dynamics for a class of general hematopoiesis model with periodic coefficients abstract sufficient conditions are obtained for the existence and global attractivity of a unique positive periodic solution x t of x t a t x t b t 1 x n t t t 0 where n 1 a and b are continuous positive periodic function also some sufficient conditions are established for oscillation of all positive solutions of about x t for the proof of existence and uniqueness of x t the method used here is better than contraction mapping principle

applications of the brauer complex card shuffling permutation statistics and dynamical systems by algebraic group theory there is a map from the semisimple conjugacy classes of a finite group of lie type to the conjugacy classes of the weyl group picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the weyl group using the brauer complex it is proved that this measure agrees with a second measure on conjugacy classes of the weyl group induced by a construction of cellini using the affine weyl group formulas for cellini s measure in type a are found this leads to new models of card shuffling and has interesting combinatorial and number theoretic consequences an analysis of type c gives another solution to a problem of rogers in dynamical systems the enumeration of unimodal permutations by cycle structure the proof uses the factorization theory of palindromic polynomials over finite fields contact is made with symmetric function theory

splitting methods for the nonlocal fowler equation we consider a nonlocal scalar conservation law proposed by andrew c fowler to describe the dynamics of dunes and we develop a numerical procedure based on splitting methods to approximate its solutions we begin by proving the convergence of the well known lie formula which is an approximation of the exact solution of order one in time we next use the split step fourier method to approximate the continuous problem using the fast fourier transform and the finite difference method our numerical experiments confirm the theoretical results

on fourier time splitting methods for nonlinear schrodinger equations in the semiclassical limit we prove an error estimate for a lie trotter splitting operator associated with the schrodinger poisson equation in the semiclassical regime when the wkb approximation is valid in finite time and so long as the solution to a compressible euler poisson equation is smooth the error between the numerical solution and the exact solution is controlled in sobolev spaces in a suitable phase amplitude representation as a corollary we infer the numerical convergence of the quadratic observables with a time step independent of the planck constant a similar result is established for the nonlinear schrodinger equation in the weakly nonlinear regime the function u e u e t x is complex valued and the space variable x belongs to r d the presence of the parameter e is motivated by the semiclassical limit e 0 physically e corresponds to a small ratio between microscopic and macroscopic quantities so the limit e 0 is expected to yield a relevant approximation see e g 18 and references therein the parameter 0 measures the strength of nonlinear interactions in the wkb regime which is recalled below the nonlinearity is negligible if 1 it has a leading order moderate influence if 1 weakly nonlinear regime and its influence is very strong in the regime e 0i f 0 the case 0 1 is not considered here but it should be considered as similar to the case 0 see 9 in this paper we consider mostly two families of nonlinearity nonlocal nonlinearity in the case 0 f k local or nonlocal nonlinearity in the case 1 the first case includes the schrodinger poisson system in space dimension d 3 f 1 r hence k x cd x d 2 the second case includes the cubic nonlinearity focusing or defocusing we will also discuss why the case of the cubic nonlinearity is not treated in the regime 0 see remark 6 3 our analysis is limited to bounded time intervals so the exponentials are controlled which is the reason why we do not keep track of such factors when they are eventually included in a uniform constant the reason why the analysis is bound to finite time intervals is in the case 0 that the solution of the euler poisson equation generically develops

strang splitting method for burgers huxley equation we derive an analytical approach to the strang splitting method for the burgers huxley equation bhe u t u u x u x x 1 u u u we proved that srtang splitting method has a second order convergence in h s r where h s r is the sobolev space and s is an arbitrary nonnegative integer we numerically solve the bhe by strang splitting method and compare the results with the reference solution

mackey glass model of hematopoiesis with monotone feedback revisited for the blood cell production modeldxdt ax t f 0 q n q n x n t t we review known results permanence of solutions oscillation and stability of a positive equilibrium or a positive periodic solution are analyzed for non autonomous generalizations of this equation with both concentrated and distributed delays

on the benney lin and kawahara equations abstract we establish global well posedness for the initial value problem ivp associated to the so called benney lin equation this model is a korteweg de vries equation perturbed by dissipative and dispersive terms which appears in fluid dynamics we also study the limiting behaviour of solutions to this ivp when the parameters of the perturbed terms approach 0

