strong allee effect in a diffusive predator prey system with a protection zone article history a reaction diffusion predator prey system with strong allee effect and a protection zone for the prey is considered dynamics and steady state solutions of the system are analyzed in particular it is shown that the overexploitation phenomenon can be avoided if the allee effect threshold is low and the protection zone is large

convergence to a steady state for asymptotically autonomous semilinear heat equations on r double struck n abstract we consider parabolic equations of the form u t u f u h x t x t r n 0 where f is a c 1 function with f 0 0 f 0 0 and h is a suitable function on r n 0 which decays to zero as t hence the equation is asymptotically autonomous we show that as t each bounded localized solution u 0 approaches a set of steady states of the limit autonomous equation u t u f u moreover if the decay of h is exponential then u converges to a single steady state we also prove a convergence result for abstract asymptotically autonomous parabolic equations

on a system of reaction diffusion equations arising from competition with internal storage in an unstirred chemostat abstract in this paper we study a system of reaction diffusion equations arising from competition of two microbial populations for a single limited nutrient with internal storage in an unstirred chemostat the conservation principle is used to reduce the dimension of the system by eliminating the equation for the nutrient the reduced system limiting system generates a strongly monotone dynamical system in its feasible domain under a partial order we construct suitable upper lower solutions to establish the existence of positive steady state solutions given the parameters of the reduced system we answer the basic questions as to which species survives and which does not in the spatial environment and determine the global behaviors the primary conclusion is that the survival of species depends on species s intrinsic biological characteristics the external environment forces and the principal eigenvalues of some scalar partial differential equations we also lift the dynamics of the limiting system to the full system

some results about perturbed maximal monotone mappings in this paper we consider a class of variational inequality problems for monotone type mappings by using one coercive assumption which is weaker than the one used in 1 we establish some existence results of solutions to variational inequality problems with generalized pseudomonotone mappings generalized pseudomonotone perturbation and t pseudomonotone perturbation of maximal monotone mappings respectively

a non autonomous seirs model with general incidence rate for a non autonomous seirs model with general incidence that admits kuniya and nakata 2012 as a very particular case we obtain conditions for extinction and strong persistence of the infectives our conditions are computed for several particular settings and extend the hypothesis of several proposed non autonomous models additionally we show that our conditions are robust in the sense that they persist under small perturbations of the parameters in some suitable family we also present some simulations that illustrate our results

a cascadic multigrid algorithm for the stokes equations a variant of multigrid schemes for the stokes problem is discussed in particular we propose and analyse a cascadic version for the stokes problem the analysis of the transfer between the grids requires special care in order to establish that the complexity is the same as that for classical multigrid algorithms

factorization of minty and stampacchia variational inequality systems abstract in this paper sufficient regularity and coercivity conditions for minty and stampacchia type variational inequality systems are offered the typical results state that if the independent inequalities are solvable and the functions involved are lower semicontinuous then the system has also a solution that is a factorization principle holds as an application a variational inequality system consisting of two partial differential inequalities is considered this result is analogous to that of chen journal of mathematical analysis and application 231 1999 177 obtained for one variational inequality

a cascadic multigrid algorithm for semilinear elliptic problems summary we propose a cascadic multigrid algorithm for a semilinear elliptic problem the nonlinear equations arising from linear finite element discretizations are solved by newton s method given an approximate solution on the coarsest grid on each finer grid we perform exactly one newton step taking the approximate solution from the previous grid as initial guess the newton systems are solved iteratively by an appropriate smoothing method we prove that the algorithm yields an approximate solution within the discretization error on the finest grid provided that the start approximation is sufficiently accurate and that the initial grid size is sufficiently small moreover we show that the method has multigrid complexity

skew product semiflows and morse decomposition abstract this paper is devoted to the investigation of the dynamics of non autonomous differential equations the description of the asymptotic dynamics of non autonomous equations lies on dynamical structures of some associated limiting non autonomous and autonomous differential equations one for each global solution in the attractor of the driving semigroup of the associated skew product semiflow in some cases we have infinitely many limiting problems in contrast with the autonomous or asymptotically autonomous case for which we have only one limiting problem that is the semigroup itself we concentrate our attention in the study of the morse decomposition of attractors for these non autonomous limiting problems as a mean to understand some of the asymptotics of our non autonomous differential equations in particular we derive a morse decomposition for the global attractors of skew product semiflows and thus for pullback attractors of non autonomous differential equations from a morse decomposition of the attractor for the associated driving semigroup our theory is well suited to describe the asymptotic dynamics of non autonomous differential equations defined on the whole line or just for positive times or for differential equations driven by a general semigroup

asymptotically autonomous semiflows chain recurrence and lyapunov functions from the work of c conley it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set here we improve a result of l markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow in the special case that there is a lyapunov function for the limiting semiflow sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the lyapunov function

