on the method of tangent hyperbolas on banach spaces abstract as an application of the technique employed by the author in a series of papers 15 18 some results are established concerning convergence of the method of tangent hyperbolas for solving nonlinear equations in banach spaces as well as existence and uniqueness of solution the results are compared with those obtained by doring 5

frequency identification of nonparametric wiener systems containing backlash nonlinearities this paper addresses the problem of identifying wiener systems constituted of nonparametric linear dynamics and backlash nonlinearities the linear subsystem is arbitrary but stable the backlash nonlinearity borders are almost arbitrary shape in particular these are allowed to be nonsmooth noninvertible and crossing a frequency identification method is developed to estimate the system frequency response function at a given set of frequencies and the backlash nonlinearity borders within a given working domain the identification method involves sine excitations and consistent estimators designed using analytic geometry tools e g fictive limit cycles informative backlash limit cycles spread and orientation compatibility

new exact periodic solitary wave solutions for the 3 1 dimensional generalized kp and bkp equations the extended homoclinic test approach ehta is a straightforward and efficient method to solve nonlinear partial differential equations in this paper new exact solutions including kinky periodic solitary wave solutions periodic soliton solutions and periodic kink solutions for the 3 1 dimensional generalized kadomtsev petviashvili kp and b type kadomtsev petviashvili bkp equations are obtained moreover the strangely mechanical features of these wave solutions are investigated these solutions enrich the variety of the dynamics of higher dimensional nonlinear wave field

a search for bilinear equations passing hirota s three soliton condition ii mkdv type bilinear equations in this paper the results of a search for complex bilinear equations with two soliton solutions are presented the following basic types are discussed a the nonlinear schrodinger equation b dx g f 0 a dx dt f f gg and b the benjamin ono equation p dx f f 0 it is found that the existence of two soliton solutions is not automatic but introduces conditions that are like the usual three and four soliton conditions the search was limited by the degree of a 2 and by degree of p 4 the main results are the following 1 iad3x dxdt idy b g f 0 d2xf f gg 2 d2x ad2y idt b g f 0 dxdy f f gg 3 iad3x d2x idt f f 0 and 4 dxdt i adx bdt f f 0

attracting cycles for the relaxed newton s method we study the relaxed newton s method applied to polynomials in particular we give a technique such that for any n 2 we may construct a polynomial so that when the method is applied to a polynomial the resulting rational function has an attracting cycle of period n we show that when we use the method to extract radicals the set consisting of the points at which the method fails to converge to the roots of the polynomial p z z m c this set includes the julia set has zero lebesgue measure consequently iterate sequences under the relaxed newton s method converge to the roots of the preceding polynomial with probability one

an extension of gander s result for quadratic equations in the study of iterative methods with high order of convergence gander provides a general expression for iterative methods with order of convergence at least three in the scalar case taking into account an extension of this result we define a family of iterations in banach spaces with r order of convergence at least four for quadratic equations

third order iterative methods with applications to hammerstein equations a unified approach the geometrical interpretation of a family of higher order iterative methods for solving nonlinear scalar equations was presented in s amat s busquier j m gutierrez geometric constructions of iterative functions to solve nonlinear equations j comput appl math 157 1 2003 197 205 this family includes as particular cases some of the most famous third order iterative methods chebyshev methods halley methods super halley methods c methods and newton type two step methods the aim of the present paper is to analyze the convergence of this family for equations defined between two banach spaces by using a technique developed in j a ezquerro m a hernandez halley s method for operators with unbounded second derivative appl numer math 57 3 2007 354 360 this technique allows us to obtain a general semilocal convergence result for these methods where the usual conditions on the second derivative are relaxed on the other hand the main practical difficulty related to the classical third order iterative methods is the evaluation of bilinear operators typically second order frechet derivatives however in some cases the second derivative is easy to evaluate a clear example is provided by the approximation of hammerstein equations where it is diagonal by blocks we finish the paper by applying our methods to some nonlinear integral equations of this type

