on the non autonomous lotka volterra n species competing systems1supported by national science foundation of education committee of people s republic of china 1 in this paper the qualitative properties of general non autonomous lotka volterra n species competing systems are studied some new criteria of uniform persistence extinction and globally asymptotical stability for partial species are established these results improve and extend the recent ones obtained by montes de oca zeeman and ahmad in proc amer math soc 117 1993 199 204 appl math comput 90 1998 155 166 proc amer math soc 124 1996 3677 3687 j math anal appl 192 1995 360 370

the dynamics of an impulsive delay predator prey model with variable coefficients abstract in this paper we formulate a new robust two species nonautonomous predator prey model with multi delays and impulsive effects and perform a systematic mathematical and ecological study our results in this paper indicate that under the appropriate linear periodic impulsive perturbations the system is permanent and has a unique positive globally attractive semi trivial periodic solution by using the brouwer fixed point theorem we prove that if the periodic system is permanent then there is at least one positive periodic solution of the system we show that the conditions for global attractivity of the positive semi trivial periodic solution and permanence of the population of the model depend on time delay so we call it profitless in this paper the main feature is that we introduce multi delays and impulses into the predator prey model exhibit a new modeling method which is applied to investigate multi species impulsive multi delays differential equations

topological conjugation and asymptotic stability in impulsive semidynamical systems we prove several results concerning topological conjugation of two impulsive semidynamical systems in particular we prove that the homeomorphism which defines the topological conjugation takes impulsive points to impulsive points it also preserves limit sets prolongational limit sets and properties as the minimality of positive impulsive orbits as well as stability and invariance with respect to the impulsive system we also present the concepts of attraction and asymptotic stability in this setting and prove some related results

asymptotically periodic solutions of n competing species problem with time delays abstract a general nonautonomous lotka volterra system of integro differential equations with infinite delay is considered and sufficient conditions for uniform stability and asymptotic stability of solutions are investigated these limiting equations are used to obtain the existence of asymptotically periodic solutions of the corresponding system of equations where lower limit in the integration is zero

poisson stability for impulsive semidynamical systems abstract in this paper the concept of poisson stability is investigated for impulsive semidynamical systems recursive properties are also investigated

a survey of exact algorithms for the simple assembly line balancing problem in this survey paper we discuss the development of the simple assembly line balancing problem salbp modifications and generalizations over time present alternate 0 1 programming formulations and a general integer programming formulation of the problem discuss other well known problems related to salbp describe and comment on a number of exact i e optimum seeking methods and present a summary of the reported computational experiences all models discussed here are deterministic i e all input parameters are assumed to be known with certainty and all the algorithms discussed are exact the problem is termed simple in the sense that no mixed models subassembly lines zoning restrictions etc are considered due to the richness of the literature we exclude from discussion here a the inexact i e heuristic approximate algorithms for salpb and b the algorithms for the general assembly line balancing problem including the stochastic models

positive solutions of p type retarded functional differential equations abstract for systems of retarded functional differential equations with unbounded delay and with finite memory sufficient and necessary conditions of existence of positive solutions on an interval of the form t 0 are derived a general criterion is given together with corresponding applications including a linear case too examples are inserted to illustrate the results

impulsive control for a class of neural networks with bounded and unbounded delays in this paper we study the problem of global exponential stability for a class of impulsive neural networks with bounded and unbounded delays and fixed moments of impulsive effect we establish stability criteria by employing lyapunov functions and razumikhin technique an illustrative example is given to demonstrate the effectiveness of the obtained results

relaxation oscillations in a class of delay differential equations we study a class of delay differential equations which have been used to model hematological stem cell regulation and dynamics under certain circumstances the model exhibits self sustained oscillations with periods which can be significantly longer than the basic cell cycle time we show that the long periods in the oscillations occur when the cell generation rate is small and we provide an asymptotic analysis of the model in this case this analysis bears a close resemblance to the analysis of relaxation oscillators such as the van der pol oscillator except that in our case the slow manifold is infinite dimensional despite this a fairly complete analysis of the problem is possible

global attractors for impulsive dynamical systems a precompact approach abstract in this work we give the definition of a global attractor of an impulsive dynamical system and obtain several important properties for this class of attractors we prove the theorem on existence of such attractors and apply it to chosen ordinary and partial differential equations with impulsive functions

