on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

critical functions for complex analytic maps critical functions measure the width of the domain of stability around a given fixed point or an invariant circle for complex analytic and area preserving maps the author discusses their dependence on the rotation number of the invariant curves and proposes some new methods to determine them based on the existence of critical points and on some properties of quasiconformal maps by means of the majorant series method some rigorous estimates are given for complex area preserving maps like the semistandard map and the modulated singular map in particular the author makes use of the brjuno function to interpolate critical maps and proves that the convergence of the brjuno function is a necessary and sufficient condition for the existence of an analytic invariant curve of a given rotation number the author also discusses the optimality of the rigorous bounds obtained

on the convergence of asynchronous iteration methods for nonlinear paracontractions and consistent linear systems abstract elsner et al l elsner i koltracht m neumann numer math 62 1992 305 319 introduced the concept of paracontracting operators for fixed point problems and their solution with asynchronous iteration methods their results are extended with respect to the properties of the pool of operators from which a common fixed point is searched also a more general kind of asynchronous iterations than theirs is presented as an application of this theory asynchronous iteration methods for consistent linear systems ax b are considered where a is a singular m matrix lubashevski and mitra b lubashevski d mitra j acm 33 1 1986 130 150 investigated asynchronous iteration methods for the perron vector problem i e determining a positive solution of tx p t x where t is an irreducible nonnegative matrix and its spectral radius is know their result can be fortified and extended to the reducible affine case by using a quite different approach the developed theory of paracontractions and confluence

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

on the asymtotic behavior of some alternate smoothing series expansion iterative methods abstract we study the asymptotic behavior of some iterative methods used in image reconstruction from projections these methods combine in each step a major iteration of a standard series expansion algorithm like art followed by the application of a smoothing matrix we prove convergence for a whole family of these methods when the matrix is symmetric and we give counterexamples for other cases

existence of homoclinic solutions for a class of second order hamiltonian systems abstract a new result for existence of homoclinic orbits is obtained for the second order hamiltonian systems under a class of new superquadratic conditions a homoclinic orbit is obtained as a limit of solutions of a certain sequence of boundary value problems which are obtained by the minimax methods

on asynchronous iterations asynchronous iterations arise naturally on parallel computers if one wants to minimize idle times this paper reviews certain models of asynchronous iterations using a common theoretical framework the corresponding convergence theory and various domains of applications are presented these include nonsingular linear systems nonlinear systems and initial value problems

models of parallel chaotic iteration methods abstract we consider two models of parallel multisplitting chaotic iterations for solving large nonsingular systems of equations ax b in the first model each processor can carry out an arbitrary number of local iterations before the next global approximation to the solution is formed in the second model any processor can update the global approximation which resides in the central processor at any time this model is a generalization of a sequential iterative scheme due to ostrowski called the free steering group jacobi iterative scheme and a chaotic relaxation point iterative scheme due to chazan and miranker we show that when a is a monotone matrix and all the splittings are weak regular both models lead to convergent schemes

a norm convergence result on random products of relaxed projections in hilbert space suppose x is a hilbert space and c 1 ldots c n are closed convex intersecting subsets with projections p 1 ldots p n suppose further r is a mapping from bbb n onto 1 ldots n that assumes every value infinitely often we prove a more general version of the following result begin quote if the n tuple c 1 ldots c n is innately boundedly regular then the sequence x n defined by x 0 in x mbox arbitrary x n 1 p r n x n mbox for all n geq 0 converges in norm to some point in bigcap i 1 n c i end quote examples without the usual assumptions on compactness are given methods of this type have been used in areas like computerized tomography and signal processing

notes on flexible sequential block iterative methods abstract we consider a flexible sequential block iterative method for the solution of consistent linear systems of equations and give its convergence analysis the method is able to use weight matrices and relaxation parameters which can be updated in each iteration whereas the most of previous studies on sequential block iterative methods considered a finite number of weight matrices furthermore we consider the constraint version of the method and give its convergence analysis for the special case of relaxation parameters and weight matrices we report on some numerical tests with examples taken from the field of image reconstruction from projections our numerical results show considerable improvement specially on noisy data compared to the other methods which use finite number of weight matrices

