existence of positive solutions for 2nth order singular sublinear boundary value problems abstract this paper investigates the existence of positive solutions for 2nth order n 1 singular sub linear boundary value problems a necessary and sufficient condition for the existence of c 2 n 2 0 1 as well as c 2 n 1 0 1 positive solutions is given by constructing lower and upper solutions and with the maximal theorem our nonlinearity f t x 1 x 2 x n may be singular at x i 0 i 1 2 n t 0 and or t 1

twin positive solutions for higher order m point boundary value problems with sign changing nonlinearities a new fixed point theorem on double cones is applied to obtain the existence of at least two positive solutions for the 2nd order m point boundary value problemy 2 n t f t y t y t y 2 n 1 t 0

reductions of integrable lattices based on the notion of darboux kp chain hierarchy and its invariant submanifolds we construct some class of constraints compatible with integrable lattices some simple examples are given

on some classes of discrete polynomials and ordinary difference equations we introduce two classes of discrete polynomials and use them to construct ordinary difference equations admitting a lax representation in terms of these polynomials we also construct lattice integrable hierarchies in their explicit form and show some examples

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

numerical methods for fourth order elliptic equations with nonlocal boundary conditions this paper is concerned with some numerical methods for a fourth order semilinear elliptic boundary value problem with nonlocal boundary condition the fourth order equation is formulated as a coupled system of two second order equations which are discretized by the finite difference method three monotone iterative schemes are presented for the coupled finite difference system using either an upper solution or a lower solution as the initial iteration these sequences of monotone iterations called maximal sequence and minimal sequence respectively yield not only useful computational algorithms but also the existence of a maximal solution and a minimal solution of the finite difference system also given is a sufficient condition for the uniqueness of the solution this uniqueness property and the monotone convergence of the maximal and minimal sequences lead to a reliable and easy to use error estimate for the computed solution moreover the monotone convergence property of the maximal and minimal sequences is used to show the convergence of the maximal and minimal finite difference solutions to the corresponding maximal and minimal solutions of the original continuous system as the mesh size tends to zero three numerical examples with different types of nonlinear reaction functions are given in each example the true continuous solution is constructed and is used to compare with the computed solution to demonstrate the accuracy and reliability of the monotone iterative schemes

jacobi polynomials bernstein type inequalities and dispersion estimates for the discrete laguerre operator abstract the present paper is about bernstein type estimates for jacobi polynomials and their applications to various branches in mathematics this is an old topic but we want to add a new wrinkle by establishing some intriguing connections with dispersive estimates for a certain class of schrodinger equations whose hamiltonian is given by the generalized laguerre operator more precisely we show that dispersive estimates for the schrodinger equation associated with the generalized laguerre operator are connected with bernstein type inequalities for jacobi polynomials we use known uniform estimates for jacobi polynomials to establish some new dispersive estimates in turn the optimal dispersive decay estimates lead to new bernstein type inequalities

a generalized multi point boundary value problem for second order ordinary differential equations let f 0 1 x r 2 r be a function satisfying caratheodory s conditions and e t l 1 0 1 let x i t j 0 1 a i b j r i 1 2 m 2 j 1 2 n 2 0 x 1 x 2 x m 2 1 0 t 1 t 2 t n 2 1 be given this paper is concerned with the problem of existence of a solution for the generalized multi point boundary value problems x t f t x t x t e t 0

solvability of some fourth and higher order singular boundary value problems abstract existence results for a variety of singular fourth order boundary value problems of the form yiv f t y y y are given here our nonlinear term f may be singular at t 0 t 1 y 0 and or y 0 for example some singularities of the type y a and y b are included also we discuss and treat the extension of these results to nth order boundary value problems

necessary and sufficient conditions for the existence of positive solutions of fourth order multi point boundary value problems abstract this paper investigates the existence of positive solutions of singular multi point boundary value problems of fourth order ordinary differential equation with p laplacian a necessary and sufficient condition for the existence of c 2 0 1 positive solution as well as pseudo c 3 0 1 positive solution is given by means of the fixed point theorems on cones

