rotation numbers eigenvalues and the poincare birkhoff theorem using the relation between rotation numbers and eigenvalues we prove the existence of nontrivial t periodic solutions having precise oscillatory properties for a class of asymptotically linear second order differential equations

geometrical aspects of stability theory for hill s equations where a and b are real parameters apparently the first stability diagram was drawn in the classical paper by b van der pol m j o sti utt 15 since then many papers and textbooks have appeared on the subject we mention j j stoker 13 j meixner f w sch fke 12 d m levy 8 j b keller 11 m i weinstein j b keller 16 17 j poschel e trubowltz 9 and v i arnold 3 5 6 this literature contains extensive estimates on the order of tangency of resonance tongues and the behavior of stability boundaries at infinity a full understanding of the stability diagrams is still lacking for instance the appearance of instability pockets in some cases see figure 1 cf 15 calls for a geometric explanation it should be noted that this phenomenon does not occur in the classical mathieu case cf 12 for a preliminary study near resonances using singularity theory we refer to afsharnejad 1 related bifurcational aspects of nonlinear perturbations of mathieu s equation near resonances were studied by broer vegter 7 the global geometry of the symplectic group was used for a stability proof of a hill equation in levi 10 we introduce the main object of study of this paper

continuity and minimization of spectrum related with the periodic camassa holm equation abstract an important point in looking for period solutions of the camassa holm equation is to understand the associated spectral problem y 1 4 y m t y the first aim of this paper is to study the dependence of eigenvalues for the periodic camassa holm equation on potentials as an infinitely dimensional parameter to be precise we prove that as nonlinear functionals of potentials eigenvalues for the periodic camassa holm equation are continuous in potentials with respect to the weak topologies in the l p lebesgue spaces the second aim of this paper is to find the optimal lower bound of the lowest eigenvalue for the periodic camassa holm equation when the l 1 norm of potentials are given in order to make our results more applicable we will find the optimal lower bound for the lowest eigenvalue in the more general setting of measure differential equations

rotation numbers for random dynamical systems on the circle in this paper we study rotation numbers of random dynamical systems on the circle we prove the existence of rotation numbers and the continuous dependence of rotation numbers on the systems as an application we prove a theorem on analytic conjugacy to a circle rotation

unconditionally convergent l1 galerkin fems for nonlinear time fractional schrodinger equations in this paper a linearized l1 galerkin finite element method is proposed to solve the multidimensional nonlinear time fractional schrodinger equation in terms of a temporal spatial error splitting argument we prove that the finite element approximations in the l 2 norm and l infty norm are bounded without any time step size conditions more importantly by using a discrete fractional gronwall type inequality optimal error estimates of the numerical schemes are obtained unconditionally while the classical analysis for multidimensional nonlinear fractional problems always required certain time step restrictions dependent on the spatial mesh size numerical examples are given to illustrate our theoretical results

the identifiability of the mixed proportional hazards competing risks model we prove identification of dependent competing risks models in which each risk has a mixed proportional hazard specification with regressors and the risks are dependent by way of the unobserved heterogeneity or frailty components we show that the conditions for identification given by heckman and honore can be relaxed we extend the results to the case in which multiple spells are observed for each subject

finite difference methods and a fourier analysis for the fractional reaction subdiffusion equation various fields of science and engineering deal with dynamical systems that can be described by fractional partial differential equations fpde for example systems biology chemistry and biochemistry applications due to anomalous diffusion effects in constrained environments however effective numerical methods and numerical analysis for fpde are still in their infancy in this paper we consider a fractional reaction subdiffusion equation fr subde in which both the motion and the reaction terms are affected by the subdiffusive character of the process using the relationship between the riemann liouville and grunwald letnikov definitions of fractional derivatives an implicit and an explicit difference methods for the fr subde are presented the stability and the convergence of the two numerical methods are investigated by a fourier analysis the solvability of the implicit finite difference method is also proved the high accuracy algorithm is structured using richardson extrapolation finally a comparison between the exact solution and the two numerical solutions is given the numerical results are in excellent agreement with our theoretical analysis

