stability in terms of two measures for difference equations this paper establishes new stability criteria in terms of two measures for difference equations via vector lyapunov functions and a new comparison theorem by employing the concept of the upper quasi monotone increasing of the function which connects the solutions of two finite difference equations and a scalar one an example is given to illustrate the results obtained

criteria on boundedness in terms of two measures for volterra type discrete systems abstract we consider the boundedness for the discrete system of volterra types we establish several criteria on boundedness in terms of two measures for this system employing razumikhin techniques

on the asymptotic stability for nonautonomous functional differential equations by lyapunov functionals sufficient conditions are given for the asymptotic stability and uniform asymptotic stability of the zero solution of the nonautonomous fde s whose right hand sides can be unbounded functions of the time the theorems are based upon lyapunov krasovskii functionals whose derivatives with respect to the equations are negative semidefinite and can vanish at long intervals the functionals and their derivatives are estimated by either x t the norm of the instantaneous value of the solutions or x t 2 the l 2 norm of the phase segment of the solutions examples are given to show that the conditions are sharp and the main theorems with the two different types of estimates are independent and improve earlier results the theorems are applied to linear and nonlinear retarded fde s with one delay and with distributed delays

error analysis of a boundary element collocation method for a screen problem in we examine the numerical approximation of the first kind integral equation on a plane rectangle defined by the single layer potential of the threedimensional laplacian the solution is approximated by nodal collocation with piecewise bilinear trial functions on a rectangular grid we prove stability and convergence of this method in the sobolev space j1 2 a key ingredient in the proof is the observation that the collocation equations define symmetric positive definite toeplitz matrices

moving least square reproducing kernel method part ii fourier analysis in part i of this work the moving least square reproducing kernel mlsrk method is formulated and implemented based on its generic construction an m consistency structure is discovered and the convergence theorems are established in this part of the work a systematic fourier analysis is employed to evaluate and further establish the method the preliminary fourier analysis reveals that the mlsrk method is stable for sufficiently dense non degenerated particle distribution in the sense that the kernel function family satisfies the riesz bound one of the novelties of the current approach is to treat the mlsrk method as a variant of the standard finite element method and depart from there to make a connection with the multiresolution approximation in the spirits of multiresolution analysis we propose the following mlsrk transformation fm k hu i 1npu k ik h x xi x w1 the highlight of this paper is to embrace the mlsrk formulation with the notion of the controlled fp approximation based on its characterization the strang fix condition for example a systematic procedure is proposed to design new window functions so they can enhance the computational performance of the mlsrk algorithm the main effort here is to obtain a constant correction function in the interior region of a general domain i e c h 1 this can create a leap in the approximation order of the mlsrk algorithm significantly if a highly smooth window function is embedded within the kernel one consequence of this development is the synchronized convergence phenomenon a unique convergence mechanism for the mlsrk method i e by properly tuning the dilation parameter the convergence rate of higher order error norms will approach the same order convergence rate of the l2 error norm they are synchronized

the bond based peridynamic system with dirichlet type volume constraint in this paper the bond based peridynamic system is analysed as a non local boundary value problem with volume constraint the study extends earlier works in the literature on non local diffusion and non local peridynamic models to include non positive definite kernels we prove the well posedness of both linear and nonlinear variational problems with volume constraints the analysis is based on some non local poincare type inequalities and the compactness of the associated non local operators it also offers careful characterizations of the associated solution spaces such as compact embedding separability and completeness in the limit of vanishing non locality the convergence of the peridynamic system to the classical navier equations of elasticity with poisson ratio is demonstrated

asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic navier equation abstract in this work we study reproducing kernel rk collocation method for peridynamic navier equation in the first part we apply a linear rk approximation to both displacement and dilatation and then back substitute dilatation and solve the peridynamic navier equation in a pure displacement form the rk collocation scheme converges to the nonlocal limit for a fixed nonlocal interaction length and also to the local limit as nonlocal interactions vanish the stability is shown by comparing the collocation scheme with the standard galerkin scheme using fourier analysis in the second part we apply the rk collocation to the quasi discrete peridynamic navier equation and show its convergence to the correct local limit when the ratio between the nonlocal length scale and the discretization parameter is fixed the analysis is carried out on a special family of rectilinear cartesian grids for the rk collocation method with a designated kernel with finite support we assume the lame parameters satisfy to avoid extra assumptions on the nonlocal kernel finally numerical experiments are conducted to validate the theoretical results

on the asymptotic convergence of spline collocation methods for partial differential equations we examine the asymptotic accuracy of the method of collocation for the approximate solution of linear elliptic partial differential equations specifically we consider the nodal collocation of a second order equation in the plane with biperiodicity conditions using tensor product smooth splines of odd degree as trial functions we prove optimal rates of convergence in l 2 for partial derivatives of the approximate solution which are of order at least two in one variable while the solution itself and its gradient converge in l 2 at rates less than the optimal approximation theoretic results