an extension of the concept of gradient semigroups which is stable under perturbation abstract in this article we introduce the concept of a gradient like nonlinear semigroup as an intermediate concept between a gradient nonlinear semigroup those possessing a lyapunov function see j k hale asymptotic behavior of dissipative systems math surveys monogr vol 25 amer math soc 1989 and a nonlinear semigroup possessing a gradient like attractor we prove that a perturbation of a gradient like nonlinear semigroup remains a gradient like nonlinear semigroup moreover for non autonomous dynamical systems we introduce the concept of a gradient like evolution process and prove that a non autonomous perturbation of a gradient like nonlinear semigroup is a gradient like evolution process for gradient like nonlinear semigroups and evolution processes we prove continuity characterization and pullback and forwards exponential attraction of their attractors under perturbation extending the results of a n carvalho j a langa j c robinson a suarez characterization of non autonomous attractors of a perturbed gradient system j differential equations 236 2007 570 603 on characterization and of a v babin m i vishik attractors in evolutionary equations stud math appl vol 25 north holland amsterdam 1992 on exponential attraction

well defined steady state response does not imply cics abstract systems for which each constant input gives rise to a unique globally attracting equilibrium are considered a counterexample is provided to show that inputs which are only asymptotically constant may not result in states converging to equilibria failure of the converging input converging state or cics property

the stability of attractors for non autonomous perturbations of gradient like systems abstract we study the stability of attractors under non autonomous perturbations that are uniformly small in time while in general the pullback attractors for the non autonomous problems converge towards the autonomous attractor only in the hausdorff semi distance upper semicontinuity the assumption that the autonomous attractor has a gradient like structure the union of the unstable manifolds of a finite number of hyperbolic equilibria implies convergence i e also lower semicontinuity provided that the local unstable manifolds perturb continuously we go further when the underlying autonomous system is itself gradient like and show that all trajectories converge to one of the hyperbolic trajectories as t in finite dimensional systems in which we can reverse time and apply similar arguments to deduce that all bounded orbits converge to a hyperbolic trajectory as t this implies that the gradient like structure of the attractor is also preserved under small non autonomous perturbations the pullback attractor is given as the union of the unstable manifolds of a finite number of hyperbolic trajectories

lipschitz perturbations of morse smale semigroups abstract in this paper we deal with lipschitz continuous perturbations of gradient morse smale semigroups all critical elements are equilibria we study the permanence of connections between equilibrium points structural stability when subjected to lipschitz perturbations to this end we extend the notions of hyperbolicity and transversality to the cases in which differentiability is no longer available

existence stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations abstract in this paper we study the existence stability and bifurcation of random complete and periodic solutions for stochastic parabolic equations with multiplicative noise we first prove the existence and uniqueness of tempered random attractors for the stochastic equations and characterize the structures of the attractors by random complete solutions we then examine the existence and stability of random complete quasi solutions and establish the relations of these solutions and the structures of tempered attractors when the stochastic equations are incorporated with periodic forcing we obtain the existence and stability of random periodic solutions for the stochastic chafee infante equation we further establish the multiplicity and stochastic bifurcation of complete and periodic solutions

non autonomous morse decomposition and lyapunov functions for gradient like processes we define time dependent morse decompositions for non autonomous evolution processes non autonomous dynamical systems and prove that a non autonomous gradient like evolution process possesses a morsedecomposition on the associated pullback attractor we also prove the existence of an associated lyapunov function which describes the gradient behavior of the system finally we apply these abstract results to non autonomous perturbations of autonomous gradient like evolution processes semigroups or autonomous dynamical systems

on the convergence of an extrapolation cascadic multigrid method for elliptic problems abstract classical cascadic multigrid method is optimal under the energy norm for h 2 regular elliptic problems this paper analyzes an extrapolation cascadic multigrid excmg method originally proposed by chen et al 2008 for solving second order elliptic equations following the idea of bornemann and deuflhard 1996 we present a superconvergence result for the excmg method which enables us to show that the excmg method with the conjugate gradient method as a basic iterative scheme is optimal for h 3 regular elliptic problems in three dimension with respect to the l 2 norm moreover we also prove the super optimality of the excmg method under the energy norm for h 2 regular 0 1 problems in both two and three dimensions with a reasonable assumption on asymptotic error expansions finally numerical results are presented to verify our theoretical analysis

input to state stability for cascade systems with multiple invariant sets abstract in a recent paper angeli and efimov 2015 the notion of input to state stability iss has been generalized for systems with decomposable invariant sets and evolving on riemannian manifolds in this work we analyze the cascade interconnection of such iss systems and we characterize the finest possible decomposition of its invariant set for three different scenarios 1 the driving system exhibits multistability convergence to fixed points only 2 the driving system exhibits multi almost periodicity convergence to fixed points as well as periodic and almost periodic orbits and the driven system is assumed to be incremental iss 3 the driving system exhibits multiperiodicity convergence to fixed points and periodic orbits whereas the driven system is iss in the sense of angeli and efimov 2015 furthermore we provide marginal results on the backward forward asymptotic behavior of incremental iss systems and on the response of a contractive system under asymptotically almost periodic forcing three examples illustrate the potentiality of the proposed framework

