pointwise lipschitzian mappings in uniformly convex and uniformly smooth banach spaces abstract let x be a uniformly convex and uniformly smooth banach space let t c c be an asymptotic pointwise nonexpansive mapping where c is a bounded closed and convex subset of x in this paper we investigate conditions sufficient for the weak convergence of the generalized mann and ishikawa processes to a fixed point of t

a note on differentiability of the cluster density for independent percolation in high dimensions the cluster density function of independent percolation in ad dimensional lattice is considered for eachn it is shown that p has finitenth leftderivative at critical probabilityp c ifd is sufficiently large this result agrees with the bethe lattice approximation where thenth one sided derivative of p is bounded atp c for alln

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

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

some computational aspects of metric fixed point theory abstract we show how bounds on asymptotic regularity for nonexpansive functions can effectively be converted into certain bounds on the convergence towards a fixed point

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

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

approximating common fixed points of asymptotically nonexpansive maps in uniformly convex banach spaces we introduce three step iterative schemes with errors for two and three nonexpansive maps and establish weak and strong convergence theorems for these schemes mann type and ishikawa type convergence results are included in the analysis of these new iteration schemes the results presented in this paper substantially improve and extend the results due to s h khan h fukhar ud din weak and strong convergence of a scheme with errors for two nonexpansive mappings nonlinear anal 8 2005 1295 1301 n shahzad approximating fixed points of non self nonexpansive mappings in banach spaces nonlinear anal 61 2005 1031 1039 w takahashi t tamura convergence theorems for a pair of nonexpansive mappings j convex anal 5 1995 45 58 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 and h f senter w g dotson approximating fixed points of nonexpansive mappings proc amer math soc 44 1974 375 380

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

a rate of convergence for asymptotic contractions abstract in p gerhardy a quantitative version of kirk s fixed point theorem for asymptotic contractions j math anal appl 316 2006 339 345 p gerhardy gives a quantitative version of kirk s fixed point theorem for asymptotic contractions this involves modifying the definition of an asymptotic contraction subsuming the old definition under the new one and giving a bound expressed in the relevant moduli and a bound on the picard iteration sequence on how far one must go in the iteration sequence to at least once get close to the fixed point however since the convergence to the fixed point needs not be monotone this theorem does not provide a full rate of convergence we here give an explicit rate of convergence for the iteration sequence expressed in the relevant moduli and a bound on the sequence we furthermore give a characterization of asymptotic contractions on bounded complete metric spaces showing that they are exactly the mappings for which every picard iteration sequence converges to the same point with a rate of convergence which is uniform in the starting point

on quantitative versions of theorems due to f e browder and r wittmann abstract this paper is another case study in the program of logically analyzing proofs to extract new typically effective information proof mining we extract explicit uniform rates of metastability in the sense of t tao from two ineffective proofs of a classical theorem of f e browder on the convergence of approximants to fixed points of nonexpansive mappings as well as from a proof of a theorem of r wittmann which can be viewed as a nonlinear extension of the mean ergodic theorem the first rate is extracted from browder s original proof that is based on an application of weak sequential compactness in addition to a projection argument wittmann s proof follows a similar line of reasoning and we adapt our analysis of browder s proof to get a quantitative version of wittmann s theorem as well in both cases one also obtains totally elementary proofs even for the strengthened quantitative forms of these theorems that neither use weak compactness nor the existence of projections anymore in this way the present article also discusses general features of extracting effective information from proofs based on weak compactness we then extract another rate of metastability of similar nature from an alternative proof of browder s theorem essentially due to halpern that already avoids any use of weak compactness the paper is concluded by general remarks concerning the logical analysis of proofs based on weak compactness as well as a quantitative form of the so called demiclosedness principle in a subsequent paper these results will be utilized in a quantitative analysis of baillon s nonlinear ergodic theorem

approximating fixed points of non self nonexpansive mappings in banach spaces abstract suppose k is a nonempty closed convex nonexpansive retract of a real uniformly convex banach space e with p as a nonexpansive retraction let t k e be a nonexpansive non self map with f t x k tx x suppose x n is generated iteratively by x 1 k x n 1 p 1 n x n n tp 1 n x n n tx n n 1 where n and n are real sequences in e 1 e for some e 0 1 1 if the dual e of e has the kadec klee property then weak convergence of x n to some x f t is proved 2 if t satisfies condition a then strong convergence of x n to some x f t is obtained

a new metastable convergence criterion and an application in the theory of uniformly convex banach spaces abstract we study a convergence criterion which generalises the notion of being monotonically decreasing and introduce a quantitative version of this criterion a so called metastable rate of asymptotic decreasingness we then present a concrete application in the fixed point theory of uniformly convex banach spaces in which we carry out a quantitative analysis of a convergence proof of kirk and sims more precisely we produce a rate of metastability in the sense of tao for the picard iterates of mappings which satisfy a variant of the convergence criterion and whose fixed point set has nonempty interior

