calculation of exact bounds for the solution set of linear interval systems abstract this paper presents some topological and graph theoretical properties of the solution set of linear algebraic systems with interval coefficients based on these properties we describe a method which in a finite number of steps either calculates exact bounds for each component of the solution set or finds a singular matrix within the interval coefficients the calculation of exact bounds of the solution set is known to be np hard our method needs p calls of a polynomial time algorithm where p is the number of nonempty intersections of the solution set with the orthants frequently due to physical or economical requirements many variables do not change the sign in those cases p is small and our method works efficiently

strong convergence of modified mann iterations abstract the mann iterations for nonexpansive mappings have only weak convergence even in a hilbert space we propose two modifications of the mann iterations in a uniformly smooth banach space one for nonexpansive mappings and the other for the resolvent of accretive operators the two modified mann iterations are proved to have strong convergence

weak almost convergence theorem without opial s condition abstract let e be banach space with property u m m 1 r m n and a uniformly gateaux differentiable norm j e e a duality mapping d a nonempty closed convex bounded subset of e and t d d a uniformly l lipschitzian asymptotically hemicontractive mapping with l

fixed point theorem for generalized φ pseudocontractive mappings abstract let e be a real banach space c be a nonempty closed convex subset of e and t c c be a continuous generalized pseudocontractive mapping it is proved that t has a unique fixed point in c

towards ideal multistart a stochastic approach for locating the minima of a continuous function inside a bounded domain a stochastic global optimization method based on multistart is presented in this the local search is conditionally applied with a probability that takes in account the topology of the objective function at the detail offered by the current status of exploration as a result the number of unnecessary local searches is drastically limited yielding an efficient method results of its application on a set of common test functions are reported along with a performance comparison against other established methods of similar nature

strong convergence of modified mann iterations for a countable family of nonexpansive mappings abstract in this paper we propose a new modified mann iteration for computing common fixed points of nonexpansive mappings in a banach space we give certain different control conditions for the modified mann iteration then we prove strong convergence theorems for a countable family of nonexpansive mappings in uniformly smooth banach spaces these results improve and extend results of kim and xu t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 2005 51 60 yao et al y yao r chen and j yao strong convergence and certain control conditions for modified mann iteration nonlinear anal 68 2008 1687 1693 qin and su x qin y su approximation of a zero point of accretive operator in banach spaces j math anal appl 329 2007 415 424 and many others

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

checking bounds on solutions of linear interval equations is np hard abstract we prove that it is np hard to decide whether the solution set of a system of linear interval equations is contained in a given interval vector even in the case that the system matrix is strongly regular

the numerical solution of diffusion problems in strongly heterogeneous non isotropic materials a new second order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non isotropic media is constructed on problems with rough coefficients or highly nonuniform grids the new algorithm is superior to all other algorithms we have compared it with for problems with smooth coefficients on smooth grids the method is comparable with other second order methods the new algorithm is formulated for logically rectangular grids and is derived using thesupport operators method a key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator defined as the material properties tensor times the gradient rather than the gradient as one of the basic first order operators in the support operators method the discrete analog of the flux operator must also be the negative adjoint of the discrete divergence in an inner product that is a discrete analog of the continuum inner product the resulting method is conservative and the discrete analog of the variable coefficient laplacian is symmetric and negative definite on nonuniform grids in addition on any grid the discrete divergence is zero on constant vectors the null space for the gradient is the constant functions and when the material properties are piecewise constant the discrete flux operator is exact for piecewise linear functions we compare the methods on some of the most difficult examples to be found in the literature

strong convergence and certain control conditions for modified mann iteration abstract in this paper we propose a new modified mann iteration for computing fixed points of nonexpansive mappings in a banach space setting this new iterative scheme combines the modified mann iteration introduced by kim and xu t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 2005 51 60 and the viscosity approximation method introduced by moudafi a moudafi viscosity approximation methods for fixed points problems j math anal appl 241 2000 46 55 we give certain different control conditions for the modified mann iteration strong convergence in a uniformly smooth banach space is established