extremal values of smallest eigenvalues of hill s operators with potentials in l 1 balls given a 1 periodic real potential q l1 r z we use 0 q to denote the smallest 1 periodic eigenvalue of the hill s equation x q t x 0 let b1 r be the ball centered at 0 of radius r in the l1 space l1 r z it is trivial that sup 0 q q b1 r r for all r 0 based on continuity of 0 q in q with the weak topology and continuous differentiability of 0 q in q with the l1 norm 1 we will apply scaling technique variational approach to extremal values in lp balls singular integrals and the limiting approach as p 1 to obtain i 0 q is bounded for q in any bounded set of l1 r z 1 and ii the minimal value l1 r inf 0 q q l1 r z q 1 r inf 0 q q l1 r z q 1 r is simply z0 1 r where z0 x 2 xtanh x 2 x 0 the extremal values of the smallest neumann eigenvalues for potentials in l1 balls are also found explicitly

analysing competing risks data with transformation models we present a flexible class of marginal models for the cumulative incidence function the semiparametric transformation model is utilized in a decomposition for the marginal failure probabilities which extends previous work on farewell s cure model novel estimation inference and prediction procedures are developed with large sample properties derived from the theory of martingales and u statistics a small simulation study demonstrates that the methods are appropriate for practical use the methods are illustrated with a thorough analysis of a prostate cancer clinical trial simple graphical displays are used to check for the goodness of fit

a generalized gronwall inequality and its application to a fractional differential equation this paper presents a generalized gronwall inequality with singularity using the inequality we study the dependence of the solution on the order and the initial condition of a fractional differential equation

multiple positive solutions for nonlinear periodic problems abstract we consider a nonlinear periodic equation driven by the scalar p laplacian and with a caratheodory asymptotically p 1 linear nonlinearity using variational methods coupled with suitable truncation techniques we show that the problem has at least two positive solutions for the semilinear case p 2 using morse theory we show that the problem has three distinct positive solutions

continuous dependence and estimates of eigenvalues for periodic generalized camassa holm equations abstract in this paper we are concerned with the spectral problem y 1 y 2 x 1 2 1 2 m 1 2 m 1 2 y 1 y 2 for the periodic generalized camassa holm equations the first aim is to study the structure of eigenvalues and prove the continuous dependence of eigenvalues on potentials we prove that as nonlinear functionals of potentials eigenvalues are continuous in potentials with respect to the weak topologies in the lebesgue space l 1 0 t the second aim is to find the estimates of the eigenvalues when the l 1 norm of potentials is given based on an application of trace formulas

time stepping discontinuous galerkin methods for fractional diffusion problems time stepping hp hp versions discontinuous galerkin dg methods for the numerical solution of fractional subdiffusion problems of order alpha with 1 alpha 0 1 0 will be proposed and analyzed generic hp hp version error estimates are derived after proving the stability of the approximate solution for h h version dg approximations on appropriate graded meshes near t 0 t 0 we prove that the error is of order o k max 2 p frac alpha 2 o kmax 2 p 2 where k k is the maximum time step size and p ge 1 p 1 is the uniform degree of the dg solution for hp hp version dg approximations by employing geometrically refined time steps and linearly increasing approximation orders exponential rates of convergence in the number of temporal degrees of freedom are shown finally some numerical tests are given

a compact finite difference scheme for the fractional sub diffusion equations in this paper a compact finite difference scheme for the fractional sub diffusion equations is derived after a transformation of the original problem the l1 discretization is applied for the time fractional part and fourth order accuracy compact approximation for the second order space derivative the unique solvability of the difference solution is discussed the stability and convergence of the finite difference scheme in maximum norm are proved using the energy method where a new inner product is introduced for the theoretical analysis the technique is quite novel and different from previous analytical methods finally a numerical example is provided to show the effectiveness and accuracy of the method

sharp error estimate of the nonuniform l1 formula for linear reaction subdiffusion equations stability and convergence of the l1 formula on nonuniform time grids are studied for solving linear reaction subdiffusion equations with the caputo derivative a discrete fractional gronwall inequality is developed for the nonuniform l1 formula by introducing a discrete convolution kernel of riemann liouville fractional integral to simplify the consistency analysis of the nonuniform l1 formula we bound the local truncation error in a discrete convolution form and consider a global convolution error involving the discrete riemann liouville integral kernel with the help of discrete fractional gronwall inequality and global consistency error analysis a sharp error estimate reflecting the regularity of solution is obtained for a simple l1 scheme numerical examples are provided to verify the sharpness of the error analysis