logical metatheorems for abstract spaces axiomatized in positive bounded logic abstract in this paper we show that normed structures which can be axiomatized in positive bounded logic in the sense of henson and iovino admit proof theoretic metatheorems as developed by the second author since 2005 on the extractability of explicit uniform bounds from proofs in the respective theories we apply this to design such metatheorems for abstract banach lattices l p and c k spaces as well as bands in l p l q bochner spaces we also show that a proof theoretic uniform boundedness principle can serve in many ways as a substitute for the model theoretic use of ultrapowers of banach spaces

weak and strong convergence theorems for three step iterations with errors for asymptotically nonexpansive mappings abstract in the present paper several weak and strong convergence theorems are established for the three step iterative schemes with errors for asymptotically nonexpansive mappings our results extend and improve the recent ones announced by tan and xu xu and noor and many others

on the critical percolation probabilities we prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c p 1 furthermore we prove the continuity of the function percolation probability

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

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

zip bifurcation in a competition model on considere le systeme s s 1 s k m 1 x 1 s a 1 s m 2 x 2 s a 2 s x 1 m 1 x 1 s a 1 s d 1 w 1 x 2 m 2 x 2 s a 2 s d 2 x 2 soit i a i d i m i d i on etudie le cas 1 2

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

effective results on compositions of nonexpansive mappings abstract this paper provides uniform bounds on the asymptotic regularity for iterations associated to a finite family of nonexpansive mappings we obtain our quantitative results in the setting of r convex spaces a class of geodesic spaces which generalizes metric spaces with a convex geodesic bicombing

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

a genus 3 topological recursion relation in this paper we give a new genus 3 topological recursion relation for gromov witten invariants of compact symplectic manifolds this formula also applies to intersection numbers on moduli spaces of spin curves a by product of the proof of this formula is a new relation in the tautological ring of the moduli space of 1 pointed genus 3 stable curves

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

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

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

effective rates of convergence for lipschitzian pseudocontractive mappings in general banach spaces abstract this paper gives an explicit and effective rate of convergence for an asymptotic regularity result t x n x n 0 due to chidume and zegeye in 2004 14 where x n is a certain perturbed krasnoselski mann iteration schema for lipschitz pseudocontractive self mappings t of closed and convex subsets of a real banach space we also give a qualitative strengthening of the theorem by chidume and zegeye by weakening the assumption of the existence of a fixed point for the bounded case our bound is polynomial in the data involved

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

perturbative series and the moduli space of riemann surfaces on utilise des techniques de la theorie quantique des champs pour calculer des quantites reliees aux groupes de symetrie de paires f g ou f est une surface et g une epine de f

effective asymptotic regularity for one parameter nonexpansive semigroups abstract we give explicit bounds on the computation of approximate common fixed points of one parameter strongly continuous semigroups of nonexpansive mappings on a subset c of a general banach space moreover we provide the first explicit and highly uniform rate of convergence for an iterative procedure to compute such points for convex c our results are obtained by a logical analysis of the proof proof mining of a theorem by t suzuki

weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups we give three equivalent conditions for weak convergence of almost orbits of an asymptotically nonexpansive commutative semigroup acting on a nonempty bounded closed convex subset of a uniformly convex banach space whose dual has the kadec property

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

on the removal of weak compactness arguments in proof mining abstract the main observation of this paper is that some sequential weak compactness arguments in hilbert space theory can be replaced by heine borel compactness arguments for the strong topology even though the latter form of compactness fails in infinite dimensional hilbert spaces it nevertheless trivializes under the so called bounded functional interpretation as a consequence the proof mining program of extracting computational bounds from ordinary proofs of mathematics can be applied to modified proofs which use these false heine borel compactness arguments additionally the bounded functional interpretation provides good logical guidance in formulating quantitative versions of analytical statements we illustrate these claims with three minings the bounded functional interpretation is here used for the first time in proof mining

random matrix singularities and open close intersection numbers the s point correlation function of a gaussian hermitian random matrix model with an external source tuned to generate a multi critical singularity provides the intersection numbers of the moduli space for the p th spin curves through a duality for one marked point the intersection numbers are expressed to all orders in the genus by bessel functions the matrix models for the lie algebras of and provide the intersection numbers of non orientable surfaces the kontsevich penner model and the higher p th airy matrix model with a logarithmic potential are investigated for the open intersection numbers which describe the topological invariants of non orientable surfaces with boundaries string equations for the open closed riemann surface are derived from the structure of the s point correlation functions the gromov witten invariants of the model are evaluated for one marked point as an application of the present method

a quantitative nonlinear strong ergodic theorem for hilbert spaces abstract we give a quantitative version of a strong nonlinear ergodic theorem for a class of possibly even discontinuous selfmappings of an arbitrary subset of a hilbert space due to r wittmann and outline how the existence of uniform bounds in such quantitative formulations of ergodic theorems can be proved by means of a general logical metatheorem in particular these bounds depend neither on the operator nor on the initial point furthermore we extract such uniform bounds in our quantitative formulation of wittmann s theorem implicitly using the proof theoretic techniques on which the metatheorem is based however we present our result and its proof in analytic terms without any reference to logic as such our bounds turn out to involve nested iterations of relatively low computational complexity while in theory these kind of iterations ought to be expected so far this seems to be the first occurrence of such a nested use observed in practice

