a short proof of zheludev s theorem we give a short proof of zheludev s theorem that states the existence of precisely one eigenvalue in sufficiently distant spectral gaps of a hill operator subject to certain short range perturbations as a by product we simultaneously recover rofe beketov s result about the finiteness of the number of eigenvalues in essential spectral gaps of the perturbed hill operator our methods are operator theoretic in nature and extend to other one dimensional systems such as perturbed periodic dirac operators and weakly perturbed second order finite difference operators we employ the trick of using a selfadjoint birman schwinger operator even in cases where the perturbation changes sign a method that has already been successfully applied in different contexts and appears to have further potential in the study of point spectra in essential spectral gaps

a general iterative method with strongly positive operators for general variational inequalities in this paper we introduce and study a general iterative method with strongly positive operators for finding solutions of a general variational inequality problem with inverse strongly monotone mapping in a real hilbert space the explicit and implicit iterative algorithms are proposed by virtue of the general iterative method with strongly positive operators under two sets of quite mild conditions we prove the strong convergence of these explicit and implicit iterative algorithms to the unique common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality problem respectively

strong convergence of a general iterative method for variational inequality problems and fixed point problems in hilbert spaces abstract in this paper we introduce a new iterative scheme to investigate the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of a variational inequality problem for a relaxed cocoercive mapping by viscosity approximate methods our results improve and extend the recent ones announced by chen et al j m chen l j zhang t g fan viscosity approximation methods for nonexpansive mappings and monotone mappings doi 10 1016 j jmaa 2006 12 088 iiduka and tahakshi h iiduka w takahashi strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings nonlinear anal 61 2005 341 350 yao and yao y h yao j c yao on modified iterative method for nonexpansive mappings and monotone mappings appl math comput doi 10 1016 j amc 2006 08 062 and many others

effective prufer angles and relative oscillation criteria abstract we present a streamlined approach to relative oscillation criteria based on effective prufer angles adapted to the use at the edges of the essential spectrum based on this we provided a new scale of oscillation criteria for general sturm liouville operators which answer the question whether a perturbation inserts a finite or an infinite number of eigenvalues into an essential spectral gap as a special case we recover and generalize the gesztesy unal criterion which works below the spectrum and contains classical criteria by kneser hartman hille and weber and the well known results by rofe beketov including the extensions by schmidt

viscosity approximation methods for nonexpansive mappings abstract viscosity approximation methods for nonexpansive mappings are studied consider a nonexpansive self mapping t of a closed convex subset c of a banach space x suppose that the set fix t of fixed points of t is nonempty for a contraction f on c and t 0 1 let x t c be the unique fixed point of the contraction x tf x 1 t tx consider also the iteration process x n where x 0 c is arbitrary and x n 1 n f x n 1 n tx n for n 1 where n 0 1 if x is either hilbert or uniformly smooth then it is shown that x t and under certain appropriate conditions on n x n converge strongly to a fixed point of t which solves some variational inequality

strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings abstract in this paper we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mapping in a hilbert space then we show that the sequence converges strongly to a common element of two sets using this result we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly monotone mapping

on the eigenvalues of operators with gaps application to dirac operators this paper is devoted to a general min max characterization of the eigenvalues in a gap of the essential spectrum of a self adjoint unbounded operator we prove an abstract theorem then we apply it to the case of dirac operators with a coulomb like potential the result is optimal for the coulomb potential

on approximation of the eigenvalues of perturbed periodic schrodinger operators this paper addresses the problem of computing the eigenvalues lying in the gaps of the essential spectrum of a periodic schrodinger operator perturbed by a fast decreasing potential we use a recently developed technique the so called quadratic projection method in order to achieve convergence free from spectral pollution we describe the theoretical foundations of the method in detail and illustrate its effectiveness by several examples

viscosity approximation methods for nonexpansive mappings and monotone mappings abstract viscosity approximation methods for nonexpansive mappings are studied consider the iteration process x n where x 0 c is arbitrary and x n 1 n f x n 1 n s p c x n n a x n f is a contraction on c s is a nonexpansive self mapping of a closed convex subset c of a hilbert space h it is shown that x n converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mapping which solves some variational inequality

eigenvalue excluding for perturbed periodic one dimensional schrodinger operators subject of investigation in this paper is a one dimensional schrodinger equation where the potential is a sum of a periodic function and a perturbation decaying at graphic 1 it is well known that the essential spectrum consists of spectral bands and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands while enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations e g by d weinstein s bounds there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps i e which identifies subregions of a gap which contain no eigenvalues here we propose such a method it makes heavy use of computer assistance nevertheless the results are completely rigorous in the strict mathematical sense because all computational errors are taken into account 1 embed inline graphic 1 gif

on the asymptotic distribution of the eigenvalue branches of the schrodinger operator h λw in a spectral gap of h on considere des operateurs de schrodinger h v dans l espace de hilbert l 2 r avec un potentiel v satisfaisant 1 vel r on suppose que l intervalle a b est contenu dans l ensemble resolvant de h on obtient des estimations asymptotiques sur le nombre des branches de valeurs propes dans a b des familles d operateurs h w 0 quand ou w est un potentiel a courte portee non negatif