large sparse linear systems arising from mimetic discretization in this work we perform an experimental study of iterative methods for solving large sparse linear systems arising from a second order 2d mimetic discretization the model problem is the 2d poisson equation with different boundary conditions we use gmres with the restarted parameter and bicgstab as iterative methods we also use various preconditioning techniques including the robust preconditioner ilut the numerical experiments consist of large sparse linear systems with up to 643200 degrees of freedom

on optimal solution of interval linear equations for interval linear algebraic systems bf a x bf b we consider the problem of component wise evaluation of the set sigma exists exists bf a bf b a 1 b mid a in bf a b in bf b formed by all solutions of ax b when a and b vary independently in bf a and bf b respectively an iterative pss algorithm is introduced that computes optimal exact componentwise estimates of sigma exists exists and its convergence is proved under fairly general conditions on the interval system we introduce the concept of a sequentially guaranteeing algorithm as a reasonable compromise between the requirements for the interval result to be guaranteed and to be obtained in a practically acceptable time

approximation of boundary conditions for mimetic finite difference methods abstract the numerical solution of partial differential equations solved with finite difference approximations that mimic the symmetry properties of the continuum differential operators and satisfy discrete versions of the appropriate integral identities are more likely to produce physically faithful results furthermore those properties are often needed when using the energy method to prove convergence and stability of a particular difference approximation unless special care is taken mimetic difference approximations derived for the interior grid points will fail to preserve the symmetries and identities between the gradient curl and divergence operators at the computational boundary in this paper we describe how to incorporate boundary conditions into finite difference methods so the resulting approximations mimic the identities for the differential operators of vector and tensor calculus the approach is valid for a wide class of partial differential equations of mathematical physics and will be described for poisson s equation with dirichlet neumann and robin boundary conditions we prove that the resulting difference approximation is symmetric and positive definite for each of these boundary conditions

strong convergence theorems for uniformly continuous pseudocontractive maps abstract let k be a nonempty closed convex subset of a real banach space e and let t k k be a uniformly continuous pseudocontraction fix any u k let x n be defined by the iterative process x 0 k x n 1 n n t x n 1 n x n 1 n u let denote the modulus of continuity of t with pseudo inverse if t t 0 t 1 and x n are bounded then under some mild conditions on the sequences n n and n n the strong convergence of x n to a fixed point of t is proved in the special case where t is lipschitz it is shown that the boundedness assumptions on t t 0 t 1 and x n can be dispensed with

retracted a one step iterative process for two multivalued nonexpansive mappings in banach spaces weak and strong convergence a b s t r a c t in this paper we introduce a new one step iterative process to compute the common fixed points of two multivalued nonexpansive mappings furthermore we also prove some strong and weak convergence theorems in uniformly convex banach spaces

iterative solution of nonlinear equations involving set valued uniformly accretive operators abstract let e be a real normed linear space and let a e 2 e be a uniformly continuous and uniformly quasi accretive multivalued map with nonempty closed values such that the range of i a is bounded and the inclusion f ax has a solution x e it is proved that ishikawa and mann type iteration processes converge strongly to x further if t e 2 e is a uniformly continuous and uniformly hemicontractive set valued map with bounded range and a fixed point x e it is proved that both the mann and ishikawa type iteration processes converge strongly to x the strong convergence of these iteration processes with errors is also proved

strong convergence of an iterative method for nonexpansive and accretive operators abstract let x be a banach space and a an m accretive operator with a zero consider the iterative method that generates the sequence x n by the algorithm x n 1 n u 1 n j r n x n where n and r n are two sequences satisfying certain conditions and j r denotes the resolvent i r a 1 for r 0 strong convergence of the algorithm x n is proved assuming x either has a weakly continuous duality map or is uniformly smooth

solving interval linear systems with linear programming techniques abstract in this paper we show how it is possible to use convex polyhedra for solving linear interval systems without using preconditioning we first show how to derive from an enclosure of a b a polyhedron which contains the convex hull of the solution set then a simplex like method enables us to find a new outer inclusion moreover the constraints obtained may be used to compute an inner inclusion of a b

existence theorems for m accretive operators in banach spaces abstract in 1985 the second author proved a surjective result for m accretive and expansive mappings for uniformly smooth banach spaces however in this case we have been able to remove the uniform smoothness of the banach space without any additional assumption