discrete painleve equations and their appearance in quantum gravity we discuss an algorithmic approach for both deriving discrete analogues of painleve equations as well as using such equations to characterize similarity reductions of spatially discrete integrable evolution equations as a concrete example we show that a discrete analogue of painleve i can be used to characterize similarity solutions of the kac moerbeke equation it turns out that these similarity solutions also satisfy a special case of painleve iv equation in addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions as an example we use the wkb method in the presence of two turning points of the third order to parametrize at the continuous limit the solution of painleve i in terms of the solution of discrete painleve i finally we show that these results are useful for investigating the partition function of the matrix model in 2d quantum gravity associated with the measure exp t1z2 t2z4 t3z6

solvability of a three point nonlinear boundary value problem for a second order ordinary differential equation in case f x u u ps x p x u pjx u with pk 0 l 4 r locally integrable the boundary value problem l lt 1 2 was studied by kiguradze and lomtatidze i the purpose of this paper is to obtain existence and uniqueness theorems for the boundary value problem 1 1 t 1 2 under natural conditions on f using degree theoretic arguments we note that if u is a solution of 1 1 1 2 then there exists at least one 5 e q l such that u c 0 accordingly the boundary value problem

error bounds for quadrature formulas near gaussian quadrature abstract let rn be the error functional of a quadrature formula qn on 1 1 using n nodes in this paper we consider estimates of the form r n c m m sup x 1 x with best possible constant cm i e c m c m r n sup m 1 r n for the error constants c2n k rgn of the gaussian quadrature formulas qgn we prove results which are asymptotically sharp when n increases and k is fixed for this latter case comparing with the corresponding error constants c2n k rn of every other quadrature formula qn we show that the order of magnitude of c2n k rgn cannot be improved in n in particular we investigate the question of minimal and maximal values of c2n k rn in the class of all quadrature formulas qn having at least algebraic degree of exactness deg qn 2n k 1

multiple positive solutions for nth order impulsive integro differential equations in banach spaces abstract in this paper the author obtains the existence of multiple positive solutions for a boundary value problem of n th order nonlinear impulsive integro differential equations on an infinite interval in a banach space by means of fixed point index theory of completely continuous operators

existence of multiple symmetric positive solutions of higher order lidstone problems this paper gives the sufficient conditions of the existence of at least three symmetric positive solutions for one type of higher order autonomous lidstone problem by applying the five functionals fixed point theorem the analogous result for higher order nonautonomous singular lidstone problem is also proved here

multiplicity results for positive solutions of some semi positone three point boundary value problems in this paper some multiplicity results for positive solutions of some singular semi positone three point boundary value problem be obtained by using the fixed point index method

monotone iterative method for the second order three point boundary value problem with upper and lower solutions in the reversed order abstract this article introduces a new concept of upper and lower solutions and studies the existence and uniqueness of solutions of second order three point boundary value problems with upper and lower solutions in the reversed order the sufficient conditions for the existence and uniqueness of solutions are obtained by using the monotone iterative method meantime the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution are given some results of previous literature are extended and improved a numerical example is also included to illustrate the effectiveness of the proposed results

the gegenbauer polynomials and typically real functions linear and nonlinear coefficient problems for some class of typically real functions are studied different inequalities for the gegenbauer polynomials appear to be very useful

positive solutions of singular sublinear emden fowler boundary value problems abstract this paper is about the dirichlet boundary value problem of singular sublinear emden fowler equations a necessary and sufficient condition for the existence of positive solutions to this problem has been obtained by using the method of lower and upper solutions the uniqueness of positive solutions has also been discussed

multiple positive solutions of a m point boundary value problem for 2nth order singular integro differential equations in banach spaces abstract in this paper we obtain the existence of multiple positive solutions of a m point boundary value problem for 2 n th order singular nonlinear integro differential equations in a banach space by means of fixed point index theory of completely continuous operators

on the geometry of darboux transformations for the kp hierarchy and its connection with the discrete kp hierarchy we tackle the problem of interpreting the darboux transformation for the kp hierarchy and its relations with the modified kp hierarchy from a geometric point of view this is achieved by introducing the concept of a darboux covering we construct a darboux covering of the kp equations and obtain a new hierarchy of equations which we call the darboux kp hierarchy dkp we employ the dkp equations to discuss the relationships among the kp equations the modified kp equations and the discrete kp equations our approach also handles the various reductions of the kp hierarchy we show that the kp hierarchy is a projection of the dkp the mkp hierarchy is a dkp restriction to a suitable invariant submanifold and that the discrete kp equations are obtained as iterations of the dkp ones

