on the numerical solution of two point discrete boundary value problems we propose several different methods to construct the solutions of linear two point discrete boundary value problems these methods are applied to solve some continuous boundary value problems which are known to be numerically unstable

block monotone iterative methods for numerical solutions of nonlinear elliptic equations two block monotone iterative schemes for a nonlinear algebraic system which is a finite difference approximation of a nonlinear elliptic boundary value problem are presented and are shown to converge monotonically either from above or from below to a solution of the system this monotone convergence result yields a computational algorithm for numerical solutions as well as an existence comparison theorem of the system including a sufficient condition for the uniqueness of the solution an advantage of the block iterative schemes is that the thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one dimensional problems the block iterative schemes are compared with the point monotone iterative schemes of picard jacobi and gauss seidel and various theoretical comparison results among these monotone iterative schemes are given these comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given

maximum principles and singular elliptic inequalities abstract in this paper we present a version of the omori yau maximum principle a liouville type result and a phragmen lindeloff type theorem for a class of singular elliptic operators on a riemannian manifold which include the p laplacian and the mean curvature operator some applications of the results obtained are discussed

abel gontscharoff interpolation continuous discrete and time scale in this paper we shall discuss the abel gontscharoff interpolation problem in the continuous discrete and time scale cases let pn 1 denote the n 1 th degree abel gontscharoff interpolating polynomial of a given function x the interpolating conditions in the continuous discrete and time scale cases are respectively given by pn 1 i ai 1 x i ai 1 ipn 1 ai 1 ix ai 1 and p in 1 ai 1 0 i n 1 where ai 1 i n are the interpolating nodes in each of the three cases we shall present the best possible error bounds under different settings of ai s furthermore as an application of the error estimates obtained criteria are developed for the right disfocality as well as disconjugacy for higher order equations in each of the three cases

strong maximum principle for general quasilinear problems in non divergence form abstract let us consider a solution u of the following i j a i j x u u 2 u x i x j f x u u in r n u x 0 x r n in this paper we give i some conditions ensuring a strong maximum principle for u ii some conditions ensuring that u has a compact support

monotone methods for higher order partial difference equations abstract a nonlinear difference equation of the order n m in two independent variables nx myu x y u together with appropriate initial and periodic boundary conditions is considered several comparison results for these problems are established these results are then used to provide sufficient conditions for the convergence of monotone iterative methods numerical illustrations dwelling upon the importance of our results are also included

monotone convergence of iterative methods for right focal point boundary value problems abstract we provide sufficient conditions so that the picard iterative scheme for the right focal point boundary value problems converges monotonically an example which dwells upon the usefulness of this technique is also included

monotone method for equations describing transport phenomena in a banach space in this paper monotone methods have been developed for equations arising in transport processes in a banach space this method proves the existence of extremal solutions for such equations the advantage of the method for such type of equations is that the successive iterates are solutions of the corresponding initial value problems

initial value methods for discrete boundary value problems motivated by the work of angel and kalaba s and golberg 8 on twopoint boundary value problems for difference equations in this paper we shall study several initial value methods for multipoint boundary value problems for linear and nonlinear difference systems we generalize method of adjoints goodman and lance 9 method of complementary functions tifford 17 d an method of particular solutions miele 111 proposed for two point continuous problems to linear multipoint problems for difference equations next we show that method of adjoint and method of complementary functions can be used in an iterative way to solve nonlinear problems to justify the results obtained here several examples are also given the methods discussed here are similar to our earlier work l 3 on continuous problems we will take up the initial adjusting method of ojika kasue and welsh 12 15 18 for the discrete case later throughout the paper we shall assume the existence uniqueness and stability properties of solutions of the problems discussed and refer to 4 for some of these questions

error and extrapolation of a compact lod method for parabolic differential equations this paper is concerned with a compact locally one dimensional lod finite difference method for solving two dimensional nonhomogeneous parabolic differential equations an explicit error estimate for the finite difference solution is given in the discrete infinity norm it is shown that the method has the accuracy of the second order in time and the fourth order in space with respect to the discrete infinity norm a richardson extrapolation algorithm is developed to make the final computed solution fourth order accurate in both time and space when the time step equals the spatial mesh size numerical results demonstrate the accuracy and the high efficiency of the extrapolation algorithm

