time dependent differential inclusions cocycle attractors and fuzzy differential equations traditional formulations of fuzzy differential equations do not reproduce the rich and varied behavior of crisp differential equations des a recent interpretation in terms of differential inclusions expressed level setwise overcomes this deficiency and opens up for profitable investigation such properties as stability attraction periodicity and the like this is especially important for investigating continuous systems which are uncertain or incompletely specified this paper studies attractors of fuzzy des in terms of cocycles and encompasses both the time dependent and autonomous cases

on multistability of equations with a distributed delay monotone production and the allee effect abstract we consider the delay population dynamics model d n d t f n t d n t with an increasing fecundity function f and any mortality function d where the allee effect or multistability can be observed the study includes non autonomous equations with a distributed delay attractivity of certain positive solutions persistence extinction of solutions with initial values below a threshold boundedness and oscillation are discussed

on a class of fuzzy functional differential equations in this paper we establish the local and global existence and uniqueness results for fuzzy functional differential equations for the local existence and uniqueness we use the method of successive approximations and for global existence and uniqueness we use the contraction principle also we apply these results to fuzzy differential equations with distributed delays and fuzzy population models

first order linear fuzzy differential equations under generalized differentiability first order linear fuzzy differential equations are investigated we interpret a fuzzy differential equation by using the strongly generalized differentiability concept because under this interpretation we may obtain solutions which have a decreasing length of their support which means a decreasing uncertainty in several applications the behaviour of these solutions better reflects the behaviour of some real world systems derivatives of the h difference and the product of two functions are obtained and we provide solutions of first order linear fuzzy differential equations using different versions of the variation of constants formula some examples show the rich behaviour of the solutions obtained

brief note on the variation of constants formula for fuzzy differential equations this note gives a theory of state transition matrices for linear systems of fuzzy differential equations this is used to give a fuzzy version of the classical variation of constants formula a simple example of a time independent control system is used to illustrate the methods while similar problems to the crisp case arise for time dependent systems in time independent cases the calculations are elementary solutions of eigenvalue eigenvector problems in particular for nonnegative or nonpositive matrices the problems at each level set can easily be solved in matlab to give the level sets of the fuzzy solution

periodic boundary value problems for first order linear differential equations with uncertainty under generalized differentiability we study the existence of solutions to a class of first order linear fuzzy differential equations subject to periodic boundary conditions from the point of view of generalized differentiability the objective of this paper is to show that fuzzy differential equations under generalized differentiability can be used in the study of periodic phenomena by considering a combination of two types of derivatives with a switching point we provide sufficient conditions which guarantee that the piecewise defined solutions match adequately and illustrate through some examples the process of construction and calculation of solutions

convergence to equilibria in scalar nonquasimonotone functional differential equations abstract we consider a class of scalar functional differential equations generating a strongly order preserving semiflow with respect to the exponential ordering introduced by smith and thieme it is shown that the boundedness of all solutions and the stability properties of an equilibrium are exactly the same as for the ordinary differential equation which is obtained by ignoring the delays the result on the boundedness of the solutions combined with a convergence theorem due to smith and thieme leads to explicit necessary and sufficient conditions for the convergence of all solutions starting from a dense subset of initial data under stronger conditions guaranteeing that the functional differential equation is asymptotically equivalent to a scalar ordinary differential equation a similar result is proved for the convergence of all solutions

global attractivity of positive periodic solution to periodic lotka volterra competition systems with pure delay abstract we consider a periodic lotka volterra competition system without instantaneous negative feedbacks i e pure delay systems x i t x i t r i t j 1 n a i j t x j t i j t i 1 2 n we establish some 3 2 type criteria for global attractivity of a positive periodic solution of the system which generalize the well known wright s 3 2 criteria for the autonomous delay logistic equation and thereby address the open problem proposed by both kuang y kuang global stability in delayed nonautonomous lotka volterra type systems without saturated equilibria differential integral equations 9 1996 557 567 and teng z teng nonautonomous lotka volterra systems with delays j differential equations 179 2002 538 561

