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

a unifying local semilocal convergence analysis and applications for two point newton like methods in banach space abstract we provide a local as well as a semilocal convergence analysis for two point newton like methods in a banach space setting under very general lipschitz type conditions our equation contains a frechet differentiable operator f and another operator g whose differentiability is not assumed using more precise majorizing sequences than before we provide sufficient convergence conditions for newton like methods to a locally unique solution of equation f x g x 0 in the semilocal case we show under weaker conditions that our error estimates on the distances involved are finer and the information on the location of the solution at least as precise as in earlier results in the local case a larger radius of convergence is obtained several numerical examples are provided to show that our results compare favorably with earlier ones as a special case we show that the famous newton kantorovich hypothesis is weakened under the same hypotheses as the ones contained in the newton kantorovich theorem

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

secant like methods for solving nonlinear integral equations of the hammerstein type we consider a one parametric family of secant type iterations for solving nonlinear equations in banach spaces we establish a semilocal convergence result for these iterations by means of a technique based on a new system of recurrence relations this result is then applied to obtain existence and uniqueness results for nonlinear integral equations of the hammerstein type we also present a numerical example where the solution of a particular hammerstein integral equation is approximated by different secant type methods

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

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

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

almost sure exponential stability of numerical solutions for stochastic delay differential equations using techniques based on the continuous and discrete semimartingale convergence theorems this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations sddes the important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of sddes directly this is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the chebyshev inequality and the borel cantelli lemma

exponential stability in p th mean of solutions and of convergent euler type solutions of stochastic delay differential equations one concept of the stability of a solution of an evolutionary equation relates to the sensitivity of the solution to perturbations in the initial data there are other stability concepts notably those concerned with persistent perturbations results are presented on the stability in p th mean of solutions of stochastic delay differential equations with multiplicative noise and of stochastic delay difference equations the difference equations are of a type found in numerical analysis and we employ our results to obtain mean square stability criteria for the solution of the euler maruyama discretization of stochastic delay differential equations the analysis proceeds as follows we show that an inequality of halanay type derivable via comparison theory can be employed to derive conditions for p th mean stability of a solution we then produce a discrete analogue of the halanay type theory that permits us to develop a p th mean stability analysis of analogous stochastic difference equations the application of the theoretical results is illustrated by deriving mean square stability conditions for solutions and numerical solutions of a constant coefficient linear test equation

extensions of some infinite algebras in space time in this paper the operator extensions for infinite algebras of residual gauge transformations and space time transformations in physical space time are discussed on general grounds possible general forms of the extensions are presented and their realizations in quantum field theories are discussed the meaning of nontriviality for the extensions will also be discussed in particular cohomology methods are used to calculate the extension of spatial diffeomorphism algebra when chiral fermions are coupled to both an external u 1 gauge field and gravity

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

an analysis of stability of milstein method for stochastic differential equations with delay this paper deals with the adapted milstein method for solving linear stochastic delay differential equations it is proved that the numerical method is mean square ms stable under suitable conditions the obtained result shows that the method preserves the stability property of a class of linear constant coefficient problems this is also verified by several numerical examples

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

euclidean space time diffeomorphisms and their fueter subgroups holomorphic fueter functions of the position quaternion form a subgroup of euclidean space time diffeomorphisms an o 4 covariant treatment of such mappings is presented with the quaternionic argument x being replaced by either px or xp involving self dual and anti self dual structures and p denoting an arbitrary euclidean time direction an infinite group the quasiconformal group is exhibited that admits the conformal group so 5 1 as a subgroup in analogy to the two dimensional case in which the mobius group so 3 1 is a subgroup of the infinite virasoro group the ensuing 3 1 covariant decomposition of diffeomorphisms suggests covariant gauges that throw the metric and the stress tensors in standard forms suitable for canonical quantization leading to improved energy momentum tensors other possible applications to current algebra and gravity will be mentioned

ms stability of the euler maruyama method for stochastic differential delay equations the aim of this paper is to investigate the stability of the euler maruyama method for the stochastic differential equations with time delay the definition of ms stability of numerical methods is established the condition of the mean square stability of analytical solution is obtained for a linear scalar system with multiplicative noise and the ms stability of the numerical scheme is proved

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

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

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

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

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

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

a discretization scheme for some conservative problems we approximate a locally unique solution of an equation in banach spaces using a newton like method of r order three then we apply this method to obtain an existence uniqueness result for a basic conservative problem given by a nonlinear boundary value problem next by means of a discretization method we approximate the solution of the conservative problem

algebras of diffeomorphisms of the n torus the general algebra d n v of infinitesimal diffeomorphisms of the n torus s1 n involving generators lv m depending on a structure vector v cn and a vector valued index m zn is constructed several results are proved for this algebra special cases of the algebra were previously presented in ramos shrock e ramos and r e shrock int j mod phys a 4 4279 1989 the concept of the q rank of the space v of structure vectors denoted rq v is defined and certain related cartan matrices are introduced it is shown that d n v as an ungraded algebra is simple if and only if rq v n i e the q rank of v is maximal a classification under isomorphisms is given for the algebra and is shown to reduce to a classification of the cartan matrices the space of structure vectors is isomorphic to the unique cartan subalgebra a number of properties concerning the central extensions of these algebras are then proved for the case dim v 1 rq v n it is shown that the central extension is unique fo