error estimates of the lumped mass finite element method for semilinear elliptic problems we are concerned with the semilinear elliptic problems we first investigate the l 2 error estimate for the lumped mass finite element method we then use the cascadic multigrid method to solve the corresponding discrete problem on the basis of the finite element error estimates we prove the optimality of the proposed multigrid method we also report some numerical results to support the theory

dynamics in a diffusive predator prey system with strong allee effect and ivlev type functional response abstract the dynamics of a kind of reaction diffusion predator prey system with strong allee effect in the prey population is considered we prove the existence and uniqueness of the solution and give a priori bound hopf bifurcation and steady state bifurcation are studied results show that the allee effect has significant impact on the dynamics

convergence on cooperative cascade systems with length one abstract motivated by the open problems on the generic convergence of cooperative systems without the assumption of irreducibility independently proposed by smith and sontag this paper investigates the generic convergence for the solutions of cooperative cascade systems with length one first by fixing a solution of a base system converging to an equilibrium we establish both the nonordering of limit sets and the limit set dichotomy for the solutions of the cascade system combining these tools with the idea of limiting equation we then prove the sequential limit set trichotomy and hence the quasiconvergence in generic meaning the generic convergent result is finally obtained by improving the limit set dichotomy

complementarity problems for multivalued monotone operator in banach spaces we utilize s park s maximal element theorem in this paper to prove the existence theorems of solutions of the complementarity problems for multivalued monotone operator in banach spaces

pullback permanence in a non autonomous competitive lotka volterra model abstract the goal of this work is to study in some detail the asymptotic behaviour of a non autonomous lotka volterra model both in the conventional sense as t and in the pullback sense starting a fixed initial condition further and further back in time the non autonomous terms in our model are chosen such that one species will eventually die out ruling out any conventional type of permanence in contrast we introduce the notion of pullback permanence and show that this property is enjoyed by our model this is not just a mathematical artifice but rather shows that if we come across an ecology that has been evolving for a very long time we still expect that both species are represented and their numbers are bounded below even if the final fate of one of them is less happy the main tools in the paper are the theory of attractors for non autonomous differential equations the sub supersolution method and the spectral theory for linear elliptic equations

global asymptotic behavior of a chemostat model with discrete delays this paper studies the global asymptotic behavior of an exploitative competition model between n species in a chemostat the model incorporates discrete time delays to describe the delay in the conversion of nutrient consumed to viable biomass and hence includes delays simultaneously in variables of nutrient and species concentrations in the case where only two species are engaged in competition it is shown that competitive exclusion holds for any monotone growth response functions sufficient conditions are also obtained for the model to exhibit competitive exclusion in the n species case in regard to the delay effects on the qualitative outcome of competition it is demonstrated that when the delays are relatively small the predictions of the model are identical with the predictions given by corresponding models without time delays however including large delays in the model may alter the predicted outcome of competition the techniques used also work when different removal rates are permitted an

uniform persistence in processes with application to nonautonomous competitive models abstract an acyclicity theorem on uniform persistence is proved for continuous processes this result is applied to two species nonautonomous competitive lotka volterra models

traveling fronts of the volume filling chemotaxis model with general kinetics this paper is devoted to the study of traveling fronts for the volume filling chemotaxis model with general kinetics by applying the perturbation method the proof relies on the fredholm theory and the banach contractive mapping principle the obtained results can easily detect the existence of traveling fronts for some models investigated in the known references

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

multivalued s 1 operators and generalized variational inequalities abstract in this paper we derive some new existence results for the generalized variational inequalities by introducing multivalued s 1 operators applications to nonlinear inequalities and the generalized complementarity problem are given

non autonomous perturbation of autonomous semilinear differential equations continuity of local stable and unstable manifolds abstract in this paper we prove a result on lower semicontinuity of pullback attractors for dynamical systems given by semilinear differential equations in a banach space the situation considered is such that the perturbed dynamical system is non autonomous whereas the limiting dynamical system is autonomous and has an attractor given as union of unstable manifold of hyperbolic equilibrium points starting with a semilinear autonomous equation with a hyperbolic equilibrium solution and introducing a very small non autonomous perturbation we prove the existence of a hyperbolic global solution for the perturbed equation near this equilibrium then we prove that the local unstable and stable manifolds associated to them are given as graphs roughness of dichotomy plays a fundamental role here moreover we prove the continuity of this local unstable and stable manifolds with respect to the perturbation with that result we conclude the lower semicontinuity of pullback attractors

