traveling waves for a diffusive sir model with delay abstract in this paper we study traveling waves connecting the infection free equilibrium state and the endemic equilibrium state for a diffusive sir model with delay and saturated incidence rate since this system does not enjoy the comparison principle we will use an iteration process to construct a pair of upper and lower solutions with the aid of the pair of upper and lower solutions we can apply the schauder fixed point theorem to construct a family of solutions of the truncated problems which via the limiting argument can generate the traveling wave indeed we show that there exists c 0 such that this system admits a traveling wave solution with speed c iff c c

extending the applicability of gauss newton method for convex composite optimization on riemannian manifolds we present a semi local convergence analysis of the gauss newton method for solving convex composite optimization problems in riemannian manifolds using the notion of quasi regularity for an initial point using a combination the l average lipszhitz condition and the center l 0 average lipschitz condition we introduce majorizing sequences for the gauss newton method that are more precise than in earlier studies consequently our semi local convergence analysis for the gauss newton method has the following advantages under the same computational cost weaker sufficient convergence conditions more precise estimates on the distances involved and an at least as precise information on the location of the solution

convergence of newton s method and inverse function theorem in banach space under the hypothesis that the derivative satisfies some kind of weak lipschitz condition a proper condition which makes newton s method converge and an exact estimate for the radius of the ball of the inverse function theorem are given in a banach space also the relevant results on premises of kantorovich and smale types are improved in this paper

convergence analysis of the secant type methods in this paper under the hypothesis that derivative satisfies some kinds of weak lipschitz condition the radius estimates of the convergence balls of secant type methods for operator equations are established in banach space some applications and numerical experiments are also given

on newton s method under holder continuous derivative a mysovskii type theorem for newton s method under k p holder continuous derivative is considered in this paper for the application studied by others the new condition is weaker than ones in the literature also we prove that the optimal convergent order is p 1 for 0 p 1

stability and traveling waves of a vaccination model with nonlinear incidence abstract in this paper we propose a spatial vaccination model with nonlinear incidence first we consider the well posedness of solutions of the model second in the case of the bounded spatial habitat r n we investigate the global stability of the model more precisely it is shown that if the threshold value r 0 1 then the disease free equilibrium e 0 is globally asymptotically stable if r 0 1 then there exists a unique disease equilibrium e which is globally asymptotically stable third in the case of the unbounded spatial habitat r n we study the existence of traveling wave solutions of the model here we show that when the threshold value r 0 1 then there exists c 0 such that there exist positive traveling wave solutions of the model connecting the two equilibria e 0 and e with speed c c and when r 0 1 there is not such a traveling wave solution with speed c c numerical simulations are performed to illustrate our analytic results our results indicate that the global dynamics of the model are completely determined by the threshold value r 0

coherent sets for nonautonomous dynamical systems we describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems in the autonomous setting such objects are variously known as almost invariant sets metastable sets persistent patterns or strange eigenmodes and have proved to be important in a variety of applications in this current work we explain how to extend existing autonomous approaches to the nonautonomous setting we call the new time dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly and remain coherent the new methods are illustrated via detailed examples in both discrete and continuous time

extended newton methods for multiobjective optimization majorizing function technique and convergence analysis we consider the extended newton method for approaching a pareto optimum of a multiobjective optimization problem establish quadratic convergence criteria and estimate a radius of convergence ball

a kantorovich type convergence analysis for the gauss newton method we present a semilocal kantorovich type convergence analysis for the gauss newton method which reduces to the wellknown newton kantorovich theorem for the newton method in a natural way additionnally a classification of the nonlinear regression problem into adequate and not adequate models is obtained

point estimation of simultaneous methods for solving polynomial equations a survey ii the construction of computationally verifiable initial conditions which provide both the guaranteed and fast convergence of the numerical root finding algorithm is one of the most important problems in solving nonlinear equations smale s point estimation theory from 1981 was a great advance in this topic it treats convergence conditions and the domain of convergence in solving an equation f z 0 using only the information of f at the initial point z 0 the study of a general problem of the construction of initial conditions of practical interest providing guaranteed convergence is very difficult even in the case of algebraic polynomials in the light of smale s point estimation theory an efficient approach based on some results concerning localization of polynomial zeros and convergent sequences is applied in this paper to iterative methods for the simultaneous determination of simple zeros of polynomials we state new improved initial conditions which provide the guaranteed convergence of frequently used simultaneous methods for solving algebraic equations ehrlich aberth s method ehrlich aberth s method with newton s correction borsch supan s method with weierstrass correction and halley like or wang zheng method the introduced concept offers not only a clear insight into the convergence analysis of sequences generated by the considered methods but also explicitly gives their order of convergence the stated initial conditions are of significant practical importance since they are computationally verifiable they depend only on the coefficients of a given polynomial its degree n and initial approximations to polynomial zeros