discrete time deterministic and stochastic models for the spread of rabies discrete spatial and temporal models for the spread and control of rabies are developed analyzed and simulated first a deterministic model is formulated then an analogous stochastic model the models are structured with respect to space m patches age juveniles and adults and disease state for each patch there are six state variables corresponding to either juveniles or adults and their disease state susceptible infected or vaccinated the models have seven stages which repeat every year the impact of different vaccination strategies on the dynamics of the deterministic and stochastic models are compared in particular the relationships among the vaccination proportion the width of the vaccination barrier the initial number infected and the transmissibility of the disease are examined an estimate for the probability of disease elimination is given for the stochastic model it is shown that in some cases where the deterministic model predicts disease persistence the stochastic model predicts a high probability of disease elimination

on the existence of solutions to periodic boundary value problems for fuzzy linear differential equations in this work we provide sufficient conditions which guarantee the existence of solutions to periodic boundary value problems for first order linear fuzzy differential equations by using generalized differentiability and switching points in comparison with some previous works we consider equations whose coefficient may change its sign a finite number of times in the interval of interest we also study the existence of solutions which are crisp or real at the switching points where the diameter of the level sets changes from nonincreasing to nondecreasing character

phase plane analysis and traveling wave solution of third order nonlinear singular problems arising in thin film evolution a nonlinear third order singular ordinary differential equation is a model for a viscous fluid which drains over a wet surface as a thin film down a vertical wall here we discuss the physical processes of a thin liquid film evolution the classical thin film equation is being derived and the series solution is obtained by employing the adomian decomposition method adm in this paper a new way for the phase planes analysis and traveling wave solution by adm is presented the proposed algorithm is implemented using mathematical software mathematica 6 0

strongly order preserving semiflows generated by functional differential equations the aim of this paper is to apply some recently developed improvements in the theory of monotone dynamical systems due to the authors 12 13 j to certain systems of functional differential equations fde s which do not enjoy the quasimonotone property considered by one of us in 9 the present work extends our previous results in this direction for scalar equations in 111 to systems in the fundamental paper s m w hirsch establishes that most orbits of a strongly monotone scmiflow on a strongly ordered space x tend to the set e of equilibria employing ideas of hirsch and of h matano 7 the authors have extended the theory to obtain some improvements in the results of hirsch and matano under matano s weaker assumption that the semiflow is strongly order preserving recently in 13 we gave sufficient conditions for most orbits of a strongly order preserving semiflow to converge to an equilibrium these conditions require additional smoothness of the semiflow and a certain strong monotonicity condition for the linearized flow determined by the variational equation about each equilibrium p policik s has also given sufficient conditions for most orbits to converge for semilinear parabolic evolution equations possessing certain strong monotonicity properties in this paper wc apply the convergence results developed in 131 to nonquasimonotone fde s in this introduction wc briefly review the application of monotone dynamical systems theory to fde s enjoying the quasimonotone property this research appeared in 9 133 then we state one of the main results of this paper which applies when the quasimonotone property fails but when another property m holds some simple examples are given which

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

on the solutions to first order linear fuzzy differential equations in this paper we study different formulations of first order linear fuzzy differential equations using the concept of generalized differentiability we present sufficient conditions for the existence of solutions and obtain the general expression of these solutions which exhibit different behavior some examples are given to illustrate our results

solution of the system of ordinary differential equations by adomian decomposition method in this article we use adomian decomposition method which is a well known method for solving functional equations now a days to solve systems of differential equations of the first order and an ordinary differential equation of any order by converting it into a system of differential of the order one theoretical considerations are being discussed and convergence of the method for theses systems is addressed some examples are presented to show the ability of the method for linear and non linear systems of differential equations

diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations abstract for abstract functional differential equations and reaction diffusion equations with delay an exponential ordering is introduced which takes into account the spatial diffusion the induced monotonicity of the solution semiflows is established and applied to describe the threshold dynamics extinction or persistence convergence to positive equilibria for a nonlocal and delayed reaction diffusion population model

