existence results for urysohn integral inclusions involving the henstock integral this paper presents two existence results for urysohn integral inclusions in banach spaces the set valued integral involved being the henstock integral in particular the existence of continuous solutions for integral inclusions of volterra and of hammerstein type is obtained our results extend the existence theorems given in literature in the single or set valued case under bochner or pettis integrability hypothesis the compactness of solutions set is also studied

continuous dependence for impulsive functional dynamic equations involving variable time scales using a known correspondence between the solutions of impulsive measure functional differential equations and the solutions of impulsive functional dynamic equations on time scales we prove that the limit of solutions of impulsive functional dynamic equations over a convergent sequence of time scales converges to a solution of an impulsive functional dynamic equation over the limiting time scale

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

a class of the first order impulsive dynamic equations on time scales abstract in this paper the first order impulsive dynamic equations on time scales is considered the existence and uniqueness of solution for a class of nonlinear impulsive dynamic equations on time scales is proved

differential equations on variable time scales abstract we introduce a class of differential equations on variable time scales with a transition condition between two consecutive parts of the scale conditions for existence and uniqueness of solutions are obtained periodicity boundedness and stability of solutions are considered the method of investigation is by means of two successive reductions b equivalence of the system e akalin m u akhmet the principles of b smooth discontinuous flows computers and mathematics with applications 49 2005 981 995 m u akhmet perturbations and hopf bifurcation of the planar discontinuous dynamical system nonlinear analysis 60 2005 163 178 m u akhmet n a perestyuk the comparison method for differential equations with impulse action differential equations 26 9 1990 1079 1086 on a variable time scale to a system on a time scale a reduction to an impulsive differential equation m u akhmet perturbations and hopf bifurcation of the planar discontinuous dynamical system nonlinear analysis 60 2005 163 178 m u akhmet m turan the differential equations on time scales through impulsive differential equations nonlinear analysis 65 2006 2043 2060 appropriate examples are constructed to illustrate the theory

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

a dynamic matrix exponential via a matrix cylinder transformation abstract in this work we develop a new matrix exponential on time scales via a cylinder transformation with a component wise locally integrable square matrix subscript our resulting matrix function can be written in terms of the matrix exponential of a lebesgue integral added to a logarithmic sum in terms of the gaps of a general time scale under strict commutativity conditions we show our dynamic matrix exponential is equivalent to the one in the standard literature finally we demonstrate that our matrix exponential satisfies a nonlinear dynamic integral equation

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

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

existence theorems of global solutions for nonlinear volterra type integral equations in banach spaces in this paper first a new fixed point theorem is established and then by the use of it the existence theorems of global solutions for nonlinear volterra type integral equations in banach spaces are investigated the results obtained in this paper generalize and improve the results corresponding to those obtained by others

boundedness of solutions of measure differential equations and dynamic equations on time scales abstract in this paper we investigate the boundedness results for measure differential equations in order to obtain our results we use the correspondence between these equations and generalized odes furthermore we prove our results concerning boundedness of solutions for dynamic equations on time scales using the fact that these equations represent a particular case of measure differential equations

existence and uniqueness of c n almost periodic solutions to some ordinary differential equations abstract in this paper we prove the existence and uniqueness of c n almost periodic solutions to the ordinary differential equation x t a t x t f t t r where the matrix a t r m k c is periodic and f r c k is c n almost periodic we also prove the existence and uniqueness of an ultra weak c n almost periodic solution in the case when a t a is independent of t finally we prove also the existence and uniqueness of a mild c n almost periodic solution of the semilinear hyperbolic equation x t a x t f t x considered in a banach space assuming f t x is c n almost periodic in t for each x x satisfies a global lipschitz condition and takes values in an extrapolation space f a 1 associated to a

existence results for dynamic inclusions on time scales with nonlocal initial conditions this paper is mainly concerned with the existence of solutions for first order dynamic inclusions on time scales with nonlocal initial conditions by using bohnenblust karlin s fixed point theorem and leray schauder nonlinear alternative for multivalued maps some sufficient conditions are established an example is also included to illustrate our results