a higher order compact adi method with monotone iterative procedure for systems of reaction diffusion equations this paper is concerned with an existing compact finite difference adi method published in the paper by liao et al 2002 3 for solving systems of two dimensional reaction diffusion equations with nonlinear reaction terms this method has an accuracy of fourth order in space and second order in time the existence and uniqueness of its solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear reaction terms the convergence of the finite difference solution to the continuous solution is proved an efficient monotone iterative algorithm is presented for solving the resulting discrete system and some techniques for the construction of upper and lower solutions are discussed an application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method

on entire solutions of degenerate elliptic differential inequalities with nonlinear gradient terms in this paper we give sufficient conditions for the nonexistence of nonnegative nontrivial entire weak solutions of p laplacian elliptic inequalities with possibly singular weights and gradient terms of the form divfg jxj jduj p 2 dug h jxj f u jxj l jduj under the main request

simple calculation of lowdin s alpha function iii procedure for calculating hn 2n i lm l 2n i successively when r a 0 lowdin s function 1 r l flm a r is expressed as 1 r l flm a r 2 lm l al 1 n l imaxi l m hn 2n i lm l 2n i a2n ig 2n i a r a 2n l where a is a separation between a new center placed on the origin and the old center located at a given point on the z axis that is the origin of the coordinate system defining a function f r yml to be expanded around the new center f r is an arbitrary radial function and yml a complex spherical harmonics and r is a distance from the new center here imax min 2n 2 l l and the function g j a is expressed as g j a d dr j f ir rl 1 r a the closed form of the coefficients hn 2n i lm l is given by eq 27 in j math phys 26 3193 1985 referred to as part ii since they are expressed in terms of the different coefficients bkk lm l independent of the form of f r hn 2n i lm l constitute a set of universal constants as do bkk lm l the explicit expression for the factor lm l in front of the summa

error inequalities for quintic and biquintic discrete hermite interpolation in this paper we shall develop a class of discrete hermite interpolates in one and two independent variables further we offer explicit error bounds in norm for the quintic and biquintic discrete hermite interpolates some numerical examples are included to illustrate the results obtained

on the numerical solution of two point boundary value problems abstract this paper presents a new numerical method for the solution of linear two point boundary value problems of ordinary differential equations after reducing the differential equation to a second kind integral equation the latter are discretized via a high order nystrom scheme a somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using o n dot p2 operations where n is the number of nodes on the interval and p is the desired order of convergence thus the advantages of the integral equation formulation small condition number insensitivity to boundary layers insensitivity to end point singularities etc are retained while achieving a computational efficiency previously available only to finite difference or finite element methods keywords two point boundary value problems integral equations chebyshev polynomials approximation theory

monotone enclosure for a class of discrete boundary value problems without monotone nonlinearities abstract a new concept of a pair of upper and lower solutions is introduced for a class of discrete boundary value problems without monotone nonlinearities some comparison results are established an existence and enclosing theorem for the solutions is given in terms of upper and lower solutions a new monotone iterative scheme is proposed the numerical example is given

on the sturm liouville type boundary value problem abstract the problems of existence uniqueness and continuous dependence on parameter of solutions of the nonlinear boundary value problem of sturm liouville type for ordinary differential equations are considered the well known banach fixed point theorem is used to establish results

asymptotic analysis for linear difference equations we are concerned with asymptotic analysis for linear difference equations in a locally convex space first we introduce the profile operator which plays a central role in analyzing the asymptotic behaviors of the solutions then factorial asymptotic expansions for the solutions are given quite explicitly finally we obtain gevrey estimates for the solutions in a forthcoming paper we will develop the theory of cohomology groups for recurrence relations the main results in this paper lay analytic foundations of such an algebraic theory while they are of intrinsic interest in the theory of finite differences

qualitative properties for solutions of singular elliptic inequalities on complete manifolds abstract we establish several qualitative properties for solutions of singular quasilinear elliptic differential inequalities on complete riemannian manifolds such as the validity of the compact support principle of the strong maximum principle existence of solutions to exterior dirichlet problems existence of dead core solutions

