a simple solution of the bratu problem a brief survey of the properties and different treatments of the one dimensional 1d and 2d bratu problems is presented different iterative treatments of the resulting nonlinear system of equations are discussed the finite difference treatment of the problem is considered nonstandard finite difference methods with a simple sinusoidal starting function having an appropriate amplitude are recommended bounds on the amplitude for yielding both lower and upper solutions are given

gaussian beams and legendre polynomials as invariants of the time reversal operator for a large rigid cylinder the dort method french acronym for decomposition of the time reversal operator is an active remote sensing technique using an array of antennas for the detection and localization of scatterers this method is based on the singular value decomposition of the interelement response matrix in this paper an analytical expression of the singular vectors due to the reflection from a large rigid cylinder is provided depending on the array aperture two asymptotic regimes are described it is shown that the singular vectors correspond to hermite gaussian modes for large apertures and legendre polynomials for small ones using perturbation theory the corresponding singular values are deduced theoretical predictions are in good agreement with experimental results

extended sufficient semilocal convergence for the secant method we establish new sufficient convergence conditions for the secant method to a locally unique solution of a nonlinear equation in a banach space using our new concept of recurrent functions and combining lipschitz and center lipschitz conditions on the divided difference operator we obtain a new semilocal convergence analysis of the secant method moreover our sufficient convergence conditions expand the applicability of the secant method in cases not covered before dennis 1971 9 hernandez et al 2005 8 laasonen 1969 15 ortega and rheinboldt 1970 11 potra 1982 5 potra 1985 7 schmidt 1978 18 yamamoto 1987 12 wolfe 1978 19 numerical examples are also provided in this study

lyapunov functions for riemann liouville like fractional difference equations abstract discrete memory effects are introduced by fractional difference operators asymptotic stabilities of nonlinear fractional difference equations are investigated in this paper a linear scalar fractional difference equality is utilized lyapunov second direct method is proposed for nonlinear discrete fractional systems asymptotic stability conditions are provided and some examples are given

stability estimates for an inverse scattering problem at high frequencies we consider an inverse scattering problem and its near field approximation at high frequencies we first prove for both problems lipschitz stability results for determining the low frequency component of the potential then we show that in the case of a radial potential supported sufficiently near the boundary infinite resolution can be achieved from measurements of the near field operator in the monotone case

a method for finding sharp error bounds for newton s method under the kantorovic assumptions this paper gives a method for finding sharpa posteriori error bounds for newton s method under the assumptions of kantorovich s theorem on the basis of this method new error bounds are derived and comparison is made among the known bounds of dennis 2 doring 4 gragg tapia 5 kantorovich 6 7 kornstaedt 9 lancaster 10 miel 11 13 moret 14 ostrowski 17 18 potra 19 and potra ptak 20

adomian decomposition method for a reliable treatment of the bratu type equations in this paper we present a framework to determine exact solutions of bratu type equations the algorithm rests mainly on the adomian decomposition method the proposed scheme is illustrated by studying two boundary value problems and an initial value problem of bratu type the first type gives a solution that blows up at the middle of the domain whereas the other equations give bounded solutions

the legendre wavelet method for solving initial value problems of bratu type the aim of this work is to study the legendre wavelets for the solution of initial value problems of bratu type which is widely applicable in fuel ignition of the combustion theory and heat transfer the properties of legendre wavelets together with the gaussian integration method are used to reduce the problem to the solution of nonlinear algebraic equations also a reliable approach for convergence of the legendre wavelet method when applied to a class of nonlinear volterra equations is discussed and an error estimation for the proposed method is also introduced illustrative examples have been discussed to demonstrate the validity and applicability of the technique and the results have been compared with the exact solution we finally show the high accuracy and efficiency of the proposed method

on integral representation of the generalized inverse at s 2 we present a general integral representation for the generalized inverse a t s 2 which extends earlier result on the moore penrose inverse weighted moore penrose inverse and drazin inverse