geometric optics and boundary layers for nonlinear schrodinger equations we justify supercritical geometric optics in small time for the defocusing semiclassical nonlinear schrodinger equation for a large class of non necessarily homogeneous nonlinearities the case of a half space with neumann boundary condition is also studied

supercritical geometric optics for nonlinear schrodinger equations we consider the small time semi classical limit for nonlinear schrodinger equations with defocusing smooth nonlinearity for a super cubic nonlinearity the limiting system is not directly hyperbolic due to the presence of vacuum to overcome this issue we introduce new unknown functions which are defined nonlinearly in terms of the wave function itself this approach provides a local version of the modulated energy functional introduced by y brenier the system we obtain is hyperbolic symmetric and the justification of wkb analysis follows

regularized numerical methods for the logarithmic schrodinger equation we present and analyze two numerical methods for the logarithmic schrodinger equation logse consisting of a regularized splitting method and a regularized conservative crank nicolson finite difference method cnfd in order to avoid numerical blow up and or to suppress round off error due to the logarithmic nonlinearity in the logse a regularized logarithmic schrodinger equation rlogse with a small regularized parameter 0 0 the time step which implies an error bound at o varepsilon tau 1 2 ln varepsilon 1 for the logse by the lie trotter splitting method in addition the cnfd is also applied to discretize the rlogse which conserves the mass and energy in the discretized level numerical results are reported to confirm our error bounds and to demonstrate rich and complicated dynamics of the logse

affine shuffles shuffles with cuts the whitehouse module and patience sorting abstract type a affine shuffles are compared with riffle shuffles followed by a cut although these probability measures on the symmetric group s n are different they both satisfy a convolution property strong evidence is given that when the underlying parameter q satisfies gcd n q 1 1 the induced measures on conjugacy classes of the symmetric group coincide this gives rise to interesting combinatorics concerning the modular equidistribution by major index of permutations in a given conjugacy class and with a given number of cyclic descents using representation theoretic work on the whitehouse module a formula is obtained for the cycle structure of a riffle shuffle followed by a cut it is proved that the use of cuts does not speed up the convergence rate of riffle shuffles to randomness generating functions for the first pile size in patience sorting from decks with repeated values are derived this relates to random matrices

mackey glass model of hematopoiesis with non monotone feedback stability oscillation and control for the blood cell production model with a unimodal hump feedback email protected t email protected q ny email protected q n y n email protected we review the known results and investigate generalizations of this equation permanence oscillation and stability of the positive equilibrium are studied for non autonomous equations including equations with a distributed delay in addition a linear control is introduced and possibilities to stabilize an otherwise unstable positive equilibrium are explored

instabilities for supercritical schrodinger equations in analytic manifolds in this paper we consider supercritical nonlinear schrodinger equations in an analytic riemannian manifold md g where the metric g is analytic using an analytic wkb method we are able to construct an ansatz for the semiclassical equation for times independent of the small parameter these approximate solutions will help to show two different types of instabilities the first is in the energy space and the second is an immediate loss of regularity in higher sobolev norms

enriched p partitions and peak algebras abstract we develop a more general view of stembridge s enriched p partitions and use this theory to outline the structure of peak algebras for the symmetric group and the hyperoctahedral group initially we focus on commutative peak algebras spanned by sums of permutations with the same number of peaks where we consider several variations on the definition of peak whereas stembridge s enriched p partitions are related to quasisymmetric functions the dual coalgebra of solomon s type a descent algebra our generalized enriched p partitions are related to type b quasisymmetric functions the dual coalgebra of solomon s type b descent algebra using these functions we move on to explore non commutative peak algebras spanned by sums of permutations with the same set of peaks while some of these algebras have been studied before our approach gives explicit structure constants with a combinatorial description

an exponential type integrator for the kdv equation we introduce an exponential type time integrator for the kdv equation and prove its first order convergence in h 1 for initial data in h 3 furthermore we outline the generalization of the presented technique to a second order method