on periodic solutions to first order linear fuzzy differential equations under differential inclusions approach in this work we study some properties of the solutions to first order linear fuzzy differential equations under differential inclusions approach we show that the differential inclusions approach generally adapts better to some properties of the solutions to ordinary differential equations sufficient conditions for the solutions under this approach to be periodic and bounded are presented we also give some examples to show the applicability of these results

on the determination of the steady film profile for a non newtonian thin droplet abstract a shooting method is used to determine a solution to a third order ode modeling the steady profile of a non newtonian thin droplet we compare a direct approach to an iterative approach using a secant method we obtain a nonlinear relationship between the contact angle and the position of the contact line r from this nonlinear relationship we use curve fitting to obtain an empirical law of the form tan r f k where k is the power law coefficient and f is a nonlinear function of k

dynamics in a periodic competitive model with stage structure in this paper we consider a time delayed periodic system which describes the competition among mature populations by appealing to theories of monotone dynamical systems periodic semiflows and uniform persistence we analyze the evolutionary behavior of the system and establish sufficient conditions for competitive coexistence and exclusion

global stability and wavefronts in a cooperation model with state dependent time delay abstract this article deals with a diffusive cooperative model with state dependent delay which is assumed to be an increasing function of the population density with lower and upper bounds for the cooperative dde system the positivity and boundedness of solutions are firstly given using the comparison principle of the state dependent delay equations obtained the stability criterion of model is analyzed both from local and global points of view when the diffusion is properly introduced the existence of traveling waves is obtained by constructing a pair of upper lower solutions and schauder s fixed point theorem calculating the minimum wave speed shows that the wave is slowed down by the state dependent delay finally the traveling wavefront solutions for large wave speed are also discussed and the fronts appear to be all monotone regardless of the state dependent time delay this is an interesting property since many findings are frequently reported that delay causes a loss of monotonicity with the front developing a prominent hump in some other delay models

initial and boundary value problems for fuzzy differential equations we prove a wintner type result for fuzzy ivps and a superlinear type result for fuzzy bvps both results rely on a generalized schauder theorem in metric spaces

existence and uniqueness of a wavefront in a delayed hyperbolic parabolic model abstract we consider a class of hyperbolic parabolic equation with time delay and a non local feedback due to the maturation for the adult population density of a single species population and we show that the wave profile is described by a hybrid system that consists of an integral transformation and an ordinary differential equation we show the existence and uniqueness of a travelling wavefront of the hyperbolic parabolic system in the so called bistable case by considering the same problem for a properly parametrized parabolic system and then by considering the continuous dependence of the wave speed on the parameter involved

calculus of fuzzy vector valued functions and almost periodic fuzzy vector valued functions on time scales abstract in this paper we establish an embedding theorem for the fuzzy multidimensional space and introduce the new types of multiplications of fuzzy vectors determined by the determinant algorithm then some fundamental results of calculus of fuzzy vector valued functions on time scales are established based on these results we introduce the concept of shift almost periodic fuzzy vector valued functions and develop a theory of almost periodic fuzzy functions on time scales this function theory can be used to study almost periodic fuzzy dynamic equations on time scales including a new type of fuzzy dynamic equations called fuzzy q dynamic equation i e quantum fuzzy dynamic equation

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

convergence in smooth strongly monotone flows defined by semilinear parabolic equations on considere des equations paraboliques semi lineaires dy dt ay f y ou a est un operateur sectoriel sur un banach x ayant une resolvante compacte et f est une c 2 application de l espace des puissances fractionnaires x 0 1 dans x soit y t une solution qui est quasi convergente mais non convergente alors il existe un voisinage u de y 0 y 0 avec la propriete suivante si z u y 0 a une trajectoire bornee et soit z y 0 soit z y 0 alors la solution partant de z converge vers un equilibre