existence of even homoclinic orbits for second order hamiltonian systems some existence theorems for even homoclinic orbits are obtained for a class of second order nonautonomous hamiltonian systems with symmetric potentials under a class of new superquadratic conditions a homoclinic orbit is obtained as a limit of solutions of a certain sequence of nil boundary value problems which are obtained by the minimax methods

existence of homoclinic solution for the second order hamiltonian systems abstract an existence theorem of homoclinic solution is obtained for a class of the nonautonomous second order hamiltonian systems u t l t u t w t u t 0 t r by the minimax methods in the critical point theory specially the generalized mountain pass theorem where l t is unnecessary uniformly positively definite for all t r and w t x satisfies the superquadratic condition w t x x 2 as x uniformly in t and need not satisfy the global ambrosetti rabinowitz condition

convergence of sequential and asynchronous nonlinear paracontractions we establish the convergence of sequential and asynchronous iteration schemes for nonlinear paracontracting operators acting in finite dimensional spaces applications to the solution of linear systems of equations with convex constraints are outlined a first generalization of one of our convergence results to an infinite pool of asymptotically paracontracting operators is also presented

perturbation resilient iterative methods with an infinite pool of mappings we study a perturbation resilient iterative method with an infinite pool of operators having a property which allows using discontinuous operators weaker than paracontracting a convergence result is given under an extra condition which is inherently present in some state of the art iterative methods to evaluate the study we develop a new algorithmic scheme which generalizes both the string averaging algorithm and the block iterative projection methods and give its convergence analysis for the consistent case

a distributed algorithm for computing a common fixed point of a finite family of paracontractions a distributed algorithm is described for finding a common fixed point of a family of m 1 nonlinear maps m i mathbb r n rightarrow mathbb r n assuming that each map is a paracontraction and that at least one such common fixed point exists the common fixed point is simultaneously computed by m agents assuming each agent i knows only m i the current estimates of the fixed point generated by its neighbors and nothing more each agent recursively updates its estimate of a fixed point by utilizing the current estimates generated by each of its neighbors neighbor relations are characterized by a time varying directed graph mathbb n t it is shown under suitably general conditions on mathbb n t that the algorithm causes all agent estimates to converge to the same common fixed point of the m nonlinear maps

generalizations of the projection methods with application to sor theory for hermitian positive semidefinite linear systems ann n complex matrixb is calledparacontracting if b 2 1 and 0 x n i b bx 2 x 2 we show that a productb b k b k 1 b1 ofk paracontracting matrices is semiconvergent and give upper bounds on the subdominant eigenvalue ofb in terms of the subdominant singular values of theb i s and in terms of the angles between certain subspaces our results here extend earlier results due to halperin and due to smith solomon and wagner we also determine necessary and sufficient conditions forn numbers in the interval 0 1 to form the spectrum of a product of two orthogonal projections and hence characterize the subdominant eigenvalue of such a product in the final part of the paper we apply the upper bounds mentioned earlier to provide an estimate on the subdominant eigenvalue of the sor iteration matrix associated with ann n hermitian positive semidefinite matrixa none of whose diagonal entries vanish

on diagonally relaxed orthogonal projection methods we propose and study a block iterative projection method for solving linear equations and or inequalities the method allows diagonal componentwise relaxation in conjunction with orthogonal projections onto the individual hyperplanes of the system and is thus called diagonally relaxed orthogonal projections drop diagonal relaxation has proven useful in accelerating the initial convergence of simultaneous and block iterative projection algorithms but until now it was available only in conjunction with generalized oblique projections in which there is a special relation between the weighting and the oblique projections drop has been used by practitioners and in this paper a contribution to its convergence theory is provided the mathematical analysis is complemented by some experiments in image reconstruction from projections which illustrate the performance of drop