the convergence of mimetic discretization for rough grids we prove that the mimetic finite difference discretizations of laplace s equation converges on rough logically rectangular grids with convex cells mimetic discretizations for the invariant operators divergence gradient and curl satisfy exact discrete analogs of many of the important theorems of vector calculus the mimetic discretization of the laplacian is given by the composition of the discrete divergence and gradient we first construct a mimetic discretization on a single cell by geometrically constructing inner products for discrete scalar and vector fields then constructing a finite volume discrete divergence and then constructing a discrete gradient that is consistent with the discrete divergence theorem this construction is then extended to the global grid we demonstrate the convergence for the two dimensional laplace equation with dirichlet boundary conditions on grids with a lower bound on the angles in the cell corners and an upper bound on the cell aspect ratios the best convergence rate to be expected is first order which is what we prove the techniques developed apply to far more general initial boundary value problems

interval analysis theory and applications we give an overview on applications of interval arithmetic among others we discuss verification methods for linear systems of equations nonlinear systems the algebraic eigenvalue problem initial value problems for odes and boundary value problems for elliptic pdes of second order we also consider the item software in this field and give some historical remarks

viscosity approximation methods for nonexpansive mappings abstract viscosity approximation methods for nonexpansive mappings are studied consider a nonexpansive self mapping t of a closed convex subset c of a banach space x suppose that the set fix t of fixed points of t is nonempty for a contraction f on c and t 0 1 let x t c be the unique fixed point of the contraction x tf x 1 t tx consider also the iteration process x n where x 0 c is arbitrary and x n 1 n f x n 1 n tx n for n 1 where n 0 1 if x is either hilbert or uniformly smooth then it is shown that x t and under certain appropriate conditions on n x n converge strongly to a fixed point of t which solves some variational inequality

high order mimetic finite difference operators satisfying the extended gauss divergence theorem abstract we present high order mimetic finite difference operators that satisfy the extended gauss theorem these operators have the same order of accuracy in the interior and at the boundary no free parameters and optimal bandwidth they are defined over staggered grids using weighted inner products with a diagonal norm we present several examples to demonstrate that mimetic finite difference schemes using these operators produce excellent results

invariant approximations and r subweakly commuting maps the existence of invariant approximations for r subweakly commuting mappings is proved our results extend most of the known results to a new class of noncommuting mappings

solving variational inequalities involving nonexpansive type mappings abstract the purpose of this paper is to investigate the asymptotic behavior of algorithms for finding solutions for a certain class of variational inequalities v i d c i f involving nonexpansive type mappings in smooth banach spaces we study the existence of solutions of variational inequalities v i d c i f when d is the set of solutions of zeros of accretive operators or the set of fixed points of nonexpansive mappings or the set of fixed points of pseudocontractive mappings our convergence analysis covers proximal point algorithm for finding zeros of accretive operators as well as functional halpern algorithm for finding fixed points of nonexpansive mappings in banach spaces our results improve a number of results concerned with viscosity approximation methods in the context of weakly contraction mappings

the mimetic methods toolkit an object oriented api for mimetic finite differences abstract we introduce the mimetic methods toolkit mtk an object oriented application programming interface implementing mimetic finite differences to assist in the development of scientific applications where the numerical solution of partial differential equations is required the mtk s design is based on a variant of the castillo grone method for the construction of discrete differential operators that mimic important properties of their continuous counterparts the mtk is built as a collection of abstract and concrete classes thus allowing for an extensible framework which fosters code reutilization while intuitively educating the user about the theoretical aspects of mimetic finite differences we present an overview to mimetic finite differences and we discuss the computational modeling of the related concepts in this way we explain how the mtk implements these methods in this article we present examples to illustrate the mtk s usage philosophy and the validity of the implementation of the mtk by comparing our results with previously studied reference solutions

strong convergence of an iterative algorithm for nonself multimaps in banach spaces abstract let e be a uniformly convex banach space having a uniformly g teaux differentiable norm d a nonempty closed convex subset of e and t d k e a nonself multimap such that f t 0 and p t is nonexpansive where f t is the fixed point set of t k e is the family of nonempty compact subsets of e and p t x u x t x x u x d x t x suppose that d is a nonexpansive retract of e and that for each v d and t 0 1 the contraction s t defined by s t x t p t x 1 t v has a fixed point x t d let n n and n be three real sequences in 0 1 satisfying approximate conditions then for fixed u d and arbitrary x 0 d the sequence x n generated by x n n u n x n 1 n p t x n n 0 converges strongly to a fixed point of t

