a deterministic analysis of stochastic approximation with randomized directions we study the convergence of two stochastic approximation algorithms with randomized directions the simultaneous perturbation stochastic approximation algorithm and the random direction kiefer wolfowitz algorithm we establish deterministic necessary and sufficient conditions on the random directions and noise sequences for both algorithms and these conditions demonstrate the effect of the random directions on the sample path behavior of the algorithms studied we discuss ideas for further research in the analysis and design of these algorithms

uniform asymptotic regularity for mann iterates abstract in numer funct anal optim 22 2001 641 656 we obtained an effective quantitative analysis of a theorem due to borwein reich and shafrir on the asymptotic behavior of general krasnoselski mann iterations for nonexpansive self mappings of convex sets c in arbitrary normed spaces we used this result to obtain a new strong uniform version of ishikawa s theorem for bounded c in this paper we give a qualitative improvement of our result in the unbounded case and prove the uniformity result for the bounded case under the weaker assumption that c contains a point x whose krasnoselski mann iteration x k is bounded we also consider more general iterations for which asymptotic regularity is known only for uniformly convex spaces groetsch we give uniform effective bounds for an extension of groetsch s theorem which generalize previous results by kirk martinez yanez and the author

the approximate fixed point property in product spaces abstract in this paper we generalize to unbounded convex subsets c of hyperbolic spaces results obtained by w a kirk and r espinola on approximate fixed points of nonexpansive mappings in product spaces c m where m is a metric space and c is a nonempty convex closed and bounded subset of a normed or a cat 0 space we extend the results further to families c u u m of unbounded convex subsets of a hyperbolic space the key ingredient in obtaining these generalizations is a uniform quantitative version of a theorem due to borwein reich and shafrir obtained by the authors in a previous paper using techniques from mathematical logic inspired by that we introduce in the last section the notion of uniform approximate fixed point property for sets c and classes of self mappings of c the paper ends with an open problem

general logical metatheorems for functional analysis in this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in nonlinear functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters this means that the bounds are uniform for all parameters meeting these weak local boundedness conditions the results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space resp a convex subset of a normed space was assumed our results treat general classes of spaces such as metric hyperbolic cat 0 normed uniformly convex and inner product spaces and classes of functions such as nonexpansive holder lipschitz uniformly continuous bounded and weakly quasi nonexpansive ones we give several applications in the area of metric fixed point theory in particular we show that the uniformities observed in a number of recently found effective bounds by proof theoretic analysis can be seen as instances of our general logical results

holder classes with boundary conditions as interpolation spaces on etudie la caracterisation de certains espaces d interpolation reels entre le domaine d un operateur differentiel elliptique avec des conditions aux limites generales et l espace de banach de fonctions continues dans lequel le domaine est plonge

inequalities in metric spaces with applications abstract analogues of the parallelogram identity and the cn inequality of bruhat and tits in uniformly convex metric spaces are established as an application of the new inequalities we prove two fixed point results for single valued and multi valued lipschitzian mappings

parabolic evolution equations and nonlinear boundary conditions on etudie le probleme aux valeurs limites et initiales u 7b a t u f t u 7b b t u g t u 0 t t u 0 u 0 qui peut etre considere comme une contrepartie abstraite a des problemes paraboliques semilineaires

one local retracts and banach operator pairs in metric spaces abstract in this paper we first introduce the concept of nr map and then use this concept to establish the existence of common fixed points for banach operator pairs in the context of uniformly convex geodesic metric spaces new proofs of main results theorems 2 1 and 3 5 of chen and li j chen z li banach operator pair and common fixed points for nonexpansive maps nonlinear anal 74 2011 3086 3090 are presented further we prove de marr s theorem for the family of symmetric banach operator pairs in metric spaces and r trees for single and multivalued mappings satisfying conditions that generalize the concept of nonexpansivity

