an iterative importance sampling algorithm which we evaluate empirically on a stochastic volatility model

stability and periodicity in discrete delay equations abstract in this research we study the stability of the zero solution of delay difference equations and the existence of unique periodic solution by utilizing fixed point theory mainly we are interested in the qualitative analysis of the complete delay difference equation x t a t x t

polar generation of random variates with the t distribution the polar method of box and muller uses two independent uniform variates in order to generate two independent normal variates it can be adapted so that two variates from student s t distribution with parameter v are generated though the two variates are now not independent an algorithm based on the polar method is exact inexpensive and valid for all v 0 box and muller s 1 polar method for generating random normal variates relies on two convenient properties of the normal distribution which we may formulate as follows i let x n o 1 then x can be regarded as the real part of a complex random variable z which has a radial distribution the contours of the density function of z form circles centered at the origin ii write z x iy reie9 then the distribution function fr 1gr of r is a simple algebraic expression so simple that it is invertible that is given g gr r we can write down a closed expression for r in term of g the aim of this article is to show that properties i and ii are shared by the student t distribution with parameter v the tv distribution defined by the density 1 ft x b v 2 1 2 i v 1 2 1 x2 v v l 2 thus we are asserting that the tv distribution like the normal has a tractable radial parent if t has the density 1 we shall write t tv many methods have been proposed for the generation of tv variates the most important ones are described in devroye 2 pp 445 450 whose masterly survey we shall not attempt to emulate the faster algorithms may require either a comparatively great programming effort or the expensive recalculation of certain quantities required by the algorithm whenever v is changed as devroye notes problems arise when v is small and the departure from normality is greatest particularly in the region 0 0 we now show how the polar method may be applied to the tv distribution to yield such an algorithm received by the editor december 23 1992 and in revised form may 12 1993 1991 mathematics subject classification primary 65c10 secondary 62e15

stochastic volatility in asset prices estimation with simulated maximum likelihood abstract the stochastic volatility model is used to estimate daily asset price dynamics the model is estimated by integrating latent volatility out of the joint density of prices and volatility to obtain the marginal density of prices due to high number of dimensions of the integral no conventional integration technique is applicable a monte carlo method called simulated maximum likelihood is used to obtain the marginal density where the latent variable is simulated conditional on available information the model is estimated by 2022 observations from the s p 500 index for comparison arch type models are estimated with the same data

rank one theorem and subgraphs of bv functions in carnot groups abstract we prove a rank one theorem a la g alberti for the derivatives of vector valued maps with bounded variation in a class of carnot groups that includes heisenberg groups h n for n 2 the main tools are properties relating the horizontal derivatives of a real valued function with bounded variation and its subgraph

exponential stability for set dynamic equations on time scales abstract a new theory known as set dynamic equations on time scales has been built the criteria for the equistability equiasymptotic stability uniform and uniformly asymptotic stability were developed in hong 2010 1 in this paper we consider the exponential stability exponentially asymptotic stability uniform and uniformly exponentially asymptotic stability for the trivial solution of set dynamic equations on time scales by using lyapunov like functions

dynamic equations on time scales and generalized ordinary differential equations abstract the aim of this paper is to show that dynamic equations on time scales can be treated in the framework of generalized ordinary differential equations as introduced by j kurzweil we also use some known results for generalized ordinary differential equations to obtain new theorems related to stability and continuous dependence on parameters for dynamic equations on time scales

qualitative analysis of nonlinear volterra integral equations on time scales using resolvent and lyapunov functionals in this paper we use the notion of the resolvent equation and lyapunov s method to study boundedness and integrability of the solutions of the nonlinear volterra integral equation on time scales x t a t t 0 t c t s g s x s s t t 0 t in particular the existence of bounded solutions with various lp properties are studied under suitable conditions on the functions involved in the above volterra integral equation at the end of the paper we display some examples on different time scales

uniformly rd piecewise almost periodic functions with applications to the analysis of impulsive δ dynamic system on time scales we introduce some new concepts of uniformly rd piecewise almost periodic functions on time scales and obtain some properties some new estimation inequalities and completely new theorems are established on time scales for impulsive dynamic systems several applications are given to illustrate our feasible results in the present work we introduce two equivalent concepts of uniformly rd piecewise almost periodic functions on time scales and their equivalence is proved based on this some basic properties of them are obtained then some new criteria of exponential dichotomy are established for homogeneous dynamic system on time scales also some completely new theorems are established on time scales for impulsive almost periodic dynamic systems such as favard s theorem and exponential dichotomy theorem as applications we provide a method to obtain an almost periodic solution for a given nonhomogeneous impulsive dynamic system furthermore we introduce an impulsive non autonomous nicholson s blowflies system model with patch structure and multiple nonlinear harvesting terms for which the existence and exponential stability of almost periodic solutions are studied which shows that our results can be applied feasibly and effectively

the stochastic conditional duration model a latent variable model for the analysis of financial durations we introduce a class of models for the analysis of durations which we call stochastic conditional duration scd models these models are based on the assumption that the durations are generated by a dynamic stochastic latent variable the model yields a wide range of shapes of hazard functions the estimation of the parameters is performed by quasi maximum likelihood and using the kalman filter the model is applied to trade price and volume durations of stocks traded at nyse we also investigate the relation between price durations spread trade intensity and volume c 2003 elsevier b v all rights reserved

