the equivalence between mann ishikawa iterations and multistep iteration abstract we show that the convergence of mann ishikawa iterations are equivalent to the convergence of a multistep iteration for various classes of operators

a new iterative scheme for numerical reckoning fixed points of suzuki s generalized nonexpansive mappings in this paper we propose a new iterative algorithm to approximate fixed point of suzuki s generalized nonexpansive mappings we establish some weak and strong convergence theorems in a uniformly convex banach space we also provide examples to illustrate the convergence behavior of the proposed algorithm

signal sampling and recovery under dependent errors the paper examines the impact of the additive correlated noise on the accuracy of the signal reconstruction algorithm originating from the whittaker shannon ws sampling interpolation formula a class of band limited signals as well as signals which are non band limited are taken into consideration the proposed reconstruction method is a smooth post filtering correction of the classical ws interpolation series we assess both the point wise and global accuracy of the proposed reconstruction algorithm for a broad class of dependent noise processes this includes short and long memory stationary errors being independent of the sampling rate we also examine a class of noise processes for which the correlation function depends on the sampling rate whereas the short memory errors have relatively small influence on the reconstruction accuracy the long memory errors can greatly slow down the convergence rate in the case of the noise model depending on the sampling rate further degradation of the algorithm accuracy is observed we give quantitative explanations of these phenomena by deriving rates at which the reconstruction error tends to zero we argue that the obtained rates are close to be optimal in fact in a number of special cases they agree with known optimal min max rates the problem of the limit distribution of the l2 distance of the proposed reconstruction algorithm is also addressed this result allows us to tackle an important problem of designing non parametric lack of fit tests the theory of the asymptotic behavior of quadratic forms of stationary sequences is utilized in this case

multiscale inference and long run variance estimation in non parametric regression with time series errors we develop new multiscale methods to test qualitative hypotheses about the function m in the non parametric regression model yt t m t t t with time series errors t in time series applications m represents a non parametric time trend practitioners are often interested in whether the trend m has certain shape properties for example they would like to know whether m is constant or whether it is increasing or decreasing in certain time intervals our multiscale methods enable us to test for such shape properties of the trend m to perform the methods we require an estimator of the long run error variance 2 l cov e0 el we propose a new difference based estimator of 2 for the case that t belongs to the class of auto regressive ar processes in the technical part of the paper we derive asymptotic theory for the proposed multiscale test and the estimator of the long run error variance the theory is complemented by a simulation study and an empirical application to climate data

from third to fourth order variant of newton s method for simple roots abstract in this paper we show one way of deriving a fourth order scheme from any third order newton type method without any additional evaluation of the function or its derivatives for this purpose the rectangular rule may be employed in newton s theorem about a special point this special point may be selected from a variant of any third order scheme two such cases are illustrated here the first case leads to a new fourth order scheme while the second case leads to the well known jarratt s method the efficiency index of the proposed scheme is 1 587 in order to compare its performance with some of the existing third and fourth order schemes several numerical examples are furnished here

new approximation schemes for general variational inequalities in this paper we suggest and consider a class of new three step approximation schemes for general variational inequalities our results include ishikawa and mann iterations as special cases we also study the convergence criteria of these schemes

permanence and positive periodic solution for the single species nonautonomous delay diffusive models abstract single species nonautonomous delay diffusion models with nonlinear growth rates are investigated some sufficient conditions are determined that guarantee the permanence of the species and the existence of a positive periodic solution which is global attractively

polynomiography for the polynomial infinity norm via kalantaris formula and nonstandard iterations survey of fixed point iteration processes extension of the pseudo newton method idea to other root finding methods modifications of the mmp methods and their convergence visualizations non trivial and intriguing polynomiographs have been obtained in this paper an iteration process referred to in short as mmp will be considered this iteration is related to finding the maximum modulus of a complex polynomial over a unit disc on the complex plane creating intriguing images kalantari calls these images polynomiographs independently from whether they are generated by the root finding or maximum modulus finding process applied to any polynomial we show that the images can be easily modified using different mmp methods pseudo newton mmp householder methods from the mmp basic mmp parametric basic or mmp eulerschrder families of iterations with various kinds of non standard iterations such images are interesting from three points of views scientific educational and artistic we present the results of experiments showing automatically generated non trivial images obtained for different modifications of root finding mmp methods the colouring by iteration reveals the dynamic behaviour of the used root finding process and its speed of convergence the results of the present paper extend kalantaris recent results in finding the maximum modulus of a complex polynomial based on newtons process with the picard iteration to other mmp processes with various non standard iterations

