the method of lower and upper solutions for 2nth order multi point boundary value problems abstract in this paper we consider the following 2 n th order multi point boundary value problems 1 n u 2 n t f t u t u t 1 i u 2 i t 1 n 1 u 2 n 2 t 0 t 1 with the 2 n m point boundary conditions u 2 j 0 u 2 j 1 0 j 0 1 2 m m n 2 u 2 j 0 a j u 2 j j 1 0 u 2 j 1 b j u 2 j j 2 0 j m 1 n 1 where a j 0 b j 0 0 j 1 j 2 1 j m 1 n 1 the existence of iterative solutions is obtained by using the lower and upper solution method for the above 2 n m point boundary value problems

multiple solutions to a three point boundary value problem for higher order ordinary differential equations in this paper we provide sufficient conditions for the existence of at least three solutions to a three point boundary value problem for higher order ordinary differential equations the nonlinear term f in the differential equation under consideration may depend on higher order derivatives of arbitrary order and this is where the main novelty of this work lies by applying the two pairs of upper and lower solutions method of henderson and thompson as well as degree theory the existence of at least three solutions of the problem is given

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

singular perturbations for third order nonlinear multi point boundary value problem abstract this paper is devoted to study the following third order multi point singularly perturbed boundary value problem x t f t x t x t x t 0 0 t 1 0 1 x 0 0 ax 0 bx 0 i 1 n 2 i x i a cx 1 dx 1 i 1 n 2 i x i b where a b c d 0 a b r a b 0 c d 0 i 0 i 0 i 1 2 n 2 0 1 2 n 2 1 and 0 1 2 n 2 1 the existence uniqueness and asymptotic estimates of solutions of the boundary value problem are give by using priori estimates differential inequalities technique and leray schauder degree theory

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

existence theorems for a second order three point boundary value problem abstract letf 0 1 r 2 r be continuous ande c 0 1 let i 0 1 ai r all of theai s having the same sign i 1 2 m 2 0 x t f t x t x t e t t 0 1 e x 0 0 x 1 i 1 m 2 a i x i bc m we discuss existence theorems for the problem e bcm under some sign condition offabout the origin our analysis is based on a nonlinear alternative of leray schauder

positive solutions to a multi point higher order boundary value problem the authors consider a higher order multi point boundary value problem some existence and nonexistence results for positive solutions of the problem are obtained by using krasnosel skii s fixed point theorem examples are included to illustrate the results

positive solutions for three point boundary value problems with dependence on the first order derivative abstract a new fixed point theorem in a cone is applied to obtain the existence of at least one positive solution for the second order three point boundary value problem x f t x x 0 0 x 0 0 x 1 x where f is a nonnegative continuous function 0 0 1

econometric analysis of multivariate realised qml estimation of the covariation of equity prices under asynchronous trading estimating the covariance between assets using high frequency data is challenging due to market microstructure effects and asynchronous trading in this paper we develop a multivariate realised quasi likelihood qml approach carrying out inference as if the observations arise from an asynchronously observed vector scaled brownian model observed with error under stochastic volatility the resulting realised qml estimator is positive semi definite uses all available data is consistent and asymptotically mixed normal the quasi likelihood is computed using a kalman filter and optimised using a relatively simple em algorithm which scales well with the number of assets we derive the theoretical properties of the estimator and prove that it achieves the efficient rate of convergence we show how to make itobtain the non parametric efficiency bound for this problem the estimator is also analysed usingmonte carlo methods and applied to equity data with varying levels of liquidity

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

positive solutions of fourth order multi point boundary value problems with bending term abstract in this paper we study m point boundary value problem for fourth order ordinary differential equation with bending term x x 4 t g t f t x t x t t 0 1 x 0 0 x 1 i 1 m 2 a i x i x 0 0 x 1 i 1 m 2 b i x i where 0 0 1 2 m 2 m 1 1 by constructing an available integral operator and combining fixed point index theorem we establish sufficient conditions for the existence of positive solutions under some conditions concerning the first eigenvalue corresponding to the relevant linear operator the interesting point of the results is that the nonlinear term g may be singular at t 0 and or t 1 moreover f t u v is also allowed to have singularity at u 0 and or v 0