characterizations of generalized differentiable fuzzy functions this article deals with generalized differentiable fuzzy functions specifically we give some characterizations of generalized hukuhara differentiable fuzzy functions through the differentiability of their endpoint functions then we introduce a differentiability concept that is more general than the generalized hukuhara differentiability and extend the above characterization to this new type of differentiability as well as to g differentiable fuzzy functions these characterizations are useful tools for the calculus of derivatives of fuzzy functions

heteroclinic orbits for retarded functional differential equations abstract suppose is a heteroclinic orbit of a ck functional differential equation x t x i with limit set and limit set being either hyperbolic equilibrium points or periodic orbits necessary and sufficient conditions are given for the existence of an ftf close to in ck with the property that x t tf x t has a heteroclinic orbit gg close to the orbits gg are obtained from the zeros of a finite number of bifurcation functions g f r d r d 1 transversality of is characterized by the nondegeneracy of the derivative d g it is also shown that the tf which have heteroclinic orbits near are on a ck submanifold of finite codimension max 0 ind or on the closure of it where ind is the index of

on the time to extinction in recurrent epidemics summary an approximation is derived for the expected time to extinction in a stochastic model for recurrent epidemics numerical illustrations indicate that the approximation is crude but that it has the correct order of magnitude the quasi stationary distribution plays an important role in the derivation approximations for the critical community size and for the persistence threshold are derived comments are made on the classical study by bartlett 1956 1960

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

fuzzy differential equations without fuzzy convexity classical fuzzy differential equations defined in terms of the hukuhara derivative depend critically on the convexity of the level sets and result in expanding level sets here hullermeier s suggestion of defining fuzzy differential equations at each level set via differential inclusions is combined with ideas of aubin on morphological equations which allow nonlocal set evolution to remove the assumption of fuzzy convexity and thus to allow fuzzy differential equations to be defined for non convex level sets this approach uses reachable sets as a more general form of set integration and in contrast to the aumann set integral does not necessarily give rise to convex sets the results presented in this paper are even more general since they concern fuzzy sets that need to be only closed without additional assumptions of convexity compactness or even normality in particular an existence and uniqueness theorem is established under the assumption that the right hand sides satisfy a one sided lipschitz condition rather than a much stronger lipschitz condition fuzzy delay differential equations are also considered from this new perspective

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

limit set dichotomy and multistability for a class of cooperative neural networks with delays recent papers have pointed out the interest to study convergence in the presence of multiple equilibrium points eps multistability for neural networks nns with nonsymmetric cooperative nonnegative interconnections and neuron activations modeled by piecewise linear pl functions one basic difficulty is that the semiflows generated by such nns are monotone but due to the horizontal segments in the pl functions are not eventually strongly monotone esm this notwithstanding it has been shown that there are subclasses of irreducible interconnection matrices for which the semiflows although they are not esm enjoy convergence properties similar to those of esm semiflows the results obtained so far concern the case of cooperative nns without delays the goal of this paper is to extend some of the existing results to the relevant case of nns with delays more specifically this paper considers a class of nns with pl neuron activations concentrated delays and a nonsymmetric cooperative interconnection matrix a and delay interconnection matrix a the main result is that when a a satisfies a full interconnection condition then the generated semiflows which are monotone but not esm satisfy a limit set dichotomy analogous to that valid for esm semiflows it follows that there is an open and dense set of initial conditions in the state space of continuous functions on a compact interval for which the solutions converge toward an ep the result holds in the general case where the nns possess multiple eps i e is a result on multistability and is valid for any constant value of the delays

asymptotic profile of the positive steady state for an sis epidemic reaction diffusion model effects of epidemic risk and population movement abstract identifying the epidemic risk for infectious disease is crucial in order to effectively perform control measures in a series of our work from an analytical aspect we study the effects of epidemic risk and population movement on the spatiotemporal transmission of infectious disease via an sis epidemic reaction diffusion model proposed by allen et al 2008 in 36 in allen et al 2008 36 peng 2009 37 it was assumed that the habitat of the populations consists of only the low and high risk areas the present paper concerns a more complicated heterogeneous environment where the moderate risk area occurs and deals with two cases i only the moderate and high risk areas exist ii the low moderate and high risk areas coexist in each case we rigorously determine the asymptotic profile of the positive steady state i e the endemic equilibrium as the migration rate of either the susceptible or infected population tends to zero our results show how epidemic risk and population movement affect the spatial distribution of infectious disease and thereby suggest important implications for predicting the patterns of disease occurrence and designing optimal control strategies numerical simulations are carried out to support the theoretical results