fixed point iterations for asymptotically nonexpansive mappings in banach spaces in this paper we suggest and analyze a three step iterative scheme for asymptotically nonexpansive mappings in banach spaces the new iterative scheme includes ishikawa type and mann type interations as special cases the results obtained in this paper represent an extension as well as refinement of previous known results

iterative construction of fixed points of asymptotically nonexpansive mappings abstract let t be a completely continuous and asymptotically nonexpansive self mapping of a nonempty closed bounded and convex subset of a hilbert space then under certain conditions the sequence defined by xn 1 ntn xn 1 n x n converges strongly to some fixed point of t a similar result is obtained for asymptotically pseudocontractive mappings whose iterates are lipschitz continuous with respect to some common lipschitz constant

fixed point iterations for certain nonlinear mappings abstract in a recent paper schu established the convergence of the mann iterates of a completely continuous asymptotically nonexpansive mapping on a hilbert space in this paper we extend the result of schu to uniformly convex spaces we also obtain the same generalization for the ishikawa iteration process

one sided cross validation abstract a new method of selecting the smoothing parameters of nonparametric regression estimators is introduced the method termed one sided cross validation oscv has the objectivity of cross validation and statistical properties comparable to those of a plug in rule the new method may be viewed as an application of the prequential model selection method of dawid as such our results identify a situation in which the prequential method is a more efficient model selector than cross validation an example simulations and theoretical results demonstrate the utility of oscv when used with local linear and kernel estimators

harmless delays for uniform persistence abstract this paper studies a two dimensional predator prey system with a finite number of discrete delays our purpose is to demonstrate that the time delays are harmless for uniform persistence of the solutions of the system the results are obtained by constructing suitable persistence functionals

a new two step biparametric family of sixth order iterative methods free from second derivatives for solving nonlinear algebraic equations in this paper a new two step biparametric family of sixth order iterative methods free from second derivatives are proposed in this paper to find a simple root of a nonlinear algebraic equation their efficiency indices are all found to be 6 1 4 convergence analysis together with numerical experiments confirms sixth order convergence and asymptotic error constants

nonparametric estimation of residual variance revisited summary several difference based estimators of residual variance are compared for finite sample size since the introduction of a rather simple estimator by gasser sroka jennensteinmetz 1986 other proposals have been made here the one given by hall kay titterington 1990 is of particular interest it minimizes the asymptotic variance unfortunately it has severe problems with finite sample bias and the estimator of gasser et al 1986 proves still to be a good choice a new estimator is introduced compromising between bias and variance

weak and strong convergence theorems for three step iterations with errors for asymptotically nonexpansive mappings abstract in the present paper several weak and strong convergence theorems are established for the three step iterative schemes with errors for asymptotically nonexpansive mappings our results extend and improve the recent ones announced by tan and xu xu and noor and many others

newton s method s basins of attraction revisited in this paper we revisit the chaotic number of iterations needed by newton s method to converge to a root here we consider a simple modified newton method depending on a parameter it is demonstrated using polynomiography that even in the simple algorithm the presence and the position of the convergent regions i e regions where the method converges nicely to a root can be complicatedly a function of the parameter

efficient estimation of conditional variance functions in stochastic regression author s fan jianqing yao qiwei abstract conditional heteroscedasticity has been often used in modelling and understanding the variability of statistical data under a general setup which includes the nonlinear time series model as a special case we propose an e cient and adaptive method for estimating the conditional variance the basic idea is to apply a local linear regression to the squared residuals we demonstrate that without knowing the regression function we can estimate the conditional variance asymptotically as well as if the regression were given this asymptotic result established under the assumption that the observations are made from a strictly stationary and absolutely regular process is also veri ed via simulation further the asymptotic result paves the way for adapting an automatic bandwidth selection scheme an application with nancial data illustrates the usefulness of the proposed techniques