partial identification and inference in censored quantile regression abstract in this paper we study partial identification and inference in a linear quantile regression where the dependent variable is subject to possibly unknown dependent censoring characterized by an archimedean copula an outer set of the identified set for the regression coefficient is characterized via inequality constraints for one parameter ordered families of archimedean copulas we construct a simple confidence set by inverting an asymptotically pivotal statistic a bootstrap confidence set is also constructed sensitivity of the identified set to possible misspecification of the true copula and the finite sample performance of the boostrap confidence set are investigated numerically

convergence and stability of compact finite difference method for nonlinear time fractional reaction diffusion equations with delay abstract this paper is concerned with numerical solutions of nonlinear time fractional reaction diffusion equations with time delay a linearized compact finite difference scheme is proposed to solve the equations in terms of a new developed fractional gronwall type inequality convergence and stability of the proposed scheme are obtained numerical experiments are given to illustrate the theoretical results

a finite difference scheme for semilinear space fractional diffusion equations with time delay a linearized quasi compact finite difference scheme is proposed for semilinear space fractional diffusion equations with a fixed time delay the nonlinear source term is discretized and linearized by taylor s expansion to obtain a second order discretization in time the space fractional derivatives are approximated by a weighted shifted grunwald letnikov formula which is of fourth order approximation under some smoothness assumptions of the exact solution under the local lipschitz conditions the solvability and convergence of the scheme are proved in the discrete maximum norm by the energy method numerical examples verify the theoretical predictions and illustrate the validity of the proposed scheme

the accuracy and stability of an implicit solution method for the fractional diffusion equation we have investigated the accuracy and stability of an implicit numerical scheme for solving the fractional diffusion equation this model equation governs the evolution for the probability density function that describes anomalously diffusing particles anomalous diffusion is ubiquitous in physical and biological systems where trapping and binding of particles can occur the implicit numerical scheme that we have investigated is based on finite difference approximations and is straightforward to implement the accuracy of the scheme is o dx 2 in the spatial grid size and o dt 1 c in the fractional time step where 0 email protected 1 is the order of the fractional derivative and c 1 is standard diffusion we have provided algebraic and numerical evidence that the scheme is unconditionally stable for 0 c 1

dependence of solutions and eigenvalues of measure differential equations on measures abstract it is well known that solutions of ordinary differential equations are continuously dependent on finite dimensional parameters in equations in this paper we study the dependence of solutions and eigenvalues of second order linear measure differential equations on measures as an infinitely dimensional parameter we will provide two fundamental results which are the continuity and continuous frechet differentiability in measures when the weak topology and the norm topology of total variations for measures are considered respectively in some sense the continuity result obtained in this paper is the strongest one as an application we will give a natural simple explanation to extremal problems of eigenvalues of sturm liouville operators with integrable potentials

extremal values of eigenvalues of sturm liouville operators with potentials in l 1 balls abstract this paper is a continuation of zhang m zhang continuity in weak topology higher order linear systems of ode sci china ser a 51 2008 1036 1058 m zhang extremal values of smallest eigenvalues of hill s operators with potentials in l 1 balls j differential equations 246 2009 4188 4220 given a potential q l p 0 1 r p 1 we use m q to denote the m th dirichlet eigenvalue of the sturm liouville operator with potential q t where m n the minimal value l m p r and the maximal value m m p r of m q with potentials q in the l p ball of radius r are well defined in this paper we will exploit the continuity of m q in q with weak topologies and the variational method to give characterizations of l m p r and m m p r when p 1 by using the limiting approach as p 1 we find that the most important extremal values l m 1 r and m m 1 r can be evaluated explicitly using some elementary functions of r the corresponding extremal problems for neumann eigenvalues and some periodic eigenvalues will be reduced to l m p r and m m p r

two fully discrete schemes for fractional diffusion and diffusion wave equations with nonsmooth data we consider initial boundary value problems for the subdiffusion and diffusion wave equations involving a caputo fractional derivative in time we develop two fully discrete schemes based on the piecewise linear galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward euler method second order backward difference method and establish error estimates optimal with respect to the regularity of problem data these two schemes are first and second order accurate in time for both smooth and nonsmooth data extensive numerical experiments for two dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity

semiparametric estimation of a proportional hazard model with unobserved heterogeneity the proportional hazard model with unobserved heterogeneity gives the hazard function of a random variable conditional on covariates and a second random variable representing unobserved heterogeneity this paper shows how to estimate the baseline hazard function and the distribution of the unobserved heterogeneity nonparametrically the baseline hazard function and heterogeneity distribution are assumed to satisfy smoothness conditions but are not assumed to belong to known finite dimensional parametric families existing estimators assume that the baseline hazard function or heterogeneity distribution belongs to a known parametric family thus the estimators presented here are more general than existing ones

estimation of semiparametric models when the criterion function is not smooth we provide easy to verify sufficient conditions for the consistency and asymptotic normality of a class of semiparametric optimization estimators where the criterion function does not obey standard smoothness conditions and simultaneously depends on some nonparametric estimators that can themselves depend on the parameters to be estimated our results extend existing theories such as those of pakes and pollard 1989 andrews 1994a and newey 1994 we also show that bootstrap provides asymptotically correct confidence regions for the finite dimensional parameters we apply our results to two examples a hit rate and a partially linear median regression with some endogenous regressors

a stable numerical method for multidimensional time fractional schrodinger equations in this paper the stability analysis is presented for a first order difference scheme applied to a nonhomogeneous time fractional schrodinger differential equation based on the z transform method stability theorems are proved for the abstract case the stability results are applied on initial boundary value problems for multidimensional time fractional schrodinger differential equations theoretical findings are validated by numerical experiments on one and two dimensional time fractional schrodinger differential equations

multiple periodic solutions of a class of p laplacian by using the recent generalization of coincidence degree method the existence of multiple periodic solutions for a class of p laplacian is obtained under the existence of strict upper and lower solutions

variational and non variational eigenvalues of the p laplacian it is well known that all the eigenvalues of the linear eigenvalue problem u q r u in rn can under appropriate conditions on q r and be characterized by minimax principles but it has been a long standing question whether that remains true for analogous equations involving the p laplacian p it will be shown that there are corresponding nonlinear eigenvalue problems pu q r u p 1sgnu in rn with 1 0 on for which not all eigenvalues are of variational type as far as we know this is the first observation of such a phenomenon and examples will be given for one and higher dimensional equations the question of exactly which eigenvalues are variational is also discussed when n 1

identification of a competing risks model with unknown transformations of latent failure times this paper is concerned with identification of a competing risks model with unknown transformations of latent failure times the model includes as special cases competing risks versions of proportional hazards mixed proportional hazards and accelerated failure time models it is shown that covariate effects on latent failure times cause specific link functions and the joint survivor function of the disturbance terms can be identified without relying on modelling the dependence between latent failure times parametrically nor using an exclusion restriction among covariates as a result the paper provides an identification result about the joint survivor function of the latent failure times conditional on covariates

error estimates for a semidiscrete finite element method for fractional order parabolic equations we consider the initial boundary value problem for a homogeneous time fractional diffusion equation with an initial condition v x and a homogeneous dirichlet boundary condition in a bounded convex polygonal domain omega we study two semidiscrete approximation schemes i e the galerkin finite element method fem and lumped mass galerkin fem using piecewise linear functions we establish almost optimal with respect to the data regularity error estimates including the cases of smooth and nonsmooth initial data i e v in h 2 omega cap h 1 0 omega and v in l 2 omega for the lumped mass method the optimal l 2 norm error estimate is valid only under an additional assumption on the mesh which in two dimensions is known to be satisfied for symmetric meshes finally we present some numerical results that give insight into the reliability of the theoretical study