output tracking for nonlinear stochastic systems by iterative learning control an iterative learning control ilc algorithm which in essence is a stochastic approximation algorithm is proposed for output tracking for nonlinear stochastic systems with unknown dynamics and unknown noise statistics the nonlinear function of the system dynamics is allowed to grow up as fast as a polynomial of any degree but the system is linear with respect to control it is proved that the ilc generated by the algorithm a s converges to the optimal one at each time t spl isin 0 1 n and the output tracking error is asymptotically minimized in the mean square sense as the number of iterates tends to infinity although the convergence rate is rather slow the only information used in the algorithm is the noisy observation of the system output and the reference signal y sub d t when the system state equation is free of noise and the system output is realizable then the exact state tracking is asymptotically achieved and the tracking error is purely due to the observation noise

compactness and the fixed point property in l1 abstract in this paper we prove that compactness can be characterized by means of the existence of a fixed point for some classes of mappings defined on convex closed subsets of the space l 1 nominally our result involves nonexpansive mappings uniformly lipschitzian mappings and cascading nonexpansive mappings we also extend the results to some more general classes of banach spaces

demiclosed principle and δ convergence theorems for total asymptotically nonexpansive mappings in cat 0 spaces abstract the purpose of this paper is to introduce the concept of total asymptotically nonexpansive mappings and to prove the demiclosed principle for this kind of mappings in cat 0 spaces as a consequence we obtain a convergence theorem of the krasnoselskii mann type iteration for total asymptotically nonexpansive mappings in this setting our results extend and improve the corresponding recent results announced by many authors

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

on uniformly lipschitzian multivalued mappings in banach and metric spaces abstract let x d be a metric space a mapping t x x is said to be uniformly lipschitzian if there exists a constant k such that d t n x t n y k d x y for all x y x and n 1 it is known that such mappings always have fixed points in certain metric spaces for k 1 provided k is sufficiently near 1 these spaces include uniformly convex metric and banach spaces as well as metric spaces having lifsic characteristic greater than 1 a uniformly lipschitzian concept for multivalued mappings is introduced in this paper and multivalued analogues of these results are obtained

a kiefer wolfowitz algorithm with randomized differences a kiefer wolfowitz or simultaneous perturbation algorithm that uses either one sided or two sided randomized differences and truncations at randomly varying bounds is given in this paper at each iteration of the algorithm only two observations are required in contrast to 2l observations where l is the dimension in the classical algorithm the algorithm given is shown to be convergent under only some mild conditions the rate of convergence and asymptotic normality of the algorithm are also established

fixed point results for a generalized nonexpansive map in uniformly convex metric spaces abstract in this paper the existence of a unique fixed point of a map satisfying a very general contractive condition on a suitable subset of a uniformly convex metric space is proved this fixed point is approximated by averaging krasnosel skii iterations of a generalized nonexpansive map our results substantially improve and extend several known results existing in the literature

on some lp inequalities in best approximation theory h x jxp p ax p is strictly increasing in the interval p j o thus h x 0 has a unique solution x p note that x l 2 1 x p is strictly increasing in p and x 0 is the unique solution of the equation p 2 xp p l f2 1 in the interval 0 6 x 1 1 1 in 11 we proved the inequality

on a class of difference equations of monotone type of concern is the existence and uniqueness of the solution to a class of abstract second order difference equations they are the discrete version of some evolution equations which are intensely studied some asymptotic behavior results are established the periodic solutions are also investigated we use the theory of the maximal monotone operators in hilbert spaces an application to a partial differential equation is given

fixed points of uniformly lipschitzian mappings in spaces with uniformly normal structure dans un espace de banach x on etudie quand reflexivite et structure normale sont suffisantes pour assurer pour de bon 1 la propriete de point fixe pour des auto applications uniformement lipschitziennes

