on the eigenvalue problem involving the weighted p laplacian in radially symmetric domains abstract we investigate the following eigenvalue problem div l x u p 2 u k x u p 2 u in a r 1 r 2 u 0 on a r 1 r 2 where a r 1 r 2 x r n r 1 x r 2 0 r 1 r 2 0 is a parameter the weights l and k are measurable with l positive a e in a r 1 r 2 and k possibly sign changing in a r 1 r 2 we prove the existence of the first eigenpair and discuss the regularity and positiveness of eigenfunctions the asymptotic estimates for u x and u x as x r 1 or r 2 are also investigated

on bendixson s criterion abstract for autonomous differential equations in r n criteria are developed which preclude the existence of invariant closed curves such as periodic or homoclinic trajectories the technique is based on the study of functionals on 2 surfaces results generalize to higher dimensions a criterion of bendixson for the nonexistence of nonconstant periodic solutions in the case n 2 as an example an application to the lorenz system in r 3 is given

complex dynamics of an seir epidemic model with saturated incidence rate and treatment abstract in this paper we describe the dynamics of an seir epidemic model with saturated incidence treatment function and optimal control rigorous mathematical results have been established for the model the stability analysis of the model is investigated and found that the model is locally asymptotically stable when r 0 1 the model is locally as well as globally asymptotically stable at endemic equilibrium when r 0 1 the proposed model may possess a backward bifurcation the optimal control problem is designed and obtained their necessary results numerical results have been presented for justification of theoretical results

stationary distribution of stochastic population systems under regime switching abstract this paper is concerned with n species model of facultative mutualism in random environments the environment variability in this study is characterized with both white noise and color noise modeled by markovian switching we established new sufficient conditions that ensuring that the system model is positive recurrent we also showed the existence of a unique ergodic stationary distribution the presented results are demonstrated by numerical simulations

high energy solutions and nontrivial solutions for fourth order elliptic equations abstract in this paper we study the existence of nontrivial solutions and infinitely many high energy solutions for a class of nonlinear fourth order elliptic equations in r n via variational methods three main theorems are obtained

estimate of morse index of cooperative elliptic systems and its application to spatial vector solitons local instability index of unstable solutions to single partial differential equations pdes by a local minimax method lmmm was established in zhou 2005 it is known that the local min orthogonal method lmom which was first proposed in zhou 2004 and then further developed in chen et al 2008 can find more general unstable solutions to both single pdes and cooperative elliptic systems this paper is to carry out instability analysis of unstable solutions by lmom to which an infinite dimensional functional space can be decomposed as a direct sum of a finite dimensional subspace and its orthogonal complement a morse index approach is developed to show that with lmom instability behavior of a solution in such infinite dimensional complement subspace can be totally determined usual instability analysis in an entire space is then reduced to analysis in its finite dimensional subspace for which a corresponding matrix decomposition is proposed to analyze a solution s instability behavior estimates of morse index are also established finally numerical examples of both 2 and 3 component cooperative systems arising in nonlinear optics are carried out for spatial vector solitons whose local instabilities are numerically confirmed by the new estimates certain important properties of the examples are also verified or presented

analyticity of the density of states in the anderson model on the bethe lattice let h 1 2 v on l2 b whereb is the bethe lattice andv x xeb are i i d r v s with common probability distribution it is shown that for distributions sufficiently close to the cauchy distribution the density of states e is analytic in a strip about the real axis

a strategy for constructing lyapunov functions for non autonomous linear differential equations a strategy is presented for determining if the growth of solutions to a non autonomous linear differential equation satisfies a given bound in the case where the given bound is negative the approach gives a constructive method for finding a lyapunov function and showing that the system is stable the strategy takes advantage of the sign pattern of the coefficient matrix and is demonstrated with an example

global stability for an sei model of infectious disease with immigration we study an sei model of disease transmission with immigration into all three classes for incidence we allow for a nonlinear response to the number of infectives including mass action and saturating incidence as special cases there is no disease free equilibrium and therefore no basic reproduction number for all parameter values the only equilibrium is an endemic equilibrium using a lyapunov function we show that this equilibrium is globally asymptotically stable