one dimensional p laplacian with a strong singular indefinite weight i eigenvalue abstract in this paper we prove the existence of eigenvalues for the problem p u t h t p u t 0 a e in 0 1 u 0 u 1 0 where p s s p 2 s p 1 is a real parameter and the indefinite weight h is a nonnegative measurable function on 0 1 which may be singular at 0 and or 1 and h 0 on any compact subinterval in 0 1 we derive similar properties of eigenvalues as known in linear case p 2 or continuous case h c 0 1 if h satisfies 0 1 t p 1 1 t p 1 h t d t when 1 p 2 and 0 1 2 p 1 s 1 2 h d d s 1 2 1 p 1 1 2 s h d d s when p 2 respectively for the result we establish the c 1 regularity of all solutions at the boundary for the above problem as well as the following problem p u t h t f u t 0 a e in 0 1 u 0 u 1 0 where f c r r s f s 0 for s 0 f is odd and f s p s is bounded above as s 0

error analysis of a finite difference method on graded meshes for a time fractional diffusion equation a reaction diffusion problem with a caputo time derivative of order alpha in 0 1 is considered the solution of such a problem is shown in general to have a weak singularity near the initial time t 0 and sharp pointwise bounds on certain derivatives of this solution are derived a new analysis of a standard finite difference method for the problem is given taking into account this initial singularity this analysis encompasses both uniform meshes and meshes that are graded in time and includes new stability and consistency bounds the final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme so one can choose an optimal mesh grading numerical results are presented that confirm the sharpness of the error analysis

an explicit finite difference method and a new von neumann type stability analysis for fractional diffusion equations a numerical method for solving the fractional diffusion equation which could also be easily extended to other fractional partial differential equations is considered in this paper we combine the forward time centered space ftcs method well known for the numerical integration of ordinary diffusion equations with the grunwald letnikov discretization of the riemann liouville derivative to obtain an explicit ftcs scheme for solving the fractional diffusion equation the stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well known von neumann method for nonfractional partial differential equations the analytical stability bounds are in excellent agreement with numerical test a comparison between exact analytical solutions and numerical predictions is made

multiple nontrivial solutions for nonlinear periodic problems with the p laplacian abstract we consider a nonlinear periodic problem driven by the scalar p laplacian with a nonsmooth potential hemivariational inequality using the degree theory for multivalued perturbations of s operators and the spectrum of a class of weighted eigenvalue problems for the scalar p laplacian we prove the existence of at least three distinct nontrivial solutions two of which have constant sign

optimal lower bound for the first eigenvalue of the fourth order equation abstract in this paper we will find optimal lower bound for the first eigenvalue of the fourth order equation with integrable potentials when the l 1 norm of potentials is known we establish the minimization characterization for the first eigenvalue of the measure differential equation which plays an important role in the extremal problem of ordinary differential equation the conclusion of this paper will illustrate a new and very interesting phenomenon that the minimizing measures will no longer be located at the center of the interval when the norm is large enough

periodic solutions for a rayleigh type p laplacian equation with sign variable coefficient of nonlinear term as p laplacian equations have been widely applied in the field of fluid mechanics and nonlinear elastic mechanics it is necessary to investigate the periodic solutions of functional differential equations involving the scalar p laplacian by using lu s continuation theorem which is an extension of manasevich mawhin we study the existence of periodic solutions for a rayleigh type p laplacian equation f p x t f x t g 1 x email protected 1 t x email protected t g 2 x email protected 2 t x e t it is significant that the growth degree with respect to the variable u in g 1 u is allowed to be greater than p 1 and the coefficient b t of g 2 x email protected 2 t x can change sign in this paper which could be achieved rarely in the previous literature

predicting survival probabilities with semiparametric transformation models abstract prediction of survival probabilities for future patients is one of the main goals of fitting survival data with regression models in this article we consider a large class of semiparametric transformation models which includes the well known proportional hazards and proportional odds models for the analysis of failure time data specifically we propose pointwise and simultaneous confidence interval procedures for the survival probability of future patients with specific covariates these procedures can be easily implemented through simulation and are illustrated with the data from two well known clinical studies

fixed point theory and nonlinear problems introduction among the most original and far reaching of the contributions made by henri poincare to mathematics was his introduction of the use of topological or qualitative methods in the study of nonlinear problems in analysis his starting point was the study of the differential equations of celestial mechanics and in particular of their periodic solutions his work on this topic began with his thesis in 1879 and was developed in detail in his great three volume work methodes nouvelles de la mechanique celeste which appeared in the early 1890s and summarized his many memoirs of the intervening period it continued until his memoir shortly before his death in 1912 in which he put forward the unproved fixed point result usually referred to as poincare s last geometric theorem the ideas introduced by poincare include the use of fixed point theorems the continuation method and the general concept of global analysis the writer s acquaintance with poincare s influence came through contact with solomon lefschetz and marston morse both of whom were very explicit as to the role of poincare as an initiator in this direction of mathematical development in 1934 in the foreword to his colloquium volume on the calculus of variations in the large morse had put this forward very forcefully in the first paragraph