bounds on kuhfittig s iteration schema in uniformly convex hyperbolic spaces abstract the purpose of this paper is to extract an explicit effective and uniform bound on the rate of asymptotic regularity of an iteration schema involving a finite family of nonexpansive mappings the results presented in this paper contribute to the general project of proof mining as developed by the second author as well as generalize and improve various classical and corresponding quantitative results in the current literature more precisely we give a rate of asymptotic regularity of an iteration schema due to kuhfittig for finitely many nonexpansive mappings in the context of uniformly convex hyperbolic spaces the rate only depends on an upper bound on the distance between the starting point and some common fixed point a lower bound 1 n n 1 n the error 0 and a modulus of uniform convexity

an actor critic algorithm with function approximation for discounted cost constrained markov decision processes we develop in this article the first actor critic reinforcement learning algorithm with function approximation for a problem of control under multiple inequality constraints we consider the infinite horizon discounted cost framework in which both the objective and the constraint functions are suitable expected policy dependent discounted sums of certain sample path functions we apply the lagrange multiplier method to handle the inequality constraints our algorithm makes use of multi timescale stochastic approximation and incorporates a temporal difference td critic and an actor that makes a gradient search in the space of policy parameters using efficient simultaneous perturbation stochastic approximation spsa gradient estimates we prove the asymptotic almost sure convergence of our algorithm to a locally optimal policy c 2010 elsevier b v all rights reserved

averaged alternating reflections in geodesic spaces abstract we study the nonexpansivity of reflection mappings in geodesic spaces and apply our findings to the averaged alternating reflection algorithm employed in solving the convex feasibility problem for two sets in a nonlinear context we show that weak convergence results from hilbert spaces find natural counterparts in spaces of constant curvature moreover in this particular setting one obtains strong convergence

nonexpansive iterations in hyperbolic spaces one of the most active research areas in nonlinear functional analysis is the asymptotics of nonexpansive mappings most of the results however have been obtained in normed linear spaces it is natural therefore to try to develop a theory of nonexpansive iterations in more general infinite dimensional manifolds this is the purpose of the present paper more specifically we propose the class of hyperbolic spaces as an appropriate background for the study of operator theory in general and of iterative processes for nonexpansive mappings in particular this class of metric spaces which is defined in section 2 includes all normed linear spaces and hadamard manifolds as well as the hilbert ball and the cartesian product of hilbert balls in section 3 we introduce co accretive operators and their resolvents and present some of their properties in the fourth section we discuss the concept of uniform convexity for hyperbolic spaces section 5 is devoted to two new geometric properties of infinite dimensional banach spaces theorem 5 6 provides a characterization of banach spaces having these properties in terms of nonlinear accretive operators in sections 6 7 and 8 we study explicit implict and continuous iterations repectively using the same approach in all three sections we illustrate this common approach with the following special case let c be a closed convex subset of a hyperbolic space x p let t c f c be a nonexpansive mapping and let x be a point in c in order to study the iteration t x n 0 1 2 we set z 1 l n x 0 l n t x k clco zj j i l and d inf p y ty y e c the first step is to show that p x k lim p x t x n d this leads to the convergence m of lz when x is uniformly convex and to the weak convergence of z when x is a banach space which is reflexive and strictly convex when t is an averaged mapping we are also able to establish the following triple equality for all k 2 1

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

accelerated randomized stochastic optimization we propose a general class of randomized gradient estimates to be employed in a recursive search for the minimum of an unknown multivariate regression function here only two observations per iteration step are used special cases include random direction stochastic approximation kushner and clark simultaneous perturbation stochastic approximation spall and a special kernel based stochastic approximation method polyak and tsybakov if the unknown regression is p smooth p 2 at the point of minimum these methods achieve the optimal rate of convergence o n p 1 2p for both the classical stochastic approximation scheme kiefer and wolfowitz and the averaging scheme ruppert and polyak the related asymptotic distributions are computed

on the solutions of the first boundary value problem for the linear parabolic equations on etudie le premier probleme aux valeurs limites pour une equation parabolique lineaire d ordre 2 sous l hypothese que le terme inhomogene est continu en espace et en temps et continu de holder seulement par rapport aux variables d espace