weak and strong convergence criteria of noor iterations for asymptotically nonexpansive mappings in this paper weak and strong convergence theorems are established for a three step iterative scheme for asymptotically nonexpansive mappings in banach spaces mann type and ishikawa type iterations are included by the new iterative scheme the results obtained in this paper extend and improve the recent ones announced by xu and noor glowinski and le tallec noor ishikawa and many others

a comparison between iterative methods by using the basins of attraction abstract there exists a real competition between authors to construct improved iterative methods for solving nonlinear equations in this paper by using computer experiment we study the basins of attraction for some of the iterative methods for solving the equation p z 0 where p c c is a complex coefficients polynomial and this allows us to compare their performances the area of convergence and theirs speed the beauty fractal pictures generated by these methods are presented too

global stability for infinite delay lotka volterra type systems abstract in this paper sufficient conditions are established for the global stability of the saturated equilibrium of an infinite delay nonautonomous lotka volterra type system the present work is distinguished from previous work principally be allowing the system to be nonautonomous and relaxing the traditional requirement that the undelayed intraspecific competition dominates both the delayed intraspecific competition as well as the interspecific interactions we require the undelayed intraspecific competition to dominate the latter but not the former

a new third order newton type iterative method for solving nonlinear equations in this paper we introduce a new third order iterative method for solving nonlinear equations this method is converging on larger intervals than some other similar known methods a comparison between the new method and other third order methods is presented by using the basins of attraction for the real roots for some test problems the numerical results are also presented

residual variance and residual pattern in nonlinear regression summary a nonparametric estimator of residual variance in nonlinear regression is proposed it is based on local linear fitting asymptotically the estimator has a small bias but a larger variance compared with the parametric estimator in linear regression finite sample properties are investigated in a simulation study including a comparison with other nonparametric estimators the method is also useful for spotting heteroscedasticity and outliers in the residuals at an early stage of the data analysis a further application is checking the fit of parametric models this is illustrated for longitudinal growth data

bandwidth choice for nonparametric regression this paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression we analyze a tapered fourier series estimate and discuss the relationship of this estimate to a kernel estimate we first consider a method based on an unbiased estimate of mean square error and show that the bandwidth thus chosen is asymptotically optimal other methods are examined as well and are shown to be asymptotically equivalent a small simulation shows however that for small or moderate sample size the methods perform quite differently

estimation of heteroscedasticity in regression analysis consider the regression model y i g t i varepsilon i 1 leq i leq n with nonrandom design variables t i and measurements y i for the unknown regression function g cdot we assume that the data are heteroscedastic i e e varepsilon 2 i sigma 2 i not equiv operatorname const and investigate how to estimate sigma 2 i if sigma 2 i sigma 2 t i with a smooth function sigma 2 cdot initial estimators tilde sigma 2 i can be improved by kernel smoothers and the resulting class of estimators is shown to be uniformly consistent these estimates can be used to improve the estimation of the regression function g itself in parametric and nonparametric models further applications are suggested

oracally efficient estimation and consistent model selection for auto regressive moving average time series with trend summary most time series that are encountered in practice contain non zero trend yet textbook approaches to time series analysis are typically focused on zero mean stationary auto regressive moving average arma processes trend is often estimated by ad hoc methods and subtracted from time series and the residuals are used as the true arma noise for data analysis and inference including parameter estimation lag selection and prediction we propose a theoretically justified two step method to analyse time series consisting of a smooth trend function and arma error term which is computationally efficient and easy for practitioners to implement the trend is estimated by b spline regression and the maximum likelihood estimator based on residuals is shown to be oracally efficient in the sense that it is asymptotically as efficient as if the true trend function were known and then removed to obtain the arma errors in addition consistency of the bayesian information criterion for model selection is established for the detrended residual sequence finite sample performance of the procedure is illustrated by simulation studies and real data analysis

a nonautonomous predator prey system with stage structure and double time delays in the present paper we study a nonautonomous predator prey model with stage structure and double time delays due to maturation time for both prey and predator we assume that the immature and mature individuals of each species are divided by a fixed age and the mature predator only attacks the immature prey based on some comparison arguments we discuss the permanence of the species by virtue of the continuation theorem of coincidence degree theory we prove the existence of positive periodic solution by means of constructing an appropriate lyapunov functional we obtain sufficient conditions for the uniqueness and the global stability of positive periodic solution two examples are given to illustrate the feasibility of our main results