majorizing sequences for newton s method from initial value problems the most restrictive condition used by kantorovich for proving the semilocal convergence of newton s method in banach spaces is relaxed in this paper providing we can guarantee the semilocal convergence in situations that kantorovich cannot to achieve this we use kantorovich s technique based on majorizing sequences but our majorizing sequences are obtained differently by solving initial value problems

on the local convergence of a family of two step iterative methods for solving nonlinear equations a local convergence analysis for a generalized family of two step secant like methods with frozen operator for solving nonlinear equations is presented unifying earlier methods such as secant s newton chebyshev like steffensen and other new variants the family of iterative schemes is built up where a profound and clear study of the computational efficiency is also carried out numerical examples and an application using multiple precision and a stopping criterion are implemented without using any known root finally a study comparing the order efficiency and elapsed time of the methods suggested supports the theoretical results claimed

auxiliary point on the semilocal convergence of newton s method abstract we use an auxiliary point on the semilocal convergence of newton s method when the majorant principle of kantorovich is applied to operators with high order derivatives satisfying a center lipschitz type condition so that we extend the classical conditions of these types that are centered at the starting point of newton s method to other points belonging to the domain of definition of the operator involved this extension provides a modification of the domain of starting points for newton s method which allows increasing the choice of starting points we illustrate this study with nonlinear fredholm integral equations

a variant of the newton kantorovich theorem for nonlinear integral equations of mixed hammerstein type abstract we study nonlinear integral equations of mixed hammerstein type using newton s method as follows we investigate the theoretical significance of newton s method to draw conclusions about the existence and uniqueness of solutions of these equations after that we approximate the solutions of a particular nonlinear integral equation by newton s method for this we use the majorant principle which is based on the concept of majorizing sequence given by kantorovich and milder convergence conditions than those of kantorovich actually we prove a semilocal convergence theorem which is applicable to situations where kantorovich s theorem is not

a semilocal convergence analysis for directional newton methods a semilocal convergence analysis for directional newton methods in n variables is provided in this study using weaker hypotheses than in the elegant related work by y levin and a ben israel and introducing the center lipschitz condition we provide under the same computational cost as in levin and ben israel a semilocal convergence analysis with the following advantages weaker convergence conditions larger convergence domain finer error estimates on the distances involved and an at least as precise information on the location of the zero of the function a numerical example where our results apply to solve an equation but not the ones in levin and ben israel is also provided in this study

an analytic approach to solve multiple solutions of a strongly nonlinear problem based on a new kind of analytic method namely the homotopy analysis method an analytic approach to solve multiple solutions of strongly nonlinear problems is described by using gelfand equation as an example its validity is verified by comparing the approximation series with the known exact solution and different from perturbation techniques this approach is independent upon any small large perturbation quantities so the basic ideas of this approach can be employed to search for multiple solutions of strongly nonlinear problems in science and engineering

the generalized polarization tensors for resolved imaging part ii shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements this paper deals with the problem of reconstructing the electromagnetic parameters and the shape of a target from multi static response matrix measurements at a single frequency the target is of characteristic size less than the operating wavelength using long wavelength asymptotic expansions of the measurements of high order we show how the electromagnetic parameters and the equivalent ellipse of the target can be reconstructed the asymptotic expansions of the measurements are written in terms of the new concept of frequency dependent polarization tensors moreover we extend the optimization approach proposed in part i 9 to image geometric details of an electromagnetic target that are finer than the equivalent ellipse the equivalent ellipse still provides a good initial guess for the optimization procedure however compared to the conductivity case in 9 the cost functional measures the discrepancy between the computed and measured high order frequency dependent polarization tensors rather than between the generalized polarization tensors the main reason for such a modification of the cost functional is the fact that the measured frequency dependent polarization tensors can be easily obtained from multistatic measurements by solving a linear system while the derivation of the generalized polarization tensors from measurements requires more delicate treatment the proposed methods are numerically implemented to demonstrate their validity and efficiency mathematics subject classification msc2000 35r30 35b30