global convergence of nonlinear cascade flows with morse bott zero dynamics abstract we derive a sufficient condition that a flow captures the dynamics on an invariant submanifold this leads to a refinement of the lasalle invariance principle as a consequence we generalize a well known global asymptotic stability result of nonlinear cascade systems to show global convergence to a compact invariant set this includes the case where a globally asymptotically stable system is coupled to a morse bott flow

numerical determination of the basin of attraction for asymptotically autonomous dynamical systems we develop a method to numerically analyse asymptotically autonomous systems of the form dot x f t x where f t x tends to g x as t the rate of convergence is not limited to exponential but may be polynomial logarithmic or any other rate for these systems we propose a transformation of the infinite time interval to a finite compact one which reflects the rate of convergence of f to g in the transformed system the origin is an asymptotically stable equilibrium which is exponentially stable in x direction weconsider a lyapunov function in this transformed system as a solution of a suitable linear first order partial differential equation and approximate it using radial basis functions

dynamics of lotka volterra competition systems with large interaction abstract we show that under certain additional hypothesis two population competing species models in bounded domains with diffusion and large interaction have simple dynamics in particular the solutions approach stationary states as t tends to infinity

almost periodic dynamics of perturbed infinite dimensional dynamical systems abstract this paper is concerned with the dynamics of an infinite dimensional gradient system under small almost periodic perturbations under the assumption that the original autonomous system has a global attractor given as the union of unstable manifolds of a finite number of hyperbolic equilibrium solutions we prove that the perturbed non autonomous system has exactly the same number of almost periodic solutions as a consequence the pullback attractor of the perturbed system is given by the union of unstable manifolds of these finitely many almost periodic solutions an application of the result to the chafee infante equation is discussed

characterization of non autonomous attractors of a perturbed infinite dimensional gradient system in this paper we determine the exact structure of the pullback attractors in non autonomous problems that are perturbations of autonomous gradient systems with attractors that are the union of the unstable manifolds of a finite set of hyperbolic equilibria we show that the pullback attractors of the perturbed systems inherit this structure and are given as the union of the unstable manifolds of a set of hyperbolic global solutions which are the non autonomous analogues of the hyperbolic equilibria we also prove again parallel to the autonomous case that all solutions converge as t infinity to one of these hyperbolic global solutions we then show how to apply these results to systems that are asymptotically autonomous as t infinity and as t infinity and use these relatively simple test cases to illustrate a discussion of possible definitions of a forwards attractor in the non autonomous case c 2007 elsevier inc all rights reserved

morse decompositions for delay differential equations abstract the attractor for the scalar delay differential equation x t f x t x t 1 is shown to possess a morse decomposition under the hypothesis of a negative feedback condition in the delay term using this decomposition we prove results on the asymptotic rate of oscillation of solutions of the initial value problem existence of both slowly and rapidly oscillating periodic solutions is also proved

a combined relaxation method for variational inequalities with nonlinear constraints a simple iterative method for solving variational inequalities with a set valued nonmonotone mapping and a convex feasible set is proposed this set can be defined by nonlinear functions the method is based on combining and extending ideas contained in various relaxation methods of nonsmooth optimization also a modification of the averaging method for the problem under consideration is proposed 1998 the mathematical programming society inc published by elsevier science b v

the cascadic multigrid method for elliptic problems the paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of 10 from the multigrid point of view no correction cycles are needed from the cascade principle view a basic iteration method without any preconditioner is used at successive refinement levels for a prescribed error tolerance on the final level more iterations must be spent on coarser grids in order to allow for less iterations on finer grids a first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of 9 in short ccg method whichused the cg method as basic iteration method on each level in 18 it has been proven that the ccg method is accurate with optimal complexity for elliptic problems in 2d and quasi uniform triangulations the present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid smoothers moreover an adaptive control strategy for the number of iterations on successive refinement levels for possibly highly non uniform grids is worked out on the basis of a posteriori estimates numerical tests confirm the efficiency and robustness of the cascadic multigrid method

semicontinuity of attractors for impulsive dynamical systems abstract in this paper we introduce the concept of collective tube conditions which assures a suitable behaviour for a family of dynamical systems close to impulsive sets using the collective tube conditions we develop the theory of upper and lower semicontinuity of global attractors for a family of impulsive dynamical systems

persistence of wavefronts in delayed nonlocal reaction diffusion equations we develop a perturbation argument based on existing results on asymptotic autonomous systems and the fredholm alternative theory that yields the persistence of traveling wavefronts for reaction diffusion equations with nonlocal and delayed nonlinearities when the time lag is relatively small this persistence result holds when the nonlinearity of the corresponding ordinary reaction diffusion system is either monostable or bistable we then illustrate this general result using five different models from population biology epidemiology and bio reactors