on a sub supersolution method for variational inequalities with leray lions operators in variable exponent spaces abstract in this paper we consider a sub supersolution method for variational inequalities with leray lions operators in sobolev spaces with variable exponents existence and qualitative properties of solutions between sub and supersolutions are established

existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations in connection with the maximizing problem for the functional r u u lq u lp in w01 p 0 we consider the equation div u p 2 u x u q 2 u x x 1 p resp q p e has no nontrivial resp nonnegative nontrivial solution belonging to the class p u w01 p lq xi u q 2u lp p 1 i 1 2 n w01 p provided that is star shaped the crucial point of the proof of our result is to obtain an l estimate of weak solutions and to verify a certain pohozaev type inequality for weak solutions belonging to p

existence results for the positive solutions of nonlinear elliptic systems in this paper we discuss the existence of solutions for nonlinear elliptic systems the positive weak solution is sought by the leray schauder degree

high energy solutions of systems of kirchhoff type equations in rn in the present paper five new critical point theorems on the product spaces are given and three existence theorems for the sequence of high energy solutions for the systems of kirchhoff type equations are studied

a variational inequality theory in reflexive smooth banach spaces and applications to p laplacian elliptic inequalities abstract we develop a theory of variational inequality for demicontinuous s contractive maps in reflexive smooth banach spaces variational inequality theorems are proved and applied to study existence of nonzero positive weak solutions for p laplacian elliptic inequalities and a population model of one species arising in mathematical biology

1d quasiperiodic operators latent symmetries a large class of discrete quasiperiodic operators is shown to be decomposed into orbits ofsl 2 z action with equal densities of states moreover under some natural assumptions all nontrivial representatives of the mentioned action transform operators with pure point spectrum into those with absolutely continuous spectrum some applications of these results are presented

positive solutions of variational inequalities a degree theoretic approach if the mapping a is strictly monotone and coercive is equivalent to a fixed point equation u m u in k cf section 1 suppose that m is continuous and maps bounded subsets of k into compact ones then due to the fact that k is a retract of x it is possible to define a fixed point index of m with properties similar to those associated with the leray schauder degree 1 sect 111 in section 1 we introduce another index which is based on the previous one but has the advantage of being directly related to inequality rather than to the fixed point equation u m u in section 2 we apply this new index in order to show that if c is a closed convex subset of k then under suitable assumptions upon f the set of solutions of the inequality

global dynamics of an seir epidemic model with vertical transmission we study a population model for an infectious disease that spreads in the host population through both horizontal and vertical transmission the total host population is assumed to have constant density and the incidence term is of the bilinear mass action form we prove that the global dynamics are completely determined by the basic reproduction number r0 p q where p and q are fractions of infected newborns from the exposed and infectious classes respectively if r 0 p q le 1 the disease free equilibrium is globally stable and the disease always dies out if r0 p q 1 a unique endemic equilibrium exists and is globally stable in the interior of the feasible region and the disease persists at an endemic equilibrium state if it initially exists the contribution of the vertical transmission to the basic reproduction number is also analyzed

existence results for some fourth order nonlinear elliptic problems of local superlinearity and sublinearity in this paper we study the existence of positive solutions for the problem 0 1 2u c u f x u in u 0 u 0in u u 0on where c 1 and f x u satisfies the local superlinearity and sublinearity condition

the existence of solutions for a class of semilinear elliptic systems in this paper some new existence theorems of weak solutions for a class of semilinear elliptic systems are obtained by means of the local linking theorem and the saddle point theorem

positive weak solutions of semilinear second order elliptic inequalities via variational inequalities existence uniqueness and convergence of approximants of positive weak solutions for semilinear second order elliptic inequalities are obtained the nonlinearities involved in these inequalities satisfy suitable upper or lower bound conditions or monotonicity conditions the lower bound conditions are allowed to contain the critical sobolev exponents the methodology is to establish variational inequality principles for demicontinuous pseudo contractive maps in hilbert spaces by considering convergence of approximants and apply them to the corresponding variational inequalities arising from the semilinear second order elliptic inequalities examples on the existence uniqueness and convergence of approximants of positive weak solutions of the semilinear second order elliptic inequalities are given