controllability of linear and nonlinear systems governed by stieltjes differential equations abstract in this manuscript controllability of vector valued g derivative stieltjes differential system is considered for linear system explicit solution of homogeneous case is obtained by constructing a generalized matrix exponential function and nonhomogeneous solution is achieved with transferring the corresponding problem into a class of stieltjes impulsive differential equations based on the analysis techniques of linear system theory the gramian criterion and the rank criterion for linear case are established respectively existence of local and global solution for nonlinear system are presented by the well knonw banach contraction principle and the controllability result for nonlinear systems is shown by krasnoselskii s fixed point theorem simulations for linear and nonlinear systems are tackled respectively to verify the efficiency of theoretical results

on first order ordinary differential equations with nonnegative right hand sides existence of extremal solutions for scalar initial value problems is investigated we shall concentrate upon nonlinearities having constant sign which lead to new existence results

calculus of fuzzy vector valued functions and almost periodic fuzzy vector valued functions on time scales abstract in this paper we establish an embedding theorem for the fuzzy multidimensional space and introduce the new types of multiplications of fuzzy vectors determined by the determinant algorithm then some fundamental results of calculus of fuzzy vector valued functions on time scales are established based on these results we introduce the concept of shift almost periodic fuzzy vector valued functions and develop a theory of almost periodic fuzzy functions on time scales this function theory can be used to study almost periodic fuzzy dynamic equations on time scales including a new type of fuzzy dynamic equations called fuzzy q dynamic equation i e quantum fuzzy dynamic equation

on the solvability of distributional and impulsive systems of cauchy problems abstract in this paper existence results are derived for the unique smallest greatest minimal and or maximal solutions of finite systems of nonlinear distributional cauchy problems the dependence of solutions on the data is also studied applications to systems of distributional differential equations with impulses and to higher order distributional cauchy problems are presented the main tools are fixed point results in function spaces and a primitive integral of distributions defined in the paper including recently introduced regulated and continuous primitive integrals on compact intervals

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

extremal solutions for nonresonance impulsive functional dynamic equations on time scales in this paper we give an existence theorem for extremal solutions of a class of first order nonresonance impulsive dynamic equations on time scales with a periodicity condition

convergence theorems for hl integrable vector valued functions with applications abstract in this paper we prove dominated and monotone convergence theorems for hl integrable banach valued functions these results and a fixed point theorem in ordered spaces are then applied to prove existence and comparison results for integral equations of fredholm type in ordered banach spaces involving kurzweil integrals or improper integrals results are used also to solve concrete second order functional boundary value problems involving discontinuities and singularities

on continuous dependence of solutions of dynamic equations the main goal of the paper is to present a new approach to the problem of continuous dependence of solutions of differential or dynamic problems on their domains this is of particular interests when we use dynamic difference in particular equations as discretization of a given one we cover a standard construction based of difference approximations for the continuous one but we are not restricted only to this case for a given differential equation we take a sequence of time scales and we study the convergence of time scales to the domain of the considered problem we choose a kind of convergence of such approximated solutions to the exact solution this is a step for creating numerical analysis on time scales and we propose to replace in such a situation the difference equations by dynamic ones in the proposed approach we are not restricted to the case of classical numerical algorithms moreover this allows us to find an exact solution for considered problems as a limit of a sequence of solutions for appropriate time scales instead of solving it analytically or calculating approximated solutions for the original problems

henstock kurzweil delta and nabla integrals we will study the henstock kurzweil delta and nabla integrals which generalize the henstock kurzweil integral many properties of these integrals will be obtained these results will enable time scale researchers to study more general dynamic equations the henstock kurzweil delta nabla integral contains the riemann delta nabla and lebesque delta nabla integrals as special cases c 2005 elsevier inc all rights reserved

lyapunov stability for measure differential equations and dynamic equations on time scales abstract in this paper we prove the stability results for measure differential equations considering more general conditions under the lyapunov functionals and concerning the functions f and g moreover we prove these stability results for the dynamic equations on time scales using the correspondence between the solutions of these last equations and the solutions of the measure differential equations