statistically stable ultrasonic imaging in random media analysis of array data from acoustic scattering in a random medium with a small number of isolated targets is performed in order to image and thereby localize the spatial position of each target because the host medium has random fluctuations in wave speed the background medium is itself a source of scattered energy it is assumed however that the targets are sufficiently larger and or more reflective than the background fluctuations so that a clear distinction can be made between targets and background scatterers in numerical simulations nonreflective boundary conditions are used so as to isolate the effects of the host randomness from those of the spatial boundaries which can then be treated in a separate analysis it is shown that the key to successful imaging is finding statistically stable functionals of the data whose extreme values provide estimates of scatterer locations the best ones are related to the eigenfunctions and eigenvalues of the array response matrix just as one might expect from prior work on array data processing in complex scattering media having homogeneous backgrounds the specific imaging functionals studied include matched field processing and linear subspace methods such as music mutiple signal classification but statistical stability is not characteristic of the frequency domain which is often the province of these methods by transforming back into the time domain after first diagonalizing the array data in the frequency domain one can take advantage of both the time domain stability and the frequency domain orthogonality of the relevant eigenfunctions

the majorant method in the theory of newton kantorovich approximations and generalized lipschitz conditions we provide a semilocal as well as a local convergence analysis for newton s and modified newton s methods in order to approximate a locally unique solution of a nonlinear equation in a banach space setting we use more precise majorizing sequences than in earlier studies such as appell et al 1997 appell et al 1991 argyros 2004 argyros and hilout 2009 kantorovich and akilov 1982 ortega and rheinboldt 1970 and generalized lipschitz continuity conditions our sufficient convergence conditions are weaker than before and our convergence analysis is tighter special cases and numerical examples are also given in this study

the iterative time reversal process analysis of the convergence the efficiency of a time reversal acoustic mirror to focus on a reflective target through an inhomogeneous media has been demonstrated in a multitarget media the ability of such a mirror to work in an iterative mode in order to focus selectively on the strongest target was shown c prada f wu and m fink j acoust soc am 90 1119 1991 the theory of how the iterative time reversal process is built is based on a matrix formalism and treats the array of l transducers in a given medium as a linear system of l inputs l outputs the system is characterized at each frequency by its transfer matrix k and the time reversal iterative process is then described by a time reversal operator k k because of the reciprocity principle this operator is hermitian the following result is shown if the scattering medium is a set of well resolved targets of different reflectivities then each eigenvector of the operator k k with nonzero eigenvalue corresponds to one of the targets in the set and provides the opt

chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one dimensional bratu equation we introduce two methods for simultaneously approximating both branches of a two branched function using chebyshev polynomials both schemes remove the pernicious convergence wrecking effects of the square root singularity at the limit point where the two branches meet the chebyshev shafer method gives the approximants as the solution to a quadratic equation the mapping algorithm makes a quadratic change of variable for both the only input information is the set of values of f x at a set of discrete points there is little to choose between the two schemes in accuracy but the single expansion mapping method is more flexible in that it can accommodate different ranges on the two branches the eigenrelation for the one dimensional bratu equation is an interesting example because the upper branch is also singular at another point besides the limit point this too can be removed by subtracting the asymptotic solution which is the lambert w function from the upper branch only when the domain of the variable is infinite the quadratic change of variable can still be applied by substituting rational chebyshev functions which are basis functions for an unbounded interval for chebyshev polynomials we illustrate this by approximating the real valued root of the brill quintic u 5 u l l which was first solved by hermite using elliptic modular functions more than a century ago

an improved imaging method for extended targets when we use direct imaging methods for solving inverse scattering problems we observe artificial lines which make it hard to determine the shape of targets we propose a signal space test to study the cause of the artificial lines and we use multiple frequency data to reduce the effect from them finally we use the active contour without edges acwe method to further improve the imaging results the final reconstruction is accurate and robust the computational cost is lower than iterative imaging methods which need to solve the forward problem in each iteration