polynomial spline approach for solving second order boundary value problems with neumann conditions abstract in this paper a new difference scheme based on quartic splines is derived for solving linear and nonlinear second order ordinary differential equations subject to neumann type boundary conditions the scheme can achieve sixth order accuracy at the interior nodal points and fourth order accuracy at and near the boundary which is superior to the well known numerov s scheme with the accuracy being fourth order convergence analysis of the present method for linear cases is discussed finally numerical results for both linear and nonlinear cases are given to illustrate the efficiency of our method

computational methods for discrete boundary value problems ii abstract for a given nonlinear discrete system together with nonlinear boundary conditions we use newton s iterative scheme and provide sufficient conditions for its convergence to the unique solution then the computational aspects of this iterative method on a floating point system are discussed this includes necessary and sufficient conditions for the convergence of the approximate newton s method an oscillatory state and the stopping criterion to make the analysis widely applicable all the results are proved component wise

a fourth order compact finite difference method for nonlinear higher order multi point boundary value problems a fourth order compact finite difference method is proposed for a class of nonlinear 2nth order multi point boundary value problems the multi point boundary condition under consideration includes various commonly discussed boundary conditions such as the three or four point boundary condition n 2 point boundary condition and 2 n m point boundary condition the existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term the convergence and the fourth order accuracy of the method are proved an efficient monotone iterative algorithm is developed for solving the resulting nonlinear finite difference systems various sufficient conditions for the construction of upper and lower solutions are obtained some applications and numerical results are given to demonstrate the high efficiency and advantages of this new approach

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

boundary value problems for general discrete systems on infinite intervals this paper continues the study of boundary value problems for a discrete system which in particular includes the prototype equation x k 1 f k x k equations with finite as well as infinite delays equations of neutral type and the discrete integral equations of volterra type while the finite discrete interval case regular as well as at resonance has been discussed in 1 2 in the present paper the infinite interval case will be addressed

monotone iterative methods for finite difference system of reaction diffusion equations this paper presents an existence comparison theorem and an iterative method for a nonlinear finite difference system which corresponds to a class of semilinear parabolic and elliptic boundary value problems the basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences from a suitable linear discrete system using upper and lower solutions as two distinct initial iterations two monotone sequences from a suitable linear system are constructed it is shown that these two sequences converge monotonically from above and below respectively to a unique solution of the nonlinear discrete equations this formulation leads to a well posed problem for the nonlinear discrete system applications are given to several models arising from physical chemical and biological systems numerical results are given to some of these models including a discussion on the rate of convergence of the monotone sequences

a higher order compact lod method and its extrapolations for nonhomogeneous parabolic differential equations abstract a higher order compact locally one dimensional lod finite difference method for two dimensional nonhomogeneous parabolic differential equations is proposed the resulting scheme consists of two one dimensional tridiagonal systems and all computations are implemented completely in one spatial direction as for one dimensional problems the solvability and the stability of the scheme are proved almost unconditionally the error estimates are obtained in the discrete h 1 l 2 and l norms and show that the proposed compact lod method has the accuracy of the second order in time and the fourth order in space two richardson extrapolation algorithms are presented to increase the accuracy to the fourth order and the sixth order in both time and space when the time step is proportional to the spatial mesh size numerical results demonstrate the accuracy of the compact lod method and the high efficiency of its extrapolation algorithms

a semi discretization method based on quartic splines for solving one space dimensional hyperbolic equations in this paper based on c 3 quartic splines a semi discretization method containing two schemes is constructed to solve one space dimensional linear hyperbolic equations it is shown that both schemes are unconditionally stable and their approximation orders are of o k 5 h 4 and of o k 7 h 4 with k and h being step sizes in time and space respectively which are much higher than those of other published schemes a numerical example is presented and the results are compared with other published numerical results

