nontrivial periodic solutions for the asymptotically linear hamiltonian systems with resonance at infinity abstract in this paper we are concerned with the existence of nontrivial periodic solutions of an asymptotically linear hamiltonian systems when a kind of resonance condition is satisfied we prove several new existence results via the morse theory the garlerkin methods and the moslov index theory

an infinite dimensional morse theory with applications in this paper we construct an infinite dimensional extraordinary cohomology theory and a morse theory corresponding to it these theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension several applications are given to hamiltonian systems the onedimensional wave equation of vibrating string type and systems of elliptic partial differential equations 0 introduction let e be a real hilbert space with an inner product and let be a twice continuously differentiable functional denote the frechet derivative and the gradient of at x by x and x respectively where as usual x y x y y e recall that a point x0 e is said to be critical if x0 0 or equivalently if x0 0 the level c r will be called regular if 1 c contains no critical points and critical if x0 0 for some x0 1 c let a b a b be two regular levels of denote m 1 a b and consider the restriction of to m in morse theory one is interested in the local topological structure of the level sets of m near a critical point and in the relation between this local structure and the topological structure of the set m to be more specific suppose that x0 m is an isolated critical point of then one defines a sequence of critical groups of at x0 by setting cq x0 hq c u u x0 q 0 1 2 0 1 where c x0 c x e x c hq is the q th singular homology group with coefficients in some field f and u is a neighbourhood of x0 define the morse index of x0 to be the maximal dimension of a subspace of e on which the quadratic form x0 y y is negative definite one shows that if x0 is a nondegenerate critical point i e if x0 e e is invertible then cq x0 f received by the editors march 20 1995 1991 mathematics subject classification primary 58e05 secondary 34c25 35j65 35l05 55n20 58f05

morse theory and asymptotic linear hamiltonian system on considere l existence de solutions 2 periodiques des systemes hamiltoniens asymptotiquement quadratiques jz h z t ou j 0 i i 0 i in la matrice identite de n n

stability of solutions of nonlinear neutral differential equations with piecewise constant delay and their discretizations this paper is devoted to stability analysis of solutions of nonlinear neutral differential equations with piecewise constant delay these results form the basis for obtaining insight into the analogous properties of numerical solutions generated by bn f stable runge kutta methods it is showed theoretically and numerically that two classes of runge kutta methods considered in literature are contractive and asymptotically stable numerical experiments also demonstrate that a class of runge kutta methods can preserve their original convergence order for odes but not the other class of runge kutta methods

stability of numerical methods for delay differential equations abstract consider the following delay differential equation dde y t y t y t t t t 0 with the initial condition y t for t t 0 where and are such that 0 1 0 2 has a unique solution y t the author gives sufficient conditions for the asymptotic stability of the equation 0 1 for which he introduces new definitions of numerical stability the approach is reminiscent of that from the nonlinear stiff ordinary differential equation ode field he investigates stability properties of the class of one point collocation rules in particular the backward euler method turns out to be stable with respect to all the given definitions

infinite dimensional cohomology groups and periodic solutions of asymptotically linear hamiltonian systems abstract in this paper we study the existence of nontrivial 2 periodic solutions of asymptotically linear hamiltonian systems we consider the case of resonance both at zero and at infinity and we permit time dependent asymptotic matrices our main tools are an infinite dimensional cohomology theory and a corresponding morse theory recently constructed by w kryszewski and the first author we develop a method to compute the new critical groups

optimal superconvergence results for volterra functional integral equations with proportional vanishing delays abstract in this paper we develop a new technique to study the optimal convergence orders of collocation methods for volterra functional integral equations with vanishing delays on quasi geometric meshes basing on a perturbation analysis we show that for m collocation points the global convergence order of the collocation solution is only m however the collocation solution may exhibit superconvergence with order m 1 at the collocation points in particular the local convergence order may attain 2 m 1 at the nodes provided that the collocation is based on the m radau ii points finally some numerical examples are performed to verify our theoretical results