a periodic age structured epidemic model with a wide class of incidence rates abstract in this paper a time delayed epidemic model is formulated to describe the dynamics of seasonal diseases with age structure by the method of the spectral radius of an integral operator we define the basic reproduction number r 0 of the model it is shown that the disease is uniformly persistent and there exists at least one positive periodic state when r 0 1 while the disease will die out if r 0 1 the presented case study not only confirms the theoretical results but also demonstrates that the epidemic peak is very sensitive to the maturation period and the magnitude of seasonality which is different from the dynamics of the model without considering age heterogeneities these findings contribute to better understanding the epidemiological properties of the disease with age structure

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

analysis of adomian decomposition applied to a third order ordinary differential equation from thin film flow abstract the adomian decomposition method is applied to the third order ordinary differential equation y y k obtained by considering a travelling wave solution admitted by a generalized thin film equation or for thin film flow down a vertical wall the adomian decomposition method leads to a power series approximation to the solution of y y k we show that the domain of convergence of the adomian decomposition solution is dependent on k we truncate the adomian decomposition solution at a suitable order to ensure that the truncated solution satisfies the contact line condition y 0 at some point x x we then determine the contact angle where tan d y d x at x x and plot the variation of contact angle with increasing k

a generalization of hukuhara difference and division for interval and fuzzy arithmetic we propose a generalization of the hukuhara difference first the case of compact convex sets is examined then the results are applied to generalize the hukuhara difference of fuzzy numbers using their compact and convex level cuts finally a similar approach is suggested to attempt a generalization of division for real intervals and fuzzy numbers applications to solving interval and fuzzy linear equations and fuzzy differential equations are shown

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

relationship between bede gal differentiable set valued functions and their associated support functions in this study we adapt the concept of the bede gal derivative which was initially suggested for fuzzy number valued functions to set valued functions we use an example to demonstrate that this concept overcomes some of the shortcomings of the hukuhara derivative we prove some properties of bede gal differentiable set valued functions we also study the relationship between a bede gal differentiable set valued function and its value s support function which we call the associated support function we provide examples of set valued functions that are not bede gal differentiable whereas their associated support functions are differentiable we also present some applications of the bede gal derivative to solving set valued differential equations

a reaction diffusion model for a single species with age structure i travelling wavefronts on unbounded domains in this paper we derive the equation for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment we show that if the mature deat

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

convergence of adomian s method applied to differential equations this paper deals with a new proof of convergence of adomian s method applied to differential equations we also give new formulae and properties and we suggest a simple computational form for adomian s polynomials

periodic boundary value problems for impulsive fuzzy differential equations the peculiarity of the hukuhara derivative makes it impossible to find periodic solutions for fuzzy differential equations with the exception of very restrictive situations in this work we consider a boundary value problem associated with an impulsive fuzzy differential equation and approximate the extremal solutions in a fuzzy functional interval using the monotone method fuzzy comparison results are useful in our procedure and the expression of the solution for some impulsive periodic linear differential problems is also provided

applications of the exponential ordering in the study of almost periodic delayed hopfield neural networks abstract this paper studies almost periodic neural networks of hopfield type described by delayed differential equations the authors introduce an exponential ordering to analyze the long term behavior of the solutions they prove some theorems of global convergence and deduce the stabilization role of the fast inhibitory self connections the proof which in the case of the neural networks considered in this paper requires uniform stability uses arguments of comparison of differential equations and methods of the non autonomous monotone theory of dynamical systems when the connections between different neurons are excitatory improved conditions of convergence are obtained and the stabilization role of strongly positive inputs is also shown the applicability of the results is illustrated with several numerical experiments based on two different families of neural networks