simple calculation of lowdin s alpha function iv procedure for evaluating the coefficients in expansion of a slater type atomic orbital by using the explicit expression for the coefficients bkk lm l eq 16 in j math phys 26 3193 1985 which is referred to hereafter as part ii a closed expression for the coefficients cij nlm l appearing in lowdin s alpha function 1 r l nlm a r for a slater type atomic orbital is obtained by manipulating its expression mathematically it was manifested that cij nlm l vanishes if i n l m l for the specific cases that each of i and j takes 0 or 1 cij nlm l take the same value from the symmetry relation on bkk lm l eq 6 in part ii it was proved that cij nlm l cji nlm l provided m l through some manipulation the recursion formula for cij nlm l in n was derived while that about m was obtained by introducing the recursion formula for bkk lm l in m eq 24 in part ii into the expression for cij nlm l for the special cases where n and m both are equal to l the expression for cij nlm l is given in a compact and symmetric form the use of this expression leads to a recursion

keller osserman conditions for diffusion type operators on riemannian manifolds abstract in this paper we obtain essentially sharp generalized keller osserman conditions for wide classes of differential inequalities of the form l u b x f u l u and l u b x f u l u g u h u on weighted riemannian manifolds where l is a non linear diffusion type operator prototypical examples of these operators are the p laplacian and the mean curvature operator the geometry of the underlying manifold is reflected via bounds for the modified bakry emery ricci curvature by growth conditions for the functions b and l a weak maximum principle which extends and improves previous results valid for the laplacian is also obtained geometric comparison results valid even in the case of integral bounds for the modified bakry emery ricci tensor are presented

optimal abel gontscharoff interpolation error bounds on measure chains in this paper we offer the best possible error bounds for abel gontscharoff interpolation on a measure chain our results unify the error analysis performed on continuous and discrete abel gontscharoff interpolation furthermore as an application of the error estimates obtained criteria are developed for the right disfocality as well as disconjugacy for a higher order equation on a measure chain

some recent developments of numerov s method abstract this paper is a survey of some recent developments of numerov s method for solving nonlinear two point boundary value problems the survey consists of three different parts the existence uniqueness of a solution computational algorithm for computing a solution and some extensions of numerov s method the sufficient conditions for the existence and uniqueness of a solution are presented some of them are best possible various iterative methods are reviewed including picard s iterative method modified newton s iterative method monotone iterative method and accelerated monotone iterative method in particular two more direct monotone iterative methods are presented to save computational work each of these iterative methods not only gives a computational algorithm for computing a solution but also leads to an existence and uniqueness theorem the estimate on the rate of convergence of the iterative sequence is given the extensions of numerov s method to a coupled problem and a general problem are addressed the numerical results are presented to validate the theoretical analysis

multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities abstract this paper is devoted to the study of nonlinear singular and non singular fourth order difference equations coupled with periodic boundary value conditions in the paper some existence and nonexistence results are given the results are deduced from krasnoselskii s fixed point theorems in cones an exhaustive study of the green s function related to a linear fourth order problem is done

higher order compact finite difference method for systems of reaction diffusion equations this paper is concerned with a compact finite difference method for solving systems of two dimensional reaction diffusion equations this method has the accuracy of fourth order in both space and time the existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term three monotone iterative algorithms are provided for solving the resulting discrete system efficiently and the sequences of iterations converge monotonically to a unique solution of the system a theoretical comparison result for the various monotone sequences is given the convergence of the finite difference solution to the continuous solution is proved and richardson extrapolation is used to achieve fourth order accuracy in time an application is given to an enzyme substrate reaction diffusion problem and some numerical results are presented to demonstrate the high efficiency and advantages of this new approach

lidstone polynomials and boundary value problems abstract we define lidstone polynomials provide their explicit representations give their relations with bernoulli polynomials obtain best possible error estimates in lidstone interpolation and establish several best possible sharp inequalities which compare favourably with the known results next we use these results to study even order differential equations together with lidstone boundary conditions

existence and uniqueness results for discrete second order periodic boundary value problems abstract we prove existence and uniqueness results for solutions of second order nonlinear difference equations on a finite discrete segment with periodic boundary conditions the results are based on the notion of upper and lower solutions