stochastic approximation for consensus a new approach via ergodic backward products this paper considers both synchronous and asynchronous consensus algorithms with noisy measurements for stochastic approximation based consensus algorithms the existing convergence analysis with dynamic topologies heavily relies on quadratic lyapunov functions whose existence however may be difficult to ensure for switching directed graphs our main contribution is to introduce a new analytical approach we first show a fundamental role of ergodic backward products for mean square consensus in algorithms with additive noise subsequently we develop ergodicity results for backward products of degenerating stochastic matrices converging to a 0 1 matrix via a discrete time dynamical system approach these results complement the existing ergodic theory of stochastic matrices and provide an effective tool for analyzing consensus problems under a joint connectivity assumption our approach may deal with switching topologies delayed measurements and correlated noises and it does not require the double stochasticity condition typically assumed for the existence of quadratic lyapunov functions

asynchronous two stage iterative methods parallel block two stage iterative methods for the solution of linear systems of algebraic equations are studied convergence is shown for monotone matrices and for h matrices two different asynchronous versions of these methods are considered and their convergence investigated

on the convergence of asynchronous paracontractions with application to tomographic reconstruction from incomplete data abstract convergence of iterative processes in c k of the form x i r i j i x 1 r i 1 1 p j i x i where j i 1 2 n i 1 2 is analyzed it is shown that if the matrices p 1 p n are paracontracting in the same smooth strictly convex norm and if the sequence j i i 1 has certain regularity properties then the above iterates converge this result implies the convergence of a parallel asynchronous implementation of the algebraic reconstruction technique art algorithm often used in tomographic reconstruction from incomplete data

a general asynchronous block iterative model with related convergence conditions abstract this paper formulates a general time varying asynchronous block iterative model a convergence condition for asynchronous block iterations based on this model is given compared to existing conditions for similar and shown to be strictly weaker

stability of asynchronous two dimensional fornasini marchesini dynamical systems abstract conditions for stability of two dimensional 2d fornasini marchesini fm models with variable shifts are derived using nonnegative matrix theory the conditions are shift independent the stability is proved under a general boundary condition bc

asynchronous parallel methods for enclosing solutions of nonlinear equations we consider interval arithmetic based parallel methods for enclosing solutions of nonlinear systems of equations where processors are allowed to proceed asynchronously we present a general study on the convergence of asynchronous iterations in interval spaces and apply them to a variety of methods for the nonlinear equations case the convergence results turn out to be very similar to those known for synchronous methods several practical examples on shared memory architectures are included the asynchronous methods sometimes perform substantially better than their synchronous counterparts

homoclinic solutions for some second order non autonomous hamiltonian systems without the globally superquadratic condition abstract this paper deals with the existence of homoclinic solutions for the following second order non autonomous hamiltonian system hs q v q t q f t where v c 1 r r n r v t q k t q w t q is t periodic in t f is aperiodic and belongs to l 2 r r n under the assumptions that k satisfies the pinching condition b 1 q 2 k t q b 2 q 2 w t q is not globally superquadratic on q and some additionally reasonable assumptions we give a new existence result to guarantee that hs has a homoclinic solution q t emanating from 0 the homoclinic solution q t is obtained as a limit of 2 k t periodic solutions of a sequence of the second order differential equations and these periodic solutions are obtained by the use of a standard version of the mountain pass theorem recent results in the literature are generalized and significantly improved

homoclinic orbits for superquadratic hamiltonian systems without a periodicity assumption abstract in this paper we consider the second order hamiltonian system q t v t q t f t where v t q k t q w t q under suitable conditions on the growth of w and k we establish the existence of a nontrivial homoclinic orbit without any assumption of periodicity on v this homoclinic orbit is obtained as a limit of solutions of a certain sequence of nil boundary value problems which are obtained by the minimax methods