convergence of composite iterative methods for finding zeros of accretive operators abstract strong convergence theorems on composite iterative schemes by the viscosity approximation methods for finding a zero of an accretive operator are established in banach spaces the main results generalize the recent corresponding results of aoyama et al k aoyama y kimura w takahashi m toyoda approximation of common fixed points of a countable family of nonexpansive mappings in banach spaces nonlinear anal 67 2007 2350 2360 ceng et al l c ceng a r khan q h ansari j c yao strong convergence of composite iterative schemes for zeros of m accretive operators in banach spaces nonlinear anal 70 2009 1830 1840 kim and xu t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 2005 51 60 and xu h k xu strong convergence of an iterative method for nonexpansive and accretive operators j math anal appl 314 2006 631 643 to viscosity methods in a strictly convex and reflexive banach space having a uniformly g teaux differentiable norm our results also improve the corresponding results of t d benavides g l acedo h k xu iterative solutions for zeros of accretive operators math nachr 248 249 2003 62 71 r chen z zhu viscosity approximation fixed points for nonexpansive and m accretive operators fixed point theory appl 2006 2006 1 10 s kamimura w takahashi approximation solutions of maximal monotone operators in hilbert spaces j approx theory 106 2000 226 240 p e mainge viscosity methods for zeroes of accretive operators j approx theory 140 2 2006 127 140 k nakajo strong convergence to zeros of accretive operators in banach spaces j nonlinear convex anal 7 2006 71 81

strong convergence theorems for multivalued nonexpansive nonself mappings in banach spaces abstract let e be a uniformly convex banach space with a uniformly g teaux differentiable norm c a nonempty closed convex subset of e and t c k e a nonexpansive mapping for u c and t 0 1 let x t be a fixed point of a contraction g t c k e defined by g t x t t x 1 t u x c it is proved that if c is a nonexpansive retract of e x t is bounded and t z z for any fixed point z of t then the strong lim t 1 x t exists and belongs to the fixed point set of t furthermore we study the strong convergence of x t with the weak inwardness condition on t in a reflexive banach space with a uniformly g teaux differentiable norm

semigroups generated by pseudo contractive mappings under the nagumo condition let x be a banach space whose dual space x is uniformly convex we demonstrate that for any demicontinuous weakly nagumo k pseudo contractive mapping t d t x x with closed domain a t i weakly generates a semigroup on d t in this paper we project the consequences of this result on fixed point theory in particular we show that if k 1 id est if t is strongly pseudo contractive then t has a unique fixed point this implies that if t is pseudo contractive k 1 and d t is closed bounded and convex then t has at least one fixed point consequently any demicontinuous pseudo contractive mapping t c c for an appropriate c has a fixed point which has been an important open question in fixed point theory for quite some time in a subsequent paper we explore the consequences of the semigroup result on the existence of solutions to certain partial differential equations the semigroup result directly implies the existence of unique global solutions to time evolution equations of the form u au where a is a combination of derivatives the fixed point results from this paper imply the existence of solutions to partial differential equations of the form lu f

strong convergence theorems for nonexpansive semigroup in banach spaces abstract let k be a nonempty closed convex subset of a reflexive and strictly convex banach space e with a uniformly g teaux differentiable norm and f t t t 0 a nonexpansive self mappings semigroup of k and f k k a fixed contractive mapping the strongly convergent theorems of the following implicit and explicit viscosity iterative schemes x n are proved x n n f x n 1 n t t n x n x n 1 n f x n 1 n t t n x n and the cluster point of x n is the unique solution to some co variational inequality

systems of linear interval equations this paper presents theory and methods for computing the exact bounds on the solution of a system of n linear equations in n variables whose coefficients and right hand sides vary in some real intervals finite and iterative methods are given based on results from linear complementarity theory also regularity conditions regularity testing and computing the exact inverse of an interval matrix are dealt with