monotone methods for a boundary value problem of second order discrete equation abstract the comparison and existence results of the upper and lower solutions are established for a boundary value problem of second order discrete equation a monotone iterative scheme is proposed the local as well as global existence and uniqueness of the solution are obtained the global convergence of the iteration is investigated the convergence rate of the iteration is analyzed and compared some examples are given

quasilinearization and approximate quasilinearization for multipoint boundary value problems quasilinearization technique has been applied to a general nonlinear lidstone boundary value problem for the construction of a sequence of its approximate solutions xn t sufficient conditions

accelerated monotone iterative methods for a boundary value problem of second order discrete equations abstract an accelerated monotone iterative method for a boundary value problem of second order discrete equations is presented this method leads to an existence comparison theorem as well as a computational algorithm for the solutions the monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function some numerical results are presented to illustrate the monotone convergence of the iterative sequences and the rate of convergence of the iterations

maximum principle and convergence analysis for the meshfree point collocation method the discrete laplacian operator is considered in the sense of the meshfree point collocation method which will be called the strong meshfree laplacian operator to define the strong meshfree laplacian operator we use the fast version of the generalized moving least square approximation which can calculate the approximated derivatives of shape functions some types of the locally layered node distribution are defined in this paper and two specific domains are constructed onto which we can distribute locally layered nodes at such nodes the discrete maximum principle can be shown to hold through the representation formula for the strong meshfree laplacian operator the discrete maximum principle together with the reproducing property of the meshfree approximations results in an a priori estimate for the strong meshfree laplacian operator in the nodal solution space furthermore the a priori estimate we have obtained guarantees the existence and the uniqueness of the numerical solution and plays a central role in achieving converged results for the poisson problem with dirichlet boundary conditions in the nodal solution space the order of convergence of the nodal solutions can be raised up to o h2 at the proposed type of nodes in specific domains for generally shaped domains immersed in the previously mentioned domains we can obtain the first order convergence result of o h

addition theorem of slater type orbitals a numerical evaluation of barnett coulson lowdin functions when using the one centre two range expansion method to evaluate multicentre integrals over slater type orbitals stos it may become necessary to compute numerical values of the corresponding fourier coefficients also known as barnett coulson lowdin functions bclfs bouferguene and jones 1998 j chem phys 109 5718 to carry out this task it is crucial to not only have a stable numerical procedure but also a fast algorithm in previous work bouferguene and rinaldi 1994 int j quantum chem 50 21 bclfs were represented by a double integral which led to a numerically stable algorithm but this turned out to be disappointingly time consuming the present work aims at exploring another path in which bclfs are represented either by an infinite series involving modified bessel functions or by an integral whose integrand is a smooth function both of these representations have the advantage of being symmetrical with respect to the cusp parameter a and the radial variable r as a consequence it is no longer necessary to split the integrals over r 0 into several components with a different analytical form in each of these a numerical study is also carried out to help select the most appropriate method to be used in practice

the steady temperature distributions with different types of nonlinearities the nonlinear two point boundary value problems are solved and the steady temperature distributions in a rod are found by considering different types of the nonlinear parts of the problems particularly in the polynomial exponential and trigonometric forms in this paper with the aid of some transformations the variational iteration method s scheme is reproduced for the nonlinear problems including two point boundary value problems the illustrative related problems are solved by means of the method scheme

a complexity analysis of the gauss bessel quadrature as applied to the evaluation of multi centre integrals over stfs in a previous work bouferguene 2005 j phys a math gen 38 3923 we have shown that in the framework of the gaussian integral transform multi centre integrals over slater type functions can be evaluated to an acceptable accuracy using a tailored gauss quadrature in which the weight function has the form w z z exp z z to be considered a solution worth implementing within a software for routine use in ab initio molecular simulations the method must also prove to be at least as efficient as those methods previously published in the literature two major results are provided in this paper firstly an improvement of the procedure used to generate the roots and weights of the gauss bessel quadrature is proposed secondly a computational cost analysis of the present method and the safouhi 2001 j phys a math gen 34 2801 based approach are compared hence proving the equivalence of the two from a complexity point of view