improved local convergence of newton s method under weak majorant condition we provide a local convergence analysis for newton s method under a weak majorant condition in a banach space setting our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than before 14 special cases and numerical examples are also provided in this study

the lyapunov exponents of generic volume preserving and symplectic maps we show that the integrated lyapunov exponents of c1 volume preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding oseledets splitting is trivial all lyapunov exponents are equal to zero or else dominated uniform hyperbolicity in the projective bundle almost everywhere we deduce a sharp dichotomy for generic volume preserving diffeomorphisms on any compact manifold almost every orbit either is projectively hyperbolic or has all lyapunov exponents equal to zero similarly for a residual subset of all c1 symplectic diffeomorphisms on any compact manifold either the diffeomorphism is anosov or almost every point has zero as a lyapunov exponent with multiplicity at least 2 finally given any set s gl d satisfying an accessibility condition for a residual subset of all continuous s valued cocycles over any measure preserving homeomorphism of a compact space the oseledets splitting is either dominated or trivial the condition on s is satisfied for most common matrix groups and also for matrices that arise from discrete schr nodinger operators

convergence and uniqueness properties of gauss newton s method abstract the generalized radius and center lipschitz conditions with l average are introduced to investigate the convergence of gauss newton s method for finding the nonlinear least squares solution of nonlinear equations the radii of the convergence ball of gauss newton s method and the uniqueness ball of the solution are estimated some applications are given

kantorovich s theorem on newton s method for solving generalized equations under the majorant condition in this paper we consider a version of the kantorovich s theorem for solving the generalized equation f x t x 0 where f is a frechet derivative function and t is a set valued and maximal monotone acting between hilbert spaces we show that this method is quadratically convergent to a solution of f x t x 0 we have used the idea of majorant function which relaxes the lipschitz continuity of the derivative f it allows us to obtain the optimal convergence radius uniqueness of solution and also to solving generalized equations under smale s condition

local convergence of the gauss newton method for injective overdetermined systems of equations under a majorant condition a local convergence analysis of the gauss newton method for solving injective overdetermined systems of nonlinear equations under a majorant condition is provided the convergence as well as results on its rate are established without a convexity hypothesis on the derivative of the majorant function the optimal convergence radius the biggest range for uniqueness of the solution along with some other special cases are also obtained

a new semilocal convergence theorem for newton s method abstract a new semilocal convergence theorem for newton s method is established for solving a nonlinear equation f x 0 defined in banach spaces it is assumed that the operator f is twice frechet differentiable and f satisfies a lipschitz type condition results on uniqueness of solution and error estimates are also given finally these results are compared with those that use kantorovich conditions

perturbations of random matrix products if x1 x2 are identically distributed independent random matrices with a common distribution then with the probability 1 the limit lambda mu mathop lim limits n to infty frac 1 n ln left x n x 1 right exists the paper treats the problem is it true that k if k in the weak sense

convergence of gauss newton s method and uniqueness of the solution abstract in this paper we study the convergence of gauss newton s method for nonlinear least squares problems under the hypothesis that derivative satisfies some kinds of weak lipschitz condition we obtain the exact estimates of the radii of convergence ball of gauss newton s method and the uniqueness ball of the solution new results can be used to determinate approximation zero of gauss newton s method

two conditions concerning newton s method in this note we compare two conditions concerning newton s method and find out that even under some weaker condition one has the convergence of the newton s method and a better error bound than the relative strong condition

dynamical behavior of an epidemic model with a nonlinear incidence rate abstract in this paper we study the global dynamics of an epidemic model with vital dynamics and nonlinear incidence rate of saturated mass action by carrying out global qualitative and bifurcation analyses it is shown that either the number of infective individuals tends to zero as time evolves or there is a region such that the disease will be persistent if the initial position lies in the region and the disease will disappear if the initial position lies outside this region when such a region exists it is shown that the model undergoes a bogdanov takens bifurcation i e it exhibits a saddle node bifurcation hopf bifurcations and a homoclinic bifurcation existence of none one or two limit cycles is also discussed