maximal inequalities for degenerate u processes with applications to optimization estimators maximal inequalities for degenerate u processes of order k k geq 1 are established the results rest on a moment inequality due to bonami for k th order forms and on extensions of chaining and symmetrization inequalities from the theory of empirical processes rates of uniform convergence are obtained the maximal inequalities can be used to determine the limiting distribution of estimators that optimize criterion functions having u process structure as an application a semiparametric regression estimator that maximizes a u process of order 3 is shown to be sqrt n consistent and asymptotically normally distributed

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

the identifiability of the competing risks model summary this paper considers the consequences for identifiability of introducing regressors into the competing risks model of multistate duration analysis we establish conditions under which access to regressors overturns the nonidentification theorem of cox and tsiatis for both proportional and accelerated failure time models

the spectrum of the periodic p laplacian we consider one dimensional p laplacian eigenvalue problems of the form pu q u p 1sgnu on 0 b together with periodic or separated boundary conditions where p 1 p is the p laplacian q c1 0 b and b 0 r it will be shown that when p 2 the structure of the spectrum in the general periodic case that is with q 0 and periodic boundary conditions can be completely different from those of the following known cases i the general periodic case with p 2 ii the periodic case with p 2 and q 0 and iii the general separated case with any p 1

on a class of non linear delay distributed order fractional diffusion equations in this paper we consider a numerical scheme for a class of non linear time delay fractional diffusion equations with distributed order in time this study covers the unique solvability convergence and stability of the resulted numerical solution by means of the discrete energy method the derivation of a linearized difference scheme with convergence order o 4 h 4 in l norm is the main purpose of this study numerical experiments are carried out to support the obtained theoretical results

periodic solutions for second order differential inclusions with the scalar p laplacian we study periodic problems driven by the scalar p laplacian with a multivalued right hand side nonlinearity we prove two existence theorems in the first we assume nonuniform nonresonance conditions between two successive eigenvalues of the negative p laplacian with periodic boundary conditions in the second we employ certain landesman lazer type conditions our approach is based on degree theory

u processes rates of convergence on introduit un nouveau processus stochastique une collection de statistiques u indicees par une famille de noyaux symetriques on obtient des conditions pour la convergence presque sure uniforme d une suite de tels processus

positive solutions for nonlinear periodic problems with concave terms abstract we consider a nonlinear periodic problem driven by the scalar p laplacian with a parametric concave term and a caratheodory perturbation whose potential primitive exhibits a p superlinear growth near without satisfying the usual in such cases ambrosetti rabinowitz condition using critical point theory and truncation techniques we prove a bifurcation type theorem describing the nonexistence existence and multiplicity of positive solutions as the parameter varies

generalizations of browder s degree theory the starting point of this paper is the recent important work of f e browder who extended degree theory to operators of monotone type the degree function of browder is generalized to maps of the form t f g where t is maximal monotone f is of class s bounded and g is an u s c compact multifunction it is also generalized to maps of the form f ng with f of class s and ng the nemitsky operator of a multifunction g x r satisfying various types of sign conditions some examples are also included to illustrate the abstract results

simulation and the asymptotics of optimization estimators on demontre un theoreme de limite centrale general pour des estimateurs definis par minimisation de la longueur d une fonction de critere aleatoire a valeurs vectorielles aucune hypothese de regularite suffisante n est imposee sur la fonction de critere pour que les resultats puissent etre appliques a une classe d estimlateurs de simulation assez large des analyses completes de deux estimateurs de simulation l un introduit par pakes l autre par mcfadden illustrent l application des theoremes generaux