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

on the cauchy problem for first order discontinuous ordinary differential equations abstract we prove the existence of extremal caratheodory solutions for the initial value problem x t f t x t for a e t 0 1 x 0 0 under weak conditions over f 0 1 r r

general existence principles for stieltjes differential equations with applications to mathematical biology abstract stieltjes differential equations which contain equations with impulses and equations on time scales as particular cases simply consist on replacing usual derivatives by derivatives with respect to a nondecreasing function in this paper we prove new existence results for functional and discontinuous stieltjes differential equations and we show that such general results have real world applications specifically we show that stieltjes differential equations are specially suitable to study populations which exhibit dormant states and or very short impulsive periods of reproduction in particular we construct two mathematical models for the evolution of a silkworm population our first model can be explicitly solved as it consists on a linear stieltjes equation our second model more realistic is nonlinear discontinuous and functional and we deduce the existence of solutions by means of a result proven in this paper

periodic averaging theorems for various types of equations abstract we prove a periodic averaging theorem for generalized ordinary differential equations and show that averaging theorems for ordinary differential equations with impulses and for dynamic equations on time scales follow easily from this general theorem we also present a periodic averaging theorem for a large class of retarded equations

impulsive retarded differential equations in banach spaces via bochner lebesgue and henstock integrals t f t xt xt0 1 1 where is a given continuous function from r 0 r 0 to a banach space x and f is a continuous map from an open set r c r 0 x to x given a continuous function x t0 r t0 a x a 0 t0 r we follow 11 to deane xt r 0 x by xt x t r 0 t t0 t0 a the initial value problem 1 1 is equivalent to the integral equation

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

measure functional differential equations and functional dynamic equations on time scales abstract we study measure functional differential equations and clarify their relation to generalized ordinary differential equations we show that functional dynamic equations on time scales represent a special case of measure functional differential equations for both types of equations we obtain results on the existence and uniqueness of solutions continuous dependence and periodic averaging

algorithm developments for discrete adjoint methods this paper presents a number of algorithm developments for adjoint methods using the discrete approach in which the discretisation of the non linear equations is linearised and the resulting matrix is then transposed with a new iterative procedure for solving the adjoint equations exact numerical equivalence is maintained between the linear and adjoint discretisations the incorporation of strong boundary conditions within the discrete approach is discussed as well as a new application of adjoint methods to linear unsteady flow in turbomachinery

on kurzweil generalized ordinary differential equations on etudie sous quelles conditions une equation differentielle generalisee de kurzweil est equivalente a une equation differentielle

the differential equations on time scales through impulsive differential equations in this paper we investigate differential equations on certain tim es cales with transition conditions detc on the basis of reduction to the impulsive differential equations ide detc are in some sense more general than dynamic equations on time scales m bohner a peterson dynamic equations on

singular fibers and kodaira dimensions let f x rightarrow mathbb p 1 be a non isotrivial semi stable family of varieties of dimension m over mathbb p 1 with s singular fibers assume that the smooth fibers f are minimal i e their canonical line bundles are semiample then kappa x le kappa f 1 if kappa x kappa f 1 then s frac 4 m 2 if kappa x ge 0 then s ge frac 4 m 2 in particular if m 1 s 6 and kappa x 0 then the family f is teichmuller

robust design of structures using optimization methods the robust design of structures with stochastic parameters is studied using optimization techniques the first two statistical moments of the stochastic parameters including design variables are considered in conjunction with the second order perturbation method for the approximation of mean value and variance of the structural response in this framework the sensitivities of the mean values and variances of the structural performance function with respect to the design variables are calculated for use in the optimization task the robust design of structures is formulated as a multi criteria optimization problem in which both the expected value and the standard deviation of the objective function are to be minimized the robustness of the feasibility is also taken into account by involving the variability of the structural response in the constraints the two criteria optimization problem is converted into a scalar one and is then solved by a gradient based optimization algorithm to demonstrate the applicability of the presented method numerical examples are given involving static and dynamic response

first and second order discontinuous functional differential equations with impulses at fixed moments abstract we give sufficient conditions for the existence of extremal solutions to discontinuous and functional differential equations with impulses our main results are new even for ordinary differential equations without impulses

upper and lower solutions with jumps abstract we study a periodic boundary value problem for a first order differential equation from a new point of view which permits us to obtain an existence result involving upper and lower solutions with jump discontinuities we present some applications of our result with illustrative examples

measure functional differential equations with infinite delay abstract we introduce measure functional differential equations with infinite delay and an axiomatically described phase space we show how to transform these equations into generalized ordinary differential equations whose solutions take values in a suitable infinite dimensional banach space even in the special case of functional equations with finite delay our result improves the existing one by imposing weaker conditions on the right hand side

new definitions of exponential hyperbolic and trigonometric functions on time scales abstract we propose two new definitions of the exponential function on time scales the first definition is based on the cayley transformation while the second one is a natural extension of exact discretizations our exponential functions map the imaginary axis into the unit circle therefore it is possible to define hyperbolic and trigonometric functions on time scales in a standard way the resulting functions preserve most of the qualitative properties of the corresponding continuous functions in particular pythagorean trigonometric identities hold exactly on any time scale dynamic equations satisfied by cayley motivated functions have a natural similarity to the corresponding differential equations the exact discretization is less convenient as far as dynamic equations and differentiation are concerned