phragmen lindelof theorems for equations with nonstandard growth abstract the phragmen lindelof theorem on unbounded domains is studied for subsolutions of variable exponent p laplace equations of homogeneous and nonhomogeneous types the discussion is illustrated by a number of examples of unbounded domains such as half space angular domains and domains narrowing at infinity our approach gives some new results also in the setting of the p laplacian and the harmonic operator

discrete polynomial interpolation green s functions maximum principles error bounds and boundary value problems abstract we construct discrete interpolating polynomials provide explicit representations of discrete green s functions give several identities and inequalities for these green s functions use the explicit forms of the interpolating polynomials and that of green s functions to establish several maximum principles further we obtain error bounds in discrete polynomial interpolation and use them to study existence and uniqueness of the discrete boundary value problems these bounds are also used to provide sufficient conditions for the convergence of the picard s method the approximate picard s method quasilinearization and the approximate quasilinearization the monotone convergence of the picard s iterative method is also analysed

a new gauss quadrature for multicentre integrals over stos in the gaussian integral transform approach when computing multicentre integrals over slater type orbitals stos by means of the shavitt and karplus gaussian integral transforms shavitt and karplus 1962 j chem phys 36 550 one usually ends up with a multiple integral of the form shavitt and karplus 1965 j chem phys 43 398 in which all the integrals are inter related the most widely used approach for computing such an integral is to apply a product of gauss legendre quadratures for the integrals over 0 1 while the semi infinite term is evaluated by a special procedure although numerous approaches have been developed to accurately perform the integration over 0 efficiently it is the aim of this work to add a new tool that could be of some benefit in carrying out the hard task of multicentre integrals over stos the new approach relies on a special gauss quadrature referred to as gauss bessel to accurately evaluate the semi infinite integral of interest in this work emphasis is put on accuracy rather than efficiency since its aim is essentially to bring a proof of concept showing that gauss bessel quadrature can successfully be applied in the context of multicentre integrals over stos the obtained accuracy is comparable to that obtained with other methods available in the literature

on discrete boundary value problems arising in transport phenomena for the discrete boundary value problems arising in transport processes we provide comparison results these results are used to develop monotone iterative methods for the construction of the maximal and minimal solutions in a sector the advantage of this technique is that the successive approximations are the solutions of the initial and terminal value problems numerical illustration showing the sharpness as well the importance of the obtained results is also included

existence of positive solutions of bvps for second order nonlinear difference systems this paper is concerned with the following system d 2u i k f i k u 1 k u 2 k u n k 0 k 0 t i 1 2 nwith the dirichlet boundary conditionu i 0 u i t 2 0 i 1 2 n some new results are obtained for the existence and multiplicity of positive solutions to the above system by using nonlinear alternative of leray schauder type krasnosel skii s fixed point theorem in a cone and leggett williams fixed point theorem in particular it proves that the above system has n positive solutions under suitable conditions where n is an arbitrary integer

criteria for right disfocality of linear difference equations abstract right disfocality and criteria for right disfocality of linear difference equations are considered the criteria require the positivity of appropriate minors of a wronskian determinant and are analogues to criteria for the disconjugacy of linear ordinary differential equations and linear difference equations

double positive solutions of n p boundary value problems for higher order difference equations abstract we shall provide existence criteria for double positive solutions of the n p boundary value problem n y f k y y n 2 y g k y y n 1 y n 1 k n i y 0 0 0 i n 2 p y n n p 0 where n 2 and 0 p n 1 is fixed upper and lower bounds for the two positive solutions are also established for a particular boundary value problem when n 2 several examples are included to dwell upon the importance of the results obtained

multiple positive solutions of a discrete difference system this paper is concerned with the following system d 2u 1 k f 1 k u 1 k u 2 k 0 k 0 t d 2u 2 k f 2 k u 1 k u 2 k 0 k 0 t u 1 0 u 1 t 2 0 u 2 0 u 2 t 2 by using leggett williams fixed point theorem sufficient conditions are obtained for the existence of three positive solutions to the above system