the upper and lower solution method for a class of singular nonlinear second order three point boundary value problems in this paper we develop the upper and lower solution method and the monotone iterative technique for a class of singular nonlinear second order three point boundary value problems a maximum principle is established and some new existence results are obtained

multiple positive solutions of a boundary value problem for a class of 2 nth order m point singular integro differential equations in banach spaces in this paper we obtain the existence of multiple positive solutions of a boundary value problem for 2nth order singular nonlinear integro differential equations in a banach space by means of fixed point index theory of completely continuous operators

extension of the discrete kp hierarchy in this paper we discuss possible generalizations of the discrete kp toda lattice hierarchy as a result we introduce the integrable hierarchy which can be considered as the proper extension of the discrete kp hierarchy such extension forces us to introduce an infinite number of additional multi times t n t n 1 x n t n 2 n 2 whereas ordinary discrete kp is relevant to t 1 it is shown that any subsystem of extended discrete kp attached to multi time t n is in fact equivalent to a bi infinite sequence of continuous kp hierarchies whose lax operators are glued together by compatible gauge transformations this paper can be thought of as a natural continuation and generalization of our previous paper svinin a k 2001 j phys a math gen 34 10559 68

on some class of reductions for the itoh narita bogoyavlenskii lattice we show a broad class of constraints compatible with the itoh narita bogoyavlenskii lattice hierarchy all these constraints can be written in the form of a discrete conservation law i i 1 i i with an appropriate homogeneous polynomial discrete function i i a

on estimates for the weights in gaussian quadrature in the ultraspherical case in this paper the christoffel numbers a 2 g for ultraspherical weight functions w a wa x 1 x2 1 2 are investigated using only elementary functions we state new inequalities monotonicity properties and asymptotic approximations which improve several known results in particular denoting by 0 i the trigonometric representation of the gaussian nodes we obtain for a e 0 11 the inequalities n v fn 2 n a 2 sin2 a ag 7r 2a1 3 av n 1 2 a a g 7r 2a a sin 0 in v n n a f n 2 n a 2sin 2 v n a i a 3 a i a 2 4 sin 2 0 a n 8 n a 4 sin 46 o n

a necessary and sufficient condition for 2nth order singular super linear m point boundary value problems abstract in this paper we study the existence of positive solutions for singular super linear m point boundary value problems of 2nth order ordinary differential equations a necessary and sufficient condition for the existence of c 2 n 2 0 1 positive solutions as well as c 2 n 1 0 1 positive solutions is given by means of the fixed point theorems on cones

r matrix approach to lattice integrable systems an r matrix formalism is applied to the construction of the integrable lattice systems and their bi hamiltonian structure miura like gauge transformations between the hierarchies are also investigated in the end the ladder of linear maps between generated hierarchies is established and described

positive solutions of nonlinear fourth order boundary value problems with local and non local boundary conditions we establish new existence results for multiple positive solutions of fourth order nonlinear equations which model deflections of an elastic beam we consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non local boundary conditions with a unified approach our method is to show that each boundary value problem can be written as the same type of perturbed integral equation in the space c 0 1 involving a linear functional u but although we seek positive solutions the functional is not assumed to be positive for all positive u the results are new even for the classic boundary conditions of clamped or hinged ends when u 0 because we obtain sharp results for the existence of one positive solution for multiple solutions we seek optimal values of some of the constants that occur in the theory which allows us to impose weaker assumptions on the nonlinear term than in previous works our non local boundary conditions contain multi point problems as special cases and for the first time in fourth order problems we allow coefficients of both signs

three positive solutions for higher order m point boundary value problems abstract in this paper we consider the existence of at least three positive solutions for the 2nth order m point boundary value problem y 2n t f t y t y t y 2 n 1 t 0 t 1 y 2i 0 0 y 2i 1 j 1 m 2 k ij y 2i j 0 i n 1 where kij 0 i 0 1 n 1 j 1 2 m 2 0

existence of solutions for 2n order boundary value problem in this paper we discuss the existence of solutions for the 2n order boundary value problem 1 nu 2 n t f t u t u t u 2 n 2 t 0r is continuous

a class of fourth order singular boundary value problems by constructing lower and upper solutions and with the maximal theorem the author gives a necessary and sufficient condition for the existence of c 2 0 1 as well as c 3 0 1 positive solutions of singular boundary value problems for a class of fourth order sublinear differential equations