quantitative image recovery theorems abstract this paper provides a quantitative version of the classical image recovery problem to find an approximate solution of the problem the rate of asymptotic regularity of the iteration schemas connected with the problem of image recovery coincides with the existing optimal and quadratic bound for krasnoselskii mann iterations we then provide explicit effective and uniform bounds on the approximate fixed points of the mappings under consideration to be an approximate solution of the image recovery problem up to a uniform change from to when combined these results provide algorithms with explicit rates of convergence for the recovery of an perturbation of the original image in different settings

fixed point theorems for uniformly lipschitzian semigroups in uniformly convex spaces for all x y in c and each n 1 2 goebel and kirk 2 first studied these mappings and they proved that such t has a fixed point if c is a closed bounded convex subset of a uniformly convex banach space x and k k k is the unique solution of the equation 1 a l k k 1 with 6 being the modulus of convexity of x lifschitz s showed that in a hilbert space i a uniformly k lipschitzian mapping t with k c 21j2 has a fixed point lim 6 7 and lim et al 8 extended goebel kirk s theorem in lp spaces recently downing and ray l and ishihara and takahashi 3 verified that lifschitz s theorem is valid for uniformly lipschitzian semigroups in hifbert spaces the purpose of this paper is to prove fixed point theorems for a uniformly k lipschitzian semigroup in uniformly convex banach spaces which is left reversible or for which the space of right uniformly continuous functions on the semigroup has a left invariant mean the results generalize those of downing and ray l lau 4 lim 6 7 and ishihara and takahashi 3

on the abstract cauchy problem of parabolic type in spaces of continuous functions on etudie l existence l unicite et les proprietes de regularite des solutions du probleme de cauchy u t au r f t o t t u o u 0 ou a d a ce e est un operateur lineaire clos dans l espace de banach e

convergence of simultaneous perturbation stochastic approximation for nondifferentiable optimization we consider simultaneous perturbation stochastic approximation for function minimization the standard assumption for convergence is that the function be three times differentiable although weaker assumptions have been used for special cases however all work that we are aware of at least requires differentiability we relax the differentiability requirement and prove convergence using convex analysis

holder regularity in variational parabolic non homogeneous equations abstract we prove optimal holder regularity results for the solutions to second order variational parabolic equations with holder continuous coefficients and non homogeneous boundary data using analytic semigroups methods

a quadratic rate of asymptotic regularity for cat 0 spaces abstract in this paper we obtain a quadratic bound on the rate of asymptotic regularity for the krasnoselski mann iterations of nonexpansive mappings in cat 0 spaces whereas previous results guarantee only exponential bounds the method we use is to extend to the more general setting of uniformly convex hyperbolic spaces a quantitative version of a strengthening of groetsch s theorem obtained by kohlenbach using methods from mathematical logic so called proof mining

brief paper recursive identification for multivariate errors in variables systems the recursive algorithm is given for estimating matrix coefficients of the multivariate errors in variables eiv systems it is shown that under mild conditions the estimate given by the algorithm converges to a limit belonging to the solution set of the yule walker equation satisfied by the true coefficients of the system the sufficient conditions guaranteeing the uniqueness of the solution to the yule walker equation are given in this case the estimate provided by the recursive algorithm is strongly consistent

distributed stochastic approximation algorithm with expanding truncations in this paper a novel distributed stochastic approximation algorithm dsaa is proposed to seek roots of the sum of local functions each of which is associated with an agent from multiple agents connected over a network at each iteration each agent updates its estimate for the root utilizing the noisy observations of its local function and the information derived from the neighboring agents the key difference of the proposed algorithm from the existing ones consists in the expanding truncations so it is called the dsaawet by which the boundedness of the estimates can be guaranteed without imposing the growth rate constraints on the local functions the estimates generated by the dsaawet are shown to converge almost surely to a consensus set which belongs to a connected subset of the root set of the sum function in comparison with the existing results we impose weaker conditions on the local functions and on the observation noise we then apply the proposed algorithm to two applications one from signal processing and the other one from distributed optimization and establish the almost sure convergence numerical simulation results are also included