extension of moment projection method to the fragmentation process the method of moments is a simple but efficient method of solving the population balance equation which describes particle dynamics recently the moment projection method mpm was proposed and validated for particle inception coagulation growth and more importantly shrinkage here the method is extended to include the fragmentation process the performance of mpm is tested for 13 different test cases for different fragmentation kernels fragment distribution functions and initial conditions comparisons are made with the quadrature method of moments qmom hybrid method of moments hmom and a high precision stochastic solution calculated using the established direct simulation algorithm dsa and advantages of mpm are drawn

testing if the market microstructure noise is fully explained by the informational content of some variables from the limit order book abstract in this paper we build tests for the presence of residual noise in a model where the market microstructure noise is a known parametric function of some variables from the limit order book the tests compare two distinct quasi maximum likelihood estimators of volatility where the related model includes a residual noise in the market microstructure noise or not the limit theory is investigated in a general nonparametric framework in the presence of residual noise we examine the central limit theory of the related quasi maximum likelihood estimation approach

existence of at least three solutions of a second order three point boundary value problem abstract we study existence of at least three solutions in the presence of two lower and two upper solutions of some second order nonlinear three point boundary value problem of the type x f t x x t i 0 1 x 0 0 x 1 x 0 1 0 1 the growth of f with respect to x is allowed to be quadratic we use some degree theory arguments to get the multiplicity result

the method of lower and upper solutions for fourth order two point boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u iv f x u x u x u 0 u 1 u 0 u 1 0 where f 0 1 r r r is continuous

solvability of multipoint boundary value problems at resonance for higher order ordinary differential equations let f 0 1 x r n r be a continuous function and e l 1 0 1 let b j 1 j m 2 r 0 h 1 h 2 h m 2 1 be given this paper is concerned with the existence of solutions for the following n t h order multipoint boundary value problems at resonance case x n t f t x t x t x n 1 t e t email protected 0 1 x 0 x 0 x n 2 0 0 x 1 email protected email protected jx h j and x n t f t x t x t x n 1 t e t email protected 0 1 x 0 x 0 x n 2 0 0 x 1 email protected email protected jx h j some existence results are obtained by using the coincidence degree theory of mawhin

positive solutions of even higher order singular superlinear boundary value problems abstract this paper investigates 2 m th order m 1 superlinear singular two point boundary value problems and obtains some necessary and sufficient conditions for existence of c 2 m 1 or c 2 m 1 positive solutions on the closed interval

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

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 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

fluctuation effects on wave propagation in a reaction diffusion process abstract the reaction diffusion process corresponding to the fisher kolmogorov equation is studied by means of a discrete multivariate master equation for travelling wave fronts the stability criterion necessary for the applicability of a system size expansion is shown to be violated due to the existence of a zero mode of the first variational equation this zero mode is connected to the translational invariance of the system performing stochastic simulations of the master equation in a wide range of parameters it is demonstrated that for finite size of the system up to about 10 7 particles in the frontal region a rather large fluctuation effect on the wave propagation speed results in general the asymptotic wave speed lies below the stable minimal speed which is given by a theorem of kolmogorov for the macroscopic equation the wave front position exhibits a diffusion type behaviour associated with translative fluctuations along the propagation direction

multiplicity of solutions to two point boundary value problems for second order impulsive differential equations abstract in this paper we obtain some sufficient condition for the existence of at least three solutions to two point boundary value problems for second order impulsive differential equations by using leray schauder degree theory we assume the existence of two pairs of strict lower and upper solutions and the nonlinear term f satisfies a nagumo s growth condition with respect to the first order derivative

a bias corrected estimator of the covariation matrix of multiple security prices when both microstructure effects and sampling durations are persistent and endogenous i propose a bias corrected non parametric estimator of the covariation matrix of log security prices designed as a convex combination of two realized kernels the estimator is simple but possesses desirable statistical properties including consistency asymptotic normality and the parametric rate of convergence in the presence of persistent diurnally heteroskedastic and endogenous microstructure effects it is robust to the asynchronous trading of multiple securities with persistent and endogenous refresh time durations i also prove the consistency of a subsampling based estimator of the asymptotic covariance matrix of the proposed estimator in simulations the non linear functions of the proposed estimator exhibit smaller bias than those based on a realized kernel while only slightly increasing the variance thereby the proposed estimator reduces the mean squared error

solutions of 2nth lidstone boundary value problems and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem 1 n u 2n x f x u x u x u 2 n 1 x 0 u 2i 0 u 2i 1 0 0 i n 1 some monotone conditions are imposed on f which enable us to apply a new maximum principle to deduce the existence of solution the emphasis here is that f depends on higher order derivatives two examples are also included to dwell upon the importance of the results obtained

upper and lower solution method for fourth order four point boundary value problems this paper is concerned with the fourth order ordinary differential equation u 4 t f t u t u t 0 0 some new existence results are obtained by developing the upper and lower solution method and the monotone iterative technique