a new existence theory for positive periodic solutions to functional differential equations abstract this paper deals with a new existence theory for positive periodic solutions to a kind of nonautonomous functional differential equation by employing the fixed point theorem in cones applying the general theorems established to several biomathematical models the paper improves some previous results and obtains some new results

operator splitting for the benjamin ono equation abstract in this paper we analyze operator splitting for the benjamin ono equation u t u u x h u x x where h denotes the hilbert transform if the initial data are sufficiently regular we show the convergence of both godunov and strang splittings

on landau damping going beyond the linearized study has been a longstanding problem in the theory of landau damping in this paper we establish exponential landau damping in analytic regularity the damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables rather than exchanges of energy phase mixing is the driving mechanism the analysis involves new families of analytic norms measuring regularity by comparison with solutions of the free transport equation new functional inequalities a control of non linear echoes sharp deflection estimates and a newton approximation scheme our results hold for any potential no more singular than coulomb or newton interaction the limit cases are included with specific technical effort as a side result the stability of homogeneous equilibria of the non linear vlasov equation is established under sharp assumptions we point out the strong analogy with the kam theory and discuss physical implications finally we extend these results to some gevrey non analytic distribution functions

existence of positive periodic solutions for a functional differential equation with a parameter abstract in this paper we study the existence multiplicity and nonexistence of positive periodic solutions for a kind of nonautonomous functional differential equation by employing the fixed point theorem in cones some new results are obtained as an application of our results we discuss the existence of a positive periodic solution for a class of models of blood cell production with a parameter which has not been investigated by using previous methods

numerical study of time splitting spectral discretizations of nonlinear schrodinger equations in the semiclassical regimes in this paper we study the performance of time splitting spectral approximations for general nonlinear schrodinger equations nls in the semiclassical regimes where the planck constant varepsilon is small the time splitting spectral approximation under study is explicit unconditionally stable and conserves the position density in l 1 moreover it is time transverse invariant and time reversible when the corresponding nls is extensive numerical tests are presented for weak strong focusing defocusing nonlinearities for the gross pitaevskii equation and for current relaxed quantum hydrodynamics the tests are geared towards the understanding of admissible meshing strategies for obtaining correct physical observables in the semiclassical regimes furthermore comparisons between the solutions of the nls and its hydrodynamic semiclassical limit are presented

a new existence theory for positive periodic solutions to functional differential equations with impulse effects the principle of this paper is to deal with a new existence theory for positive periodic solutions to a kind of nonautonomous functional differential equations with impulse actions at fixed moments easily verifiable sufficient criteria are established the approach is based on the fixed point theorem in cones the paper extends some previous results and obtains some new results

fractional error estimates of splitting schemes for the nonlinear schrodinger equation abstract we investigate the lie and the strang splitting for the cubic nonlinear schrodinger equation on the full space and on the torus in up to three spatial dimensions we prove that the strang splitting converges in l 2 with order 1 for initial values in h 2 2 with 0 1 and that the lie splitting converges with order one for initial values in h 2

an accurate approximation algorithm for burgers equation in the presence of small viscosity abstract most of the existing numerical schemes developed to solve burgers equation cannot exhibit its correct physical behavior for very small values of viscosity this difficulty can be overcome by using splitting methods derived for near integrable system this class of methods has positive real coefficients and can be used for non reversible systems such as burgers equation it also has the advantage of being able to account small viscosity in the accuracy the algorithm is based on the combination of implicit explicit finite difference schemes to solve each simplified problem and filtering technique to treat nonlinear instability the resulting algorithm is accurate efficient and easy to implement the new numerical results are compared with numerical and exact solutions reported in the literature and found that they are very accurate for small values of the viscosity

order estimates in time of splitting methods for the nonlinear schrodinger equation in this paper we consider the nonlinear schrodinger equation u t i delta u f u 0 in two dimensions we show by an operator theoretic proof that the well known lie and strang formulae which are splitting methods are approximations of the exact solution of order 1 and 2 in time

positive periodic solutions of nonautonomous functional differential systems abstract considered is the periodic functional differential system with a parameter x t a t x t x t f t x t using the eigenvalue problems of completely continuous operators we establish some criteria on the existence of positive periodic solutions moreover we apply the results to a couple of population models and obtain sufficient conditions for the existence of positive periodic solutions which are compared with existing ones