dissipativity of θ methods for a class of nonlinear neutral delay differential equations abstract this paper is concerned with the dissipativity of theoretical solution and numerical solution of a class of nonlinear neutral delay differential equations at first a sufficient condition for the dissipativity of theoretical solution of the mentioned problem above is given then the dissipativity results are obtained for one leg and linear methods when they are applied to these problems

state and bias estimation for soil moisture profiles by an ensemble kalman filter effect of assimilation depth and frequency 1 an ensemble kalman filter for state estimation and a bias estimation algorithm were applied to estimate individual soil moisture profiles in a small corn field with the clm2 0 model through the assimilation of measurements from capacitance probes both without and with inclusion of bias correction the effect of the assimilation frequency the assimilation depth and the number of observations assimilated per profile were studied assimilation of complete profiles had the highest impact on deeper soil layers and the optimal assimilation frequency was about 1 2 weeks if bias correction was applied the optimal assimilation depth depended on the calibration results assimilation in the surface layer had typically less impact than assimilation in other layers through bias correction the soil moisture estimate greatly improved in general the correct propagation of the innovations for both the bias blind state and bias filtering from any layer to other layers was insufficient the approximate estimation of the a priori bias error covariance and the choice of a zero initialized persistent bias model made it impossible to estimate the bias in layers for which no observations were available

convergence analysis of fitted numerical method for a singularly perturbed nonlinear volterra integro differential equation with delay abstract in this paper the initial value problem for a quasilinear singularly perturbed delay volterra integro differential equation was considered by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder term in integral form a fitted difference scheme is constructed and analysed it is shown that the method displays first order uniform convergence in perturbation parameter some numerical results are given to confirm the theoretical analysis

numerical modelling in biosciences using delay differential equations our principal purposes here are i to consider from the perspective of applied mathematics models of phenomena in the biosciences that are based on delay differential equations and for which numerical approaches are a major tool in understanding their dynamics ii to review the application of numerical techniques to investigate these models we show that there are prima facie reasons for using such models i they have a richer mathematical framework compared with ordinary differential equations for the analysis of biosystem dynamics ii they display better consistency with the nature of certain biological processes and predictive results we analyze both the qualitative and quantitative role that delays play in basic time lag models proposed in population dynamics epidemiology physiology immunology neural networks and cell kinetics we then indicate suitable computational techniques for the numerical treatment of mathematical problems emerging in the biosciences comparing them with those implemented by the bio modellers

stability analysis of numerical methods for linear neutral volterra delay integro differential system this paper is concerned with the stability of numerical methods for linear neutral volterra delay integro differential system a sufficient condition such that the system is asymptotically stable is derived furthermore it is proved that every linear q method with email protected 1 2 1 and a stable bdf method preserve the delay independent stability of its exact solutions finally the numerical experiments are given to demonstrate the conclusions

global dissipativity for a stable methods this paper concerns the discretization of the initial value problem u t f u under the three structural conditions i f bbb c n longrightarrow bbb c n re e langle f u u rangle leq a b u 2 a geq 0 b 0 for all u in bbb c n ii f bbb c n longrightarrow bbb c n re e langle f u u rangle 0 iii f w longrightarrow h re e langle f w w rangle h leq a b w h 2 a geq 0 b 0 for all w in w for complex hilbert spaces w subseteq h dahlquist s g stability theory is used to show that linear multistep and one leg methods yield dissipative discretizations for all f satisfying i if and only if the method rho sigma is a stable extensions of g stability theory are made to find necessary and sufficient conditions on rho sigma for similar properties to hold in cases ii and iii in every case conditions are found for the strict contractivity of solutions for large initial data and bounds for the asymptotic rate of decay are calculated in cases i and iii