fixed point theory and nonlinear problems introduction among the most original and far reaching of the contributions made by henri poincare to mathematics was his introduction of the use of topological or qualitative methods in the study of nonlinear problems in analysis his starting point was the study of the differential equations of celestial mechanics and in particular of their periodic solutions his work on this topic began with his thesis in 1879 and was developed in detail in his great three volume work methodes nouvelles de la mechanique celeste which appeared in the early 1890s and summarized his many memoirs of the intervening period it continued until his memoir shortly before his death in 1912 in which he put forward the unproved fixed point result usually referred to as poincare s last geometric theorem the ideas introduced by poincare include the use of fixed point theorems the continuation method and the general concept of global analysis the writer s acquaintance with poincare s influence came through contact with solomon lefschetz and marston morse both of whom were very explicit as to the role of poincare as an initiator in this direction of mathematical development in 1934 in the foreword to his colloquium volume on the calculus of variations in the large morse had put this forward very forcefully in the first paragraph

stability and bifurcation in plant pathogens interactions we consider a plant pathogen interaction model and perform a bifurcation analysis at the threshold where the pathogen free equilibrium loses its hyperbolicity we show that a stimulatory inhibitory host response to infection load may be responsible for the occurrence of multiple steady states via backward bifurcations we also find sufficient conditions for the global stability of the pathogen present equilibrium in case of null or linear inhibitory host response the results are discussed in the framework of the recent literature on the subject

on the use of the geometric approach to global stability for three dimensional ode systems a bilinear case abstract in this paper we consider a general bilinear three dimensional ode system whose structure generalizes many mathematical models of biological interest including many from epidemics our main goal is to find sufficient conditions expressed in terms of the parameters of the system ensuring that the geometric approach to global stability analysis due to m y li j s muldowney a geometric approach to global stability problems siam j math anal 27 4 1996 1070 1083 may be successfully applied we completely determine the dynamics of the general system including thresholds and global stability of the nontrivial equilibrium the obtained result is applied to several epidemic models we further show how the role of new parameters on stability of well established models may be emphasized

random schrodinger operators arising from lattice gauge fields i existence and examples we consider new models of ergodic schrodinger operators whose existence relies on a cohomological theorem of feldman and moore in ergodic theory these operators generalize the harper operator which describes the case of a constant magnetic field an example is the case when the magnetic field is given by independent random variables attached to the lattice plaquettes a generalization of the feldman moore theorem by lind to non abelian groups also allows us to consider schrodinger operators obtained from non abelian lattice gauge fields the existence result extends to more general graphs like to operators on tilings and to higher dimensions we compute some moment expansions for the density of states for example for independent identically and uniformly distributed magnetic fields a model which has been studied at least since 1970 and whose existence can also be seen without involving the above mentioned existence theorem we show that the nth moment is the number of closed paths in the two dimensional lattice starting at the origin for which the winding number vanishes at each plaquette point this goes beyond the brinkman rice self retracing path approximation other examples are a higher dimensional example a one dimensional anderson model which can be treated in this framework as well as the hofstadter model with constant magnetic field where one averages over all possible magnetic fields we also reprove a result of jitomirskaya mandelshtam stating that the deterministic aharonov bohm model is a compact perturbation of the free laplacian

anderson localization for the almost mathieu equation ii point spectrum for lambda 2 we prove that for any 2 and a e the pure point spectrum of the almost mathieu operator h n n 1 n 1 cos 2 n n contains the essential closure hat sigma of the spectrum corresponding eigenfunctions decay exponentially the singular continuous component if it exists is concentrated on a set of zero measure which is nowhere dense in hat sigma

global dynamics of a diffusive and delayed viral infection model with cellular infection and nonlinear infection rate abstract a diffusive and delayed viral dynamics model which incorporates cell to cell transmission cell mediated immune responses and general nonlinear incidence is investigated by constructing lyapunov functionals it is shown that the global threshold dynamics of the model is fully determined by the basic reproduction numbers of the virus and the immune response r 0 and r 1 the result implies that both the basic reproduction number of the virus and the immune response might be underevaluated without considering either the virus to cell transmission or cell to cell transmission the main results obtained in this paper extend some known results