on the asymptotic convergence of collocation methods we prove quasioptimal and optimal order estimates in various sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines the analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications equations to which the analysis is applied include fredholm integral equations of the second kind certain first kind fredholm equations singular integral equations involving cauchy kernels a variety of integro differential equations and two point boundary value problems for ordinary differential equations the error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard galerkin methods we compare the collocation method with a standard galerkin method using splines of the same degree showing that the galerkin method is quasioptimal in a sobolev space of lower index and furnishes optimal order approximation for a range of sobolev indices containing and extending below that for the collocation method and so the standard galerkin method achieves higher rates of convergence

on the stability in terms of two measures of perturbed neutral functional differential equations different concepts of stability are presented in the literature to recall a few we name the usual stability partial stability and the conditional stability the overall concepts of stability can be put forward in the notion of stability in terms of two different measures following movchan 12 many authors have successfully developed the theory of stability in terms of two measures which became important in the investigation of converse theorems see 1 3 8 9 10 13

perturbing families of lyapunov functions and stability in terms of two measures on considere les proprietes de stabilite non uniforme en termes de deux mesures en employant des familles perturbantes de fonctions de lyapunov

perturbation and stability analysis of strong form collocation with reproducing kernel approximation solving partial differential equations using strong form collocation with nonlocal approximation functions such as orthogonal polynomials and radial basis functions offers an exponential convergence but with the cost of a dense and ill conditioned linear system in this work the local approximation functions based on reproducing kernel approximation are introduced for strong form collocation method called the reproducing kernel collocation method rkcm we perform the perturbation and stability analysis of rkcm and estimate the condition numbers of the discrete equation our stability analyses validated with numerical tests show that this approach yields a well conditioned and stable linear system similar to that in the finite element method we also introduce an effective condition number where the properties of both matrix and right hand side vector of a linear system are taken into consideration in the measure of conditioning we first derive the effective condition number of the linear systems resulting from rkcm and show that using the effective condition number offers a tighter estimation of stability of a linear system the mathematical analysis also suggests that the effective condition number of rkpm does not grow with model refinement the numerical results are also presented to validate the mathematical analysis copyright 2011 john wiley sons ltd

stability and convergence in strongly monotone dynamical systems soit x un systeme dynamique x est un espace topologique et le flot t t r est une famille d applications continues de sous ensembles ouverts de x dans x on decrit qualitativement le comportement asymptotique quand t des trajectoires t t x pour les etats initiaux x dans le cas de systemes ayant un principe de comparaison fort

attractors in strongly monotone flows where f is a c vector field in r and all the off diagonal terms of its jacobian matrix are nonnegative the well known kamke theorem implies that the flow 4 i ir0 generated by 1 has the property it preserves the partial ordering 0 another system which has this property is the semilinear parabolic equation on a compact n dimensional submanifold solutions of its dirichlet or neumann problem determine a monotone flow in appropriate function space see 11 or 51 in general a flow which preserves a partial ordering on the state space is called monotone m w hirsch has studied this flow in a series of papers 14 io he 2 31 has studied the limiting behavior and convergence of solutions of system 1 he 1 4 lo has investigated this flow in the considerably abstract state space in chapter iii of cl he has used two sections to discuss the properties of attractors for monotone flows and obtained some very important results he has shown that a monotone flow cannot have an attracting cycle see l corollary 2 41 and that every attractor must contain an equilibrium see cl theorem 3 11 in particular if the flow is strongly monotone he has shown that every attractor k contains an order stable equilibrium see 1 theorem 4 11 meanwhile if we suppose z is attracted to k but is not quasiconvergent then k contains two order stable equilibria p q such that p o z 4 q see cl theorem 4 31 the purpose of this paper is further to discuss the properties of attractors for strongly monotone flows we shall study when an o limit set can be an 210 0022 247x 91 3 00