the decomposition method for ordinary differential equations with discontinuities in this paper the decomposition method is applied to the initial value problem ivp for ordinary differential equations with discontinuities for both linear and nonlinear cases

monotone semiflows in scalar non quasi monotone functional differential equations abstract by introducing a stronger than pointwise ordering conditions are found under which scalar functional differential equations generate monotone semiflows even if they are not quasi monotone typically the maximum delay must be the smaller the more quasi monotonicity is violated the theory of monotone semiflows is used to show that most solutions converge to equilibrium and that stability of equilibria is essentially the same as for ordinary differential equations

new approach for studying nonlocal problems related to differential systems and partial differential equations in generalized fuzzy metric spaces abstract this study considers the solvability of nonlocal problems for fuzzy differential systems under gh differentiability using some linear transforms we convert partial differential equations into fuzzy integral systems before some similar results are obtained for fuzzy wave equations with nonlocal conditions as a consequence we give some further estimates of the convergence rates of approximation schemes to exact solutions the main tool used in our proofs is based on perov s fixed point principle which is applied to vector valued integral operators these operators appear as the sum of two integral operators where one is fredholm type and the other is of a volterra type that depends on the restriction of the domain the novel feature of this study is that we combine the matrix convergent to zero technique with calculations of fuzzy valued functions in generalized fuzzy metric spaces some examples are presented to demonstrate our theoretical results

travelling waves for delayed reaction diffusion equations with global response we develop a new approach to obtain the existence of travelling wave solutions for reaction diffusion equations with delayed non local response the approach is based on an abstract formulation of the wave profile as a solution of an operational equation in a certain banach space coupled with an index formula of the associated fredholm operator and some careful estimation of the nonlinear perturbation the general result relates the existence of travelling wave solutions to the existence of heteroclinic connecting orbits of a corresponding functional differential equation and this result is illustrated by an application to a model describing the population growth when the species has two age classes and the diffusion of the individual during the maturation process leads to an interesting non local and delayed response for the matured population

thin films with high surface tension this paper is a review of work on thin fluid films where surface tension is a driving mechanism its aim is to highlight the substantial amount of literature dealing with relevant physical models and also analytic work on the resultant equations in general the introduction of surface tension into standard lubrication theory leads to a fourth order nonlinear parabolic equation pad h t pad x left c frac h 3 3 frac partial 3 h partial x 3 f h h x h xx right 0 label abeq1 where h h x t is the fluid film height for steady situations this equation may be integrated once and a third order ordinary differential equation is obtained appropriate forms of this equation have been used to model fluid flows in physical situations such as coating draining of foams and the movement of contact lenses in the introduction a form of the above equation is derived for flow driven by surface tension surface tension gradients gravity and long range molecular forces modifications to the equation due to slip the effect of two free surfaces two phase fluids and higher dimensional forms are also discussed the second section of this paper describes physical situations where surface tension driven lubrication models apply and the governing equations are given the third section reviews analytical work on the model equations as well as the generalized lubrication equation pad h t pad x left h n h xxx right 0 in particular the discussion focusses on asymptotic results travelling waves stability and similarity solutions numerical work is also discussed while for analytical results the reader is directed to existing literature

convergence and stability for essentially strongly order preserving semiflows abstract this paper is concerned with a class of essentially strongly order preserving semiflows which are defined on an ordered metric space and are generalizations of strongly order preserving semiflows for essentially strongly order preserving semiflows we prove several principles which are analogues of the nonordering principle for limit sets the limit set dichtomy and the sequential limit set trichotomy for strongly order preserving semiflows then under certain compactness hypotheses we obtain some results on convergence quasiconvergence and stability in essentially strongly order preserving semiflows finally some applications are made to quasimonotone systems of delay differential equations and reaction diffusion equations with delay and the main advantages of our results over the classical ones are that we do not require the delicate choice of state space and the technical ignition assumption

new generic quasi convergence principles with applications in this paper essentially strongly order preserving and conditionally set condensing semiflows are considered obtained is a new type of generic quasi convergence principles implying the existence of an open and dense set of stable quasi convergent points when the state space is order bounded the generic quasi convergence principles are then applied to essentially cooperative and irreducible systems in the forms of ordinary differential equations and delay differential equations giving some results of theoretical and practical significance