convergence theorems for nonexpansive semigroups in cat 0 spaces abstract let c be a convex and compact subset of a cat 0 space x in this paper we consider the following iterative scheme for a one parameter nonexpansive semigroup t t t 0 on c x 1 c x n 1 t t n x n 1 x n where 0 1 and t n 0 and we prove that under certain conditions x n converges to a common fixed point of the semigroup t t t 0

characteristic inequalities of uniformly convex and uniformly smooth banach spaces abstract let x be a real banach space with dual x and moduli of convexity and smoothness x e and x respectively for 1 p denotes the duality mapping from x into 2 x with gauge function t p 1 and j p denotes an arbitrary selection for j p let a r 0 0 t is strictly increasing and there exists c 0 such that t c x t 2 and f r 0 0 t is convex nondecreasing and there exists k 0 such that k x it is proved that x is uniformly convex if and only if there is a a such that x y p x p p j p x y x y x y x and x is uniformly smooth if and only if there is a f such that x y p x p p j p x y x y x y x where for given function f f x y is defined by f x y p 0 1 x ty x p t f t y x ty x dt these inequalities which have various applications can be regarded as general banach space versions of the well known polarization identity occurring in hilbert spaces

an lp inequality and its applications to fixed point theory and approximation theory on etablit une inegalite l p pour 1 p 2 et on donne des applications aux points fixes des applications uniformement lipschitziennes et aux meilleures approximations fortement uniques

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

general bounds and finite time improvement for the kiefer wolfowitz stochastic approximation algorithm we consider the kiefer wolfowitz kw stochastic approximation algorithm and derive general upper bounds on its mean squared error the bounds are established using an elementary induction argument and phrased directly in the terms of tuning sequences of the algorithm from this we deduce the nonnecessity of one of the main assumptions imposed on the tuning sequences by kiefer and wolfowitz kiefer j j wolfowitz 1952 stochastic estimation of the maximum of a regression function ann math statist 23 3 462 466 and essentially all subsequent literature the optimal choice of sequences is derived for various cases of interest and an adaptive version of the kw algorithm scaled and shifted kw or sskw is proposed with the aim of improving its finite time behavior the key idea is to dynamically scale and shift the tuning sequences to better match them with characteristics of the unknown function and noise level and thus improve algorithm performance numerical results are provided that illustrate that the proposed algorithm retains the convergence properties of the original kw algorithm while dramatically improving its performance in some cases

simultaneous perturbation stochastic approximation for tracking under unknown but bounded disturbances multi dimensional stochastic optimization plays an important role in analysis and control of many technical systems to solve the challenging multidimensional problems of nonstationary optimization it is suggested to use a stochastic approximation algorithm like spsa with perturbed input and constant step size which has simple form we get a finite bound of residual between estimates and time varying unknown parameters when observations are made under an unknown but bounded noise applications of the algorithm are considered for a random walk an optimization of uav s flight and a load balancing problem

asymptotic behavior of bounded solutions to a nonhomogeneous second order evolution equation of monotone type abstract by using previous results of djafari rouhani b djafari rouhani ergodic theorems for nonexpansive sequences in hilbert spaces and related problems ph d thesis yale university 1981 part i pp 1 76 b djafari rouhani asymptotic behaviour of quasi autonomous dissipative systems in hilbert spaces j math anal appl 147 1990 465 476 b djafari rouhani asymptotic behaviour of almost nonexpansive sequences in a hilbert space j math anal appl 151 1990 226 235 for dissipative systems we study the asymptotic behavior of solutions to the following system of second order nonhomogeneous evolution equations u t a u t f t a e t 0 u 0 u 0 sup t 0 u t where a is a maximal monotone operator in a real hilbert space h and f r h is a suitable function we investigate weak and strong convergence theorems for solutions to this system our results extend previous results of morosanu g morosanu nonlinear evolution equations and applications editura academiei romane and d reidel publishing company bucharest 1988 g morosanu asymptotic behaviour of solutions of differential equations associated to monotone operators nonlinear anal 3 1979 873 883 and mitidieri e mitidieri some remarks on the asymptotic behaviour of the solutions of second order evolution equations j math anal appl 107 1985 211 221 who studied the case f 0 by assuming that a 1 0 as well as previous results of the authors b djafari rouhani h khatibzadeh asymptotic behavior of bounded solutions to some second order evolution systems rocky mountain j math in press who studied the homogeneous case without this additional assumption