monotone iterative methods for a general class of discrete boundary value problems abstract in this paper we shall offer comparison results as well as monotone iterative schemes for the construction of solutions to a very general class of discrete boundary value problems the discrete system we consider includes in particular the n th order prototype systems finite as well as infinite discrete delay equations and discrete integral equations further the boundary conditions we consider include the initial terminal periodic and transport type problems numerical examples illustrating the usefulness of the proposed schemes to a variety of boundary value problems are also included

parallel monotone iterative relaxation methods for a class of two dimensional discrete boundary value problems abstract a parallel monotone iterative relaxation method for a class of two dimensional discrete boundary value problems is established and the sequence of iterations is shown to converge monotonically either from above or below to a solution of the problem this monotone convergence results yields a parallel computational algorithm as well as an existence comparison result for the solutions to compute the sequence of iterations the thomas algorithm can be used in the same fashion as for one dimensional problem the existence and comparison results of the upper and lower solutions are given the local as well as global existence uniqueness of the solution are obtained the global convergence of the iterations is investigated and the influence of the parameters on the rate of convergence of the iterations is analyzed numerical results are given to corroborate the analytical results

maximum principle in some unbounded domains for an elliptic operator with unbounded drift abstract we study the maximum principle for wg domain which can be unbounded the wg condition was introduced by a vitolo and the maximum principle was proved for the uniformly elliptic operator with bounded coefficients using safonov s growth lemma we treat the operator when the first order coefficients belong to n integrable lebesgue space some examples including infinite cones various unbounded coefficients are presented

a classification of the solutions of a difference equation according to their behavior at infinity on considere l equation aux differences lineaires d ordre n ly t p t y t 0 ou p t est de signe defini sur a a a 1 a est un entier

numerov s method for strongly nonlinear two point boundary value problems abstract by a proper transformation a class of strongly nonlinear two point boundary value problems are transformed into semilinear problems so that the well known numerov s method can be applied some applications and numerical results are presented to demonstrate the efficiency of the approach

monotone convergence of iterative methods for n p and p n boundary value problems abstract we provide sufficient conditions so that the picard iterative scheme for n p and p n boundary value problems converges monotonically an example which illustrates the usefulness of this technique is also included

simple calculation of lowdin s alpha function ii easier procedure for evaluating bk k l m l and vanishing of hn 2n i l m l for special values of i and n this paper subsequent to the one j math phys 25 1133 1984 referred to as part i presents the following new results it is found out that for m l and l 1 the coefficients bk k l m l in lowdin s function have properties other than manifested in part i the expression for bk k l m l is shown to be equivalent to the one into which sharma s expression obtained in a different manner from that in part i is simplified by rashid the use of rashid s expression leads to the recurrence formula for bk k l m l with respect to m only this formula and the expression for the bk k l m l with m l provide an easier procedure for successively evaluating bk k l m l than in part i furthermore it is proved that the coefficients hn 2n i l m l in the asymptotic form of the function vanish for i l m and for n l

focal boundary value problems for nonlinear difference equations ii on considere l unicite des solutions impliquant l existence des solutions de problemes aux valeurs limites de type focal pour l equation aux differences non lineaire d ordre n u t n f t u t u t n 1 ou f a r n r est continue et l equation u n 1 f t u 1 u n peut etre resolue pour u 1 comme fonction continue des variables t u 2 u u 1

accelerated monotone iterations for numerical solutions of nonlinear elliptic boundary value problems abstract an accelerated monotone iterative scheme for numerical solutions of a class of nonlinear elliptic boundary value problems is presented the mathematical analysis is devoted to a system of discretized equations of the elliptic boundary value problem by the finite difference or finite element method it is shown that the sequence of iterations from a linear iteration process converges monotonically and quadratically to a unique solution in a sector between a pair of upper and lower solutions this result is then used to show the quadratic convergence of the iterations to a maximal solution and a minimal solution when the nonlinear discrete system possesses multiple solutions an application is given to a tabular reactor model from chemical engineering for numerical solutions and the number of iterations are compared with that by the regular monotone iterative scheme

nonlinear variation of parameter methods for summary difference equations in several independent variables in this paper we shall present two new nonlinear variation of parameters formulae for the summary difference equations in several independent variables a corrected variation of parameters formula for the corresponding integrodifferential equation is also provided some applications that dwell upon the importance of these new results are also included