the mathematics of infectious diseases many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases threshold theorems involving the basic reproduction number r 0 the contact number sigma and the replacement number r are reviewed for the classic sir epidemic and endemic models similar results with new expressions for r 0 are obtained for mseir and seir endemic models with either continuous age or age groups values of r 0 and sigma are estimated for various diseases including measles in niger and pertussis in the united states previous models with age structure heterogeneity and spatial structure are surveyed

hilbert space lyapunov exponent stability we study cocycles of compact operators acting on a separable hilbert space and investigate the stability of the lyapunov exponents and oseledets spaces when the operators are subjected to additive gaussian noise we show that as the noise is shrunk to 0 the lyapunov exponents of the perturbed cocycle converge to those of the unperturbed cocycle and the oseledets spaces converge in probability to those of the unperturbed cocycle this is to our knowledge the first result of this type with cocycles taking values in operators on infinite dimensional spaces the infinite dimensionality gives rise to a number of substantial difficulties that are not present in the finite dimensional case

wave propagation in an infectious disease model abstract this paper is devoted to the study of the wave propagation in a reaction convection infectious disease model with a spatio temporal delay previous numerical studies have demonstrated the existence of traveling wave fronts for the system and obtained a critical value c which is the minimal wave speed of the traveling waves in the present paper we provide a complete and rigorous proof to overcome the difficulty due to the lack of monotonicity for the system we construct a pair of upper and lower solutions and then apply the schauder fixed point theorem to establish the existence of a nonnegative solution for the wave equation on a bounded interval moreover we use a limiting argument and in turn generate the solution on the unbounded interval r in particular by constructing a suitable lyapunov functional we further show that the traveling wave solution converges to the epidemic equilibrium point as t

newton s method for overdetermined systems of equations complexity theoretic aspects of continuation methods for the solution of square or underdetermined systems of polynomial equations have been studied by various authors in this paper we consider overdetermined systems where there are more equations than unknowns we study newton s method for such a system

the guaranteed convergence of laguerre like method abstract the construction of initial conditions that provide a guaranteed convergence of zero finding methods has attracted a great deal of attention for many years in this paper we consider convergent properties of the laguerre like method of the fourth order for the simultaneous approximation of polynomial zeros using a procedure based on smale s point estimation theory and some recent results concerned with localization of complex polynomial zeros we state initial conditions which enable both the guaranteed and fast convergence of this method these conditions are computationally verifiable since they depend only on initial approximations polynomial coefficients and polynomial degree which is of practical importance

approximate zeros of quadratically convergent algorithms smale s condition for a point to be an approximate zero of a func tion for newton s method is extended to the general quadratically convergent iterative algorithm it is shown in which way the bound in the condition is affected by the characteristics of the algorithm this puts the original condition of smale for newton s method in a more general perspective the results are also discussed in the light of numerical evidence

convergence criterion of newton s method for singular systems with constant rank derivatives abstract the present paper is concerned with the convergence problem of newton s method to solve singular systems of equations with constant rank derivatives under the hypothesis that the derivatives satisfy a type of weak lipschitz condition a convergence criterion based on the information around the initial point is established for newton s method for singular systems of equations with constant rank derivatives applications to two special and important cases the classical lipschitz condition and the smale s assumption are provided the latter in particular extends and improves the corresponding result due to dedieu and kim in j p dedieu m kim newton s method for analytic systems of equations with constant rank derivatives j complexity 18 2002 187 209

kantorovich s type theorems for systems of equations with constant rank derivatives the famous newton kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of newton s method to a solution of an equation here we present a kantorovich type convergence analysis for the gauss newton s method which improves the result in w m hauszler a kantorovich type convergence analysis for the gauss newton method numer math 48 1986 119 125 and extends the main theorem in i k argyros on the newton kantorovich hypothesis for solving equations j comput appl math 169 2004 315 332 furthermore the radius of convergence ball is also obtained

mixed types of waves in a discrete diffusive epidemic model with nonlinear incidence and time delay abstract in this paper we investigate the traveling wave solutions of a discrete diffusive epidemic model with nonlinear incidence and time delay employing the method of upper and lower solutions schauder s fixed point theorem and a limiting approach we prove the existence of bounded super critical and critical traveling wave solutions moreover we obtain the positiveness and asymptotic boundary of the traveling wave solutions which guarantee that the traveling wave solutions are non trivial the existence results show that the traveling waves are mixed of front type and pulse type by way of contradiction and two sided laplace transform we derive the non existence of non trivial positive and bounded traveling wave solutions it is the first time to apply the method of upper and lower solutions together with schauder s fixed point theorem and two sided laplace transform to investigate the existence and non existence of traveling wave solutions for discrete diffusive epidemic models respectively