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

econometrics of co jumps in high frequency data with noise abstract we establish estimation methods to determine co jumps in multivariate high frequency data with non synchronous observations and market microstructure a rate optimal estimator of the entire quadratic covariation of an ito semimartingale is constructed by a locally adaptive spectral approach thresholding allows to disentangle the co jump from the continuous part we derive a feasible limit theorem for a truncated estimator of integrated covolatility which facilitates asymptotically efficient co volatility estimation in the presence of jumps a test for common jumps is presented simulations and an empirical application to intra day tick data from eurex futures demonstrate the practical value of the approach

occupation density estimation for noisy high frequency data abstract this paper studies the nonparametric estimation of occupation densities for semimartingale processes observed with noise as leading examples we consider the stochastic volatility of a latent efficient price process the volatility of the latent noise that separates the efficient price from the actually observed price and nonlinear transformations of these processes our estimation methods are decidedly nonparametric and consist of two steps the estimation of the spot price and noise volatility processes based on pre averaging techniques and in fill asymptotic arguments followed by a kernel type estimation of the occupation densities our spot volatility estimates attain the optimal rate of convergence and are robust to leverage effects price and volatility jumps general forms of serial dependence in the noise and random irregular sampling the convergence rates of our occupation density estimates are directly related to that of the estimated spot volatilities and the smoothness of the true occupation densities an empirical application involving high frequency equity data illustrates the usefulness of the new methods in illuminating time varying risks market liquidity and informational asymmetries across time and assets

partially stirred reactor model analytical solutions and numerical convergence study of a pdf monte carlo method we investigate the partially stirred reactor pasr which is based on a simplified joint composition probability density function pdf transport equation analytical solutions for the first four moments of the mass density function mdf obtained from the pasr model are presented the monte carlo particle method with first order time splitting algorithm is implemented to obtain the first four moments of the mdf numerically the dynamics of the stochastic particle system is determined by inflow outflow chemical reaction and mixing events three different inflow outflow algorithms are investigated an algorithm based on the inflow outflow event modeled as a poisson process an inflow outflow algorithm mentioned in the literature and a novel algorithm derived on the basis of analytical solutions it is demonstrated that the inflow outflow algorithm used in the literature can be explained by considering a deterministic waiting time parameter of a corresponding stochastic process and also forms a specific case of the new algorithm the number of particles in the ensemble n the nondimensional time step delta t ratio of the global time step to the characteristic time of an event and the number of independent simulation trials l are the three sources of the numerical error the split analytical solutions and the numerical experiments suggest that the systematic error converges as delta t and n 1 the statistical error scales as l 1 2 and n 1 2 the significance of the numerical parameters and the inflow outflow algorithms is also studied by applying the pasr model to a practical case of premixed kerosene and air combustion

dependent microstructure noise and integrated volatility estimation from high frequency data abstract in this paper we develop econometric tools to analyze the integrated volatility iv of the efficient price and the dynamic properties of microstructure noise in high frequency data under general dependent noise we first develop consistent estimators of the variance and autocovariances of noise using a variant of realized volatility next we employ these estimators to adapt the pre averaging method and derive consistent estimators of the iv which converge stably to a mixed gaussian distribution at the optimal rate n 1 4 to improve the finite sample performance we propose a multi step approach that corrects the finite sample bias which turns out to be crucial in applications our extensive simulation studies demonstrate the excellent performance of our multi step estimators in an empirical study we analyze the dependence structures of microstructure noise and provide intuitive economic interpretations we also illustrate the importance of accounting for both the serial dependence in noise and the finite sample bias when estimating iv

positive solutions of m point boundary value problems for higher order ordinary differential equations abstract in this paper we study m point boundary value problems for higher order ordinary differential equation y 2 n t f t y t y t y 2 n 1 t 0 t 1 y 2 i 0 0 y 2 i 1 j 1 m 2 k i j y 2 i j 0 i n 1 where f is allowed to change sign and 0 0 1 2 m 2 m 1 1 we show sufficient conditions for the existence of at least two positive solutions by applying a new fixed point theorem in cones and the associated green s function in particular the second positive solutions for the above problem is not concave

on numerical studies of multi point boundary value problem and its fold bifurcation based upon shooting method we set up a numerical method for approximating solutions and fold bifurcation solutions of second order multi point boundary value problems we prove the equivalence of the regularity of solutions and simple fold bifurcation solutions between the numerical method and the multi point boundary value problems many numerical experiments are carried out to illustrate the theoretical analysis from which we explore that the fold bifurcation phenomenon are generic in multi point boundary value problems although there is no theoretical analysis in the literature so far