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

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

expanding the applicability of newton s method using smale s α theory we present a tighter convergence analysis than earlier studies such as in cianciaruso 2007 guo 2007 shen and li 2010 smale 1986 1987 wang and zhao 1995 wang 1999 wang and han 1990 of newton s method using smale s a theory by introducing the notion of the center c 0 condition in particular in the semilocal convergence case we show that if the center c 0 condition is smaller than the c condition then the new majorizing sequence is tighter than the old majorizing sequence the new convergence criteria are weaker than the older convergence criteria furthermore in the local convergence case we obtain a larger radius of convergence and tighter error estimates on the distances involved these improvements are obtained under the same computational cost numerical examples and applications are also provided in this study to show that the older results cannot apply but the new results apply to solve equations

concerning the terra incognita between convergence regions of two newton methods abstract the majorizing principle is used to show local and semilocal convergence of newton methods to a locally unique solution of a nonlinear operator in a banach space when the frechet derivative of the operator involved satisfies a center holder and a holder continuity condition then we investigate an unknown area terra incognita between the convergence regions of newton s method and the corresponding modified newton s method our approach compares favorably with other corresponding ones in this direction

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

a steffensen s type method in banach spaces with applications on boundary value problems in this paper a modified steffensen s type iterative scheme for the numerical solution of a system of nonlinear equations is studied two convergence theorems are presented the numerical solution of boundary value problems by the multiple shooting method using the proposed iterative scheme is analyzed

split step theta method for stochastic delay integro differential equations with mean square exponential stability abstract in this paper we propose the split step theta method for stochastic delay integro differential equations by the lagrange interpolation technique and investigate the mean square exponential stability of the proposed scheme it is shown that the split step theta method can inherit the mean square exponential stability of the continuous model under the linear growth condition and the proposed stability condition by the delayed differential and difference inequalities established in the paper a numerical example is given at the end of the paper to illustrate the method and conclusion of the paper in addition the convergence of the split step theta method is proved in the appendix

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

weak convergence conditions for the newton s method in banach space using general majorizing sequences using the majorizing principle we develop sufficient convergence conditions for the semilocal convergence of newton s method in a banach space setting the involved majorizing sequences are very general and as special cases include most relevant sequences from the published literature our convergence conditions can be weaker and our error analysis can be finer than before

on a class of newton like methods for solving nonlinear equations we provide a semilocal convergence analysis for a certain class of newton like methods considered also in i k argyros a unifying local semilocal convergence analysis and applications for two point newton like methods in banach space j math anal appl 298 2004 374 397 i k argyros computational theory of iterative methods in c k chui l wuytack eds series studies in computational mathematics vol 15 elsevier publ co new york usa 2007 j e dennis toward a unified convergence theory for newton like methods in l b rall ed nonlinear functional analysis and applications academic press new york 1971 in order to approximate a locally unique solution of an equation in a banach space using a combination of lipschitz and center lipschitz conditions instead of only lipschitz conditions f a potra sharp error bounds for a class of newton like methods libertas math 5 1985 71 84 we provide an analysis with the following advantages over the work in f a potra sharp error bounds for a class of newton like methods libertas math 5 1985 71 84 which improved the works in w e bosarge p l falb a multipoint method of third order j optimiz theory appl 4 1969 156 166 w e bosarge p l falb infinite dimensional multipoint methods and the solution of two point boundary value problems numer math 14 1970 264 286 j e dennis on the kantorovich hypothesis for newton s method siam j numer anal 6 3 1969 493 507 j e dennis toward a unified convergence theory for newton like methods in l b rall ed nonlinear functional analysis and applications academic press new york 1971 h j kornstaedt ein allgemeiner konvergenzstaz fu r verscha rfte newton verfahrem in isnm vol 28 birkhau ser verlag basel and stuttgart 1975 pp 53 69 p laasonen ein uberquadratisch konvergenter iterativer algorithmus ann acad sci fenn ser i 450 1969 1 10 f a potra on a modified secant method l analyse numerique et la theorie de l approximation 8 2 1979 203 214 f a potra an application of the induction method of v ptak to the study of regula falsi aplikace matematiky 26 1981 111 120 f a potra on the convergence of a class of newton like methods in iterative solution of nonlinear systems of equations in lecture notes in mathematics vol 953 springer verlag new york 1982 f a potra v ptak nondiscrete induction and double step secant method math scand 46 1980 236 250 f a potra v ptak on a class of modified newton processes numer funct anal optim 2 1 1980 107 120 f a potra sharp error bounds for a class of newton like methods libertas math 5 1985 71 84 j w schmidt untere fehlerschranken fur regula falsi verfahren period math hungar 9 3 1978 241 247 j w schmidt h schwetlick ableitungsfreie verfhren mit hoherer konvergenzgeschwindifkeit computing 3 1968 215 226 j f traub iterative methods for the solution of equations prentice hall englewood cliffs new jersey 1964 m a wolfe extended iterative methods for the solution of operator equations numer math 31 1978 153 174 larger convergence domain and weaker sufficient convergence conditions numerical examples further validating the results are also provided

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

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

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

convergence of newton like methods for singular operator equations using outer inverses we present a semilocal kantorovich type analysis for newton like methods for singular operator equations using outer inverses we establish sharp generalizations of the kantorovich theory and the mysovskii theory for operator equations when the derivative is not necessarily invertible the results reduce in the case of an invertible derivative to well known theorems of kantorovich and mysovskii with no additional assumptions unlike earlier theorems which impose strong conditions the strategy of the analysis is based on banach type lemmas and perturbation bounds for outer inverses which show that the set of outer inverses to a given bounded linear operator admits selections that behave like bounded linear inverses in contrast to inner inverses or generalized inverses which do not depend continuously on perturbations of the operator we give two examples to illustrate our results and compare them with earlier results and another numerical example to relate our results to computational issues

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