keller osserman type conditions for differential inequalities with gradient terms on the heisenberg group abstract we study the qualitative behavior of non negative entire solutions of differential inequalities with gradient terms on the heisenberg group we focus on two classes of inequalities u f u l u and u f u h u g u where f l h g are non negative continuous functions satisfying certain monotonicity properties the operator called the laplacian generalizes the p laplace operator considered by various authors in this setting we prove some liouville theorems introducing two new keller osserman type conditions both extending the classical one which appeared long ago in the study of the prototype differential inequality u f u in r m we show sharpness of our conditions when we specialize to the p laplacian while proving these results we obtain a strong maximum principle for which to the best of our knowledge seems to be new our results continue to hold with the obvious minor modifications also for euclidean space

on the omori yau maximum principle and its applications to differential equations and geometry abstract in this paper we prove a generalised version of the omori yau maximum principle and describe some applications to problems in geometry and differential equations

the strong maximum principle revisited in this paper we rst present the classical maximum principle due to e hopf together with an extended commentary and discussion of hopf s paper we emphasize the comparison technique invented by hopf to prove this principle which has since become a main mathematical tool for the study of second order elliptic partial dierential equations and has generated an enormous number of important applications while hopf s principle is generally understood to apply to linear equations it is in fact also crucial in nonlinear theories such as those under consideration here in particular we shall treat and discuss recent generalizations of the strong maximum principle and also the compact support principle for the case of singular quasilinear ellip tic dierential inequalities under generally weak assumptions on the quasilinear operators and the nonlinearities involved our principal interest is in necessary and sucient con ditions for the validity of both principles in exposing and simplifying earlier proofs of corresponding results and in extending the conclusions to wider classes of singular oper ators than previously considered the results have unexpected rami cations for other problems as will develop from the exposition e g i two point boundary value problems for singular quasilinear ordinary dierential equations sections 3 4 ii the exterior dirichlet boundary value problem section 5 iii the existence of dead cores and compact support solutions i e dead cores at in nity section 7 iv euler lagrange inequalities on a riemannian manifold section 9 v comparison and uniqueness theorems for solutions of singular quasilinear dierential inequalities section 10 the case of p regular elliptic inequalities is briesy considered in section 11

a fourth order compact finite difference method for higher order lidstone boundary value problems a compact finite difference method is proposed for a general class of 2nth order lidstone boundary value problems the existence and uniqueness of the finite difference solution is investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term a monotone iteration process is provided for solving the resulting discrete system efficiently and a simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations the convergence of the finite difference solution and the fourth order accuracy of the proposed method are proved numerical results demonstrate the high efficiency and advantages of this new approach

existence of positive solutions of boundary value problem for a discrete difference system in this paper we consider the following boundary value problem of discrete system d 2u 1 k f 1 k u 1 k u 2 k 0 k 0 t d 2u 2 k f 2 k u 1 k u 2 k 0 k 0 t with the dirichlet boundary value conditionu 1 0 u 1 t 2 0 u 2 0 u 2 t 2 some sufficient conditions are obtained for the existence of one or two positive solutions to the above system by using nonlinear alternative of leray schauder type and krasnosel skii s fixed point theorem in a cone

fourth order compact finite difference method for fourth order nonlinear elliptic boundary value problems a compact finite difference method with non isotropic mesh is proposed for a two dimensional fourth order nonlinear elliptic boundary value problem the existence and uniqueness of its solutions are investigated by the method of upper and lower solutions without any requirement of the monotonicity of the nonlinear term three monotone and convergent iterations are provided for resolving the resulting discrete systems efficiently the convergence and the fourth order accuracy of the proposed method are proved numerical results demonstrate the high efficiency and advantages of this new approach

monotone method for nonlinear boundary value problems arising in transport processes recently monotone iterative methods have been successfully employed to prove existence of multiple solutions and point wise bounds on solutions of nonlinear boundary value problems for both ordinary and partial differential equations because of the importance of this problem in other physical applications we extend in this paper the monotone technique to a general class of nonlinear boundary value problem which includes the transport problem treated as a special case