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

estimating the quadratic covariation matrix from noisy observations local method of moments and efficiency an efficient estimator is constructed for the quadratic covariation or integrated covolatility matrix of a multivariate continuous martingale based on noisy and non synchronous observations under high frequency asymptotics our approach relies on an asymptotically equivalent continuous time observation model where a local generalised method of moments in the spectral domain turns out to be optimal asymptotic semiparametric efficiency is established in the cramer rao sense main findings are that non synchronicity of observation times has no impact on the asymptotics and that major efficiency gains are possible under correlation simulations illustrate the finite sample behaviour

positive solutions of 2nth order ordinary differential equations with multi point boundary conditions in this paper we study the existence of positive solutions for 2nth order n 1 boundary value problems 1 nu 2n t f t u 0 t 1 subject to the 2 n m point boundary value conditions u 2j 0 u 2j 1 0 j 0 1 2 m m n 2 u 2j 0 bju 2j j 1 0 u 2j 1 dju 2j j 2 0 j m 1 n 1 where bj 0 dj 0 0 j 1 j 2 1 j m 1 n 1 by means of the fixed point theorems and the lower and upper solution method some results on the existence of positive solutions to the above 2 n m point boundary value problems are obtained

a moment projection method for population balance dynamics with a shrinkage term a new method of moments for solving the population balance equation is developed and presented the moment projection method mpm is numerically simple and easy to implement and attempts to address the challenge of particle shrinkage due to processes such as oxidation evaporation or dissolution it directly solves the moment transport equation for the moments and tracks the number of the smallest particles using the algorithm by blumstein and wheeler 1973 41 the performance of the new method is measured against the method of moments mom and the hybrid method of moments hmom the results suggest that mpm performs much better than mom and hmom where shrinkage is dominant the new method predicts mean quantities which are almost as accurate as a high precision stochastic method calculated using the established direct simulation algorithm dsa

positive solutions for some second order four point boundary value problems abstract in this paper the second order four point boundary value problem x t h t f t x t 0 0 t 1 x 0 a x x 1 b x is studied where 0 1 0 a b 1 and h 0 1 0 f 0 1 0 0 are nonnegative continuous functions by the use of the property of the corresponding green s function fixed point index theory leray schauder degree and upper and lower solution method some existence nonexistence and multiplicity results of positive solutions are acquired

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

a rigorous derivation of the chemical master equation it is widely believed that the chemical master equation has no rigorous microphysical basis and hence no a priori claim to validity this view is challenged here through arguments purporting to show that the chemical master equation is exact for any gas phase chemical system that is kept well stirred and thermally equilibrated

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

efficient asymptotic variance reduction when estimating volatility in high frequency data abstract this paper shows how to carry out efficient asymptotic variance reduction when estimating volatility in the presence of stochastic volatility and microstructure noise with the realized kernels rk from barndorff nielsen et al 2008 and the quasi maximum likelihood estimator qmle studied in xiu 2010 to obtain such a reduction we chop the data into b blocks compute the rk or qmle on each block and aggregate the block estimates the ratio of asymptotic variance over the bound of asymptotic efficiency converges as b increases to the ratio in the parametric version of the problem i e 1 0025 in the case of the fastest rk tukey hanning 16 and 1 for the qmle the impact of stochastic sampling times and jump in the price process is examined carefully the finite sample performance of both estimators is investigated in simulations while empirical work illustrates the gain in practice

an improved stochastic algorithm for temperature dependent homogeneous gas phase reactions we propose an improved stochastic algorithm for temperature dependent homogeneous gas phase reactions by combining forward and reverse reaction rates a significant gain in computational efficiency is achieved two modifications of modelling the temperature dependence with and without conservation of enthalpy are introduced and studied quantitatively the algorithm is tested for the combustion of n heptane which is a reference fuel component for internal combustion engines the convergence of the algorithm is studied by a series of numerical experiments and the computational cost of the stochastic algorithm is compared with the dae code dassl if less accuracy is needed the stochastic algorithm is faster on short simulation time intervals the new stochastic algorithm is significantly faster than the original direct simulation algorithm in all cases considered