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

global properties of a general dynamic model for animal diseases a case study of brucellosis and tuberculosis transmission abstract animal diseases such as brucellosis and tuberculosis can be transmitted through an environmentally mediated mechanism but the topics of most modeling work are based on infectious contact and direct transmission which leads to the limited understanding of the transmission dynamics of these diseases in this paper we propose a new deterministic model which incorporates general incidences various stages of infection and a general shedding rate of the pathogen to analyze the dynamics of these diseases under the biologically motivated assumptions we derive the basic reproduction number r 0 show the uniqueness of the endemic equilibrium and prove the global asymptotically stability of the equilibria some specific examples are used to illustrate the utilization of our results in addition we elaborate the epidemiological significance of these results which are very important for the prevention and control of animal diseases

solution method for a boundary value problem with fuzzy forcing function in this paper we present a new approach to a non homogeneous fuzzy boundary value problem we consider a linear differential equation with real coefficients but with a fuzzy forcing function and fuzzy boundary values we assume that the forcing function is a triangular fuzzy function unlike previous studies we look for a solution that is a fuzzy set of real functions not a fuzzy valued function each of these real functions satisfies the boundary value problem with some membership degree we have developed a method that finds this solution and demonstrated its effectiveness using a test example to show that the approach can be extended to other types of fuzzy numbers we extended it to the trapezoidal case for a particular example we used the product t norm to demonstrate how a new solution type can be obtained

stability and convergence in strongly monotone dynamical systems soit x un systeme dynamique x est un espace topologique et le flot t t r est une famille d applications continues de sous ensembles ouverts de x dans x on decrit qualitativement le comportement asymptotique quand t des trajectoires t t x pour les etats initiaux x dans le cas de systemes ayant un principe de comparaison fort

generalized hukuhara differentiability of interval valued functions and interval differential equations abstract in the present paper we use a recently proposed generalization of the hukuhara difference for compact convex sets to introduce and study a generalization of the hukuhara differentiability to the case of interval valued functions we also consider several possible definitions for the derivative of an interval valued function recently proposed in the literature and we study connections between them and their properties using these concepts we study interval differential equations local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ode systems several examples are also shown

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

a periodic epidemic model with age structure in a patchy environment in this paper a periodic epidemic model with age structure in a patchy environment is introduced we investigate its global dynamics in term of the basic reproduction number mathcal r 0 and show that there exists at least one positive periodic state and the disease persists when mathcal r 0 1 while the disease will die out if mathcal r 0 1 some numerical examples are given to confirm our analytic results and to show that the age and spatial heterogeneities are important factors for the global dynamics

spatial structures and periodic travelling waves in an integro differential reaction diffusion population model an integro differential reaction diffusion equation is proposed as a model for populations where local aggregation is advantageous but intraspecific competition increases as global populations increase it is claimed that this is inherently more realistic than the usual kind of reaction diffusion model for mobile populations three kinds of bifurcation from the uniform steady state solution are considered i to steady spatially periodic structures ii to periodic standing wave solutions and iii to periodic travelling wave solutions these correspond to aggregation and motion of populations

qualitative analysis of a prey predator model with stage structure for the predator in this paper we propose a diffusive prey predator model with stage structure for the predator we first analyze the stability of the nonnegative steady states for the reduced ode system and then study the same question for the corresponding reaction diffusion system with homogeneous neumann boundary conditions we find that a hopf bifurcation occurs in the ode system but no turing pattern happens in the reaction diffusion system however when a natural cross diffusion term is included in the model we can prove the emergence of stationary patterns i e nonconstant positive stationary solutions for this system moreover these stationary patterns do not exist in the considered parameter regime when there is no cross diffusion