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

periodic solutions for dynamic equations on time scales massera type criteria are established for the existence of periodic solutions of linear and nonlinear dynamic equations on time scales some interesting properties of the exponential function on time scales are presented furthermore a sufficient condition guaranteeing the boundedness of the solutions of the equation is presented

a new result on impulsive differential equations involving non absolutely convergent integrals in this paper we obtain as an application of a darbo type theorem global solutions for differential equations with impulse effects under the assumption that the function on the right hand side is integrable in the henstock sense we thus generalize several previously given results in literature for ordinary or impulsive equations

impulsive nonlocal differential equations through differential equations on time scales abstract we propose a non standard approach to impulsive differential equations in banach spaces by embedding this type of problems into differential dynamic problems on time scales we give an existence result for dynamic equations and as a consequence we obtain an existence result for impulsive differential equations

arakelov parshin rigidity of towers of curve fibrations arakelov parshin rigidity is concerned with varieties mapping rigidly to the moduli stack mathfrak m h of canonically polarized manifolds affirmative answer for any class of maps implies finiteness of the given class this article studies arakelov parshin rigidity on an open subspace of mathfrak m h on the locus mathfrak k mathfrak f h of iterated kodaira fibrations first we prove rigidity for all complete curves mapping finitely onto mathfrak k mathfrak f h then for generic affine curves mapping into mathfrak k mathfrak f h rigidity is shown when deg h 2 the method used in the latter part is showing that the iterated kodaira spencer map is injective

extremal solutions of measure differential equations abstract we investigate the existence of the least and greatest solutions to measure differential equations as well as the relation between the extremal solutions and lower or upper solutions along the way we prove a fairly general local existence theorem and an analogue of peano s uniqueness theorem for measure differential equations finally we show that the general theory is applicable in the study of differential equations with impulses or dynamic equations on time scales

generalized elementary functions abstract we use the theory of generalized linear differential equations to introduce new definitions of the exponential hyperbolic and trigonometric functions we derive some basic properties of these generalized functions and show that the time scale elementary functions with lebesgue integrable arguments represent a special case of our definitions

a density matching approach for optimization under uncertainty abstract modern computers enable methods for design optimization that account for uncertainty in the system so called optimization under uncertainty ouu we propose a metric for ouu that measures the distance between a designer specified probability density function of the system response the target and the system response s density function at a given design we study an ouu formulation that minimizes this distance metric over all designs we discretize the objective function with numerical quadrature and we approximate the response density function with a gaussian kernel density estimate we offer heuristics for addressing issues that arise in this formulation and we apply the approach to a cfd based airfoil shape optimization problem we qualitatively compare the density matching approach to a multi objective robust design optimization to gain insight into the method

efficient global reliability analysis for nonlinear implicit performance functions many engineering applications are characterized by implicit response functions that are expensive to evaluate and sometimes nonlinear in their behavior making reliability analysis difficult this paper develops an efficient reliability analysis method that accurately characterizes the limit state throughout the random variable space the method begins with a gaussian process model built from a very small number of samples and then adaptively chooses where to generate subsequent samples to ensure that the model is accurate in the vicinity of the limit state the resulting gaussian process model is then sampled using multimodal adaptive importance sampling to calculate the probability of exceeding or failing to exceed the response level of interest by locating multiple points on or near the limit state more complex and nonlinear limit states can be modeled leading to more accurate probability integration by concentrating the samples in the area where accuracy is important i e in the vicinity of the limit state only a small number of true function evaluations are required to build a quality surrogate model the resulting method is both accurate for any arbitrarily shaped limit state and computationally efficient even for expensive response functions this new method is applied to a collection of example problems including one that analyzes the reliability of a microelectromechanical system device that current available methods have difficulty solving either accurately or efficiently

