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

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

the root and bell s disk iteration methods are of the same error propagation characteristics in the simultaneous determination of the zeros of a polynomial part ii round off error analysis by use of interval arithmetic in part i ikhile 2008 4 it was established that the root and bell s disk point iteration methods with or without correction term are of the same asymptotic error propagation characteristics in the simultaneous determination of the zeros of a polynomial this concluding part of the investigation is a study in round offs its propagation and its effects on convergence employing interval arithmetic means the purpose is to consequently draw attention on the effects of round off errors introduced from the point arithmetic part on the rate of convergence of the generalized root and bell s simultaneous interval iteration algorithms and its enhanced modifications introduced in part i for the numerical inclusion of all the zeros of a polynomial simultaneously the motivation for studying the effects of round off error propagation comes from the fact that the readily available computing devices at the moment are limited in precision more so that accuracy expected from some programming or computing environments or from these numerical methods are or can be machine dependent in fact a part of the finding is that round off propagation effects beyond a certain controllable order induces overwhelmingly delayed or even a severely retarded convergence speed which manifest glaringly as poor accuracy of these interval iteration methods in the computation of the zeros of a polynomial simultaneously however in this present consideration and even in the presence of overwhelming influence of round offs we give conditions under which convergence is still possible and derive the error round off relations along with the order r order of convergence of these methods with the results extended to similar interval iteration methods for computing the zeros of a polynomial simultaneously especially to bell s interval methods for refinement of zeros that form a cluster our findings are instructive and quite revealing and supported by evidence from numerical experiments the analysis is preferred in circular interval arithmetic

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

some logical metatheorems with applications in functional analysis in previous papers we have developed proof theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis here uniform means independence from parameters in compact spaces a recent case study in fixed point theory systematically yielded uniformity even w r t parameters in metrically bounded but noncompact subsets which had been known before only in special cases in the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all our theorems guarantee under general logical conditions such strong uniform versions of non uniform existence statements moreover they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result our metatheorems deal with general classes of spaces like metric spaces hyperbolic spaces cat 0 spaces normed linear spaces uniformly convex spaces as well as inner product spaces

a concept of convergence in geodesic spaces abstract a cat 0 space is a geodesic space for which each geodesic triangle is at least as thin as its comparison triangle in the euclidean plane a notion of convergence introduced independently several years ago by lim and kuczumow is shown in cat 0 spaces to be very similar to the usual weak convergence in banach spaces in particular many banach space results involving weak convergence have precise analogues in this setting at the same time many questions remain open

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

on δ convergence theorems in cat 0 spaces by using the concept of convergence introduced by lim t c lim remarks on some fixed point theorems proc amer math soc 60 1976 179 182 we are able to give the cat 0 space analogs of results on weak convergence of the picard mann and ishikawa iterates proved in uniformly convex banach spaces by opial z opial weak convergence of the sequence of successive approximations for nonexpansive mappings bull amer math soc 73 1967 591 597 ishikawa s ishikawa fixed points and iteration of a nonexpansive mapping in a banach space proc amer math soc 59 1976 65 71 and tan and xu k k tan h k xu approximating fixed points of nonexpansive mappings by the ishikawa iteration process j math anal appl 178 1993 301 308

convergence of a general algorithm of asymptotically nonexpansive maps in uniformly convex hyperbolic spaces abstract in this paper we establish convergence theorems for a general algorithm of an asymptotically nonexpansive map in a uniformly convex hyperbolic space our results generalize simultaneously the approximation results of rhoades 1994 18 suantai 2005 20 and xu and noor 2002 26 on a nonlinear domain our results are refinements and generalizations of the corresponding ones in uniformly convex banach spaces and cat 0 spaces

some improved inclusion methods for polynomial roots with weierstrass corrections one decade ago the third order method without derivatives for the simultaneous inclusion of simple zeros of a polynomial was proposed in 1 following nourein s idea 2 some modifications of this method with the increased convergence are proposed the acceleration of convergence is attained by using weierstrass corrections without additional calculations which provides a high computational efficiency of the modified methods it is proved that their r orders of convergence are asymptotically greater than 3 5 the presented interval methods are realized in circular complex arithmetic