towards fuzzy differential calculus part 3 differentiation this paper deals successively with differentiation of ordinary functions at a fuzzy point and with differentiation of fuzzy valued functions at a nonfuzzy point in both cases the concept of differentiation is extended and the problem of the linearity is addressed in the second case practical methods of computation are presented in practical situations the available knowledge about mappings may be fuzzy it is the motivation of this work

stage structured models of intra and inter specific competition within age classes abstract in some species larvae and adults experience competition in completely different ways simple stage structured models without larval competition usually yield a single delay equation for the adults using an age structured system incorporating competition among both larvae and adults we derive a system of distributed delay equations for the numbers of larvae and adults the system is neither cooperative nor reducible to a single equation for either variable positivity boundedness and uniform strong persistence are established linear stability analysis of equilibria is difficult due to the strong coupling but results are proved for small delays using monotone systems theory and exponential ordering for small delay we prove a theorem on generic convergence to equilibria which does not directly follow from standard theory but can be proved indirectly using comparison arguments finally we consider an extension to two strain competition and prove theorems on the linear stability of the boundary equilibria

almost periodic fuzzy number valued functions in this paper we develop a theory of almost periodic fuzzy functions i e of the almost periodic functions of real variable and with values fuzzy real numbers although the class of fuzzy real numbers does not form a linear normed space the majority of main properties of almost periodic functions with values in banach spaces are extended to this case applications to fuzzy differential equations and to fuzzy dynamical systems are given

solution of linear differential equations with fuzzy boundary values we investigate linear differential equations with boundary values expressed by fuzzy numbers in contrast to most approaches which search for a fuzzy valued function as the solution we search for a fuzzy set of real functions as the solution we define a real function as an element of the solution set if it satisfies the differential equation and its boundary values are in intervals determined by the corresponding fuzzy numbers the membership degree of the real function is defined as the lowest value among membership degrees of its boundary values in the corresponding fuzzy sets to find the fuzzy solution we use a method based on the properties of linear transformations we show that the fuzzy problem has a unique solution if the corresponding crisp problem has a unique solution we prove that if the boundary values are triangular fuzzy numbers then the value of the solution at a given time is also a triangular fuzzy number the defined solution is the same as one of the solutions obtained by zadeh s extension principle for a second order differential equation with constant coefficients the solution is expressed in analytical form examples are given to describe the proposed approach and to compare it to a method that uses the generalized hukuhara derivative which demonstrates the advantages of our method

monotone semiflows generated by functional differential equations abstract this obtains sufficient condition for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space this allows the application to functional differential equations of very powerful recent results on strongly monotone semiflows due to hirsch and matano in addition a very striking relationship is established between such functional differential equations and corresponding ordinary differential equations an example involving a biochemical feedback loop is considered keywords stability steady states

convergence of solutions for an equation with state dependent delay abstract a result of smith and thieme shows that if a semiflow is strongly order preserving then a typical orbit converges to the set of equilibria for the equation with state dependent delay t x t f x t r x t where 0 and f and r are smooth real functions with f 0 0 and f 0 we construct a semiflow which is monotone but not strongly order preserving we prove a convergence result under a monotonicity condition different from the strong order preserving property and apply it to the above equation to obtain generic convergence

the decomposition method for stiff systems of ordinary differential equations in this paper the decomposition method is applied to initial value problems for systems of ordinary differential equations in both linear and nonlinear cases focusing our interest in stiff problems

a numerical and asymptotic study of some third order ordinary differential equations relevant to draining and coating flows some draining or coating fluid flow problems in which surface tension forces are important can be described by third order ordinary differential equations accurate computations are provided here

on the periodic solutions of periodic multi species competitive systems with delays the paper considers general n species periodic kolmogorov competitive systems of functional differential equations with finite or infinite delays applying schauder s fixed point theorem a criterion on the existence of the positive periodic solutions is established as applications the existence and global asymptotic stability of the single species logistic periodic model and n species lotka volterra type periodic model with delays are discussed