on the r order of the halley method an r order bound for the halley method is obtained in this work where an analysis of the convergence of the method is also presented under mild differentiability conditions to do this a new technique is developed where the involved operator must satisfy some recurrence relations

convergence analysis of recursive identification algorithms based on the nonlinear wiener model recursive identification algorithms based on the nonlinear wiener model are presented a recursive identification algorithm is first derived from a general parameterization of the wiener model using a stochastic approximation framework local and global convergence of this algorithm can be tied to the stability properties of an associated differential equation since inversion is not utilized noninvertible static nonlinearities can be handled which allows a treatment of for example saturating sensors and blind adaptation problems gauss newton and stochastic gradient algorithms for the situation where the static nonlinearity is known are then suggested in the single input single output case the proposed methods can outperform conventional linearizing inversion of the nonlinearity when measurement disturbances affect the output signal for fir finite impulse response models it is also proved that global convergence of the schemes is tied to sector conditions on the static nonlinearity in particular global convergence of the stochastic gradient method is obtained provided that the nonlinearity is strictly monotone the local analysis performed for iir infinite impulse response models illustrates the importance of the amplitude contents of the exciting signals

parameter estimation error bounds for hammerstein nonlinear finite impulsive response models abstract this paper presents a parameter estimation algorithm for a class of hammerstein nonlinear systems input nonlinear fir finite impulse response models and studies in detail the convergence properties of the proposed identification algorithm in the stochastic framework and derives the upper and lower bounds of the parameter estimation errors pee from the available input output data the analysis indicates that the mean square pee upper and lower bounds of the algorithm approach zero as the data length increases a simulation example is given

maximum likelihood identification of wiener models the wiener model is a block oriented model having a linear dynamic system followed by a static nonlinearity the dominating approach to estimate the components of this model has been to minimize the error between the simulated and the measured outputs we show that this will in general lead to biased estimates if there are other disturbances present than measurement noise the implications of bussgang s theorem in this context are also discussed for the case with general disturbances we derive the maximum likelihood method and show how it can be efficiently implemented comparisons between this new algorithm and the traditional approach confirm that the new method is unbiased and also has superior accuracy

combined parametric nonparametric identification of hammerstein systems a novel parametric nonparametric methodology for hammerstein system identification is proposed assuming random input and correlated output noise the parameters of a nonlinear static characteristic and finite impulse response system dynamics are estimated separately each in two stages first the inner signal is recovered by a nonparametric regression function estimation method stage 1 and then system parameters are solved independently by the least squares stage 2 convergence properties of the scheme are established and rates of convergence are given

a note on halley s method we introduce the degree of logarithmic convexity which provides a measure of the convexity of a function at each point making use of this concept we obtain a new theorem of global convergence for halley s method

an improvement of the region of accessibility of chebyshev s method from newton s method a simple modification of chebyshev s method is presented so that the region of accessibility is extended to the one of newton s method

newton type methods of high order and domains of semilocal and global convergence we present the geometric construction of some classical iterative methods that have global convergence and infinite speed of convergence when they are applied to solve certain nonlinear equations f t 0 in particular for nonlinear equations with the degree of logarithmic convexity of f l f t f t f t f t 2 is constant a family of newton type iterative methods of high orders of convergence is constructed we see that this family of iterations includes the classical iterative methods the convergence of the family is studied in the real line and the complex plane and domains of semilocal and global convergence are located

a new concept of convergence for iterative methods restricted global convergence abstract we introduce a new concept of convergence for iterative methods named restricted global convergence that consists of locating a solution and obtaining a domain of global convergence as a consequence results of semilocal convergence and local convergence are obtained for this we use auxiliary points and obtain balls of convergence the study is illustrated with chebyshev s method

parameter bounds evaluation of wiener models with noninvertible polynomial nonlinearities in this paper a three stage procedure is presented for deriving parameters bounds of siso wiener models when the nonlinear block is modeled by a possibly noninvertible polynomial and the output measurement errors are bounded first using steady state input output data parameters of the nonlinear part are bounded by a tight orthotope then given the estimated uncertain nonlinearity and the output measurements collected exciting the system with an input dynamic signal bounds on the unmeasurable inner signal are computed finally such bounds together with noisy output measurements are used for bounding the parameters of the linear block