a family of fifth order iterations composed of newton and third order methods abstract in this paper we present a family of fifth order methods formed by the composition of newton and third order methods for solving non linear equations per iteration the new methods require two evaluations of the function one of its first derivative and one of its second derivative analysis of efficiency shows that the new methods can compete with newton s method and the classical third order methods numerical results show that the methods have definite practical utility

kernel regression in the presence of correlated errors it is a well known problem that obtaining a correct bandwidth and or smoothing parameter in nonparametric regression is difficult in the presence of correlated errors there exist a wide variety of methods coping with this problem but they all critically depend on a tuning procedure which requires accurate information about the correlation structure we propose a bandwidth selection procedure based on bimodal kernels which successfully removes the correlation without requiring any prior knowledge about its structure and its parameters further we show that the form of the kernel is very important when errors are correlated which is in contrast to the independent and identically distributed i i d case finally some extensions are proposed to use the proposed criterion in support vector machines and least squares support vector machines for regression

local polynomial regression with correlated errors in random design and unknown correlation structure summaryautomated or data driven bandwidth selection methods tend to break down in the presence of correlated errors while this problem has previously been studied in the fixed design setting for kernel regression the results were applicable only when there is knowledge about the correlation structure this article generalizes these results to the random design setting and addresses the problem in situations where no prior knowledge about the correlation structure is available we establish the asymptotic optimality of our proposed bandwidth selection criterion based on kernels k satisfying k 0 0

bandwidth selection for kernel regression with long range dependent errors summary we investigate the effect of long range dependence on bandwidth selection for kernel regression with the plug in method of herrmann gasser kneip 1992 a new bandwidth estimator is proposed to allow for long range dependence properties of the proposed estimator are investigated theoretically and via simulation we find that the proposed estimator performs well in terms of integrated squared error of the estimated trend allowing us to incorporate both deterministic nonlinear features having an unknown structure and long range dependence into a single model the method is illustrated using biweekly measurements of the volume of the great salt lake

on the rate of convergence of mann ishikawa noor and sp iterations for continuous functions on an arbitrary interval in this paper we propose a new iteration called the sp iteration for approximating a fixed point of continuous functions on an arbitrary interval then a necessary and sufficient condition for the convergence of the sp iteration of continuous functions on an arbitrary interval is given we also compare the convergence speed of mann ishikawa noor and sp iterations it is proved that the sp iteration is equivalent to and converges faster than the others our results extend and improve the corresponding results of borwein and borwein d borwein j borwein fixed point iterations for real functions j math anal appl 157 1991 112 126 qing and qihou y qing l qihou the necessary and sufficient condition for the convergence of ishikawa iteration on an arbitrary interval j math anal appl 323 2006 1383 1386 rhoades b e rhoades comments on two fixed point iteration methods j math anal appl 56 1976 741 750 and many others moreover we also present numerical examples for the sp iteration to compare with the mann ishikawa and noor iterations

comparison of two bandwidth selectors with dependent errors for nonparametric regression in the case of dependent observations cross validation is known to be severely affected by dependence this effect is precisely quantified through a limiting distribution for the cross validated bandwidth the performance of two methods the leave 2e 1 out version of cross validation and partitioned cross validation which adjust for the dependence effect on bandwidth selection is investigated the bandwidths produced by these two methods are analyzed by further limiting distributions which reveal significantly different characteristics simulations demonstrate that the asymptotic effects hold for reasonable sample sizes

detecting dependencies in smooth regression models summary a class of simple estimators for the correlation of the errors in nonparametric regression models is proposed for the fixed design case these estimators are based on squared differences of various spans of the data and are consistent so long as the regression function is lipschitz continuous the limiting distribution is established and applied to derive asymptotic global and local tests for different null hypotheses of uncorrelatedness the finite sample behaviour of estimators and tests is investigated in a monte carlo study a more general class of estimators using generalized difference schemes is also introduced

permanence and global attractivity of delay diffusive prey predator systems with the michaelis menten functional response in this paper the author investigates two species nonautonomous delay diffusive preypredator models with michaelis menten functional response sufficient conditions are derived for permanence of the species further the author also establishs the sufficient conditions that guarantee the arbitrary positive periodic solution of the systems is global attractive