attractors for impulsive non autonomous dynamical systems and their relations in this work we deal with several different notions of attractors that may appear in the impulsive non autonomous framework and we explore their relationships to obtain properties regarding the different scenarios of asymptotic dynamics such as the cocycle attractor the uniform attractor and the global attractor for the impulsive skew product semiflow lastly we illustrate our theory by exhibiting an example of a non classical non autonomous parabolic equation with subcritical nonlinearity and impulses

persistence and extinction in two species reaction diffusion systems with delays abstract both uniform persistence and global extinction are established for two species predator prey and competition reaction diffusion systems with delays in terms of the principal eigenvalues of the scalar elliptic eigenvalue problems by appealing to the theories of abstract persistence asymptotically autonomous semiflows and monotone dynamical systems

minimal time control of the two tanks gradostat model under a cascade input constraint we study the minimum time control problem of a series of two interconnected chemostats under the input constraint u2 u1 where ui are the respective dilution rates in the tanks this constraint brings controllability issues in the study of the optimal strategies we encounter this difficulty by splitting the state domain into two sub domains one with no lack of controllability of the target and its complementary is such that any optimal control fulfills u1 u2 we explicit the complete optimal synthesis that depends on the position of the target with respect to semi permeable curve that passes through a steady state singular point

asymptotic behavior of an age structured population model and optimal maturation age abstract a continuous age structured population model with two population groups juveniles and adults and with a dynamics for the resource is considered the only density dependence is through a uniform increase of the death rates a complete description of the asymptotic behavior of the population dynamics system is obtained showing under natural hypotheses the existence of a nontrivial ecological equilibrium which attracts any initial nonzero population with nonzero initial amount of resources assuming that the offspring production rate is an increasing function of the maturation age the evolutionarily stable value of this variable is briefly discussed

classification of radial solutions for semilinear elliptic systems with nonlinear gradient terms abstract we are concerned with the classification of positive radial solutions for the system u v p v f u where p 0 and f c 1 0 is a nondecreasing function such that f t 0 for all t 0 we show that in the case where the system is posed in the whole space r n such solutions exist if and only if 1 0 s f t d t p 2 p 1 d s this is the counterpart of the keller osserman condition for the case of single semilinear equation similar optimal conditions are derived in case where the system is posed in a ball of r n if f t t q q 1 using dynamical system techniques we are able to describe the behaviour of solutions at infinity in case where the system is posed in the whole r n or around the boundary in case of a ball

generalized variational and quasi variational inequalities with discontinuous operators abstract we study the existence of solutions to generalized variational and generalized quasi variational inequalities with discontinuous operators we obtain results which generalize and extend previously known theorems we also compare our new continuity condition on the operator in the variational inequality to previously used continuity conditions we then apply our results to generalized variational inequalities which involve pseudo monotone operators

stable coexistence and bi stability for competitive systems on ordered banach spaces abstract autonomous and asymptotically autonomous semiflows modeling two species competition are studied which are strongly order preserving on a convex subset of a banach space with competitive order conditions are derived for stable coexistence bi stability and competitive exclusion a complete classification of all possible outcomes is obtained in case that there is at most one positive steady state representing coexistence of both species

distributed event triggered cooperative attitude control of multiple groups of rigid bodies on manifold so 3 in this note the distributed event triggered cooperative attitude control of multiple groups of rigid bodies evolved on nonlinear manifold so 3 is investigated the distributed event triggered mechanism is introduced into the controller design to reduce the communication pressure it is verified that the orientations of rigid bodies in each group converge to a common value with zero angular velocities and these common values in all groups achieve an anti consensus configuration under the proposed intra and inter group cooperative control law simulation results show the effectiveness of the proposed method

the stability of stochastic coupled systems with time varying coupling and general topology structure we introduce a class of novel stochastic coupled systems in which the coupling structure is time varying and the topology structure is not strongly connected and first establish the system on a digraph with a time varying weight matrix motivated by du and li 2014 we give a hierarchical method to deal with digraphs without strong connectivity and establish the corresponding hierarchical algorithm to realize this approach also an example is given to illustrate our hierarchical algorithm and its feasibility in the sequel based on the theory of asymptotically autonomous systems kirchhoff s matrix tree theorem and lyapunov method several moment exponential stability criteria are presented including a lyapunov type theorem and a coefficient type criterion furthermore theoretical results are applied to stochastic coupled oscillators with time varying coupling structure sctcs and the stability criterion of sctcs is obtained finally the effectiveness of theoretical results is illustrated by two numerical examples