an efficient stochastic algorithm for simulating nano particle dynamics nano particles produced at high temperatures often undergo rapid coalescence with complex associated rate laws in this paper we develop and study the numerical properties of a stochastic algorithm for the modelling of nano particle dynamics in the free molecular regime following the work in a eibeck and w wagner 2000 siam j sci comput 22 3 802 821 we model the system as a markov process and introduce a new majorant kernel that enables us to extend the use of fictitious jumps to a wider class of problems we also include a source term we study the convergence properties of the algorithm the systematic error decreases as 1 n we then examine the efficiency of the algorithm by comparing it to a direct simulation monte carlo dsmc algorithm that described by d t gillespie 1975 j atmos sci 32 10 1977 1989 we also compare the efficiency of our new majorant with the linear majorant used in a eibeck and w wagner 2001 ann appl prob 11 4 1137 1165 the improved stochastic algorithm compares very favourably with the dsmc algorithm the cpu time required for simulation is orders of magnitude lower and for low particle numbers the cpu time increases linearly with particle number rather than as the square of the particle number as with the dsmc algorithm our majorant kernel enables us to simulate solutions to a wider variety of problems than the linear majorant and also gives a significant gain in efficiency the results of this report promise excellent efficiency of simulation of problems such as soot formation and synthesis of fumed silica and also for extension to a more general class of problems in which the population balance equation occurs

optimal sparse volatility matrix estimation for high dimensional ito processes with measurement errors stochastic processes are often used to model complex scientiflc problems in flelds ranging from biology and flnance to engineering and physical science this paper investigates rate optimal estimation of the volatility matrix of a high dimensional it o process observed with measurement errors at discrete time points the minimax rate of convergence is established for estimating sparse volatility matrices by combining the multi scale and threshold approaches we construct a volatility matrix estimator to achieve the optimal convergence rate the minimax lower bound is derived by considering a subclass of it o processes for which the minimax lower bound is obtained through a novel equivalent model of covariance matrix estimation for independent but non identically distributed observations and through a delicate construction of the least favorable parameters in addition a simulation study was conducted to test the flnite sample performance of the optimal estimator and the simulation results were found to support the established asymptotic theory

numerical study of a stochastic particle method for homogeneous gas phase reactions abstract in this paper we study a stochastic particle system that describes homogeneous gas phase reactions of a number of chemical species first we introduce the system as a markov jump process and discuss how relevant physical quantities are represented in terms of appropriate random variables then we show how various deterministic equations used in the literature are derived from the stochastic system in the limit when the number of particles goes to infinity finally we apply the corresponding stochastic algorithm to a toy problem a simple formal reaction mechanism and a real combustion problem this problem is given by the isothermal combustion of a homogeneous mixture of heptane and air modelled by a detailed reaction mechanism with 107 chemical species and 808 reversible reactions heptane as described in this chemical mechanism serves as model fuel for different types of internal combustion engines in particular we study the order of convergence with respect to the number of simulation particles and illustrate the limitations of the method

triple positive solutions for nonlinear boundary value problems in banach space in this paper by applying two pairs of lower and upper solutions method and the topological degree theory of strict set contractions the existence of at least three positive solutions to m point boundary value problems for second order ordinary differential equations in banach spaces is obtained

on an m point boundary value problem for second order ordinary differential equations where q e 0 1 is given we obtain conditions for the existence and uniqueness of a solution for the boundary value problem 2 using the leray schauder continuation theorem 2 we give an example of a three point boundary value problem where the existence condition is not satisfied and no solution exists gupta 3 recently studied the boundary value problem 2 when q 1 our results on the three point boundary value problem 2 extend the results of gupta 3 to the case of general cy see also 4 51 we use the classical spaces c o 11 c o 11 lk o 11 and l o l of continuous k times continuously differentiable measurable real valued functions whose kth power of the absolute value is lebesgue integrable on 0 11 or measurable functions that are essentially bounded

the iterative solutions of 2nth order nonlinear multi point boundary value problems abstract the aim of this paper is to investigate the existence of iterative solutions for a class of 2 n th order nonlinear multi point boundary value problems the multi point boundary condition under consideration includes various commonly discussed boundary conditions such as three or four point boundary condition n 2 point boundary condition and 2 n m point boundary condition the existence problem is based on the method of upper and lower solutions and its associated monotone iterative technique a monotone iteration is developed so that the iterative sequence converges monotonically to a maximal solution or a minimal solution depending on whether the initial iteration is an upper solution or a lower solution two examples are presented to illustrate the results