starting points for newton s method under a center lipschitz condition for the second derivative abstract we analyze the semilocal convergence of newton s method under a center lipschitz condition for the second derivative of the operator involved different from that used by other authors until now in particular we propose to center the lipschitz condition for the second derivative in a different point from that where newton s method starts this allows us to obtain different starting points for newton s method and modify the domain of starting points

extending the newton kantorovich hypothesis for solving equations the famous newton kantorovich hypothesis kantorovich and akilov 1982 3 argyros 2007 2 argyros and hilout 2009 7 has been used for a long time as a sufficient condition for the convergence of newton s method to a solution of an equation in connection with the lipschitz continuity of the frechet derivative of the operator involved here using lipschitz and center lipschitz conditions and our new idea of recurrent functions we show that the newton kantorovich hypothesis can be weakened under the same information moreover the error bounds are tighter than the corresponding ones given by the dominating newton kantorovich theorem argyros 1998 1 2 7 ezquerro and hernandez 2002 11 3 proinov 2009 2010 16 17 numerical examples including a nonlinear integral equation of chandrasekhar type chandrasekhar 1960 9 as well as a two boundary value problem with a green s kernel argyros 2007 2 are also provided in this study

an efficient method for solving bratu equations in this paper we present a numerical technique for solving bratu equation it is based on the laplace adomain decomposition method which produces an implicit equation in two variables we used the predictor corrector technique to trace the solution curve generated from this equation numerical results and conclusions will be presented

kantorovich s type theorems for systems of equations with constant rank derivatives the famous newton kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of newton s method to a solution of an equation here we present a kantorovich type convergence analysis for the gauss newton s method which improves the result in w m hauszler a kantorovich type convergence analysis for the gauss newton method numer math 48 1986 119 125 and extends the main theorem in i k argyros on the newton kantorovich hypothesis for solving equations j comput appl math 169 2004 315 332 furthermore the radius of convergence ball is also obtained

on the newton kantorovich hypothesis for solving equations the famous newton kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of newton s method to a solution of an equation in connection with the lipschitz continuity of the frechet derivative of the operator involved here using lipschitz and center lipschitz conditions we show that the newton kantorovich hypothesis can be weakened the error bounds obtained under our semilocal convergence result are more precise than the corresponding ones given by the dominating newton kantorovich theorem

a convergence theorem for newton like methods in banach spaces on the basis of the results obtained in a series of papers 25 28 a convergence theorem for newton s method in banach spaces is given which improves the theorems of kantorovich 4 lancaster 8 and ostrowski 10 the error bounds obtained improve the recent results of potra 17

boundedness and convergence on fractional order systems we establish conditions to guarantee boundedness and convergence of signals described by non integer order equations using caputo derivatives the case of linear time varying unforced equations is first studied and later results for linear time varying forced equations and time varying unforced non linear equations are presented and discussed

imaging of location and geometry for extended targets using the response matrix in this paper we study how to image both the location and the shape of extended targets using the response matrix obtained from inter element response of an active array of transducers in particular the time reversal technique is used for efficient initial localization of the target and the level set method is used for shape reconstruction we then show how to couple the location estimation and shape reconstruction in a complementary way to improve accuracy for range estimation resolution analysis for active arrays in remote sensing regime is also presented we illustrate with numerical experiments which show that the method is capable of imaging objects with complicated shapes and that the method is robust with respect to noisy data

newton methods on banach spaces with a convergence structure and applications abstract in this study we use inexact newton like methods to find solutions of nonlinear operator equations on banach spaces with a convergence structure our technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered banach space in this way the metric properties of the examined problem can be analyzed more precisely moreover this approach allows us to derive from the same theorem on the one hand semilocal results of kantorovich type and on the other hand global results based on monotonicity considerations by imposing very general lipschitz like conditions on the operators involved on the one hand we cover a wider range of problems and on the other hand by choosing our operators appropriately we can find sharper error bounds on the distances involved than before finally our methods are used to solve integral equations that cannot be solved with existing methods