approximate solutions for abstract inequality systems we consider conic inequality systems of the type f x ge k 0 with approximate solution x 0 associated to a parameter tau where f is a twice frechet differentiable function between hilbert spaces x and y and ge k is the partial order in y defined by a nonempty convex not necessarily closed cone k subseteq y we prove that under the suitable conditions the system f x ge k 0 is solvable and the ratio of the distance from x 0 to the solution set s over the distance from f x 0 to the cone k has an upper bound given explicitly in terms of tau and x 0 we show that the upper bound is sharp applications to analytic function inequality equality systems on euclidean spaces are given and the corresponding results of dedieu siam j optim 11 2000 pp 411 425 are extended and significantly improved

convergence of the family of the deformed euler halley iterations under the holder condition of the second derivative the convergence problem of the family of the deformed euler halley iterations with parameters for solving nonlinear operator equations in banach spaces is studied under the assumption that the second derivative of the operator satisfies the holder condition a convergence criterion of the order 2 p of the iteration family is established an application to a nonlinear hammerstein integral equation of the second kind is provided

threshold dynamics of a diffusive siri model with nonlinear incidence rate abstract this work is concerned with the dynamics of a diffusive siri epidemic model with nonlinear incidence rate we first establish the well posedness of the model then we show the basic reproduction number r 0 is a threshold parameter for the stability of the model in fact the disease free equilibrium is globally asymptotically stable when r 0 1 while the phenomena of uniform persistence occurs when r 0 1 if r 0 1 the disease free equilibrium is globally asymptotically stable under some assumptions we also clarify the relationship between r 0 and the local basic reproduction number moreover we establish the local and global stability of the endemic equilibrium provided that all parameters of this model are constants

semilocal convergence for halley s method under weak lipschitz condition the semilocal convergence properties of halley s method for nonlinear operator equations are studied under the hypothesis that the second derivative satisfies some weak lipschitz condition the method employed in the present paper is based on a family of recurrence relations which will be satisfied by the involved operator an application to a nonlinear hammerstein integral equation of the second kind is provided

newton s method for solving generalized equations kantorovich s and smale s approaches abstract in this paper we study newton type methods for solving generalized equations involving set valued maps in banach spaces kantorovich type theorems both local and global versions are proved as well as the quadratic convergence of the newton sequence we also extend smale s classical theory to generalized equations these results are new and can be considered as an extension of many known ones in the literature for classical nonlinear equations our approach is based on tools from variational analysis the metric regularity concept plays an important role in our analysis

an improved convergence theorem for a class of secant like methods an improved semilocal convergence theorem for a class of secant like methods in banach spaces is provided for solving a nonlinear operator equation it is assumed that the first order divided difference of the nonlinear operator is holder continuous our convergence conditions strategically proposed are weaker than some existing ones and are easily satisfied finally two simple examples are provided to show that our results apply while earlier ones fail

a convergence theorem for the newton like methods under some kind of weak lipschitz conditions we provide sufficient conditions for the convergence of the newton like methods in the assumption that the derivative satisfies some kind of weak lipschitz conditions consequently some important convergence theorems follow from our main result in this paper

proper analytic free maps this paper concerns analytic free maps these maps are free analogs of classical analytic functions in several complex variables and are defined in terms of non commuting variables amongst which there are no relations they are free variables analytic free maps include vector valued polynomials in free non commuting variables and form a canonical class of mappings from one non commutative domain d in say g variables to another non commutative domain d in g variables as a natural extension of the usual notion an analytic free map is proper if it maps the boundary of d into the boundary of d assuming that both domains contain 0 we show that if f d d is a proper analytic free map and f 0 0 then f is one to one moreover if also g g then f is invertible and f 1 is also an analytic free map these conclusions on the map f are the strongest possible without additional assumptions on the domains d and d

super critical and critical traveling waves in a three component delayed disease system with mixed diffusion abstract this paper deals with traveling waves of a three component delayed disease system with mixed diffusion when the basic reproduction number of the corresponding spatial homogenous delayed differential system r 0 1 and the wave velocity c c c is critical velocity we obtain that the system admits a positive traveling wave solution when r 0 1 and c r or r 0 1 and c c we prove that the system has no positive traveling wave solutions our theoretical results may be conducive to disease prevention and control