semisimple conjugacy classes and classes in the weyl group abstract we discuss a map from the semisimple conjugacy classes of a finite group gf of lie type to the f conjugacy classes of its weyl group we obtain two expressions for the number of semisimple classes mapped by into a given f conjugacy class of w the first involves distinguished coset representatives in the affine weyl group and the second is the number of elements in the coroot lattice satisfying certain conditions the brauer complex plays a key role in the proof the map has recently proved of interest in connection with probabilistic and combinatorial group theory

positive periodic solutions of functional differential equations abstract we consider the existence multiplicity and nonexistence of positive periodic solutions for the periodic equation x t a t g x x t b t f x t t where a b c r 0 are periodic 0 a t dt 0 0 b t dt 0 f g c 0 0 and f u 0 for u 0 g x is bounded t is a continuous periodic function define f 0 lim u 0 f u u f lim u f u u i0 number of zeros in the set f 0 f and i number of infinities in the set f 0 f we show that the equation has i0 or i positive periodic solution s for sufficiently large or small 0 respectively

operator splitting methods for generalized korteweg de vries equations we apply the method of operator splitting on the generalized korteweg de vries kdv equation ut f u x uxxx 0 by solving the nonlinear conservation law ut f u x 0 and the linear dispersive equation ut uxxx 0 sequentially we prove that if the approximation obtained by operator splitting converges then the limit function is a weak solution of the generalized kdv equation convergence properties are analyzed numerically by studying the effect of combining different numerical methods for each of the simplified problems

on splitting methods for schrodinger poisson and cubic nonlinear schrodinger equations we give an error analysis of strang type splitting integrators for nonlinear schrodinger equations for schrodinger poisson equations with an h 4 regular solution a first order error bound in the h 1 norm is shown and used to derive a second order error bound in the l 2 norm for the cubic schrodinger equation with an h 4 regular solution first order convergence in the h 2 norm is used to obtain second order convergence in the l 2 norm basic tools in the error analysis are lie commutator bounds for estimating the local error and h m conditional stability for error propagation where m 1 for the schrodinger poisson system and m 2 for the cubic schrodinger equation

on time splitting spectral approximations for the schrodinger equation in the semiclassical regime in this paper we study time splitting spectral approximations for the linear schrodinger equation in the semiclassical regime where the planck constant e is small in this regime the equation propagates oscillations with a wavelength of o e and finite difference approximations require the spatial mesh size h o e and the time step k o e in order to obtain physically correct observables much sharper mesh size constraints are necessary for a uniform l2 approximation of the wave function the spectral time splitting approximation under study will be proved to be unconditionally stable time reversible and gauge invariant it conserves the position density and gives uniform l2 approximation of the wave function for k o e and h o e extensive numerical examples in both one and two space dimensions and analytical considerations based on the wigner transform even show that weaker constraints e g k independent of e and h o e are admissible for obtaining correct observables finally we address the application to nonlinear schrodinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies

long range scattering and modified wave operators for some hartree type equations iii gevrey spaces and low dimensions abstract we study the theory of scattering for a class of hartree type equations with long range interactions in arbitrary space dimension n 1 including the case of hartree equations with time dependent potential v t x t x with 0 1 and 0 n this includes the case of potential v x x and can be extended to the limiting case of nonlinear schrodinger equations with cubic nonlinearity t n u 2 u using gevrey spaces of asymptotic states and solutions we prove the existence of modified local wave operators at infinity with no size restriction on the data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators thereby extending the results of previous papers which covered the range 0 1 but only 0 n 2 and were therefore restricted to space dimension n 3

a semi analytical fourier spectral method for the swift hohenberg equation abstract the swift hohenberg sh equation has been widely used as a model for the study of pattern formation the sh equation is a fourth order nonlinear partial differential equation and cannot generally be solved analytically therefore computer simulations play an essential role in understanding of nonequilibrium processing the aim of this paper is to present first and second order semi analytical fourier spectral methods as an accurate and efficient approach for solving the sh equation the methods are based on the operator splitting method and are to split the sh equation into linear and nonlinear subequations the linear and nonlinear subequations have closed form solutions in the fourier and physical spaces respectively the methods are simple to implement and computationally cheap to achieve high order time accuracy numerical experiments are presented demonstrating the accuracy and efficiency of proposed methods

explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case a scalar linear differential equation with time dependent delay x t a t x t t t is considered where t i t 0 t 0 r a i r 0 is a continuous function and t i r is a continuous function such that t t t t 0 t t 0 if t t 0 the goal of our investigation is to give sufficient conditions for the existence of positive solutions as t in the critical case in terms of inequalities on a and t a generalization of one known final in a certain sense result is given for the case of t being not a constant analysing this generalization we show e g that it differs from the original statement with a constant delay since it does not give the best possible result this is demonstrated on a suitable example

well posedness for the fifth order shallow water equations abstract the cauchy problems for some kind of fifth order shallow water equations t u x 5 u x 3 u x u f u x u x 2 u 0 x t r r are considered by the fourier restriction norm method where nonlinear terms f u x u x 2 u are x u k k 2 3 u x 2 u or x u x 2 u respectively the local well posedness is established for data in h s r with s 7 4 for the kawahara equation f x u 2 and is established for data in h s r with s 1 4 for the modified kawahara equation f x u 3 respectively moreover the local result is established for data in h s r with s 0 if f u x 2 u and is established for data in h s r with s 1 4 if f x u x 2 u respectively

existence and multiplicity of positive periodic solutions for a class of higher dimension functional differential equations with impulses this paper deals with the existence of multiple periodic solutions for n dimensional functional differential equations with impulses by employing the krasnoselskii fixed point theorem we obtain some easily verifiable sufficient criteria which extend previous results

a split step fourier discontinuous galerkin scheme for the kadomtsev petviashvili equation abstract in this paper we propose a method to solve the kadomtsev petviashvili equation based on splitting the linear part of the equation from the nonlinear part the linear part is treated using ffts while the nonlinear part is approximated using a semi lagrangian discontinuous galerkin approach of arbitrary order we demonstrate the efficiency and accuracy of the numerical method by providing a range of numerical simulations in particular we find that our approach can outperform the numerical methods considered in the literature by up to a factor of five although we focus on the kadomtsev petviashvili equation in this paper the proposed numerical scheme can be extended to a range of related models as well

existence nonexistence and multiplicity of periodic solutions for a kind of functional differential equation with parameter abstract in this paper we study the existence multiplicity and nonexistence of positive periodic solutions for a kind of second order delay functional differential equation with parameter u t a t u t f t u t 0 t u t 1 t u t n t by employing the krasnoselskii fixed point theorem for a cone map where 0 is a parameter some new results are obtained as an application of our results we discuss an example

multiple periodic solutions for a nonlinear first order functional differential equations with applications to population dynamics abstract in this work using legget williams multiple fixed point theorem we have obtained different sufficient conditions for the existence of at least three nonnegative periodic solutions for the first order functional differential equations of the form y t a t y t f t y h t using legget williams multiple fixed point theorem

the peak algebra of the symmetric group revisited abstract the linear span p n of the sums of all permutations in the symmetric group sn with a given set of peaks is a sub algebra of the symmetric group algebra due to nyman this peak algebra is a left ideal of the descent algebra d n and the direct sum p of all p n is a hopf sub algebra of the direct sum d of all d n dual to the stembridge algebra of peak functions in our self contained approach peak counterparts of several results on the descent algebra are established including a simple combinatorial characterization of the algebra p n an algebraic characterization of p n based on the action on the poincare birkhoff witt basis of the free associative algebra the display of peak variants of the classical lie idempotents an eulerian type sub algebra of p n a description of the jacobson radical of p n and its nil potency index of the principal indecomposable and irreducible p n modules and of the cartan matrix of p n furthermore it is shown that the primitive lie algebra of p is free and that p is its enveloping algebra

operator splitting for two dimensional incompressible fluid equations we analyze splitting algorithms for a class of two dimensional fluid equations which includes the incompressible navier stokes equations and the surface quasi geostrophic equation our main result is that the godunov and strang splitting methods converge with the expected rates provided the initial data are sufficiently regular