analysis of the response matrix for an extended target in this paper we study the response matrix obtained from the interelement response of an active array of transducers that can send out signals and record reflected signals in particular we analyze the eigenvalues and eigenvectors of the response matrix corresponding to the acoustic field reflected by an extended target the size of which is comparable to the wavelength we show that the eigenvalues are not well separated for a single extended target in general however when both the size of the target and the size of the active array are small compared to the distance from the array to the target it is shown that the eigenvalues are well separated and that the leading eigenvalues and eigenvectors can be characterized in terms of the location and dimension of the target numerical experiments are presented to verify the analysis

numerical continuation and the gelfand problem this paper examines a semilinear elliptic equation with nonlinear forcing known as the gelfand equation information about the solution set in terms of the parameter is determined we are specifically interested in determining the first two turning points of the solution continuum the primary technique is a numerical method known as pseudo arclength continuation this is applied to the problem to determine the bifurcation diagram for three dimensional domains

letter to the editor on the integral solution of the one dimensional bratu problem a brief survey of the properties and different treatments of the one dimensional 1d planar bratu problem is presented the treatment using green s function is considered different iterative treatments of the resulting nonlinear system of equations are discussed starting the solution with simple sinusoidal function having appropriate amplitude is recommended then the resulting system is solved using a nonlinear conjugate gradient nl cg method the importance of the proper choice of the starting function as well as the stopping criterion of the iterative solution are stressed

a newton kantorvich theorem for equations involving m frechet differentiable operators and applications in radiative transfer abstract in this study we approximate a locally unique solution of a nonlinear operator equation in banach space using newton s method a complete error analysis showing the quadratic convergence of our method is also given our new theorem uses lipschitz or holder continuity assumptions on m frechet differentiable operators where m 2 is a positive integer a numerical example is given to show that our results provide a better information on the location of the solution as well as finer error bounds on the distances involved than earlier results a second numerical example shows how to solve a nonlinear integral equation appearing in radiative transfer

third order convergence theorem by using majorizing function for a modified newton method in banach space abstract following the ideas of frontini and sormani we present a modified newton method in banach space which is used to solve the nonlinear operator equation we establish the newton kantorovich convergence theorem for the modified newton method with third order convergence in banach space by using majorizing function we also get the error estimate finally two examples are provided to show the application of our theorem

weak convergence conditions for inexact newton type methods abstract we establish a new semilocal convergence results for inexact newton type methods for approximating a locally unique solution of a nonlinear equation in a banach spaces setting we show that our sufficient convergence conditions are weaker and the estimates of error bounds are tighter in some cases than in earlier works 15 31 special cases and numerical examples are also provided in this study

on the sensitivity of time reversal imaging techniques to model perturbations time reversal tr techniques involve physical or synthetic retransmission of signals acquired by a set of transceivers in a time reversed fashion and can be used for a host of applications including ultrawideband imaging of obscured targets this work investigates the performance of some tr based techniques under non ideal conditions and is divided into three main parts in the first part we investigate the robustness of tr based imaging techniques via the decomposition of tr operator dort algorithm and the multiple signal classification music algorithm under clutter and additive noise in the second part we examine the effect of tr invariance breaking due to losses in the intervening media on the performance of both dort and tr based music in the third part effects of translational perturbations of the tr array are examined as a class of model perturbations these ensuing effects are then exploited in a controlled manner as a means to extract second order spatial statistics of the probed media

time reversal imaging for sensor networks with optimal compensation in time using extensive numerical simulations several distributed sensor imaging algorithms for localized damage in a structure are analyzed given a configuration of ultrasonic transducers a full response matrix for the healthy structure is assumed known it is used as a basis for comparison with the response matrix that is recorded when there is damage numerical simulations are done with the wave equation in two dimensions the healthy structure contains many scatterers the aim is to image point like defects with several regularly distributed sensors because of the complexity of the environment the recorded traces have a lot of delay spread and travel time migration does not work so well instead the traces are back propagated numerically assuming that there is some knowledge of the background since the time at which the back propagated field will focus on the defects is unknown the shannon entropy or the bounded variation norm of the image is computed and the time where it is minimal is picked this imaging method performs well because it produces a tight image near the location of the defects at the time of refocusing when there are several defects the singular value decomposition of the response matrix is also carried out