iterative algorithm for multi valued pseudocontractive mappings in banach spaces abstract let d be nonempty open convex subset of a real banach space e let t d k c e be a continuous pseudocontractive mapping satisfying the weakly inward condition and let u d be fixed then for each t 0 1 there exists y t d satisfying y t t t y t 1 t u if in addition e is reflexive and has a uniformly g teaux differentiable norm and is such that every closed convex bounded subset of d has fixed point property for nonexpansive self mappings then t has a fixed point if and only if y t remains bounded as t 1 in this case y t converges strongly to a fixed point of t as t 1 moreover an explicit iteration process which converges strongly to a fixed point of t is constructed in the case that t is also lipschitzian

a hybrid global optimization method the multi dimensional case we extend the hybrid global optimization method proposed by xu j comput appl math 147 2002 301 314 for the one dimensional case to the multi dimensional case the method consists of two basic components local optimizers and feasible point finders local optimizers guarantee efficiency and speed of producing a local optimal solution in the neighbourhood of a feasible point feasible point finders provide the theoretical guarantee for the new method to always produce the global optimal solution s correctly if a nonlinear nonconvex inverse problem has multiple global optimal solutions our algorithm is capable of finding all of them correctly three synthetic examples which have failed simulated annealing and genetic algorithms are used to demonstrate the proposed method

equivalent theorems of the convergence between proximal type algorithms abstract the main aim of this work is to give the equivalence of the convergence between proximal type algorithms our convergence analysis covers rockafellar type proximal point algorithm with a weak contraction for finding zeros of accretive operators as well as browder and halpern type iterations and viscosity approximation methods for finding fixed points of nonexpansive type mappings in banach spaces we also provide simple proofs of some results of wong sahu and yao n c wong d r sahu j c yao solving variational inequalities involving nonexpansive type mappings nonlinear anal 69 12 2008 4732 4753

strong convergence theorems for a common zero of a finite family of m accretive mappings suppose k is a closed convex subset of a strictly convex real banach space e which has a uniformly g teaux differentiable norm suppose that every nonempty closed convex bounded subset of e has the fixed point property for nonexpansive mappings a strong convergence theorem is proved for a common zero of a family of m accretive mappings from k to e as a consequence an iterative method is constructed to converge to a common fixed point assuming existence of a family of pseudocontractive mappings from k to e under certain mild condition

a stochastic method for global optimization a stochastic method for global optimization is described and evaluated the method involves a combination of sampling clustering and local search and terminates with a range of confidence intervals on the value of the global optimum computational results on standard test functions are included as well

strong convergence of iterative methods by strictly pseudocontractive mappings in banach spaces abstract in this paper we deal with fixed point computational problems by strongly convergent methods involving strictly pseudocontractive mappings in smooth banach spaces first we prove that the s iteration process recently introduced by sahu in 14 converges strongly to a unique fixed point of a mapping t where t is strongly pseudocontractive mapping from a nonempty closed and convex subset c of a smooth banach space into itself it is also shown that the hybrid steepest descent method converges strongly to a unique solution of a variational inequality problem with respect to a finite family of i strictly pseudocontractive mappings from c into itself our results extend and improve some very recent theorems in fixed point theory and variational inequality problems particularly the results presented here extend some theorems of reich 1980 1 and yamada 2001 15 to a general class of strictly pseudocontractive mappings in uniformly smooth banach spaces

strong convergence theorems for accretive operators in banach spaces abstract let e be a reflexive banach space with a uniformly g teaux differentiable norm and let a e e be m accretive assume that s cl d a e is bounded strongly accretive and continuous for x e and t 0 let xt be the unique solution of the inclusion x sxt taxt it is proved that if there exists a nonexpansive retraction p of e onto cl d a then the strong limt 0xt exists moreover it is shown that if every weakly compact convex subset of e has the fixed point property for nonexpansive mappings and 0 r a then the strong limt xt exists and belongs to a 10 an interesting application to a convergence result for an implicit iterative scheme is also included

convergence of iterative algorithms for multivalued mappings in banach spaces abstract we show strong and weak convergence for mann iteration of multivalued nonexpansive mappings t in a banach space furthermore we give a strong convergence of the modified mann iteration which is independent of the convergence of the implicit anchor like continuous path z t t u 1 t t z t