topological theory of non autonomous parabolic evolution inclusions on a noncompact interval and applications in this paper we consider the topological structure of the solution set of non autonomous parabolic evolution inclusions with time delay defined on non compact intervals the result restricted to compact intervals is then extended to non autonomous parabolic control problems with time delay moreover as the applications of the information about the structure we establish the existence result of global integral solutions for non autonomous cauchy problems subject to nonlocal condition and prove the invariance of a reachability set for non autonomous control problems under single valued nonlinear perturbations finally some illustrating examples are supplied

sufficient and necessary criteria for existence of pullback attractors for non compact random dynamical systems abstract we study pullback attractors of non autonomous non compact dynamical systems generated by differential equations with non autonomous deterministic as well as stochastic forcing terms we first introduce the concepts of pullback attractors and asymptotic compactness for such systems we then prove a sufficient and necessary condition for existence of pullback attractors we also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors for random systems containing periodic deterministic forcing terms we show the pullback attractors are also periodic under certain conditions as an application of the abstract theory we prove the existence of a unique pullback attractor for reaction diffusion equations on r n with both deterministic and random external terms since sobolev embeddings are not compact on unbounded domains the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions

on synchronous steady states and internal models of diffusively coupled systems we investigate the problem of synchronizing nonidentical nonlinear dynamical systems by means of generalized diffusive couplings the focus is not on the actual solution to the problem but on the derivation of necessary conditions for the existence of such a solution despite the systems possessing nonidentical models we show that for the problem to be solvable a synchronous steady state needs to exist this condition leads to the requirement that all individual system models need to embed an internal model of some common endosystem the latter condition is expressed in terms of nonlinear partial differential equations the conditions derived in this paper are related to those known from the theory of output regulation

generalized multivalued quasi variational inequalities in this paper we introduce and study a new class of variational inequalities known as generalized multivalued quasi variational inequalities this class is the most general and unifying one using essentially the projection technique we establish the equivalence between this class and the implicit wiener hopf equations this equivalence is used to suggest a number of iterative algorithms for solving the generalized multivalued quasi variational inequalities the convergence analysis of these algorithms is also studied the results proved in this paper represent a significant improvement and refinement of the previously known results in this area

pullback attractors and extremal complete trajectories for non autonomous reaction diffusion problems abstract we analyse the dynamics of the non autonomous nonlinear reaction diffusion equation u t u f t x u subject to appropriate boundary conditions proving the existence of two bounding complete trajectories one maximal and one minimal our main assumption is that the nonlinear term satisfies a bound of the form f t x u u c t x u 2 d t x u where the linear evolution operator associated with c t x is exponentially stable as an important step in our argument we give a detailed analysis of the exponential stability properties of the evolution operator for the non autonomous linear problem u t u c t x u between different l p spaces

coordinated motion design on lie groups the present paper proposes a unified geometric framework for coordinated motion on lie groups it first gives a general problem formulation and analyzes ensuing conditions for coordinated motion then it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integrator dynamics it thereby shows that coordination can be studied in a systematic way once the lie group geometry of the configuration space is well characterized applying the proposed general methodology to particular examples allows to retrieve control laws that have been proposed in the literature on intuitive grounds a link with brockett s double bracket flows is also made the concepts are illustrated on so 3 se 2 and se 3

nonautonomous semilinear second order evolution equations with generalized wentzell boundary conditions abstract of concern are the initial boundary value problems for nonautonomous semilinear second order evolution equations with generalized wentzell boundary conditions we succeed in establishing a global wellposedness theorem in a classical sense for these problems via a specifically designed operator theoretic approach moreover we obtain sharp estimations for the evolution of the solution along the characteristic curves which enable us to derive the uniform exponential decay of the associated energies the results obtained in this paper are still new even for the autonomous case

an estimate on the fractal dimension of attractors of gradient like dynamical systems abstract this paper is dedicated to estimate the fractal dimension of exponential global attractors of some generalized gradient like semigroups in a general banach space in terms of the maximum of the dimension of the local unstable manifolds of the isolated invariant sets lipschitz properties of the semigroup and the rate of exponential attraction we also generalize this result for some special evolution processes introducing a concept of morse decomposition with pullback attractivity under suitable assumptions if a a is an attractor repeller pair for the attractor a of a semigroup t t t 0 then the fractal dimension of a can be estimated in terms of the fractal dimension of the local unstable manifold of a the fractal dimension of a the lipschitz properties of the semigroup and the rate of the exponential attraction the ingredients of the proof are the notion of generalized gradient like semigroups and their regular attractors morse decomposition and a fine analysis of the structure of the attractors as we said previously we generalize this result for some evolution processes using the same basic ideas

dynamics of a periodically pulsed bio reactor model abstract global attractivity and uniform persistence are established for both single species growth and two species competition in a periodically pulsed bio reactor model in terms of principal eigenvalues of the periodic parabolic eigenvalue problem by appealing to the theories of monotone discrete dynamical systems abstract persistence asymptotically periodic semiflows and perturbation of global attractors