discretization and dynamic consistency of a delayed and diffusive viral infection model abstract a diffusive and delayed viral infection model with nonlinear incidence has been studied and the global dynamical behaviors of the original model is investigated by constructing lyapunov functionals furthermore the analysis is carried out for the discrete model which is obtained by applying the nonstandard finite difference nsfd scheme to the original continuous model the global stability for the corresponding equilibria is investigated by constructing discrete lyapunov functionals as well as the positivity and boundedness of solutions of the corresponding continuous model the results imply that the discretization scheme can efficiently preserves the qualitative properties of solutions for the original continuous model numerical experiments are carried out to support the theoretical results

enclosure results for quasilinear systems of variational inequalities abstract in this paper we consider systems of quasilinear elliptic variational inequalities and prove the existence of minimal and maximal in the set theoretical sense solutions within some ordered interval of an appropriately defined pair of sub and supersolutions we show that the notion of sub and supersolutions of variational inequalities introduced here is consistent with the usual notion of sub supersolutions for variational equations for weakly coupled quasimonotone systems of variational inequalities the existence of smallest and greatest solutions is proved

global stability for a class of mass action systems allowing for latency in tuberculosis a very general compartmental model of the spread of an infectious disease with mass action incidence is given the global stability of this system is completely determined using lyapunov functions the general system exhibits the traditional threshold behaviour the dimension of the system is arbitrary allowing in particular for detailed modelling of the distribution of latency times for tuberculosis

permanence of a regime switching malaria transmission model abstract malaria transmission is a complex system and lots of malaria transmission models have been showed by determined differential equations considering that variability of the environment may have an important impact on its transmission we introduce the regime switching to a malaria transmission model with susceptible infectious recovered susceptible sirs structure for human population and susceptible infectious si structure for mosquito population first we show the existence of positive solutions for the model then the dynamics of the model is carried out by constructing a suitable lyapunov function we show that there exists a stationary distribution and so the threshold for the disease to prevail is given at last an example with two states is given to illustrate the results

a host vector model for malaria with infective immigrants abstract this paper considers a host vector mathematical model for the spread of malaria that incorporates recruitment of human population through a constant immigration with a fraction of infective immigrants the model analysis is carried out to find the steady states and their stability it is found that in the presence of infective immigrant humans there is no disease free equilibrium point however the model exhibits a unique endemic equilibrium state if the fraction of the infective immigrants is positive when the fraction of infective immigrants approaches a small value there is sharp threshold for which the disease can be reduced in the community the unique endemic equilibrium for which there is a fraction of infective immigrants is globally asymptotically stable

global stability for an hiv 1 infection model including an eclipse stage of infected cells abstract we consider the mathematical model for the viral dynamics of hiv 1 introduced in rong et al 2007 37 one main feature of this model is that an eclipse stage for the infected cells is included and cells in this stage may revert to the uninfected class the viral dynamics is described by four nonlinear ordinary differential equations in rong et al 2007 37 the stability of the infected equilibrium has been analyzed locally here we perform the global stability analysis using two techniques the lyapunov direct method and the geometric approach to stability based on the higher order generalization of bendixson s criterion we obtain sufficient conditions written in terms of the system parameters numerical simulations are also provided to give a more complete representation of the system dynamics

multiple solutions for some fourth order nonlinear elliptic variational inequalities abstract we consider the solvability for fourth order nonlinear elliptic variational inequalities some existence results are given by the variational methods

existence results for some nonlinear elliptic systems abstract the paper is concerned with some nonlinear elliptic systems by using variational methods we obtain the existence of weak solutions for the semilinear elliptic system and the quasilinear elliptic system

multiple nontrivial solutions for some fourth order semilinear elliptic problems abstract in this paper we consider the existence of multiple nontrivial solutions for some fourth order semilinear elliptic boundary value problems the weak solutions are sought by means of morse theory and local linking

on the origin of integrability in matrix models the matrix integrals involved in 2d lattice gravity are studied at finiten the integrable systems that arise in the continuum theory are shown to result directly from the formulation of the matrix integrals in terms of orthogonal polynomials the partition function proves to be a tau function of the toda lattice hierarchy the associated linear problem is equivalent to finding the polynomial basis which diagonalizes the partition function the cases of one hermitian matrix one unitary matrix and hermitian matrix chains all fall within the toda framework