convergences of nonexpansive iteration processes in banach spaces abstract let e be a reflexive banach space with a uniformly g teaux differentiable norm and s be a mapping of the form s 0 i 1 t 1 2 t 2 k t k where i 0 0 0 i 0k i 1 and t i e e i 1 2 k is a nonexpansive mapping for an arbitrary x0 e let xn be a sequence in e defined by an iteration xn 1 sxn n 0 1 2 we establish a dual weak almost convergence result of xn in a reflexive banach space with a uniformly g teaux differentiable norm as a consequence of the result a weak convergence result of xn is also given

a unified hybrid iterative method for solving variational inequalities involving generalized pseudocontractive mappings we study in this paper the existence and the approximation of solutions of variational inequalities involving generalized pseudocontractive mappings in banach spaces the convergence analysis of a proposed hybrid iterative method for approximating common zeros or fixed points of a possibly infinitely countable or uncountable family of such operators will be conducted within the conceptual framework of the viscosity approximation technique in reflexive banach spaces with uniform g teaux differentiable norms this technique should make existing or new results in solving variational inequalities more applicable

variational inequalities for φ pseudo contractive mappings abstract usually results on strongly accretive operators involve various assumptions on the function such as continuity being positive and or being strictly increasing however by redefining this function we are able to eliminate all these additional assumptions and leave only the positiveness condition t 0 for t 0 this fact allows us to extend a theorem of deimling and obtain a general result for the solvability of a variational inequality for strongly pseudo contractive mappings

global descent method for global optimization this paper develops a novel method the global descent method for solving a general class of global optimization problems this method moves from one local minimizer of the objective function f to a better one at each iteration with the help of an auxiliary function termed the global descent function the global descent function is not only guaranteed to have a local minimizer x over the problem domain in mathbb r n but also ensures that each of its local minimizers is located in some neighborhoods of a better minimizer of f with f x f x these features of the global descent function enable a global descent to be achieved at each iteration using only local descent methods computational experiments conducted on several test problems with up to 1000 variables demonstrate the applicability of the proposed method furthermore numerical comparison experiments carried out with gams baron on several test problems also justify the efficiency and effectiveness of the proposed method

best simultaneous approximations asymptotically nonexpansive mappings and variational inequalities in banach spaces abstract the existence of common fixed points is established for three mappings where t is either generalized f g nonexpansive or asymptotically f g nonexpansive on a nonempty subset of a banach space as applications the invariant best simultaneous approximation results are proved and the existence of solution of variational inequalities is obtained our results unify and substantially improve several recent results existing in the current literature

viscosity approximation methods for fixed points problems the aim of this work is to propose viscosity approximation methods which amounts to selecting a particular fixed point of a given nonexpansive self mapping

viscosity approximation methods for a family of finite nonexpansive mappings in banach spaces viscosity approximation methods for a family of finite nonexpansive mappings are established in banach spaces the main theorems extend the main results of moudafi viscosity approximation methods for fixed points problems j math anal appl 241 2000 46 55 and xu viscosity approximation methods for nonexpansive mappings j math anal appl 298 2004 279 291 to the case of finite mappings our results also improve and unify the corresponding results of bauschke the approximation of fixed points of compositions of nonexpansive mappings in hilbert space j math anal appl 202 1996 150 159 browder convergence of approximations to fixed points of nonexpansive mappings in banach spaces archiv ration mech anal 24 1967 82 90 cho et al some control conditions on iterative methods commun appl nonlinear anal 12 2 2005 27 34 ha and jung strong convergence theorems for accretive operators in banach spaces j math anal appl 147 1990 330 339 halpern fixed points of nonexpansive maps bull amer math soc 73 1967 957 961 jung iterative approaches to common fixed points of nonexpansive mappings in banach spaces j math anal appl 302 2005 509 520 jung et al iterative schemes with some control conditions for a family of finite nonexpansive mappings in banach space fixed point theory appl 2005 2 2005 125 135 jung and kim convergence of approximate sequences for compositions of nonexpansive mappings in banach spaces bull korean math soc 34 1 1997 93 102 lions approximation de points fixes de contractions c r acad sci ser a b paris 284 1977 1357 1359 o hara et al iterative approaches to finding nearest common fixed points of nonexpansive mappings in hilbert spaces nonlinear anal 54 2003 1417 1426 reich strong convergence theorems for resolvents of accretive operators in banach spaces j math anal appl 75 1980 287 292 shioji and takahashi strong convergence of approximated sequences for nonexpansive mappings in banach spaces proc amer math soc 125 12 1997 3641 3645 takahashi and ueda on reich s strong convergence theorems for resolvents of accretive operators j math anal appl 104 1984 546 553 wittmann approximation of fixed points of nonexpansive mappings arch math 59 1992 486 491 xu iterative algorithms for nonlinear operators j london math soc 66 2 2002 240 256 and zhou et al strong convergence theorems on an iterative method for a family nonexpansive mappings in reflexive banach spaces appl math comput in press among others