iterative schemes for computing fixed points of nonexpansive mappings in banach spaces abstract let x be a real banach space with a normalized duality mapping uniformly norm to weak continuous on bounded sets or a reflexive banach space which admits a weakly continuous duality mapping j with gauge let f be an contraction and t n a sequence of nonexpansive mappings we study the strong convergence of explicit iterative schemes 1 x n 1 n f x n 1 n t n x n with a general theorem and then recover and improve some specific cases studied in the literature k aoyoma y kimura w takahashi m toyoda approximation of common fixed point of a countable family of nonexpansive mappings nonlinear anal 67 8 2007 2350 2360 g lopez v martin h k xu perturbation techniques for nonexpansive mappings with applications nonlinear anal real world appl in press available online 4 may 2008 h k xu viscosity approximation methods for nonexpansive mappings j math anal appl 298 1 2004 279 291 t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 1 2 2005 51 60 y song r chen strong convergence theorems on an iterative method for a family of finite nonexpansive mappings appl math comput 180 2006 275 287 y song r chen viscosity approximation methods for nonexpansive nonself mappings j math anal appl 321 1 2006 316 326 j chen l zhang t fan viscosity approximation methods for nonexpansive mappings and monotone mappings j math anal appl 334 2 2007 1450 1461 y kimura w takahashi m toyoda convergence to common fixed points of a finite family of nonexpansive mappings arch math 84 2005 350 363

critical coupling constants and eigenvalue asymptotics of perturbed periodic sturm liouville operators perturbations of asymptotic decay c r 2 arise in the partial wave analysis of rotationally symmetric partial differential operators we show that for each end point 0 of the spectral bands of a perturbed periodic sturm liouville operator there is a critical coupling constant c crit such that eigenvalues in the spectral gap accumulate at 0 if and only if c c crit 1 the oscillation theoretic method used in the proof also yields the asymptotic distribution of the eigenvalues near 0

efficient and doubly robust imputation for covariate dependent missing responses in this article we study a well known response missing data problem missing data is an ubiquitous problem in medical and social science studies imputation is one of the most popular methods for dealing with missing data the most commonly used imputation that makes use of covariates is regression imputation in which the regression model can be parametric semiparametric or nonparametric parametric regression imputation is efficient but is not robust against misspecification of the regression model although nonparametric regression imputation such as nearest neighbor imputation and kernel regression imputation is model free it is not efficient especially if the dimension of covariate vector is high the well known problem of curse of dimensionality semiparametric regression imputation such as partially linear regression imputation can reduce the dimension of the covariate in nonparametric regression fitting but is not robust against misspecification of the linear component in the regression

viscosity approximation methods for fixed points problems the aim of this work is to propose viscosity approximation methods which amounts to selecting a particular fixed point of a given nonexpansive self mapping

relative oscillation theory weighted zeros of the wronskian and the spectral shift function we develop an analog of classical oscillation theory for sturm liouville operators which rather than measuring the spectrum of one single operator measures the difference between the spectra of two different operators

estimation with missing data beyond double robustness summary we propose an estimator that is more robust than doubly robust estimators based on weighting completecasesusingweightsotherthaninverseprobabilitywhenestimatingthepopulationmean of a response variable subject to ignorable missingness we allow multiple models for both the propensity score and the outcome regression our estimator is consistent if any of the multiple models is correctly specified such multiple robustness against model misspecification is a significant improvement over double robustness which allows only one propensity score model and one outcome regression model our estimator attains the semiparametric efficiency bound when one propensity score model and one outcome regression model are correctly specified without requiring knowledge of which models are correct

on the existence of eigenvalues of the schrodinger operatorh λw in a gap of σ h the authors prove a number of results asserting the existence of eigenvalues of the schrodinger operatorh w in a gap of h whereh v andv andw are bounded the existence of these eigenvalues is an important element in the theory of the colour of crystals the basic theorems are proved in v stronger results forv 1 are also presented

a general iterative method for equilibrium problems and fixed point problems in hilbert spaces abstract the purpose of this paper is to study the strong convergence of a general iterative scheme to find a common element of the set of fixed points of a nonexpansive mapping the set of solutions of variational inequality for a relaxed cocoercive mapping and the set of solutions of an equilibrium problem our results extend the recent results of takahashi and takahashi s takahashi w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 331 2007 506 515 marino and xu g marino h k xu a general iterative method for nonexpansive mappings in hilbert spaces j math anal appl 318 2006 43 52 combettes and hirstoaga p l combettes s a hirstoaga equilibrium programming in hilbert spaces j nonlinear convex anal 6 2005 486 491 iiduka and takahashi h iiduka w takahashi strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings nonlinear anal 61 2005 341 350 and many others

a distributional approach for causal inference using propensity scores drawing inferences about the effects of treatments and actions is a common challenge in economics epidemiology and other fields we adopt rubin s potential outcomes framework for causal inference and propose two methods serving complementary purposes one can be used to estimate average causal effects assuming no confounding given measured covariates the other can be used to assess how the estimates might change under various departures from no confounding both methods are developed from a nonparametric likelihood perspective the propensity score plays a central role and is estimated through a parametric model under the assumption of no confounding the joint distribution of covariates and each potential outcome is estimated as a weighted empirical distribution expectations from the joint distribution are estimated as weighted averages or equivalently to first order regression estimates the likelihood estimator is at least as efficient and the regression estimator is at least as efficient and r

on relaxed viscosity iterative methods for variational inequalities in banach spaces in this paper we suggest and analyze a relaxed viscosity iterative method for a commutative family of nonexpansive self mappings defined on a nonempty closed convex subset of a reflexive banach space we prove that the sequence of approximate solutions generated by the proposed iterative algorithm converges strongly to a solution of a variational inequality our relaxed viscosity iterative method is an extension and variant form of the original viscosity iterative method the results of this paper can be viewed as an improvement and generalization of the previously known results that have appeared in the literature