majorizing sequences for iterative methods we provide convergence results for very general majorizing sequences of iterative methods using our new concept of recurrent functions we unify the semilocal convergence analysis of newton type methods ntm under more general lipschitz type conditions we present two very general majorizing sequences and we extend the applicability of ntm using the same information before chen and yamamoto 1989 13 deuflhard 2004 16 kantorovich and akilov 1982 19 miel 1979 20 miel 1980 21 and rheinboldt 1968 30 applications special cases and examples are also provided in this study to justify the theoretical results of our new approach

on newton s method for solving equations containing frechet differentiable operators of order at least two we provide sufficient conditions for the semilocal convergence of newton s method to a locally unique solution of a nonlinear operator equation containing operators that are frechet differentiable of order at least two in a banach space setting numerical examples are also provided to show that our results apply to solve nonlinear equations in cases earlier ones cannot j m gutierrez a new semilocal convergence theorem for newton s method j comput appl math 79 1997 131 145 z huang a note of kantorovich theorem for newton iteration j comput appl math 47 1993 211 217 f a potra sharp error bounds for a class of newton like methods libertas mathematica 5 1985 71 84

on fractional extensions of barbalat lemma a b s t r a c t this paper presents barbalat like lemmas for fractional order integrals which can be used to conclude about the convergence of a function to zero based on some conditions upon its fractional integral some examples in the context of asymptotic behaviour of solutions of fractional order differential equations indicate the potential application of these lemmas in control theory

analysis of the time reversal operator for scatterers of finite size recently it was shown that the time reversal operator for a single small spherical scatterer could have up to four distinguishable eigenstates chambers and gautesen j acoust soc am 109 2616 2624 2001 in this paper that analysis is generalized for scatterers of arbitrary shape and larger size it is shown that the time reversal operator may have an indefinitely large number of distinguishable eigenstates with the exact number depending on the nature of the scatterer and the geometry of the time reversal mirror in addition the case of a multiple number of well separated scatterers is investigated with the result that the total spectrum is the direct combination of the eigenstates associated with each scatterer as an example the singular value spectrum of the time reversal operator for a linear array is calculated explicitly for bubbles and hard rubber spheres of finite size both resonance peaks and apparent crossing points can be observed in the spectrum as the size of the scatterer increases

extending the applicability of newton s method for k frechet differentiable operators in banach spaces we extend the applicability of newton s method for k frechet differentiable operators in a banach space setting by using a more flexible way of computing upper bounds on the inverses of the operators involved in particular we improve and extend the recent works by ezquerro et al 2012 2013 13 15 moreover we illustrate our study with some numerical examples involving hammerstein integral equations

general 2 2 system of nonlinear integral equations and its approximate solution abstract in this note we consider a general 2 2 system of nonlinear volterra type integral equations the modified newton method modified nm is used to reduce the nonlinear problems into 2 2 linear system of algebraic integral equations of volterra type the latter equation is solved by discretization method nystrom method with gauss legendre quadrature is applied for the kernel integrals and newton forwarded interpolation formula is used for finding values of unknown functions at the selected node points existence and uniqueness solution of the problems are proved and accuracy of the quadrature formula together with convergence of the proposed method are obtained finally numerical examples are provided to show the validity and efficiency of the method presented numerical results reveal that the proposed methods is efficient and accurate comparisons with other methods for the same problem are also presented

on the lyapunov theory for fractional order systems abstract we provide the main features of lyapunov theory when it is formulated for fractional order systems we give consistent extensions of lyapunov lasalle and chetaev classical theorems to the case of fractional order systems we give examples to illustrate the applications of the concepts and propositions introduced