asymptotic properties of a hiv 1 infection model with time delay based on some important biological meanings a class of more general hiv 1 infection models with time delay is proposed in the paper in the hiv 1 infection model time delay is used to describe the time between infection of uninfected target cells and the emission of viral particles on a cellular level as proposed by herz et al a v m herz s bonhoeffer r m anderson r m may m a nowak viral dynamics in vivo limitations on estimates of intracellular delay and virus decay proc natl acad sci usa 93 1996 7247 7251 then the effect of time delay on stability of the equilibria of the hiv 1 infection model has been studied and sufficient criteria for local asymptotic stability of the infected equilibrium and global asymptotic stability of the viral free equilibrium are given

maximizing the production rate in simple assembly line balancing a branch and bound procedure abstract in this paper a branch and bound procedure for the simple assembly line balancing problem type 2 salbp 2 is described this np hard problem consists of assigning tasks to a given number of work stations of a paced assembly line so that the production rate is maximized besides possible precedence constraints between the tasks have to be considered existing solution procedures for salbp 2 are mainly based on repeatedly solving instances of the closely related salbp 1 which is to minimize the number of stations for a given production rate the proposed branch and bound procedure directly solves salbp 2 by using a new enumeration technique the local lower bound method which is complemented by a number of bounding and dominance rules computational results indicate that the new procedure is very efficient

discontinuous local semiflows for kurzweil equations leading to lasalle s invariance principle for differential systems with impulses at variable times abstract in this paper we consider an initial value problem for a class of generalized odes also known as kurzweil equations and we prove the existence of a local semidynamical system there under certain perturbation conditions we also show that this class of generalized odes admits a discontinuous semiflow which we shall refer to as an impulsive semidynamical system as a consequence we obtain lasalle s invariance principle for such a class of generalized odes due to the importance of lasalle s invariance principle in studying stability of differential systems we include an application to autonomous ordinary differential systems with impulse action at variable times

spatial segregation limit of a non autonomous competition diffusion system abstract this paper is concerned with the spatial behavior of the non autonomous competition diffusion system arising in population ecology the limiting profile of the system is given as the competition rate tends to infinity our result shows that two competing species spatially segregate as the competition rates become large moreover for the case of the same non autonomous terms we obtain the uniform convergence result

robust stability of uncertain impulsive dynamical systems abstract this paper studies robust stability of uncertain impulsive dynamical systems by introducing the concepts of uniformly positive definite matrix functions and hamilton jacobi riccati inequalities several criteria on robust stability robust asymptotic stability and robust exponential stability are established an example is also worked through to illustrate our results

negative trajectories in impulsive semidynamical systems abstract the theory of negative semisolutions is well defined in continuous dynamical systems in this paper we present a method to construct negative semisolutions for impulsive semidynamical systems we study topological properties of the negative limit set and the first negative prolongational set we also present a result about weak minimal sets

invariant probability measures and non wandering sets for impulsive semiflows we consider impulsive dynamical systems defined on compact metric spaces and their respective impulsive semiflows we establish sufficient conditions for the existence of probability measures which are invariant under such impulsive semiflows under these conditions we also deduce the forward invariance of their non wandering sets except the discontinuity points

asymptotic stability of competitive systems with delays and impulsive perturbations abstract a general impulsive nonautonomous lotka volterra system of integro differential equations with infinite delay is considered the impulses are realized at fixed moments of time sufficient conditions for uniform stability and asymptotic stability of solutions are investigated the main results are obtained by using the comparison principle and the lyapunov method

global dynamics of the periodic logistic system with periodic impulsive perturbations this paper studies the global behaviors of the periodic logistic system with periodic impulsive perturbations the results of d d bainov and p s simeonov 1993 are extended and dynamics different from the corresponding continuous system are found it is shown that the system may have a unique positive periodic solution which is globally asymptotically stable or go extinct when the two periods are rational dependent when they are rational independent the system has no periodic solutions however still has a global attractor or go extinct under some conditions

the influence of differencing and cfl number on implicit time dependent non linear calculations abstract this paper is concerned with the stability and precision of transient implicit finite difference algorithms the test problem is the burgers equation the influence of space discretisation and initial values is studied linear analysis provides incomplete stability conditions for both continuous and weak solutions and we derive a simple criterion of augmented stability taking into account the non linearity of the problem the real stability limits appear to be a combination of the linear stability analysis criterion and of this augmented stability criterion