on the semigroup property for some structured iterations abstract nonlinear matrix equations play a crucial role in science and engineering problems however solutions of nonlinear matrix equations cannot in general be given analytically one standard way of solving nonlinear matrix equations is to apply the fixed point iteration with usually only the linear convergence rate to advance the existing methods we exploit in this work one type of semigroup property and use this property to propose a technique for solving the equations with the speed of convergence of any desired order we realize our way by starting with examples of solving the scalar equations and also connect this method with some well known equations including but not limited to the stein matrix equation the generalized eigenvalue problem the generalized nonlinear matrix equation the discrete time algebraic riccati equations to express the capacity of this method

brief identification of linear systems with hard input nonlinearities of known structure this paper studies identification of systems with input nonlinearities of known structure for input nonlinearities parameterized by one parameter a deterministic approach is proposed based on the idea of separable least squares the identification problem is shown to be equivalent to an one dimensional minimization problem the method is very effective for several common static and nonstatic input nonlinearities for a general input nonlinearity a correlation analysis based identification algorithm is presented which is shown to be convergent

new traveling waves solutions to generalized kaup kupershmidt and ito equations in this paper we consider a special fifth order kdv equation with constant coefficients and we obtain traveling wave solutions for it using the projective riccati equation method by mean of a scaling exact solutions to general kaup kupershmidt kk equation are obtained as a particular case exact solutions to standard kk equation can be derived using the same method we obtain exact solutions to standard ito equation by mean of scaling new exact solutions to general ito equation are formally derived

breather type soliton and two soliton solutions for modified korteweg de vries equation in this letter different kinds of solutions including breather type soliton and two soliton solutions are obtained for the modified korteweg de vries m kdv equation by using bilinear form the extended homoclinic test approach and dependent variable transformations moreover we point out that the author did not obtain so called periodic two soliton solutions in w long in press 1

frequency domain identification of wiener models this paper proposes a frequency domain algorithm for wiener model identifications based on exploring the fundamental frequency and harmonics generated by the unknown nonlinearity the convergence of the algorithm is established in the presence of white noise no a priori knowledge of the structure of the nonlinearity is required and the linear part can be nonparametric

applications of hta and ehta to ytsf equation new approaches homoclinic test approach hta and extended homoclinic test approach ehta are proposed to seek solitary wave solution of high dimensional nonlinear wave system exact periodic solitary wave periodic soliton cross solitary wave and doubly periodic wave solutions for ytsf equation are obtained using hta and ehta respectively

identification of iir wiener systems with spline nonlinearities that have variable knots an algorithm is developed for the identification of wiener systems linear dynamic elements followed by static nonlinearities in this case the linear element is modeled using a recursive digital filter while the static nonlinearity is represented by a spline of arbitrary but fixed degree the primary contribution in this note is the use of variable knot splines which allow for the use of splines with relatively few knot points in the context of wiener system identification the model output is shown to be nonlinear in the filter parameters and in the knot points but linear in the remaining spline parameters thus a separable least squares algorithm is used to estimate the model parameters monte carlo simulations are used to compare the performance of the algorithm identifying models with linear and cubic spline nonlinearities with a similar technique using polynomial nonlinearities

exact periodic solitary wave solutions for the 2 1 dimensional boussinesq equation abstract exact periodic solitary wave solutions for the 2 1 dimensional boussinesq equation are obtained by using the extended ansatz function method detailed behavior of the propagation of the periodic solitary wave solutions for the 2 1 dimensional boussinesq equation is illustrated by using the method of figure analysis the result shows that it is entirely possible for the 2 1 dimensional integrable equations or non integrable equations that there exist periodic solitary waves in the different direction the propagation of the periodic solitary waves is actually phase shifts of solitons and the amplitudes of non singular periodic solitary waves depend on frequency and wave number of periodic wave