the dynamical systems approach to differential equations bulletin new series of the american mathematical society volume 11 number 1 july 1984 the dynamical systems approach to differential equations by morris w hirsch 1 this harmony that human intelligence believes it discovers in nature does it exist apart from that intelligence no without doubt a reality completely independent of the spirit which conceives it sees it or feels it is an impossibility a world so exterior as that even if it existed would be forever inaccessible to us but what we call objective reality is in the last analysis that which is common to several thinking beings and could be common to all this common part we will see can be nothing but the harmony expressed by mathematical laws h poincare la valeur de la science p 9 ignorance of the roots of the subject has its price no one denies that modern formulations are clear elegant and precise it s just that it s impossible to comprehend how any one ever thought of them m spivak a comprehensive introduction to differential geometry introduction the mathematical subject we call dynamical systems was fathered by poin care developed sturdily under birkhoff and has enjoyed a vigorous new growth for the last twenty years as i try to show in chapter i it is interesting to look at this mathematical development as the natural outcome of a much broader theme which is as old as science itself the universe is a sytem which changes in time every mathematician has a particular way of thinking about mathematics but rarely makes it explicit yet such perspectives in any particular field can be of great value to nonexperts who must apply the results of the field or who want to learn them in some fields the points of view of its practitioners cluster around several comparatively clearly defined concepts and insights dynamical systems theory is one of them the basic concepts go back to poincare the main thrust of chapters i and ii is to explain rather discursively some of received by the editors december 14 1983 1980 mathematics subject classification primary 00a25 00a99 01a45 01a55 01a60 34a40 34c35 34d10 34d30 35b05 35b35 35b40 35b50 35b30 35k55 54h20 58f25 58f40 58f12 58f10 58d25 58d07 46a40 46e10 47h07 47h20 92a15 92a17 secondary 06f30 35b65 58f13 35j60 46e05 92 03 90 03 90a16 supported in part by national science foundation and institute for advanced study 1984 american mathematical society 0273 0979 84 1 00 25 per page

continuous and discontinuous finite element methods for a peridynamics model of mechanics in contrast to classical partial differential equation models the recently developed peridynamic nonlocal continuum model for solid mechanics is an integro differential equation that does not involve spatial derivatives of the displacement field as a result the peridynamic model admits solutions having jump discontinuities so that it has been successfully applied to fracture problems the peridynamic model features a horizon which is a length scale that determines the extent of the nonlocal interactions based on a variational formulation continuous and discontinuous galerkin finite element methods are developed for the peridynamic model discontinuous discretizations are conforming for the model without the need to account for fluxes across element edges through a series of simple one dimensional computational experiments we investigate the convergence behavior of the finite element approximations and compare the results with theoretical estimates one issue addressed is the effect of the relative sizes of the horizon and the grid for problems with smooth solutions we find that continuous and discontinuous piecewise linear approximations result in the same accuracy as that obtained by continuous piecewise linear approximations for classical models piecewise constant approximations are less robust and require the grid size to be small with respect to the horizon we then study problems having solutions containing jump discontinuities for which we find that continuous approximations are not appropriate whereas discontinuous approximations can result in the same convergence behavior as that seen for smooth solutions in case a grid point is placed at the locations of the jump discontinuities such results are directly obtained in the general case we show that such results can be obtained through a simple automated abrupt local refinement of elements containing the discontinuity in order to reduce the number of degrees of freedom while preserving accuracy we also briefly consider a hybrid discretization which combines continuous discretizations in regions where the solution is smooth with discontinuous discretizations in small regions surrounding the jump discontinuities

improved one point quadrature algorithms for two dimensional peridynamic models based on analytical calculations the peridynamics theory is a reformulation of the classical theory of continuum mechanics based on long range interactions suitable for the description of material failure and damage integration plays a central role in this theory we study one point quadrature algorithms for the discretization of two dimensional integrals in peridynamics these algorithms are closely related to meshfree methods they assume an underlying reference lattice where each lattice point within a body is assigned a cell a main challenge in such algorithms is the accurate estimation of the area of the intersecting region between the neighborhood of a point and a neighbor cell we provide a classification of the different types of intersecting regions in square lattices and present analytical derivations for the exact calculation of their areas leading to improved integration accuracy to address convergence issues geometric centers of intersecting regions are taken as quadrature points replacing commonly used cell centers we present analytical derivations for the exact calculation of those geometric centers based on a decomposition of intersecting regions into subdomains of simple geometry by using exact values for the areas and geometric centers of intersecting regions we achieve an asymptotically monotonic convergence in numerical integration in contrast other algorithms from the literature exhibit a highly oscillatory behavior numerical results compare the proposed algorithms with others from the literature for different quantities of interest and demonstrate their improved accuracy and convergence error estimates for the quadrature algorithms are derived and extended and hybrid algorithms are discussed additional numerical studies using influence functions with a finite support and a controlled regularity suggest an alternative means to improve the convergence of discretization algorithms

analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations we consider the numerical solution of nonlocal constrained value problems associated with linear nonlocal diffusion and nonlocal peridynamic models two classes of discretization methods are presen

stability criteria in terms of two measures for discrete systems abstract this paper studies stability problems for two kinds of discrete systems in terms of two different measures the basic idea of this approach is to discuss the qualitative behaviour of a map h along solutions of discrete systems whose initial value are measured by a second map h 0 by doing this one may deal with in a unified way several concepts and associated problems which are usually considered separately stability criteria in terms of two measures are established employing the method of lyapunov functions and razumikhin techniques