focusing a mechanism for instability of nonlinear finite difference equations abstract a mechanism for the destabilization of numerical algorithms for partial differential equations is suggested the novelty of the work is that it attempts to explain the dynamical process by which noise can localize on a spatial grid and cause finite amplitude instability thresholds to be exceeded at distinct locations

weakly nonlinear high frequency waves abstract in this paper we derive a method for finding small amplitude high frequency solutions to hyperbolic systems of quasilinear partial differential equations our solution is the sum of two parts i a superposition of small amplitude high frequency waves ii a slowly varying mean solution each high frequency wave displays nonlinear distortion of the wave profile and shocks may form shock conditions are derived for conservative systems different high frequency waves do not interact provided the frequencies and wave numbers of two waves are not linearly related to those of a third the mean solution is found by solving a linear partial differential equation this method generalizes whitham s nonlinearization technique 9 for single waves to problems where many waves are present we obtain these results by generalizing a scheme first proposed by choquet bruhat 1 which employs the method of multiple scales author

convergence for non autonomous semidynamical systems with impulses abstract the present paper deals with impulsive non autonomous systems with convergence we show that the structure of the levinson center of a compact dissipative system is preserved under homomorphism in impulsive convergent systems also we present some criteria of convergence using lyapunov functions

topological entropy for discontinuous semiflows abstract we study two variations of bowen s definitions of topological entropy based on separated and spanning sets which can be applied to the study of discontinuous semiflows on compact metric spaces we prove that these definitions reduce to bowen s ones in the case of continuous semiflows as a second result we prove that our entropies give a lower bound for the entropy defined by alves carvalho and vasquez 2015 finally we prove that for impulsive semiflows satisfying certain regularity condition there exists a continuous semiflow defined on another compact metric space which is related to the first one by a semiconjugation and whose topological entropy equals our extended notion of topological entropy by using separated sets for the original semiflow

permanence and global stability for nonautonomous n species lotka volterra competitive system with impulses and infinite delays abstract in this paper we consider a class of nonautonomous n species lotka volterra competitive systems with impulses and infinite delays by developing the methods given in teng 2002 25 we give sufficient conditions for permanence and global attractivity of the system

absalom balancing assembly lines with assignment restrictions abstract assembly line balancing problems albps arise whenever an assembly line is configured redesigned or adjusted an albp consists of distributing the total workload for manufacturing products among the work stations along the line on the one hand research has focussed on developing effective and fast solution methods for exactly solving the simple assembly line balancing problem salbp on the other hand a number of real world extensions of salbp have been introduced but solved with straight forward and simple heuristics in many cases therefore there is a lack of procedures for exactly solving such generalized albp in this paper we show how to extend the well known solution procedure s alome scholl a klein r 1997 s alome a bidirectional branch and bound procedure for assembly line balancing informs j comput 9 319 334 which is able to solve even large salbp instances in a very effective manner to a problem extension with different types of assignment restrictions called aralbp the extended procedure referred to as a bsalom employs a favorable branching scheme an arsenal of bounding rules and a variety of logical tests using ideas from constraint programming computational experiments show that a bsalom is a very promising exact solution approach although the additional assignment restrictions complicate the problem considerably and necessitate a relaxation of some components of s alome

on jack hale s problem for impulsive systems abstract an interesting problem of jack hale deals with the existence of a maximal compact invariant set in discrete dynamical systems a solution for this problem is known for locally bounded dynamical systems following this line of research we consider in this paper a class of systems whose continuous dynamics are interrupted by abrupt changes of state and we present sufficient conditions to obtain the existence of a maximal compact invariant set for a system in this class we use the theory of asymptotic compactness to get the results

a two process implicit enumeration algorithm for the simple assembly line balancing problem abstract we present an implicit enumeration algorithm for the simple assembly line balancing problem with the objective of minimizing the number of stations along the line for a fixed cycle time the algorithm traverses two branch and bound trees one assigning tasks that appear earlier in the precedence network at lower levels of the tree the other assigning later tasks at lower levels the two processes alternately expand nodes of their respective trees and share information about each other s progress in the form of global bounds computational results are reported on several well known problems from the literature