disturbance smoother for state space models summary this paper develops a method to evaluate the smoothed estimator of the disturbance vector in a state space model together with its mean squared error matrix this disturbance smoother also leads to an efficient smoother for the state vector applications include a method to calculate auxiliary residuals for unobserved components time series models and an em algorithm for estimating covariance parameters in a state space model

phase spaces and periodic solutions of set functional dynamic equations with infinite delay abstract very recently a new theory known as set dynamic equations on time scales has been built in this paper a phase space is built for set functional dynamic equations with infinite delay on time scales and sufficient criteria are established for the existence of periodic solutions of such equations which generalize and incorporate as special cases some known results for set differential equations and for set difference equations when the time scale is the real number set or the integer set respectively moreover for differential inclusions and difference inclusions if the variable under consideration is a single valued mapping our results show that one can unify the study of some continuous or discrete problems in the sense of set dynamic equations on general time scales

floquet theory based on new periodicity concept for hybrid systems involving q difference equations using the new periodicity concept based on shifts we construct a unified floquet theory for homogeneous and nonhomogeneous hybrid periodic systems on domains having continuous discrete or hybrid structure new periodicity concept based on shifts enables the construction of floquet theory on hybrid domains that are not necessarily additive periodic this makes periodicity and stability analysis of hybrid periodic systems possible on non additive domains in particular this new approach can be useful to know more about floquet theory for linear q difference systems defined on q z q n n z 0 where q 1 by constructing the solution of matrix exponential equation we establish a canonical floquet decomposition theorem determining the relation between floquet multipliers and floquet exponents we give a spectral mapping theorem on closed subsets of reals that are periodic in shifts finally we show how the constructed theory can be utilized for the stability analysis of dynamic systems on periodic time scales in shifts

basic qualitative and quantitative results for solutions to nonlinear dynamic equations on time scales with an application to economic modelling abstract this article investigates both basic qualitative and basic quantitative properties of solutions to first and higher order dynamic equations on time scales and thus provides a foundation and framework for future advanced nonlinear studies in the field particular focus lies in the existence uniqueness dependency approximation and explicit representation of solutions to nonlinear initial value problems the main tools used are from modern areas of nonlinear analysis including the fixed point theorems of banach and schafer the method of successive approximations a novel definition of measuring distance in metric spaces and normed spaces and a separation of variables technique is introduced to the general time scale setting the new results compliment and extend those of stefan hilger s seminal paper of 1990 as an application of the new results we present and analyse a simple model from economics known as the keynesian cross model with lagged income in the general time scale environment ideas suggesting further applications and possible new directions for the novel results are also presented

on the exponential stability of dynamic equations on time scales in this paper we deal with some theorems on the exponential stability of trivial solution of time varying non regressive dynamic equation on time scales with bounded graininess in particular well known perron s theorem is generalized on time scales under rather restrictive condition that is integral boundedness of coefficient operators we obtain a characterization of the uniformly exponential stability

stability criteria of delay impulsive systems on time scales abstract in this paper we study stability of delay impulsive systems on time scales by using lyapunov functional and analysis method some stability and instability criteria for the delay system with impulses at fixed times on time scales are obtained two examples are presented to illustrate the efficiency of results proposed in this paper

variation of lyapunov s method for dynamic systems on time scales abstract a new comparison theorem that connects the solutions of perturbed and unperturbed dynamic systems in a manner useful to the theory of perturbations is given and this comparison theorem is employed as a stability criterion to compare the asymptotic behaviors of perturbed and unperturbed systems it is further shown by means of both theory and numerical computation that time scales do offer a unification in order to emphasize the better asymptotic behavior of perturbed systems in both continuous and discrete cases

existence of periodic solutions for a periodic mutualism model on time scales abstract by using mawhin s continuation theorem of coincidence degree theory sufficient criteria are obtained for the existence of periodic solutions of the mutualism model x t r 1 t k 1 t 1 t exp y t 2 t y t 1 exp y t 2 t y t exp x t 1 t x t y t r 2 t k 2 t 2 t exp x t 1 t x t 1 exp x t 1 t x t exp y t 2 t y t where r i k i i c t r i k i i 1 2 i i c t r t i 1 2 r i k i i i i i 1 2 are functions of period 0

global behavior of solutions of a nonlinear difference equation our aim in this paper is to investigate the persistence and the global attractivity of solutions of the equationx n 1 f x n x n 1 x n k forn 0 1 where f is a continuous positive and decreasing function on 0 k 1 0 the results are applied to the difference equationy n 1 ay p n 1y q n kforn 0 1 where a p email protected 0 email protected 2 3

stability by linear approximation for time scale dynamical systems abstract we study systems on time scales that are generalizations of classical differential or difference equations and appear in numerical methods in this paper we consider linear systems and their small nonlinear perturbations in terms of time scales and of eigenvalues of matrices we formulate conditions sufficient for stability by linear approximation for non periodic time scales we use techniques of central upper lyapunov exponents a common tool of the theory of linear odes to study stability of solutions also time scale versions of the famous chetaev s theorem on conditional instability are proved in a nutshell we have developed a completely new technique in order to demonstrate that methods of non autonomous linear ode theory may work for time scale dynamics

periodicity in a nonlinear predator prey system on time scales with state dependent delays abstract with the help of a continuation theorem based on gaines and mawhin s coincidence degree easily verifiable criteria are established for the global existence of positive periodic solutions of a nonlinear state dependent delays predator prey system on time scales