on 2nth order lidstone boundary value problems this paper is concerned with the solutions of a class of 2nth order lidstone boundary value problems sufficient conditions for the existence and uniqueness of a solution are given a monotone iteration is developed so that the iteration sequence converges monotonically to a maximal or a minimal solution the approach to the problem is by the method of upper and lower solutions with a new maximum principle

triple solutions for second order three point boundary value problems abstract we establish the existence of at least three positive solutions to the second order three point boundary value problem u f t u 0 u 0 0 u u 1 where 0 lt f 0 1 0 0 is continuous we accomplish this by making growth assumptions on f which can apply to many more cases than the sublinear and superlinear ones discussed in recent works

asymptotic equivalence for inference on the volatility from noisy observations we consider discrete time observations of a continuous martingale under measurement error this serves as a fundamental model for high frequency data in finance where an efficient price process is observed under microstructure noise it is shown that this nonparametric model is in le cam s sense asymptotically equivalent to a gaussian shift experiment in terms of the square root of the volatility function and a nonstandard noise level as an application new rate optimal estimators of the volatility function and simple efficient estimators of the integrated volatility are constructed

uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations abstract for the nth order differential equation y n f x y y y n 1 we consider uniqueness implies existence results for solutions satisfying certain nonlocal k 2 point boundary conditions 1 k n 1 uniqueness of solutions when k n 1 is intimately related to uniqueness of solutions when 1 k n 2 these relationships are investigated as well

positive solution for m point boundary value problems of fourth order in this paper we consider the existence of positive solutions to the fourth order boundary value problem u u u f t u 0 t 1 u 0 i 1m 2aiu i u 1 i 1m 2biu i u 0 i 1m 2aiu i u 1 i 1m 2biu i where r i 0 1 ai bi 0 for i 1 2 m 2 are given constants satisfying some suitable conditions the proofs are based on the fixed point index theorem in cones

general lidstone problems multiplicity and symmetry of solutions abstract for the 2mth order lidstone boundary value problem y 2 m t f y t y t y 2 i t y 2 m 1 t t 0 1 y 2 i 0 y 2 i 1 0 0 i m 1 where 1 mf r m 0 thinsp is continuous growth conditions are imposed on f which yield the existence of at least three symmetric positive solutions this generalizes earlier papers which have applied avery s generalization of the leggett williams theorem to lidstone problems we then prove the analogous result for difference equations

a three point boundary value problem for nonlinear fourth order differential equations abstract in this paper the authors consider the nonlinear fourth order ordinary differential equation e u t g t f u 0 with the boundary conditions b u 0 u 1 u 0 u p u 1 0 some results on the existence and nonexistence of positive solutions to problem e b are obtained results on the existence of infinitely many positive solutions are also presented examples are included to demonstrate that the results are sharp

upper and lower solution method for fourth order four point boundary value problems this paper is concerned with the fourth order ordinary differential equation u 4 t f t u t u t 0 t 1 with the four point boundary conditions u 0 u 1 0 au 1 bu 1 0

on positive solutions of some nonlinear fourth order beam equations abstract the existence uniqueness and multiplicity of positive solutions of the following boundary value problem is considered u 4 t f t u t 0 for 0 where 0 is a constant f 0 1 0 0 is continuous

existence of multiple solutions for second order boundary value problems we give conditions on f involving pairs of lower and upper solutions which lead to the existence of at least three solutions of the two point boundary value problem y f x y y 0 x epsilon 0 1 y 0 0 y 1 in the special case f x y y f y greater than or equal to 0 we give growth conditions on f and apply our general result to show the existence of three positive solutions we give an example showing this latter result is sharp our results extend those of avery and of lakshmikantham et al

a class of 2nth order singular boundary value problems abstract this paper investigates the existence of positive solutions for 2 n th order n 1 singular sub linear boundary value problems first of all we establish the maximal principle and some important lemmas then we define a partial ordering in c 2 n 2 0 1 c 2 n 0 1 and construct lower and upper solutions to give 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 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

the method of lower and upper solutions for second third fourth and higher order boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u n t t u t u i a u i b i r i 0 n 1 where is a caratheodory function we obtain necessary and sufficient conditions in to guarantee the existence of solutions between a lower solution and an upper solution for n 2 if n 3 either or and n 4 if furthermore we obtain sufficient conditions in for n 2 k 6 when