exact soliton solutions for the fifth order sawada kotera equation exact soliton solutions for the fifth order sawada kotera equation are obtained by using the hirota bilinear method these solutions include one soliton solutions periodic two soliton solutions and singular periodic soliton solutions the results show that there exist periodic two soliton solutions and singular periodic soliton solutions for the fifth order sawada kotera equation

indices of convexity and concavity application to halley method we define an index to measure the convexity of a convex function f at each point we use this index to establish conditions on the convergence of the halley method in the complex plane and in banach spaces

brief paper computational complexity analysis of set membership identification of hammerstein and wiener systems this paper analyzes the computational complexity of set membership identification of hammerstein and wiener systems its main results show that even in cases where a portion of the plant is known the problems are generically np hard both in the number of experimental data points and in the number of inputs wiener or outputs hammerstein of the nonlinearity these results provide new insight into the reasons underlying the high computational complexity of several recently proposed algorithms and point out the need for developing computationally tractable relaxations

a note on the kantorovich theorem for newton iteration abstract in this paper a new theorem for the newton method convergence is obtained its condition is different from that of the kantorovich theorem and therefore it has theoretical and practical value

an analytic geometry approach to wiener system frequency identification this paper addresses the problem of wiener system identification the underlying linear subsystem is stable but not necessarily parametric the nonlinear element in turn is allowed to be nonparametric noninvertible and nonsmooth as wiener models are uniquely defined up to an uncertain multiplicative factor it makes sense to start the frequency identification process estimating the system phase which is common to all models to this end a consistent estimator is designed using analytic geometry tools accordingly the system frequency behavior is characterized by a family of lissajous curves interestingly all these curves are candidates to modelling the system nonlinearity but the most convenient one is the less spread of them finally the frequency gain is in turn consistently estimated optimizing an appropriate cost function involving the obtained phase and nonlinearity estimates

multiple soliton solutions for the generalized 1 1 dimensional and the generalized 2 1 dimensional ito equations abstract the generalized 1 1 dimensional and the generalized 2 1 dimensional ito equations obtained by introducing a new type of bilinearism are studied the tanh coth method is used to obtain single soliton solutions and periodic solutions for the ito equations the hirota bilinear method is used to determine the multiple soliton solutions of sech squared type for these equations new entirely solutions are obtained the analysis emphasizes the power of the two methods and its capability of handling completely integrable equations

methods with prefixed order for approximating square roots with global and general convergence a family of newton like methods is constructed to approximate the square root of a positive real number the iterative methods of the family are global or generally convergent depending on the prefixed order of convergence is even or odd we show the dynamical behaviour of some of these methods by means of several julia sets and the intricate fractal structures which arise from the order of the iterative methods are displayed

on the convergence of an iterative algorithm used for hammerstein system identification an analytical counterexample to the general convergence of an interative algorithm used for identification of hammerstein systems is given numerical illustration is also included

on a characterization of some newton like methods of r order at least three in this paper we extend to banach spaces the result given by gander in amer math monthly 92 1985 131 to obtain a characterization of newton like iterative process with r order of convergence at least three to do this we consider the construction and the study of the semilocal convergence of a multiparametric family of iterative processes in banach spaces for solving the nonlinear equation f x 0

new periodic solitary wave solutions for the benjiamin ono equation in this paper the new periodic solitary wave and doubly periodic solutions for 1 1 dimensional benjiamin ono equation are obtained using the bilinear method and extended homoclinic test approach these results demonstrate that the integrable system has richly dynamical behavior even if it is 1 1 dimensional

on the series expansion approach to the identification of hammerstein systems a polynomial identification algorithm for recovering a nonlinearity in the hammerstein system is proposed the estimate employs the legendre orthogonal system with adaptively selected number of terms the global consistency along with rates of convergence are established no assumptions concerning continuity of the nonlinearity or its functional form are made a data driven method using the cross validation technique for selecting the number of terms in the estimate is presented