improving the convergence of the backpropagation algorithm using learning rate adaptation methods this article focuses on gradient based backpropagation algorithms that use either a common adaptive learning rate for all weights or an individual adaptive learning rate for each weight and apply the goldstein armijo line search the learning rate adaptation is based on descent techniques and estimates of the local lipschitz constant that are obtained without additional error function and gradient evaluations the proposed algorithms improve the backpropagation training in terms of both convergence rate and convergence characteristics such as stable learning and robustness to oscillations simulations are conducted to compare and evaluate the convergence behavior of these gradient based training algorithms with several popular training methods

iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive banach spaces abstract let e be a real reflexive banach space which admits a weakly sequentially continuous duality mapping from e to e and k be a nonempty closed convex subset of e suppose that t n n 1 2 is a uniformly asymptotically regular sequence of nonexpansive mappings from k into itself such that f n 1 f t n 0 for arbitrary initial value x 0 k and fixed contractive mapping f k k define iteratively a sequence x n as follows x n 1 n 1 f x n 1 n 1 t n 1 x n n 0 where n 0 1 satisfies lim n n 0 and n 1 n we prove that x n converges strongly to p f as n where p is the unique solution in f to the following variational inequality i f p j p u 0 for all u f t our results extend and improve the corresponding ones given by o hara et al j g o hara p pillay h k xu iterative approaches to finding nearest common fixed point of nonexpansive mappings in hilbert spaces nonlinear anal 54 2003 1417 1426 j s jung iterative approaches to common fixed points of nonexpansive mappings in banach spaces j math anal appl 302 2005 509 520 h k xu viscosity approximation methods for nonexpansive mappings j math anal appl 298 2004 279 291 and o hara et al j g o hara p pillay h k xu iterative approaches to convex feasibility problem in banach space nonlinear anal available online 20 october 2005 doi 10 1016 j na 2005 07 36

a new alternating direction method for solving variational inequalities we introduce a modified alternating direction method for structured monotone variational inequalities by performing an additional projection step at each iteration and another optimal step length is employed to reach substantial progress in each iteration this method only needs functional values for given variables in the solution process and avoids the task of estimating the co coercive modulus all the computing process are easily implemented and the global convergence is also presented under mild assumptions some preliminary computational results are given

on reich s strong convergence theorems for resolvents of accretive operators let e be a real banach space with norm i then an operator a c e x e with domain d a and range r a is said to be accretive if xi xz 0 an accretive operator a c e x e is m accretive if r i ra e for all r 0 where z is the identity if a is accretive we can define for each positive r a singlevalued mapping j r z ra d a by j z ra it is called the resolvent of a in 6 reich proved the following theorem let e be a uniformly smooth banach space and let a c e x e be m accretive if 0 e r a then for each x in e the strong lim z x exists and belongs to a io he remarked also that the assumption that a is m accretive can be replaced with the assumption that cl a the closure of d a is convex and that a satisfies the range condition r z ra 1 cl d a for all r 0 in this paper we first prove a theorem that generalizes simultaneously the above results furthermore we generalize another reich strong convergence theorem asserting that the strong lim z x exists in uniformly convex and uniformly smooth banach spaces though our proofs are similar to those in 6 they are slightly simple on account of using banach limits