epidemic waves of a spatial sir model in combination with random dispersal and non local dispersal abstract in this paper a spatial sir model in combination with random dispersal and non local dispersal is proposed we establish the existence and non existence of the traveling wave solutions connecting the disease free equilibrium and the endemic equilibrium for the model the main difficulties lie in the fact that the semiflow generated here does not admit the order preserving property and the solutions lack of regularity we use a proper iteration scheme to construct a pair of upper and lower solutions and then apply the schauder s fixed point theorem polar coordinates transform to study the threshold dynamics of the model that is we show that if the basic reproduction number of the model r 0 1 there is a positive constant critical number c such that for each c c the model admits a non trivial and positive traveling wave with wave speed c and if r 0 1 and 0

traveling waves in a kermack mckendrick epidemic model with diffusion and latent period abstract this paper is devoted to the study of a kermack mckendrick epidemic model with diffusion and latent period we first consider the well posedness of solutions of the model furthermore using the schauder fixed point theorem and laplace transform we show that if the threshold value r 0 1 then there exists c 0 such that for every c c the model admits a traveling wave solution and if r 0 1 and c 0 or r 0 1 and c 0 c then the model admits no traveling wave solutions hence the existence and non existence of traveling wave solutions is determined completely by r 0 and the constant c is the minimum speed for the existence of traveling wave solutions of the model

stochastic stability of lyapunov exponents and oseledets splittings for semi invertible matrix cocycles we establish i stability of lyapunov exponents and ii convergence in probability of oseledets spaces for semi invertible matrix cocycles subjected to small random perturbations the first part extends results of ledrappier and young to the semi invertible setting the second part relies on the study of evolution of subspaces in the grassmannian the analysis developed here is based on higher dimensional mobius transformations and is likely to be of wider interest

expanding the applicability of newton s method using smale s α theory we present a tighter convergence analysis than earlier studies such as in cianciaruso 2007 guo 2007 shen and li 2010 smale 1986 1987 wang and zhao 1995 wang 1999 wang and han 1990 of newton s method using smale s a theory by introducing the notion of the center c 0 condition in particular in the semilocal convergence case we show that if the center c 0 condition is smaller than the c condition then the new majorizing sequence is tighter than the old majorizing sequence the new convergence criteria are weaker than the older convergence criteria furthermore in the local convergence case we obtain a larger radius of convergence and tighter error estimates on the distances involved these improvements are obtained under the same computational cost numerical examples and applications are also provided in this study to show that the older results cannot apply but the new results apply to solve equations

loi des grands nombres et perturbations pour des produits reductibles de matrices aleatoires independantes soit x x n n 1 une suite de matrices aleatoires reelles d d independantes de meme loi on etudie le comportement asymptotique de la suite 1 n log x n x 1 x n

a new fourth order family of simultaneous methods for finding polynomial zeros using a suitable fixed point relation with a complex parameter a new one parameter family of simultaneous methods of the fourth order for finding complex zeros of a polynomial is derived in ordinary complex arithmetic convergence analysis of the presented family is performed under computationally verifiable initial conditions which depend only on polynomial coefficients and initial approximations further improvements of the proposed family of methods are discussed and a modification for finding multiple zeros is presented some numerical results for various values of the parameter are given

the convergence of a family of parallelzero finding methods abstract one of the crucial problems in solving polynomial equations is to state some kind of quantitative conditions that provide both safe and fast convergence in the spirit of smale s point estimation theory introduced in 1981 we construct in this paper computationally verifiable initial conditions which guarantee the convergence of the recently proposed one parameter family of methods of the fourth order for the simultaneous finding polynomial zeros these conditions are computationally verifiable since they depend only on initial approximations polynomial coefficients and polynomial degree which is of significant practical importance

critical traveling waves in a diffusive disease model abstract in this paper the existence of a non trivial positive and bounded critical traveling wave solution of a diffusive disease model whose reaction system has infinitely many equilibria is obtained for the first time this gives an affirmative answer to an open problem left in wang et al 2012 19 our result shows that the critical traveling wave in this model is a mix of front s component and pulse i component type

convergence ball of a modified secant method with convergence order 1 839 abstract a local convergence theorem on a modified secant method with convergence order 1 839 for solving nonlinear equations is established under the hypotheses that the second order and third order derivative of the function involved are bounded an estimate of the radius of the convergence ball of this method is obtained moreover an error estimate is provided which matches the convergence order of the method