global dynamics of an sveir epidemic model with distributed delay and nonlinear incidence a novel delay sveir epidemic model is formulated the uniform persistence and global stability of the system are discussed some good results and efficient approach are presented an sveir epidemic model with imperfect vaccination and nonlinear incidence and a general latent distribution is formulated by constructing lyapunov functionals it is shown that the disease will die out if the vaccination reproduction number r v a c 1 and the disease becomes endemic if r v a c 1 furthermore vaccination effects are analyzed two special forms the probability of remaining in latent class are discussed when the probability is negatively exponentially distributed we present an efficient approach of proving global stability of the endemic equilibrium of the sveir system of ordinary differential equations odes which may improve some known approaches when the probability is a step function the delay differential equation dde system derived is used to study the impacts of vaccination and saturated incidence on the mumps transmission

multiple nontrivial solutions for some fourth order quasilinear elliptic problems abstract in this paper we consider the existence of multiple nontrivial solutions for some fourth order quasilinear elliptic boundary value problems the weak solutions are sought by using the morse theory and local linking

a variational approach to subquadratic perturbations of elliptic systems abstract we consider subquadratic perturbations of semilinear elliptic systems which are in variational form a unified approach to both the cooperative and noncooperative problems is presented and a condition of nonquadraticity at infinity is introduced to treat the resonant cases through minimax techniques

nonlinear versions of stampacchia and lax milgram theorems and applications to p laplace equations we obtain the nonlinear versions of the stampacchia theorem and the lax milgram theorem our results are stronger than the classical ones even in the linear case applying these theorems we get nontrivial solutions of p laplace elliptic and pseudo plaplace problems c 2007 elsevier ltd all rights reserved msc 49j40 35j20 35j60 58e05

principal frequency and existence of solutions for quasilinear elliptic obstacle problems in this paper we are interested in the first eigenvalue of p laplacian and the relation between the first eigenvalue and the existence or nonexistence of nontrivial positive solution for quasilinear elliptic obstacle problems utilizing the fact that obstacle problem have consanguineous relations with corresponding equation we get a simple approach to study the properties of solutions of obstacle problems such as existence and nonexistence regularity and stability etc in this paper we are mainly concerned with the existence and nonexistence

global results for an epidemic model with vaccination that exhibits backward bifurcation vaccination of both newborns and susceptibles is included in a transmission model for a disease that confers immunity the interplay of the vaccination strategy together with the vaccine efficacy and waning is studied in particular it is shown that a backward bifurcation leading to bistability can occur under mild parameter constraints compound matrices are used to show that each orbit limits to an equilibrium in the case of bistability this global result requires a novel approach since there is no compact absorbing set

on the solvability of degenerated quasilinear elliptic equations of higher order abstract we prove the existence result for nonlinear degenerated elliptic boundary value problems of higher order the weak solution is sought in a suitable weighted sobolev space using the degree theory

uniform convexity of musielak orlicz sobolev spaces and applications abstract this paper deals with uniform convexity of musielak orlicz sobolev spaces and its applications to variational problems some sufficient conditions and examples for uniform convexity of musielak orlicz sobolev spaces are given some special properties relative to the uniformly convex modular for uniformly convex musielak orlicz sobolev spaces are presented as an application of these abstract results the local minimizers and the mountain pass type critical point of an integral functional with more complicated growth than the p x growth are studied

asymptotic properties of hybrid diffusion systems in response to the increasing needs for control and optimization of hybrid systems this work is concerned with such asymptotic properties as recurrence also known as weak stochastic stability in the literature and ergodicity of regime switching diffusions using liapunov functions necessary and sufficient conditions for positive recurrence are developed then ergodicity of positive recurrent regime switching diffusions is obtained by constructing cycles using the associated discrete time markov chains