convergence theorems for fixed points of uniformly quasi lipschitzian mappings in convex metric spaces abstract in this paper we consider an ishikawa type iteration process with errors to approximate the fixed point of two uniformly quasi lipschitzian mappings in convex metric spaces some results have been obtained which generalize and unify many important known results in recent literature

improved halley like methods for the inclusion of polynomial zeros improved iterative methods of halley s type for the simultaneous inclusion of all simple complex zeros of a polynomial are proposed the presented convergence analysis which uses the concept of the r order of convergence of mutually dependent sequences shows that the convergence rate of the basic fourth order method is increased to 5 and 6 using newton s and halley s corrections respectively the proposed inclusion methods possess a high computational efficiency since the increase of the convergence is attained without additional calculations this advantage together with very fast convergence make that the presented methods are ranking among the most powerful inclusion methods for polynomial zeros in order to demonstrate convergence properties of the proposed methods two numerical examples are given

mann iteration process for asymptotic pointwise nonexpansive mappings in metric spaces abstract let m d be a complete 2 uniformly convex metric space let c be a nonempty bounded closed and convex subset of m and let t c c be an asymptotic pointwise nonexpansive mapping in this paper we prove that the modified mann iteration process defined by x n 1 t n t n x n 1 t n x n converges in a weaker sense to a fixed point of t

convergence of an ishikawa type iterative scheme for asymptotically quasi nonexpansive mappings in convex metric spaces the ishikawa iteration process and the ishikawa iteration process with errors is defined for asymptotically quasi nonexpansive mappings it is proved some sufficient and necessary conditions for the iterative scheme converges to the fixed point of the asymptotically quasi nonexpansive mappings these results generalize and unify many important known results in recent literature

on an iterative method for simultaneous inclusion of polynomial complex zeros abstract starting from disjoint discs which contain polynomial complex zeros the iterative interval method of the third order for the simultaneous finding inclusive discs for complex zeros is formulated the lagrangean interpolation formula and complex circular arithmetic are used the convergence theorem and the conditions for convergence are considered the proposed method has been applied for solving an algebraic equation

approximating fixed points of a pair of contractive type mappings in generalized convex metric spaces in this paper we consider an ishikawa type iteration process with errors which converges to the unique common fixed point of a pair of contractive type mappings in complete generalized convex metric spaces furthermore we obtain the corresponding results in banach spaces our results extend and generalize many known results

strong convergence of a general iteration scheme in cat 0 spaces abstract we introduce and study strong convergence of a general iteration scheme for a finite family of asymptotically quasi nonexpansive maps in convex metric spaces and c a t 0 spaces the new iteration scheme includes modified mann and ishikawa iterations the three step iteration scheme of xu and noor and the scheme of khan domlo and fukhar ud din as special cases in banach spaces our results are refinements and generalizations of several recent results from the current literature

the root and bell s disk iteration methods are of the same error propagation characteristics in the simultaneous determination of the zeros of a polynomial part i correction methods in this paper we consider the error propagation of the root and bell s disk iteration methods enhanced by incorporating a correction term and a choice of a disk inversion formula in the methods for the simultaneous computation of the zeros of a polynomial the asymptotic error propagation is proved to be the same in both methods this result is important considering the fact that these methods are in popular usage in the simultaneous computation of the zeros of a polynomial the proof of the results herein follows the ideas of m s petkovic c carstensen some improved inclusion methods for polynomial roots with weierstrass corrections comput math appl 25 3 1993 59 67 when the refinement process of correction is efficient it is this mode of correction we have desired to propose

numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors we study numerically the complex domains of validity for kam theory in generalized standard mappings we compare methods based on pade approximants and methods based on the study of periodic orbits

strong convergence theorems with a noor type iterative scheme in convex metric spaces the paper introduces some new mappings in convex metric spaces and then it considers a noor type iterative scheme to approximate common fixed points of an infinite family of uniformly quasi sup f n lipschitzian mappings and an infinite family of g n expansive mappings in convex metric spaces its results generalize improve and unify some results in chang et al 2010 13 fukhar ud din and khan 2007 14 liu et al 2010 17 tian 2005 7 wang and liu 2009 19 and wang et al 2009 20 under some appropriate conditions