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

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

effective metastability of halpern iterates in cat 0 spaces abstract this paper provides an effective uniform rate of metastability in the sense of tao on the strong convergence of halpern iterations of nonexpansive mappings in cat 0 spaces the extraction of this rate from an ineffective proof due to saejung is an instance of the general proof mining program which uses tools from mathematical logic to uncover hidden computational content from proofs this methodology is applied here for the first time to a proof that uses banach limits and hence makes a substantial reference to the axiom of choice

weak and strong convergence of a scheme with errors for two nonexpansive mappings abstract in this paper we are concerned with the study of an iterative scheme with errors involving two nonexpansive mappings we approximate the common fixed points of these two mappings by weak and strong convergence of the scheme in a uniformly convex banach space under a condition weaker than compactness

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

punctures and p spin curves from matrix models this article investigates the intersection numbers of the moduli space of p spin curves with the help of matrix models the explicit integral representations that are derived for the generating functions of these intersection numbers exhibit p stokes domains labelled by a spin component l taking values l 1 0 1 2 p 2 earlier studies concerned integer values of p but the present formalism allows one to extend our study to half integer or negative values of p which turn out to describe new types of punctures or marked points on the riemann surface they fall into two classes ramond l 1 absent for positive integer p and neveu schwarz l ne 1 the intersection numbers of both types are computed from the integral representation of the n point correlation functions in a large n scaling limit we also consider a supersymmetric extension of the random matrix formalism to show that it leads naturally to an additional logarithmic potential open boundaries on the surface or admixtures of r and ns punctures may be handled by this extension

the euler characteristic of the moduli space of curves let fg 1 g 1 be the mapping class group consisting of all isotopy classes of base point and orientation preserving homeomorphisms of a closed oriented surface f of genus g let 1 be its euler characteristic in the sense of wall that is z f i fgl f l z e f where f is any torsion free subgroup of finite index in f 1 and e is a contractible space on which f acts freely and properly discontinuously an example of such a space is the teichmiiller space 1 and g f can be interpreted as the orbifold euler characteristic of f i join the moduli space of curves of genus g with base point the purpose of this paper is to prove the following formula for f i

the scaling limit of the incipient infinite cluster in high dimensional percolation i critical exponents this is the first of two papers on the critical behavior of bond percolation models in high dimensions in this paper we obtain strong joint control of the critical exponents and for the nearest neighbor model in very high dimensions d 6 and for sufficiently spread out models in all dimensions d 6 the exponent describes the low frequency behavior of the fourier transform of the critical two point connectivity function while describes the behavior of the magnetization at the critical point our main result is an asymptotic relation showing that in a joint sense 0 and 2 the proof uses a major extension of our earlier expansion method for percolation this result provides evidence that the scaling limit of the incipient infinite cluster is the random probability measure on d known as integrated super brownian excursion ise in dimensions above 6 in the sequel to this paper we extend our methods to prove that the scaling limits of the incipient infinite cluster s two point and three point functions are those of ise for the nearest neighbor model in dimensions d 6

a quantitative version of kirk s fixed point theorem for asymptotic contractions abstract in w a kirk fixed points of asymptotic contractions j math anal appl 277 2003 645 650 w a kirk introduced the notion of asymptotic contractions and proved a fixed point theorem for such mappings using techniques from proof mining we develop a variant of the notion of asymptotic contractions and prove a quantitative version of the corresponding fixed point theorem

local stability of ergodic averages we consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages it is not difficult to construct an example of a computable lebesgue measure preserving transformation of 0 1 and a characteristic function f xa such that the ergodic averages a n f do not converge to a computable element of l 2 0 1 in particular there is no computable bound on the rate of convergence for that sequence on the other hand we show that for any nonexpansive linear operator t on a separable hilbert space and any element f it is possible to compute a bound on the rate of convergence of a n f from t f and the norm f of the limit in particular if t is the koopman operator arising from a computable ergodic measure preserving transformation of a probability space x and f is any computable element of l 2 x then there is a computable bound on the rate of convergence of the sequence anf the mean ergodic theorem is equivalent to the assertion that for every function k n and every e 0 there is an n with the property that the ergodic averages a m f are stable to within e on the interval n k n even in situations where the sequence a n f does not have a computable limit one can give explicit bounds on such n in terms of k and f e this tells us how far one has to search to find an n so that the ergodic averages are locally stable on a large interval we use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem and we show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to bishop and ivanov finally we explain how our positive results can be viewed as an application of a body of general proof theoretic methods falling under the heading of proof mining

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

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