the implicit midpoint method for the modified anomalous sub diffusion equation with a nonlinear source term in this paper the implicit midpoint method is used to solve the semi discrete modified anomalous sub diffusion equation with a nonlinear source term and the weighted and shifted grunwald letnikov difference operator and the compact difference operator are applied to approximate the riemann liouville fractional derivative and space partial derivative respectively then the new numerical scheme is constructed the stability and the convergence of this method are analyzed numerical experiment demonstrates the high accuracy of this method and confirm our theoretical results

continuity in weak topology and extremal problems of eigenvalues of the p laplacian we will study the dependence of eigenvalues of the one dimensional p laplacian on potentials or weights two results are obtained one is the continuity of eigenvalues in potentials with respect to the weak topologies of l spaces 1 and the other is the continuous differentiability of eigenvalues in potentials with respect to l norms as applications we will study some extremal problems of eigenvalues by developing some analytical methods

nonuniform nonresonance of semilinear differential equations abstract consider the dirichlet problem of nonlinear differential equations with the principal part the p laplacian when the nonlinearity satisfies some semilinearity conditions the usual nonuniform nonresonance conditions are obtained by comparing nonlinear equations with the classical eigenvalues in this article we will introduce some weighted eigenvalues the nonuniform nonresonance conditions proved in this article using weighted eigenvalues will improve the usual ones significantly

existence and uniqueness results for some nonlinear boundary value problems u a s u b s0 dirichlet 2 u9 a s u9 b s 0 neumann 3 u a s u b and u9 a s u9 b periodic 4 w x here f r a r is an odd increasing homeomorphism k a b a r and w x 2 py2 f a b r a r equation 1 with f s s s s where p 1 and k t s n y 1 rt arises in the study of radial solutions for the p laplacian equation on an annular domain in n dimensions py2 n div u u q f x u u s 0 a x b x g r 5 w x and has been studied extensively see for instance 1 4 6 in this paper we shall establish existence results for 1 with each of the above boundary conditions 2 3 and 4 under various growth conditions on f in particular our results give the existence of radial solutions

semiparametric estimation of a panel data proportional hazards model with fixed effects this paper considers a panel duration model that has a proportional hazards specification with fixed effects the paper shows how to estimate the baseline and integrated baseline hazard functions without assuming that they belong to known finite dimensional families of functions existing estimators assume that the baseline hazard function belongs to a known parametric family therefore the estimators presented here are more general than existing ones this paper also presents a method for estimating the parametric part of the proportional hazards model with dependent right censoring under which the partial likelihood estimator is inconsistent the paper presents some monte carlo evidence on the small sample performance of the new estimators

numerical analysis of nonlinear subdiffusion equations we present a general framework for the rigorous numerical analysis of time fractional nonlinear parabolic partial differential equations with a fractional derivative of order alpha in 0 1 in time it relies on three technical tools a fractional version of the discrete gronwall type inequality discrete maximal regularity and regularity theory of nonlinear equations we establish a general criterion for showing the fractional discrete gronwall inequality and verify it for the l1 scheme and convolution quadrature generated by backward difference formulas further we provide a complete solution theory e g existence uniqueness and regularity for a time fractional diffusion equation with a lipschitz nonlinear source term together with the known results of discrete maximal regularity we derive pointwise l 2 omega norm error estimates for semidiscrete galerkin finite element solutions and fully discrete solutions which are of order o h 2 up to a logarithmic factor and o tau alpha

lyapunov inequalities and stability for linear hamiltonian systems abstract in this paper we will establish several lyapunov inequalities for linear hamiltonian systems which unite and generalize the most known ones for planar linear hamiltonian systems the connection between lyapunov inequalities and estimates of eigenvalues of stationary dirac operators will be given and some optimal stability criterion will be proved

analysis of transformation models with censored data summary in this paper we consider a class of semi parametric transformation models under which an unknown transformation of the survival time is linearly related to the covariates with various completely specified error distributions this class of regression models includes the proportional hazards and proportional odds models inference procedures derived from a class of generalised estimating equations are proposed to examine the covariate effects with censored observations numerical studies are conducted to investigate the properties of our proposals for practical sample sizes these transformation models coupled with the new simple inference procedures provide many useful alternatives to the cox regression model in survival analysis