existence and global attractivity of unique positive periodic solution for a model of hematopoiesis sufficient conditions are obtained which guarantee the uniform persistence and global attractivity of solutions for the model of hematopoiesis then some criteria are established for the existence uniqueness and global attractivity of almost periodic solutions of almost periodic system

higher order functional differential systems with parameter dependence abstract in this paper we consider a class of high order functional differential systems depending on a parameter by using the fixed point index theory we prove that there exists a positive number separating 0 into two disjoint subintervals 1 and 2 such that the system has at least two positive periodic solutions for 1 one positive periodic solution for and no positive periodic solution for 2 our results extend and improve some results in the literature as an application an example is given to demonstrate our main results

three periodic solutions for a nonlinear first order functional differential equation this paper is concerned with the existence of three positive t periodic solutions of the first order functional differential equations of the formx t a t x t email protected t f t x h t where a t b t and t t are positive t periodic functions and l is a positive parameter leggett williams multiple fixed point theorem has been used to prove our results our results extend and improve some results in the literature

existence of positive periodic solutions for a class of functional differential equations in this article we investigate the existence of positive periodic solutions for a class of functional differential equations by using fixed point index theory we establish sufficient criteria that are easily verifiable and that generalize and improve related studies in the literature also we give some examples in bio mathematics to show that our main results are valid

the existence of multiple positive periodic solutions for functional differential equations abstract the existence of multiple positive solutions for the integral equation x t t t g t s b s f s x s 1 s x s n s ds is studied by using fixed point index theory using these results we obtain new results on the existence of multiple positive periodic solutions for some first order periodic functional differential equations

existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects abstract this paper deals with a new existence theory for single and multiple positive periodic solutions to a kind of nonautonomous functional differential equations with impulse actions at fixed moments by employing a fixed point theorem in cones easily verifiable sufficient criteria are established the paper extends some previous results and reports some new results about impulsive functional differential equations

positive solutions for boundary value problems of functional differential equations abstract we study the existence of positive solutions of the second order boundary values problems of functional differential equations x t f t x t 0 0 t t x 0 x t a where f 0 t c r r is a continuous function c r c r 0 and a r the proof of our main result is based upon the fixed point theorem in cones

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

strang splitting method to benjamin bona mahony type equations we provide an error analysis of operator splitting for equations of the type u t x 2 u t a u 1 2 x u 2 where a is an unbounded linear differential operator such that the equation is well posed two particular examples are generalized benjamin bona mahony and kdv bbm equations a second order error bound of the strang splitting method in time is proved under suitable regularity assumptions on the exact solution finally the orders of convergence are checked by two numerical experiments and the errors are presented

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

on fourier time splitting methods for nonlinear schrodinger equations in the semi classical limit ii analytic regularity we consider the time discretization based on lie trotter splitting for the nonlinear schrodinger equation in the semi classical limit with initial data under the form of wkb states we show that both the exact and the numerical solutions keep a wkb structure on a time interval independent of the planck constant we prove error estimates which show that the quadratic observables can be computed with a time step independent of the planck constant the functional framework is based on time dependent analytic spaces in order to overcome a previously encountered loss of regularity phenomenon

globally dynamical behaviors for a class of nonlinear functional difference equation with almost periodic coefficients abstract a new approach will be used to obtained the existence and uniqueness of positive almost periodic solution x n of the following nonlinear functional difference equation x n a n x n f n x n n also some sufficient conditions are established for global attractivity of x n for the proof of existence and uniqueness of x n the method used here is better than contraction mapping principle meanwhile when our main results applied to some famous discrete models lasota wazewska model and hematopoiesis model some new statements will be obtained and these results complement existing ones

periodic solutions for functional differential equations of mixed type in this paper we study the existence of periodic solutions for some functional differential equations of mixed type among the main results are the averaging principle and existence theorems for some equations with homogeneous nonlinearities we use here the coincidence degree theory of mawhin

operator splitting for the kdv equation we provide a new analytical approach to operator splitting for equations of the type u t au b u where a is a linear operator and b is quadratic a particular example is the korteweg de vries kdv equation u t uu x u xxx 0 we show that the godunov and strang splitting methods converge with the expected rates if the initial data are sufficiently regular