on the generalized nonlinear variational inequalities and quasi variational inequalities the purpose of this paper is to investigate the existence of solutions for a class of generalized nonlinear inequalities gnvi and generalized nonlinear quasi variational inequalities gnqvi with non monotone set valued operators defined on non compact sets in locally convex hausdorff topological vector spaces results that generalize and unify corresponding recent results of ding and tarafdar are derived

on multivalued complementarity problems in banach spaces in this paper we prove a new existence result for multivalued complementarity problems in banach spaces our result represents a refinement and improvement of the previously known results

spatial spread of rabies revisited influence of age dependent diffusion on nonlinear dynamics we consider the spatio temporal patterns of disease spread involving structured populations we start with a general model framework in population biology and spatial ecology where the individual s spatial movement behaviors depend on its maturation status and we show how delayed reaction diffusion equations with nonlocal interactions arise naturally we then consider the impact of this delayed nonlocal interaction on the disease spread by revisiting the spatial spread of rabies in continental europe during the period between 1945 and 1985 we show how the distinction of territorial patterns between juvenile and adult foxes the main carriers of the rabies under consideration yields a class of partial differential equations involving delayed and nonlocal terms that are implicitly defined by a hyperbolic parabolic equation and we show how incorporating this distinction into the model leads to a formula describing the relation of the minimal wave speed and the maturation time of foxes we show how the ho

dynamics and pattern formation in a diffusive predator prey system with strong allee effect in prey abstract the dynamics of a reaction diffusion predator prey system with strong allee effect in the prey population is considered nonexistence of nonconstant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as nonconstant steady state solutions are studied these results show that the impact of the allee effect essentially increases the system spatiotemporal complexity

asymptotic behavior of large radial solutions of a polyharmonic equation with superlinear growth this paper concerns the blow up behavior of large radial solutions of polyharmonic equations with power nonlinearities and positive radial weights specifically we consider radially symmetric solutions of mu c x u p on an annulus x rn x with 0 and 0 that diverge to infinity as x here n m n p 1 and c is a positive continuous function on the interval letting r qc r q for r with q 2m p 1 q q q 1 q 2m 1 1 p 1 and c c 1 p 1 we show that as x the ratio u x x remains between positive constants that depend only on m and p extending well known results for the second order problem we prove in the fourth order case that u x x 1 as x and obtain precise asymptotic expansions if c is sufficiently smooth at in certain higher order cases we find solutions for which the ratio u x x does not converge but oscillates about 1 with non vanishing amplitude

browder hartman stampacchia variational inequalities for multi valued monotone operators abstract an existence theorem of browder hartman stampacchia variational inequalities is extended to multi valued monotone operators surjectivity for multi valued monotone operators and properties of solution sets are discussed

existence results for the nonlinear complementarity problem and applications to nonlinear analysis abstract we produce an existence result for the nonlinear complementarity problem associated to a weakly locally compact convex cone k in a banach space e and to the difference j f of two nonlinear operators we specialize our result to the case where k is a so called galerkin cone j is the duality mapping and f satisfies a suitable growth condition applications to the fixed point theory and to eigenvalue problems conclude the paper

the complete classification for dynamics in a nine dimensional west nile virus model bowman et al bull math biol 67 2005 pp 1107 1133 proposed a nine dimensional system of ordinary differential equations modelling west nile virus in a mosquito bird human community and presented some mathematical analysis and its biological explanation jiang et al bull math biol 2008 doi 10 1007 s11538 008 9374 6 continued to study the existence and classification of all equilibria and all their local stability and dealt with saddle node bifurcation of the system the previous investigation shows that the unique positive equilibrium is globally asymptotically stable if the basic reproduction number is greater than one and the bird death rate is suitably small but numerical simulations suggest that the unique endemic equilibrium is globally asymptotically stable even if for a large value of the bird death rate so they all leave it an open problem the present paper is to provide a thorough classification of dynamics for this system in particular if the reproduction number is greater

equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains we consider the dirichlet problem for a class of semilinear parabolic equations on a bounded domain which is reflectionally symmetric about a hyperplane h the equations consist of a symmetric time autonomous part and a nonsymmetric perturbation which decays to zero as time approaches infinity in our first theorem we prove the asymptotic symmetry of each bounded positive solution of this asymptotically symmetric problem the novelty of this result is that the solutions considered are not assumed uniformly positive which prevents one from applying common techniques based on the method of moving hyperplanes in our second main theorem we classify the positive entire solutions of the unperturbed time autonomous problems in particular we characterize all entire solutions which are not symmetrically decreasing in the direction orthogonal to h as connecting orbits from an equilibrium with a nontrivial nodal set to another invariant set

