rates of convergence for iterative solutions of equations involving set valued accretive operators abstract this paper studies proofs of strong convergence of various iterative algorithms for computing the unique zeros of set valued accretive operators that also satisfy some weak form of uniform accretivity at zero more precisely we extract explicit rates of convergence from these proofs which depend on a modulus of uniform accretivity at zero a concept first introduced by a koutsoukou argyraki and the first author in 2015 our highly modular approach which is inspired by the logic based proof mining paradigm also establishes that a number of seemingly unrelated convergence proofs in the literature are actually instances of a common pattern

sobolev type embedding and quasilinear elliptic equations with radial potentials abstract we study the existence of nontrivial radial solutions for quasilinear elliptic equations with unbounded or decaying radial potentials the existence results are based upon several new embedding theorems we establish in the paper for radially symmetric functions

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

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

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

a mixed integer programming model for multiple stage adaptive testing abstract the last decade has seen paper and pencil p p tests being replaced by computerized adaptive tests cats within many testing programs a cat may yield several advantages relative to a conventional p p test a cat can determine the questions or test items to administer allowing each test form to be tailored to a test taker s skill level subsequent items can be chosen to match the capability of the test taker by adapting to a test taker s ability a cat can acquire more information about a test taker while administering fewer items a multiple stage adaptive test mst provides a means to implement a cat that allows review before the administration the mst format is a hybrid between the conventional p p and cat formats this paper presents mixed integer programming models for mst assembly problems computational results with commercial optimization software will be given and advantages of the models evaluated

branch and price column generation for solving huge integer programs we discuss formulations of integer programs with a huge number of variables and their solution by column generation methods i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality at a node of the branch and bound tree we present classes of models for which this approach decomposes the problem provides tighter lp relaxations and eliminates symmetry we then discuss computational issues and implementation of column generation branch and bound algorithms including special branching rules and efficient ways to solve the lp relaxation we also discuss the relationship with lagrangian duality

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

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

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

the fixed point property for mappings admitting a center abstract we introduce a class of nonlinear continuous mappings in banach spaces which allow us to characterize the banach spaces without noncompact flat parts in their spheres as those that have the fixed point property for this type of mapping later on we give an application to the existence of zeroes for certain kinds of accretive operators

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

the asymptotic behavior of the solutions of the cauchy problem generated by ϕ accretive operators abstract the purpose of this paper is to study the asymptotic behavior of the solutions of certain type of differential inclusions posed in banach spaces in particular we obtain the abstract result on the asymptotic behavior of the solution of the boundary value problem u t p u u 1 u f on 0 u u on 0 u 0 x u 0 x in where is a bounded open domain in r n with smooth boundary f t x is a given l 1 function on 0 1 and 1 p p represents the p laplacian operator is the associated neumann boundary operator and a maximal monotone graph in r r with 0 0

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

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

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

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

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

existence of multiple solutions for quasilinear systems via fibering method existence and multiplicity results for a general class of radial quasilinear equations are proved using the fibering method introduced and developed by s i pohozaev

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

determining targets for multi stage adaptive tests using integer programming this paper considers a multi stage adaptive test mst where the testlets at each stage are determined prior to the administration the assembly of a mst requires target information and target response functions for the mst design the targets are chosen to create tests with accurate scoring and high utilization of items in an operational pool forcing all msts to have information and response function plots to be within an interval about the targets will yield parallel msts in the sense that standardized paper and pencil tests are considered parallel the objective of this paper is to present a method to determine targets for the mst design based on an item pool and an assumed distribution of examinee ability the approach is applied to a skills readiness inventory test designed to identify logical reasoning deficiencies of examinees this method can be applied to obtain item response theory targets for a linear test as this is a special case of a mst