on a two step optimal steffensen type method relaxed local and semi local convergence analysis and dynamical stability abstract in this paper we study on the local and semi local convergence and dynamic of an iterative optimal two step steffensen s method we study local convergence by using lipschitz conditions and continuity of differentiable function f we also obtain semi local convergence by using majorizing sequences by this approach we can obtain the convergence of the method by minimum of auxiliary sequences and few initial conditions moreover we analyze the stability of the method via julia sets and parameter planes that is a novelty for steffensen type methods we extend this method for obtaining the sign function of a matrix numerical examples are given to confirm our theoretical results

decomposition of the time reversal operator detection and selective focusing on two scatterers the decomposition of the time reversal operator dort method is a selective detection and focusing technique using an array of transmit receive transducers it relies on the theory of iterative time reversal mirrors which was presented by prada et al c prada j l thomas and m fink j acoust soc am 97 62 71 1995 the time reversal operator was defined as k k where is the frequency means complex conjugate and k is the transfer matrix of the array of l transducers insonifying a time invariant scattering medium it was shown that this time reversal operator can be diagonalized and that for ideally resolved scatterers of different reflectivities each of its eigenvectors of nonzero eigenvalue provides the phase law to be applied to the transducers in order to focus on one of the scatterers the dort method consists in determining these eigenvectors and using them for the selective focusing this paper presents a complete analysis of this method in the case of two scatterers the

on the local convergence of fast two step newton like methods for solving nonlinear equations a local convergence analysis is presented for a fast two step newton like method tsnlm for solving nonlinear equations in a banach space setting the tsnlm unifies earlier methods such as newton s secant newton like chebyshev secant chebyshev newton steffensen stirling s and other single or multistep methods numerical examples and a comparative study of these methods validating our theoretical results are also given in the concluding section of this paper

a weak kantorovich existence theorem for the solution of nonlinear equations the kantorovich theorem is a fundamental tool in nonlinear analysis for proving the existence and uniqueness of solutions of nonlinear equations arising in various fields this theorem was weakened recently by argyros who used a combination of lipschitz and center lipschitz conditions in place of the lipschitz conditions of the kantorovich theorem in the present paper we prove a weak kantorovich type theorem that gives the same conclusions as the previous two results under weaker conditions illustrative examples are provided in the paper

extending the applicability of the local and semilocal convergence of newton s method we present a local as well a semilocal convergence analysis for newton s method in a banach space setting using the same lipschitz constants as in earlier studies we extend the applicability of newton s method as follows local case a larger radius is given as well as more precise error estimates on the distances involved semilocal case the convergence domain is extended the error estimates are tighter and the information on the location of the solution is at least as precise as before numerical examples further justify the theoretical results

extending the domain of starting points for newton s method under conditions on the second derivative abstract in this paper we propose a center lipschitz condition for the second frechet derivative together with the use of restricted domains in order to improve the domain of starting points for newton s method in addition we compare the new result with an older one and see that the former improves the latter

dynamics of the solution of bratu s equation abstract in the present report we examine the dynamics exhibited by the solution of bratu s equation it represents a one dimensional map with control parameter for certain values of the parameter it exhibits successive bifurcations and shows chaotic regimes this behaviour was confirmed by calculating the corresponding lyapunov exponent power spectra and cobweb diagrams indicating similarities with other well known one dimensional maps

local convergence of inexact newton methods a simple affine invariant condition which ensures local convergence of inexact newton methods is provided this condition is closely linked with the characterization of superlinear convergence given by dennis and more radius of convergence and rate of convergence results are derived and we show the theory is in some sense sharp we apply the theory to various inexact newton methods including difference newton like methods and the general newton like method in which the iterates are perturbed by rounding errors in the computation

a phase and space coherent direct imaging method a direct imaging algorithm for point and extended targets is presented the algorithm is based on a physical factorization of the response matrix of a transducer array the factorization is used to transform a passive target problem to an active source problem and to extract principal components tones in a phase consistent way the multitone imaging function can superpose multiple tones spatial diversity aperture of the array and frequencies bandwidth of the probing signal based on phase coherence the method is a direct imaging algorithm that is simple and efficient since no forward solver or iteration is needed robustness of the algorithm with respect to noise is demonstrated via numerical examples