the mann process for perturbed m accretive operators in banach spaces this work was carried out while this author was visiting the university ofalabama in huntsville underthe nancial support by the lg yonam foundation 1998 and he would like to thank professor claudio h morales for his hospitality in the department of mathematics 0362 546x 01 see front matter 2001 elsevier science ltd all rights reserved pii s0362 546x 00 00115 2

on kirk s strong convergence theorem for multivalued nonexpansive mappings on cat 0 spaces abstract we prove a strong convergence theorem for multivalued nonexpansive mappings which includes kirk s convergence theorem on cat 0 spaces the theorem properly contains a result of jung for hilbert spaces we then apply the result to approximate a common fixed point of a countable family of single valued nonexpansive mappings and a compact valued nonexpansive mapping

approximation of fixed points of uniformly r subweakly commuting mappings we introduce a new class of uniformly r subweakly commuting mappings and then using this class study the problem of approximation of common fixed points of asymptotically s nonexpansive mappings in a banach space with uniformly g teaux differentiable norm

bounding the search space for global optimization of neural networks learning error an interval analysis approach training a multilayer perceptron mlp with algorithms employing global search strategies has been an important research direction in the field of neural networks despite a number of signifcant results an important matter concerning the bounds of the search region typically defined as a box where a global optimization method has to search for a potential global minimizer seems to be unresolved the approach presented in this paper builds on interval analysis and attempts to define guaranteed bounds in the search space prior to applying a global search algorithm for training an mlp these bounds depend on the machine precision and the term guaranteed denotes that the region defined surely encloses weight sets that are global minimizers of the neural network s error function although the solution set to the bounding problem for an mlp is in general non convex the paper presents the theoretical results that help deriving a box which is a convex set this box is an outer approximation of the algebraic solutions to the interval equations resulting from the function implemented by the network nodes an experimental study using well known benchmarks is presented in accordance with the theoretical results

on viscosity iterative methods for variational inequalities abstract let j be a commutative family of nonexpansive mappings of a closed convex subset c of a reflexive banach space x such that the set of common fixed point is nonempty in this paper we suggest and analyze a new viscosity iterative method for a commutative family of nonexpansive mappings we also prove that the approximate solution obtained by the proposed method converges to a solution of a variational inequality our method of proof is simple and different from the other methods results proved in this paper may be viewed as an improvement and refinement of the previously known results

viscosity approximation methods for countable families of nonexpansive mappings in banach spaces abstract let c be a nonempty closed convex subset of a banach space e and let s n be a family of nonexpansive mappings of c into itself such that the set of common fixed points of s n is nonempty we first introduce a sequence x n of c defined by x 1 x c and x n 1 n f x n 1 n s n x n for all n n where n 0 1 and f is a contraction of c into itself further we give the conditions of n and s n under which x n converges strongly to a common fixed point of s n this result generalizes the strong convergence theorem for nonexpansive mappings by suzuki t suzuki a sufficient and necessary condition for halpern type strong convergence to fixed points of nonexpansive mappings proc amer math soc 135 2007 99 106 and the strong convergence theorem for accretive operators by kamimura and takahashi s kamimura w takahashi weak and strong convergence of solutions to accretive operator inclusions and applications set valued anal 8 2000 361 374 simultaneously using this result we improve and extend the two above mentioned results

parallel preconditioning with sparse approximate inverses a parallel preconditioner is presented for the solution of general sparse linear systems of equations a sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method the computation of the preconditioner is inherently parallel and its application only requires a matrix vector product the sparsity pattern of the approximate inverse is not imposed a priori but captured automatically this keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system and the proximity of the approximate to the true inverse is estimated an extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory

on relaxed viscosity iterative methods for variational inequalities in banach spaces in this paper we suggest and analyze a relaxed viscosity iterative method for a commutative family of nonexpansive self mappings defined on a nonempty closed convex subset of a reflexive banach space we prove that the sequence of approximate solutions generated by the proposed iterative algorithm converges strongly to a solution of a variational inequality our relaxed viscosity iterative method is an extension and variant form of the original viscosity iterative method the results of this paper can be viewed as an improvement and generalization of the previously known results that have appeared in the literature