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

rates of convergence and metastability for abstract cauchy problems generated by accretive operators abstract we extract rates of convergence and rates of metastability in the sense of tao for convergence results regarding abstract cauchy problems generated by accretive at zero operators a d a x 2 x where x is a real banach space proved in 8 by proof theoretic analysis of the proofs in 8 and having assumed a new stronger notion of uniform accretivity at zero yielding a new notion of modulus of accretivity we compute the rate of metastability for the convergence of the solution of the abstract cauchy problem generated by a uniformly accretive at zero operator to the unique zero of a via proof mining based on a result by the first author finally we apply our results to a special class of cauchy problems considered in 8 this work is the first application of proof mining to the theory of partial differential equations

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

non radial solutions for a non homogeneous henon equation abstract in this paper we study a henon like equation see eq 1 where the nonlinearity f t is not homogeneous i e it is not a power by minimization on the nehari manifold we prove that for large values of the parameter there is a breaking of symmetry and non radial solutions appear this holds for sub and super critical growth of the nonlinearity 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

on a class of nonhomogeneous equations of henon type symmetry breaking and non radial solutions abstract in this work we study the following henon type equation div u p 2 u x a p x f u in b u 0 in b u 0 on b where b x r n x 1 is a ball centered at the origin the parameters verify the inequalities 0 a n p p n 4 0 2 p n p p n p a 1 and the nonlinearity f is nonhomogeneous by minimization on the nehari manifold we prove that for large values of the parameter there is a symmetry breaking and non radial solutions appear

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

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

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

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

quantitative results for halpern iterations of nonexpansive mappings abstract we give a rate of metastability for halpern s iteration relative to a rate of metastability for the resolvent for nonexpansive mappings in uniformly smooth banach spaces extracted from a proof due to xu in hilbert space the latter is known so we get an explicit rate of metastability we also extract a rate of asymptotic regularity for general normed spaces such rates have already been extracted by kohlenbach and leustean for different and incomparable conditions on n the proof analyzed in this paper is also more effective than the proofs treated by kohlenbach and leustean in that it does not use banach limits or weak compactness which makes the extraction particularly efficient moreover we also give an equivalent axiomatization of uniformly smooth banach spaces this paper is part of an ongoing case study of proof mining in nonlinear fixed point theory

existence of positive solutions for the henon equation involving critical sobolev terms abstract let n 3 2 2 n n 2 and r n be a bounded domain with a smooth boundary and 0 our purpose in this paper is to consider the existence of solutions of henon equation u x x u x 2 1 in u 0 in u 0 on where 0

standing waves with a critical frequency for nonlinear schrodinger equations this paper is concerned with the existence and qualitative property of standing wave solutions for the nonlinear schrodinger equation with e being a critical frequency in the sense that we show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of v and whose amplitude goes to 0 as moreover depending upon the local behaviour of the potential function v x near the minimum points the limiting profile of the standing wave solutions will be shown to exhibit quite different characteristic features this is in striking contrast with the non critical frequency case which has been extensively studied in recent years

binary programming and test design an algorithmic approach to test design using information functions is presented the approach uses a special branch of linear programming i e binary programming in addition results of some benchmark problems are presented within the same framework it is also possible to formulate the problem of individualized testing

existence and multiplicity results for quasilinear elliptic exterior problems with nonlinear boundary conditions abstract in this paper we study a class of quasilinear elliptic exterior problems with nonlinear boundary conditions existence of ground states and multiplicity results are obtained via variational methods

achievement test construction using 0 1 linear programming in educational testing the work of professional test agencies has shown a trend towards item banking achievement test construction is viewed as selecting items from a test item bank such that certain specifications are met as the number of possible tests is large and practice usually imposes various constraints on the selection process a mathematical programming approach is obvious in this paper it is shown how to formulate achievement test construction as a 0 1 linear programming problem a heuristic for solving the problem is proposed and two examples are given it is concluded that a 0 1 linear programming approach fits the problem of test construction in an appropriate way and offers test agencies the possibility of computerizing their services

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

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

on the asymptotic behavior of odd operators we give quantitative versions of strong convergence results due to baillon bruck and reich for iterations of nonexpansive odd and more general operators in uniformly convex banach spaces