estimation of semivarying coefficient time series models with arma errors serial correlation in the residuals of time series models can cause bias in both model estimation and prediction however models with such serially correlated residuals are difficult to estimate especially when the regression function is nonlinear existing estimation methods require strong assumption for the relation between the residuals and the regressors which excludes the commonly used autoregressive models in time series analysis by extending the whittle likelihood estimation this paper investigates in details a semi parametric autoregressive model with arma sequence of residuals asymptotic normality of the estimators is established and a model selection procedure is proposed numerical examples are employed to illustrate the performance of the proposed estimation method and the necessity of incorporating the serial correlation in the residuals

asymptotically optimal difference based estimation of variance in nonparametric regression summary we define and compute asymptotically optimal difference sequences for estimating error variance in homoscedastic nonparametric regression our optimal difference sequences do not depend on unknowns such as the mean function and provide substantial improvements over the suboptimal sequences commonly used in practice for example in the case of normal data the usual variance estimator based on symmetric second order differences is only 64 efficient relative to the estimator based on optimal second order differences the efficiency of an optimal mth order difference estimator relative to the error sample variance is 2m 2m 1 again this is for normal data and increases as the tails of the error distribution become heavier

some higher order modifications of newton s method for solving nonlinear equations in this paper we consider constructing some higher order modifications of newton s method for solving nonlinear equations which increase the order of convergence of existing iterative methods by one or two or three units this construction can be applied to any iteration formula and per iteration the resulting methods add only one additional function evaluation to increase the order some illustrative examples are provided and several numerical results are given to show the performance of the presented methods

using difference based methods for inference in nonparametric regression with time series errors we show that difference based methods can be used to construct simple and explicit estimators of error covariance and autoregressive parameters in nonparametric regression with time series errors when the error process is gaussian our estimators are efficient but they are available well beyond the gaussian case as an illustration of their usefulness we show that difference based estimators can be used to produce a simplified version of time series cross validation this new approach produces a bandwidth selector that is equivalent to both first and second orders to that given by the full time series cross validation algorithm other applications of difference based methods are to variance estimation and construction of confidence bands in nonparametric regression

estimating the hausdorff besicovitch dimension of boundary of basin of attraction in helicopter trim helicopter trim involves solution of nonlinear force equilibrium equations as in many nonlinear dynamic systems helicopter trim problem can show chaotic behavior this chaotic behavior is found in the basin of attraction of the nonlinear trim equations which have to be solved to determine the main rotor control inputs given by the pilot this study focuses on the boundary of the basin of attraction obtained for a set of control inputs we analyze the boundary by considering it at different magnification levels the magnified views reveal intricate geometries it is also found that the basin boundary exhibits the characteristic of statistical self similarity which is an essential property of fractal geometries these results led the authors to investigate the fractal dimension of the basin boundary it is found that this dimension is indeed greater than the topological dimension from all the observations it is evident that the boundary of the basin of attraction for helicopter trim problem is fractal in nature c 2012 elsevier inc all rights reserved

choice of bandwidth for kernel regression when residuals are correlated summary standard techniques for selecting the bandwidth of a kernel estimator from the data in a nonparametric regression model perform badly when the errors are correlated in this paper we propose a modified version of an existing plug in bandwidth selector the method is generalized to stationary error variables by estimating a functional of the residual covariance function the proposed bandwidth selector shows good properties in asymptotic theory and in simulations without assuming a parametric model for the error process

eighth order methods with high efficiency index for solving nonlinear equations in this paper we derive a new family of eighth order methods for solving simple roots of nonlinear equations by using weight function methods per iteration these methods require three evaluations of the function and one evaluation of its first derivative which implies that the efficiency indexes are 1 682 numerical comparisons are made to show the performance of the derived methods as shown in the illustration examples

some fourth order modifications of newton s method abstract in this paper we construct some fourth order modifications of newton s method for solving nonlinear equations any two existing fourth order methods can be effectively used to give rise to new fourth order methods per iteration the new methods require two evaluations of the function and one of its first derivative numerical examples are given to show the performance of the presented methods