a novel two grid method for semilinear elliptic equations a new finite element discretization technique based on two coarse and fine subspaces is presented for a semilinear elliptic boundary value problem the solution of a nonlinear system on the fine space is reduced to the solution of two small one linear and one nonlinear systems on the coarse space and a linear system on the fine space it is shown both theoretically and numerically that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation as a result the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

a new maximal element theorem in h space with applications abstract in this paper a new maximal element theorem is proved in h space and as applications the browder hartman stampacchia variational inequalities and the complementarity problem are discussed in banach spaces and locally convex space respectively

global bifurcation analysis and pattern formation in homogeneous diffusive predator prey systems abstract the dynamics of a general diffusive predator prey system is considered existence and nonexistence of non constant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation bifurcations of spatially homogeneous and nonhomogeneous periodic solutions as well as non constant steady state solutions are studied

consensus optimization on manifolds the present paper considers distributed consensus algorithms that involve n agents evolving on a connected compact homogeneous manifold the agents track no external reference and communicate their relative state according to a communication graph the consensus problem is formulated in terms of the extrema of a cost function this leads to efficient gradient algorithms to synchronize i e maximizing the consensus or balance i e minimizing the consensus the agents a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time varying the cost function is linked to a specific centroid definition on manifolds introduced here as the induced arithmetic mean that is easily computable in closed form and may be of independent interest for a number of manifolds the special orthogonal group so n and the grassmann manifold text it grass p n are treated as original examples a link is also drawn with the many existing results on the circle

on the convergence of a cascadic multigrid method for semilinear elliptic problem a cascadic multigrid method for semilinear elliptic problem is analysed it has been proved that the algorithm has optimal order of convergence in energy norm and quasi optimal computational complexity under more general conditions

synchronization and balancing on the n torus in this paper we study the behavior of a network of n agents each evolving on the circle we propose a novel algorithm that achieves synchronization or balancing in phase models under mild connectedness assumptions on the possibly time varying and unidirectional communication graphs the global convergence analysis on the n torus is a distinctive feature of the present work with respect to previous results that have focused on convergence in the euclidean space

some estimates of the rate of convergence for the cascadic conjugate gradient method abstract the paper deals with a cascadic conjugate gradient method shortly called the ccg algorithm which was proposed by p deuflhard and can be considered as a simpler version of a multigrid multilevel method we define it recurrently for discrete self adjoint positive definite problems on a sequence of grids on the coarsest grid the linear discrete algebraic system is solved directly on the finer grids the system is iteratively solved by the conjugate gradient method where the starting guess is an interpolation of the approximated solution on the previous grid any preconditioning or restriction to coarser grids is not implemented nevertheless the ccg algorithm has the same optimal property compared to multigrid methods namely the algorithm converges with a rate which is independent of the number of unknowns and the number of grids as an example this property is proved for elliptic second order dirichlet problems in two dimensional convex polygonal bounded domains for ensuring convergence the number of iterations on each grid level has to increase from finer to coarser grids the optimal dependence of these numbers is established with respect to the mesh size and the number of unknowns the theory has been presented in an abstract setting which allows the application to both finite element and finite difference methods

structure of attractors for skew product semiflows in this work we study the continuity and structural stability of the uniform attractor associated with non autonomous perturbations of differential equations by a careful study of the different definitions of attractor in the non autonomous framework we introduce the notion of lifted invariance on the uniform attractor which becomes compatible with the dynamics in the global attractor of the associated skew product semiflow and allows us to describe the internal dynamics and the characterization of the uniform attractors the associated pullback attractors and their structural stability under perturbations will play a crucial role

stability analysis of a bioreactor model for biodegradation of xenobiotics we consider an ecological model for biodegradation of toxic substances in aquatic and atmospheric biotic systems the model which is described by a nonlinear system of four ordinary differential equations is known to be experimentally validated we compute the equilibrium points of the model and study their asymptotic stability basic properties of the solutions like uniform boundedness and uniform persistence are established global asymptotic results are also developed numerical simulation results are presented to demonstrate the theoretical studies

on the continuity of pullback attractors for evolution processes in this paper we give general results on the continuity of pullback attractors for nonlinear evolution processes we then revisit results of d li p e kloeden equi attraction and the continuous dependence of pullback attractors on parameters stoch dyn 4 3 2004 373 384 which show that under certain conditions continuity is equivalent to uniformity of attraction over a range of parameters equi attraction we are able to simplify their proofs and weaken the conditions required for this equivalence to hold generalizing a classical autonomous result a v babin m i vishik attractors of evolution equations north holland amsterdam 1992 we give bounds on the rate of convergence of attractors when the family is uniformly exponentially attracting to apply these results in a more concrete situation we show that a non autonomous regular perturbation of a gradient like system produces a family of pullback attractors that are uniformly exponentially attracting these attractors are therefore continuous and we can give an explicit bound on the distance between members of this family c 2009 elsevier ltd all rights reserved