existence results for second order boundary value problem of impulsive dynamic equations on time scales abstract in this paper schaefer s fixed point theorem and nonlinear alternative of leray schauder type are used to investigate the existence of solutions for second order boundary value problem for impulsive dynamic equations on time scales

closure properties for integral problems driven by regulated functions via convergence results abstract in this paper we give necessary and sufficient conditions for the convergence of kurzweil stieltjes integrals with respect to regulated functions using the notion of asymptotical equiintegrability one thus generalizes several well known convergence theorems as applications we provide existence and closure results for integral problems driven by regulated functions both in single and set valued cases in the particular setting of bounded variation functions driving the equations we get features of the solution set of measure integrals problems

existence results for second order impulsive boundary value problems on time scales abstract in this paper using a fixed point theorem due to krasnoselskii and zabreiko and the leggett williams fixed point theorem respectively we investigate the existence of one solution and three nonnegative solutions to second order impulsive equations on time scales

positivity of relative canonical bundles and applications given a family f mathcal x to s of canonically polarized manifolds the unique kahler einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle mathcal k mathcal x s we use a global elliptic equation to show that this metric is strictly positive on mathcal x unless the family is infinitesimally trivial for degenerating families we show that the curvature form on the total space can be extended as a semi positive closed current by fiber integration it follows that the generalized weil petersson form on the base possesses an extension as a positive current we prove an extension theorem for hermitian line bundles whose curvature forms have this property this theorem can be applied to a determinant line bundle associated to the relative canonical bundle on the total space as an application the quasi projectivity of the moduli space mathcal m text can of canonically polarized varieties follows the direct images r n p f omega p mathcal x s mathcal k mathcal x s otimes m m 0 carry natural hermitian metrics we prove an explicit formula for the curvature tensor of these direct images we apply it to the morphisms s p mathcal t s to r p f lambda p mathcal t mathcal x s that are induced by the kodaira spencer map and obtain a differential geometric proof for hyperbolicity properties of mathcal m text can

well posedness results for abstract generalized differential equations and measure functional differential equations abstract in the first part of the paper we consider nonlinear generalized ordinary differential equations whose solutions take values in infinite dimensional banach spaces and prove new theorems concerning the existence of solutions and continuous dependence on initial values and parameters in the second part we apply these results in the study of nonlinear measure functional differential equations and impulsive functional differential equations with infinite delay

weak solutions for the dynamic cauchy problem in banach spaces abstract this paper is devoted to unify and extend the results of the existence of the weak solutions of continuous and discrete cauchy problem in banach spaces we offer the existence of the weak solution of dynamic cauchy problem on an infinite time scale the measure of weak noncompactness and the fixed point theorem of kubiaczyk are used to prove the main result

on functional improper volterra integral equations and impulsive differential equations in ordered banach spaces in this paper we derive existence and comparison results for discontinuous improper functional integral equations of volterra type in an ordered banach space which has a regular order cone for this purpose we prove dominated and monotone convergence theorems for improper integrals the obtained results are then applied to first order impulsive differential equations concrete examples are also solved by using symbolic programming

on nonlinear distributional and impulsive cauchy problems abstract in this paper existence results are derived for the unique smallest greatest minimal and maximal solutions of nonlinear distributional cauchy problems dependence of solutions on the data is also studied the obtained results are applied to impulsive differential equations main tools are fixed point results in function spaces and recently introduced concepts of regulated and continuous primitive integrals of distributions

applied smoothing techniques for data analysis the kernel approach with s plus illustrations 1 density estimation for exploring data 2 density estimation for inference 3 nonparametric regression for exploring data 4 inference with nonparametric regression 5 checking parametric regression models 6 comparing regression curves and surfaces 7 time series data 8 an introduction to semiparametric and additive models references

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

difference equations in banach spaces difference equations which discretely approximate boundary value problems for second order ordinary differential equations are analysed it is well known that the existence of solutions to the continuous problem does not necessarily imply existence of solutions to the discrete problem and even if solutions to the discrete problem are guaranteed they may be unrelated and inapplicable to the continuous problem analogues to theorems for the continuous problem regarding a priori bounds and existence of solutions are formulated for the discrete problem solutions to the discrete problem are shown to converge to solutions of the continuous problem in an aggregate sense an example which arises in the study of the finite deflections of an elastic string under a transverse load is investigated the earlier results are applied to show the existence of a solution the sufficient estimates on the step size are presented c 2003 elsevier science ltd all rights reserved