the breather type and periodic type soliton solutions for the 2 1 dimensional breaking soliton equation exact breather type and periodic type soliton solutions including the double breather type soliton solutions the breather type periodic soliton solutions and breather type two soliton solutions and the periodic type two soliton and three soliton solutions for the 2 1 dimensional breaking soliton equation are obtained using the extended three wave method etm the results show that the etm may provide us with a straightforward and effective mathematical tool for seeking multi wave solutions of higher dimensional nonlinear evolution equations

frequency domain identification of hammerstein models discusses hammerstein model identification in the frequency domain using sampled input output data by exploring the fundamental frequency and harmonics generated by the unknown nonlinearity we propose a frequency domain approach and show its convergence for both the linear and nonlinear subsystems in the presence of noise no a priori knowledge of the structure of the nonlinearity is required and the linear part can be nonparametric

nonparametric identification of wiener systems a wiener system i e a system in which a linear dynamic part is followed by a nonlinear and memoryless one is identified no parametric restriction is imposed on the functional form of the nonlinear characteristic of the memoryless subsystem and a nonparametric algorithm recovering the characteristic from input output observations of the whole system is proposed its consistency is shown and the rate of convergence is given an idea for identification of the impulse response of the linear subsystem is proposed results of numerical simulation are also presented

extend three wave method for the 1 2 dimensional ito equation abstract in this work extend three wave method etm is used to construct the novel multi wave solutions of the 1 2 dimensional ito equation as a result three soliton solution doubly periodic solitary wave solutions periodic two solitary wave solutions are obtained it is shown that the extend three wave method may provide us with a straightforward and effective mathematical tool for seeking multi wave solutions of higher dimensional nonlinear evolution equations

a modification of cauchy s method for quadratic equations the plan of this paper is to obtain one point iterative methods with any r order of convergence when they are applied to approximate solutions of quadratic equations in banach spaces to do this we consider real cauchy s method and under certain natural modifications it is extended to banach spaces some applications are also provided

the ehta for nonlinear evolution equations abstract two types of important nonlinear evolution equations are investigated by using the extended homoclinic test approach ehta some exact soliton solutions including breather type of soliton periodic type of soliton and two soliton solutions are obtained these results show that the extended homoclinic test technique together with the bilinear method is a simple and effective method to seek exact solutions for nonlinear evolution equations

frequency domain analysis and identification of block oriented nonlinear systems abstract block oriented nonlinear models including wiener models hammerstein models and wiener hammerstein models etc have been extensively applied in practice for system identification signal processing and control in this study analytical frequency response functions including generalized frequency response functions gfrfs and nonlinear output spectrum of block oriented nonlinear systems are developed which can demonstrate clearly the relationship between frequency response functions and model parameters and also the dependence of frequency response functions on the linear part of the model the nonlinear part of these models can be a more general multivariate polynomial function these fundamental results provide a significant insight into the analysis and design of block oriented nonlinear systems effective algorithms are therefore proposed for the estimation of nonlinear output spectrum and for parametric or nonparametric identification of nonlinear systems compared with some existing frequency domain identification methods the new estimation algorithms do not necessarily require model structure information not need the invertibility of the nonlinearity and not restrict to harmonic inputs simulation examples are given to illustrate these new results

letter to the editor accelerated convergence in newton s method for approximating square roots in this paper we construct a modification of newton s method to accelerate the convergence of this method to the approximation of the positive square root of a positive real number from this modification we can define a new iterative process with prefixed order q n q 2

parameter identification of discontinuous hammerstein systems the paper deals with the parameter identification of hammerstein systems having discontinuous asymmetric nonlinearities two segment piecewise linear with preload and dead zone application of a simple decomposition technique provides a special form of hammerstein model with a minimum number of parameters the identification is carried out with only one record of measured input output data despite the dead zone to demonstrate the feasibility of the identification method some illustrative examples are included

recursive identification of errors in variables wiener systems this paper considers the recursive identification of errors in variables eiv wiener systems composed of a linear dynamic system followed by a static nonlinearity both the system input and output are observed with additive noises being arma processes with unknown coefficients by a stochastic approximation incorporated with the deconvolution kernel functions the recursive algorithms are proposed for estimating the coefficients of the linear subsystem and for the values of the nonlinear function all the estimates are proved to converge to the true values with probability one a simulation example is given to verify the theoretical analysis