a high order schema for the numerical solution of the fractional ordinary differential equations in this paper we present a general technique to construct high order schemes for the numerical solution of the fractional ordinary differential equations fodes this technique is based on the so called block by block approach which is a common method for the integral equations in our approach the classical block by block approach is improved in order to avoiding the coupling of the unknown solutions at each block step with an exception in the first two steps while preserving the good stability property of the block by block schemes by using this new approach we are able to construct a high order schema for fodes of the order a a 0 the stability and convergence of the schema is rigorously established we prove that the numerical solution converges to the exact solution with order 3 a for 0 1 a series of numerical examples are provided to support the theoretical claims

uniform convergence of compact and bdf methods for the space fractional semilinear delay reaction diffusion equations abstract in this article two classes of finite difference methods are constructed to solve the space fractional semilinear delay reaction diffusion equations firstly fractional centered finite difference method in space and backward differential formula bdf2 in time are employed to discrete the original equations secondly the existence uniqueness and convergence of the numerical method are analyzed at length by g norm technique and the convergence order is o 2 h 2 then the compact technique of finite difference method is used to further increase the accuracy in spatial direction and a method with fourth accuracy in spatial dimension is obtained the convergence orders are proved to be o 2 h 4 in the sense of three classes of norms by ga norm technique especially we obtain the uniform convergence for the error estimation which demonstrates competitive performance compared with the preceding work in the related references finally two numerical examples are provided to verify our theoretical results and demonstrate the effectiveness of both methods when applied to simulate space fractional delay diffusive nicholson s blowflies equation

positivity and boundedness preserving schemes for space time fractional predator prey reaction diffusion model the semi implicit schemes for the nonlinear predator prey reaction diffusion model with the space time fractional derivatives are discussed where the space fractional derivative is discretized by the fractional centered difference and wsgd scheme the stability and convergence of the semi implicit schemes are analyzed in the l norm we theoretically prove that the numerical schemes are stable and convergent without the restriction on the ratio of space and time stepsizes and numerically further confirm that the schemes have first order convergence in time and second order convergence in space then we discuss the positivity and boundedness properties of the analytical solutions of the discussed model and show that the numerical solutions preserve the positivity and boundedness the presented numerical examples confirm the theoretical results and convergence orders

nonlinear resonant periodic problems with concave terms abstract we consider a nonlinear periodic problem driven by the scalar p laplacian with a concave term and a caratheodory perturbation we assume that this perturbation f t x is p 1 linear at and resonance can occur with respect to an eigenvalue m 1 m 2 of the negative periodic scalar p laplacian using a combination of variational techniques based on the critical point theory with morse theory we establish the existence of at least three nontrivial solutions useful in our considerations is an alternative minimax characterization of 1 0 the first nonzero eigenvalue that we prove in this work

nonparametric estimation of a generalized additive model with an unknown link function mean of a random scalar y conditional on a random vector x e rd let h x e yix x denote the conditional mean function this paper is concerned with estimating h under weak assumptions about its functional form when x is a multidimensional continuous random variable the paper describes a model that nests single index additive and multiplicative models and that can be estimated with a faster rate of convergence in probability than can a fully nonparametric model nonparametric estimation of h is usually unattractive when x is multidimensional the rate of convergence of nonparametric estimators decreases rapidly as the dimension of x increases and nonparametric estimates can be very imprecise with samples of practical size when x is multidimensional the curse of dimensionality this problem can be overcome through the use of dimension reduction methods they reduce the effective dimension of the estimation problem by making assumptions about the functional form of h that are stronger than those of a fully nonparametric estimator but weaker than those of a finite dimensional parametric model one method of dimension reduction is semiparametric single index modeling the best known and most easily estimated semiparametric single index model assumes that 1 1 h x g 3 x where 3 is an unknown d x 1 vector and g is an unknown function the vector 3 can be estimated with an n 1 2 rate of convergence in probability which is the rate that would be obtained if g were known han 1987 hardle and stoker 1989 horowitz and hardle 1996 ichimura 1993 klein and spady 1993 newey and stoker 1993 powell stock and stoker 1989 sherman 1993 stoker 1991 g can be estimated with the rate of convergence n r 2r 1 where r is an integer that depends on the smoothness of g but not on d dimension reduction can also be achieved by using a nonparametric generalized additive model in such a model