fourth order two point boundary value problems estimates by two sided bounds on presente une methode simple pour transformer des problemes aux deux points limites d ordre 4 en problemes d ordre 2

the method of lower and upper solutions for fourth order singular m point boundary value problems abstract this paper investigates the existence of positive solutions for fourth order singular m point boundary value problems firstly we establish a comparison theorem then we define a partial ordering in c 2 0 1 c 4 0 1 and construct lower and upper solutions to give a necessary and sufficient condition for the existence of c 2 0 1 as well as c 3 0 1 positive solutions our nonlinearity f t x y may be singular at x y t 0 and or t 1

a necessary and sufficient condition for positive solutions for fourth order multi point boundary value problems with p laplacian abstractinthispaper weconsidertheexistenceofpositivesolutionsforaclassofsingularfourth order p laplacianequationwithmulti point boundary conditions by means of a monotone iterative technique a necessary and suf cient condition for the existence of apseudo c 3 0 1 positive solution is established also the existence and uniqueness of the pseudo c 3 0 1 positive solution theiterative sequence of the solution and an error estimation for the pseudo c 3 0 1 positive solution are obtained c 2007 elsevier ltd all rights reserved msc 34b15 34b25keywords necessary and suf cient condition p laplacian pseudo c 3 0 1 positive solutions cone 1 introductionin this paper we consider the existence of positive solutions for the singular fourth order multi point boundaryvalue problem with p laplacian p u 00 t 00 f 0 1u 0 m x 2i 1 a i u u 1 0 u 00 0 m x 2i 1 b u 00 u 00 1 0 1 1 where p t t p 2 t p 2 0 a i b i i

inequalities for ultraspherical polynomials and application to quadrature abstract in this paper for the ultraspherical polynomials p n we investigate inequalities of the type sin p n cos m 1 applying the results to linear functionals we obtain new estimates in particular for the remainder term in gaussian quadrature

symmetries conserved quantities and hierarchies for some lattice systems with soliton structure basic invariants such as conserved quantities symmetries mastersymmetries and recursion operators are explicitly constructed for the following nonlinear lattice systems the modified korteweg de vries lattice the ablowitz ladik lattice the brusci ragnisco lattice the ragnisco tu lattice and some cases of the class of integrable systems introduced by bogoyavlensky the algorithmic basis for obtaining these quantities is described and the interrelation between the underlying mastersymmetry approach and the lax pair analysis is discussed by explicit presentation of the higher order members of the corresponding hierarchies new completely integrable lattice flows are found for all systems multi hamiltonian formulations are given

on some class of homogeneous polynomials and explicit form of integrable hierarchies of differential difference equations we introduce two classes of homogeneous polynomials and show their role in construction of integrable hierarchies for some integrable lattices

some extensions of w gautschi s inequalities for the gamma function it has been shown by w gautschi that if 0 i xi s f x exp i s x 1 the following closer bounds are proved exp i s 4 x 12 f exp i s x s i f x s 2 and x 2 t x 2 s 4 these are compared with each other and with inequalities given by t erber and j d keckic and p m vasic

existence and uniqueness results for the bending of an elastic beam equation at resonance etude d un probleme non lineaire aux valeurs limites pour la flexion d une poutre elastique en appui simple aux deux extremites dans le cas de resonance theoremes d existence et d unicite

positive solutions of m point boundary value problems the second order m point boundary value problem x h x f x 0 0 x 1 0 0 1 i 1m 2ai i is considered under some conditions concerning the first eigenvalue of the relevant linear operator where i 0 1 with 0 1 2 m 2 1 ai 0 with i 1m 2ai 1 h x is allowed to be singular at x 0 and x 1 the existence of positive solutions and multiple positive solutions is obtained by means of fixed point index theory similarly conclusions hold for some other m point boundary value conditions

multiple sign changing solutions for fourth order m point boundary value problems abstract in this paper we study the existence and multiplicity of nontrivial solutions for the fourth order m point boundary value problems making use of the theory of the fixed point index in a cone and the leray schauder degree under general conditions on nonlinearity we prove that there exist at least six different nontrivial solutions for the fourth order m point boundary value problems furthermore if the nonlinearity is odd we obtain that there exist at least eight different nontrivial solutions

positive solutions for singular m point boundary value problems with positive parameter in this paper some new existence results for positive solutions for a class of singular m point boundary value problems with parameter be obtained