new exact solutions to the 2 1 dimensional ito equation extended homoclinic test technique exact soliton solutions to the 2 1 dimensional ito equation are studied based on the idea of extended homoclinic test and bilinear method some explicit solutions such as triangle function solutions soliton solutions doubly periodic wave solutions and periodic solitary wave solutions are obtained it shows that the 2 1 dimensional ito equation has richer solutions besides the elastic interactions of the solutions and their corresponding physical meaning are discussed

homoclinic breather wave solutions and doubly periodic wave solutions for coupled kdv equations abstract in this paper we first find out a proper variable transformation by the ideas of painleve expansion then we apply the extended homoclinic test approach to obtain two cycle breathing places wave solutions of eq 1 which describe interactions between two physical waves and these special solutions can be applied to explain the structure of certain physical phenomena thus this method can be applied to the study of other similar nonlinear coupled system

newton kantorovich theorem for a family of modified halley s method under holder continuity conditions in banach space abstract in this study under p holder continuity conditions a family of new modified halley s method with 2 p order is proposed in banach space which is used to solve the nonlinear operator equations the newton kantorovich convergence theorem for the family of new modified halley s method is established under holder continuity conditions the error estimate also is gotten finally two examples are provided to show the application of our theorem

parameter identification of wiener systems with multisegment piecewise linear nonlinearities the paper deals with the identification of wiener systems having multisegment piecewise linear nonlinearities a special form of nonlinearity representation is used in the wiener system description the resulting output equation contains the least possible number of parameters required for the blocks of given system the proposed iterative method enables simultaneous estimation of both the linear block parameters and all the parameters characterizing the nonlinearity i e the slopes of linear segments and the constants determining the partition of domain limits for the constants are the only a priori knowledge required

identification of hammerstein nonlinear armax systems two identification algorithms an iterative least squares and a recursive least squares are developed for hammerstein nonlinear systems with memoryless nonlinear blocks and linear dynamical blocks described by armax carma models the basic idea is to replace unmeasurable noise terms in the information vectors by their estimates and to compute the noise estimates based on the obtained parameter estimates convergence properties of the recursive algorithm in the stochastic framework show that the parameter estimation error consistently converges to zero under the generalized persistent excitation condition the simulation results validate the algorithms proposed

identification of wiener systems with binary valued output observations this work is concerned with identification of wiener systems whose outputs are measured by binary valued sensors the system consists of a linear fir finite impulse response subsystem of known order followed by a nonlinear function with a known parametrization structure the parameters of both linear and nonlinear parts are unknown input design identification algorithms and their essential properties are presented under the assumptions that the distribution function of the noise is known and the nonlinearity is continuous and invertible it is shown that under scaled periodic inputs identification of wiener systems can be decomposed into a finite number of core identification problems the concept of joint identifiability of the core problem is introduced to capture the essential conditions under which the wiener system can be identified with binary valued observations under scaled full rank conditions and joint identifiability a strongly convergent algorithm is constructed the algorithm is shown to be asymptotically efficient for the core identification problem hence achieving asymptotic optimality in its convergence rate for computational simplicity recursive algorithms are also developed

combined frequency prediction error identification approach for wiener systems with backlash and backlash inverse operators wiener systems identification is studied in the presence of possibly infinite order linear dynamics and memory nonlinear operators of backlash and backlash inverse types the latter is laterally bordered with polynomial lines of arbitrary shape it turns out that the borders are allowed to be noninvertible and crossing making possible to account within a unified theoretical framework for memory and memoryless nonlinearities moreover the prior knowledge of the nonlinearity type being backlash or backlash inverse or memoryless is not required using sine excitations and getting benefit from model plurality the initial complex identification problem is made equivalent to two tractable though still nonlinear prediction error problems these are coped with using linear and nonlinear least squares estimators which all are shown to be consistent