model problems in numerical stability theory for initial value problems in the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics whilst this has resulted in a coherent and self contained body of knowledge it has not thoroughly addressed the problems of real interest in applications recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics this on going work is unified and possible directions for future work are outlined in particular striking similarities between this new developing stability theory and the classical non linear stability theory are emphasised the classical theories of a b and algebraic stability for runge kutta methods are briefly reviewed and it is emphasised that the classes of equations to which these theories apply linear decay and contractive problems only admit trivial dynamics four other categories of equations gradient dissipative conservative and hamiltonian systems are considered relationships and differences between the possible dynamics in each category which range from multiple competing equilibria to fully chaotic solutions are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications runge kutta schemes which preserve the dynamical structure of the underlying problem are sought and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given algebraic stability in particular is seen to play a central role the effects of error control are considered and multi step methods are discussed briefly finally various open problems are described

a finite difference method for a singularly perturbed delay integro differential equation we consider the singularly perturbed initial value problem for a linear first order volterra integro differential equation with delay our purpose is to construct and analyse a numerical method with uniform convergence in the perturbation parameter the numerical solution of this problem is discretized using implicit difference rules for differential part and the composite numerical quadrature rules for integral part on a layer adapted mesh error estimations for the approximate solution are established numerical examples supporting the theory are presented

dissipativity of one leg methods for a class of nonlinear functional integro differential equations abstract this paper is concerned with the dissipativity of a class of nonlinear functional integro differential equations fides the dissipativity result of the theoretical solution for this class problem is presented a type of extended one leg methods is suggested for the fides it is shown under suitable condition that a g c p 0 algebraically stable one leg method is dissipative when applied to the above problem numerical examples are given to illustrate the correctness of our theoretical results

nontrivial periodic solutions for asymptotically linear hamiltonian systems with resonance abstract in this paper we establish some existence results of nontrivial 1 periodic solutions to the first order asymptotically linear hamiltonian systems under the assumptions that the linear operators have different maslove indices at the origin and at infinity the results obtained in this paper are also valued for the case that both of the asymptotic matrices may be degenerate resonant type and time dependent

generalized halanay inequalities for dissipativity of volterra functional differential equations abstract this paper is concerned with the dissipativity of theoretical solutions to nonlinear volterra functional differential equations vfdes at first we give some generalizations of halanay s inequality which play an important role in study of dissipativity and stability of differential equations then by applying the generalization of halanay s inequality the dissipativity results of vfdes are obtained which provides unified theoretical foundation for the dissipativity analysis of systems in ordinary differential equations odes delay differential equations ddes integro differential equations ides volterra delay integro differential equations vdides and vfdes of other type which appear in practice

stability of θ methods for delay integro differential equations stability of methods for delay integro differential equations dides is studied on the basis of the linear equation du dt u t u t k t t u d where k are complex numbers and is a constant delay it is shown that every a stable method possesses a similar stability property to p stability i e the method preserves the delay independent stability of the exact solution under the condition that k is real and h is an integer where h is a step size it is also shown that the method does not possess the same property if h is not an integer as a result no method can possess a similar stability property to gp stability with respect to dides

stability analysis of volterra delay integro differential equations and their backward differentiation time discretization we study the stability of the analytical solutions of initial value problems of a general class of systems of volterra delay integro differential equations numerical methods based on backward differentiation formulae and repeated quadrature formulae are suggested nonlinear and linear stability conditions for the presented methods are derived

dissipativity and asymptotic stability of nonlinear neutral delay integro differential equations this paper is concerned with the dissipativity and asymptotic stability of the theoretical solutions of a class of nonlinear neutral delay integro differential equations ndides we first give a generalization of the halanay inequality which plays an important role in the study of dissipativity and stability of differential equations then we apply the generalization of the halanay inequality to ndides and the dissipativity and the asymptotic stability results of the theoretical solution of ndides are obtained from a numerical point of view it is important to study the potential of numerical methods in preserving the qualitative behavior of the analytical solutions therefore the results presented in this paper provide the theoretical foundation for analyzing the dissipativity and the asymptotic stability of the numerical methods when they are applied to these systems