stability analysis of a model for hbv infection with cure of infected cells and intracellular delay a viral infection model of hbv infection of hepatocytes with cure of infected cells and intracellular delay is studied the delay corresponds to the time necessary for a newly produced virion to become infectious particles we prove that the stability is completely determined by the basic reproductive number r 0 if r 0 1 the infection free steady state is globally asymptotically stable if r 0 1 then infection free steady state becomes unstable and a unique infected steady state exists and is locally asymptotically stable on the other hand we derive sufficient conditions for the global asymptotic stability of the infected steady state numerical simulations are presented to illustrate the results

discrete magnetic laplacian we consider a 2 dimensional discrete operator which we call the discrete magnetic laplacian dml it is an analogue of the magnetic schrodinger operator it follows from well known arguments that dml has the same spectrum as a subset inr as the almost mathieu operator am they also have the same integrated density of states ids which is known to be continuous we show that dml is an element in a ii1 factor and its ids can be expressed through the trace in the ii1 factor it follows that dml never has anyl2 eigenfunctions i e has no point spectrum then we formulate a natural algebraic conjecture which implies that the spectrum of dml hence the spectrum of am is a cantor set

a remark on schrodinger operators on aperiodic tilings we prove that for a large class of schrodinger operators on aperiodic tilings the spectrum and the integrated density of states are the same for all tilings in the local isomorphism class i e for all tilings in the orbit closure of one of the tilings this generalizes the argument in earlier work from discrete strictly ergodic operators onl2 d to operators on thel2 spaces of sets of vertices of strictly ergodic tilings

analyticity of the density of states and replica method for random schrodinger operators on a lattice we analyze the density of states and some aspects of the replica method for anderson s tight binding model on a lattice of arbitrary dimension with diagonal disorder we give heuristic arguments for the conjectures that the classical value of the exponent of the localization length is 1 2 and that the upper critical dimension d c loc is bounded by 4 d c loc 6

a non standard finite difference scheme for a delayed and diffusive viral infection model with general nonlinear incidence rate abstract a non standard finite difference scheme is proposed to solve a delayed and diffusive viral infection model with general nonlinear incidence rate the results show that the discrete model preserves the positivity and boundedness of solutions in order to ensure the well posedness of the problem moreover this method preserves all equilibria of the original continuous model by constructing lyapunov functionals we show that the global stability of equilibria is completely determined by the basic reproduction number 0 which implies that the proposed discrete model can efficiently blue preserve the global stability of equilibria of the corresponding continuous model numerical experiments are carried out to support the theoretical results

infinitely many solutions for fourth order elliptic equations with general potentials abstract this paper is concerned with the following fourth order elliptic equations 2 u u v x u f x u in r n u h 2 r n where u r n r under general assumption on potential v x and f x u is indefinite sign and sublinear as u we establish the existence of infinitely many solutions by using the genus properties in critical point theory recent results from the literature are extended

critical point theory for indefinite functionals with symmetries letxbe a hilbert space and c1 x r be strongly indefinite assume in addition that a compact lie groupgacts orthogonally onxand that is invariant in order to find critical points of we develop the limit relative category of fournieret al in the equivariant context we use this to prove two generalizations of the symmetric mountain pass theorem and a linking theorem in the case of the mountain pass theorem the mountain range is allowed to lie in a subspace of infinite codimension also other conditions of the classical symmetric mountain pass theorem forg z 2 due to ambrosetti and rabinowitz can be weakened considerably for example we are able to deal with infinite dimensional fixed point spaces the proofs consist of a direct reduction to a relative borsuk ulam type theorem this provides a new proof even for the classical mountain pass theorem the abstract critical point theorems are applied to an elliptic system with dirichlet boundary conditions we only need a weak version of the usual superquadraticity condition the linking theorem can be applied to asymptotically linear hamiltonian systems which are symmetric with respect to a generalized symplectic group action

analysis of the spectrum of a particle on a triangular lattice with two magnetic fluxes by algebraic and numerical methods the authors consider an electron on a triangular lattice with two magnetic fluxes using a c algebra formalism a perturbation theory around rational fluxes is developed this semiclassical type of analysis leads to the wilkinson rammal formula for the energy eigenvalues the analytic result is verified numerically with high accuracy up to second order crossing of eigenvalues near band edges is shown analytically as well as numerically