multivariate stochastic approximation using a simultaneous perturbation gradient approximation the problem of finding a root of the multivariate gradient equation that arises in function minimization is considered when only noisy measurements of the function are available a stochastic approximation sa algorithm for the general kiefer wolfowitz type is appropriate for estimating the root the paper presents an sa algorithm that is based on a simultaneous perturbation gradient approximation instead of the standard finite difference approximation of keifer wolfowitz type procedures theory and numerical experience indicate that the algorithm can be significantly more efficient than the standard algorithms in large dimensional problems

asymptotic behavior of averaged and firmly nonexpansive mappings in geodesic spaces abstract we further study averaged and firmly nonexpansive mappings in the setting of geodesic spaces with a main focus on the asymptotic behavior of their picard iterates we use methods of proof mining to obtain an explicit quantitative version of a generalization to geodesic spaces of a result on the asymptotic behavior of picard iterates for firmly nonexpansive mappings proved by reich and shafrir from this result we obtain effective uniform bounds on the asymptotic regularity for firmly nonexpansive mappings besides this we derive effective rates of asymptotic regularity for sequences generated by two algorithms used in the study of the convex feasibility problem in a nonlinear setting

on an inequality of bynum and drew an extension of the bynum drew inequality in the abstract lp space 1 p 2 is presented the extended inequality is applied to show the strong uniqueness of best approximations and existence of fixed points for uniformly lipschitzian mappings and to prove an estimate for the self jung constants

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

regularizing effects for ut aϑ u 0 in l1 abstract various initial boundary value problems and cauchy problems can be written in the form du dt a u 0 where r r is nondecreasing and a is the linear generator of strongly continuous nonexpansive semigroup e ta in an l1 space for example if a subject perhaps to suitable boundary conditions we obtain equations arising in flow in a porous medium or plasma physics depending on the choice of while if a x acting in l1 r we have a scalar conservation law in this paper we show that if m m 0 and m 2 m 2 where 1 1 then roughly speaking the norm of t du dt may be estimated in terms of the initial data u0 in l1 such estimates give information about the regularity of solutions asymptotic behaviour etc in applications side issues such as the introduction of sufficiently regular approximate problems on which estimates can be made and the assignment of a precise meaning to the operator a are also dealt with these considerations are of independent interest

δ convergence theorems for multi valued nonexpansive mappings in hyperbolic spaces the purpose of this paper is to introduce the mixed agarwal o regan sahu type iterative scheme agarwal et al 2007 for finding a common fixed point of the multi valued nonexpansive mappings in the setting of hyperbolic spaces under suitable conditions some convergence theorems of the iterative sequence generated by the proposed scheme to approximate a common fixed point of multi valued nonexpansive mappings are proved the results presented in the paper extend and improve some recent results announced in the current literature

firmly nonexpansive mappings in classes of geodesic spaces firmly nonexpansive mappings play an important role in metric fixed point theory and optimization due to their correspondence with maximal monotone operators in this paper we do a thorough study of fixed point theory and the asymptotic behaviour of picard iterates of these mappings in different classes of geodesic spaces such as uniformly convex w hyperbolic spaces busemann spaces and cat 0 spaces furthermore we apply methods of proof mining to obtain effective rates of asymptotic regularity for the picard iterations msc primary 47h09 47h10 53c22 secondary 03f10 47h05 90c25 52a41