dissipativity of runge kutta methods for neutral delay integro differential equations this paper is concerned with the numerical dissipativity of a class of nonlinear neutral delay integro differential equations the dissipativity results are obtained for algebraically stable runge kutta methods when they are applied to above problems

computations of cohomology groups and nontrivial periodic solutions of hamiltonian systems by computing the e critical groups at and infinity of the corresponding functional of hamiltonian systems we proved the existence of nontrivial periodic solutions for the systems which may be resonant at and infinity under some new conditions some results in the literature are extended and some new type of theorems are proved the main tool is the e morse theory developed by kryszewski and szulkin

runge kutta methods for dissipative and gradient dynamical systems the numerical approximation of dissipative initial value problems by fixed time stepping runge kutta methods is considered and the asymptotic features of the numerical and exact solutions are compared a general class of ordinary differential equations for which dissipativity is induced through an inner product is studied throughout this class arises naturally in many finite dimensional applications such as the lorenz equations and also from the spatial discretization of a variety of partial differential equations arising in applied mathematics it is shown that the numerical solution defined by an algebraically stable method has an absorbing set and is hence dissipative for any fixed step size h 0 the numerical solution is shown to define a dynamical system on the absorbing set if h is sufficiently small and hence a global attractor a h exists upper semicontinuity of a h at h 0 is established which shows that for h small every point on the numerical attractor is close to a point on the true global attractor a under the additional assumption that the problem is globally lipschitz it is shown that if h is sufficiently small any method with positive weights defines a dissipative dynamical system on the whole space and upper semicontinuity of a h at h 0 is again established for gradient systems with globally lipschitz vector fields it is shown that any runge kutta method preserves the gradient structure for h sufficiently small for general dissipative gradient systems it is shown that algebraically stable methods preserve the gradient structure within the absorbing set for h sufficiently small convergence of the numerical attractor is studied and for a dissipative gradient system with hyperbolic equilibria lower semicontinuity at h 0 is established thus for such a system a h converges to a in the hausdorff metric as h 0

dissipativity of θ methods for nonlinear volterra delay integro differential equations this paper deals with the analytic and numerical dissipativity of nonlinear volterra delay integro differential equations vdides a dissipativity criteria for vdides is given we investigate the dissipativity properties of one leg q methods and linear q methods for the underlying systems it is shown that for q 12 1 any one leg q method and linear q method can inherit the dissipativity of the underlying systems

stability of runge kutta methods for delay integro differential equations we study stability of runge kutta rk methods for delay integro differential equations with a constant delay on the basis of the linear equation du dt lu t mu t k t t u d where l m k are constant complex matrices in particular we show that the same result as in the case k 0 koto bit 34 1994 262 267 holds for this test equation i e every a stable rk method preserves the delay independent stability of the exact solution whenever a step size of the form h m is used where m is a positive integer

an analysis of delay dependent stability for ordinary and partial differential equations with fixed and distributed delays this paper is concerned with the study of the stability of ordinary and partial differential equations with both fixed and distributed delays and with the study of the stability of discretizations of such differential equations we start with a delay dependent asymptotic stability analysis of scalar ordinary differential equations with real coefficients we study the exact stability region of the continuous problem as a function of the parameters of the model next it is proved that a time discretization based on the trapezium rule can preserve the asymptotic stability for the considered set of test problems in the second part of the paper we study delay partial differential equations the stability region of the fully continuous problem is analyzed first then a semidiscretization in space is applied it is shown that the spatial discretization leads to a reduction of the stability region when the standard second order central difference operator is employed to approximate the diffusion operator finally we consider the delay dependent stability of the fully discrete problem where the partial differential equation is discretized both in space and in time some numerical examples and further discussions are given