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

flows of characteristic 0 in impulsive semidynamical systems abstract in this paper as in e m bonotto m federson topological conjugation and asymptotic stability in impulsive semidynamical systems j math anal appl 2006 doi 10 1016 j jmaa 2006 03 042 we continue to study the dynamics of flows defined in impulsive semidynamical systems x m i where x is a metric space x is a semidynamical system m denotes an impulsive set and i is an impulsive operator we generalize some results of non impulsive flows of characteristic 0 ch 0 for systems with impulses in particular we state conditions so that the limit set of an impulsive system of ch 0 is either a periodic orbit or a single rest point we also give conditions for a subset h in x m i to be globally asymptotically stable in the impulsive system provided the flow is of ch 0

on impulsive semidynamical systems ii recursive properties it is now realized that the theory of impulsive differential equations is an important area of investigation in a recent book l the theory of such equations is studied in detail in 4 isolating the ideas underlying the theory of impulsive differential equations we initiated the study of impulsive semidynamical systems a study on similar lines except that the underlying structure is a dynamical system has also been conducted by rozhko 5 61 in 4 we studied periodicity properties of an impulsive semidynamical system in this paper we study what may be called recursive properties we also introduce here a discrete semidynamical system naturally associated with a given impulsive semidynamical system and study the relationship between the dynamical properties of the two systems this yields among other results a new characterization of weakly almost periodic points of a semidynamical system defined in 6 in terms of what we call strong almost periodicity of the associated semidynamical system

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

weak almost periodic motions minimality and stability in impulsive semidynamical systems abstract in this paper we study topological properties of semidynamical systems whose continuous dynamics are interrupted by abrupt changes of state first we establish results which relate various concepts as stability of lyapunov weakly almost periodic motions recurrence and minimality in the sequel we study the stability of zhukovskij for impulsive systems and we obtain some results about uniform attractors

positive almost periodic solutions of a class of lotka volterra type competitive system with delays and feedback controls abstract in this paper we study the positive almost periodic solutions for a class of almost periodic lotka volterra type system with delays and feedback controls applying schauder s fixed point theorem a criterion on the existence of the positive almost periodic solution of the system is obtained our new criterion which improves and generalizes some well known results can be easily checked

positive solutions of retarded functional differential equations abstract for systems of retarded functional differential equations with unbounded delay and with finite memory sufficient conditions of existence of positive solutions on an interval of the form t 0 are derived

limit sets and the poincare bendixson theorem in impulsive semidynamical systems we consider semidynamical systems with impulse effects at variable times and we discuss some properties of the limit sets of orbits of these systems such as invariancy compactness and connectedness as a consequence we obtain a version of the poincare bendixson theorem for impulsive semidynamical systems

global stability of impulsive fractional differential equations abstract in the present paper the global stability of systems of impulsive differential equations of fractional order with impulse effect at fixed moments of time is considered using piecewise continuous functions of the type of lyapunov s functions and a new fractional comparison principle sufficient conditions for various types of global stability are proved

equilibrium states for impulsive semiflows abstract we consider impulsive semiflows defined on compact metric spaces and give sufficient conditions both on the semiflows and the potential functions for the existence and uniqueness of equilibrium states we also generalize the classical notion of topological pressure to our setting of discontinuous semiflows and prove a variational principle

positive almost periodic solution for a class of lasota wazewska model with infinite delays abstract by utilizing a fixed theorem in cones we study the existence of a unique positive almost periodic solution for a generalized lasota wazewska model with infinite delays some sufficient conditions which ensure the existence of a unique positive almost periodic solution are derived and it cannot be obtained by the contraction mapping principle furthermore under proper conditions we establish some criteria to ensure that all solutions of this model converge exponentially to a positive almost periodic solution an example is provided to illustrate the effectiveness of the proposed result

nonautonomous lotka volterra systems with delays abstract the paper studies the general nonautonomous lotka volterra multispecies systems with finite delays the ultimate boundedness permanence global attractivity and existence and uniqueness of strictly positive solutions positive periodic solutions and almost periodic solutions are obtained these results are basically an extension of the known results for nonautonomous lotka volterra multispecies systems without delay to systems with delay

sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models by using the theory of coincidence degree the existence of positive periodic solutions is studied for the neutral model with periodic delaysn t n t a t email protected t n t email protected i 1 nb i t n email protected i t email protected i 1 nc i t n email protected i t our results extend and improve existing results and have further applications in population dynamics

boundary approximations for implicit schemes for one dimensional inviscid equations of gasdynamics the applicability to practical calculations of recent theoretical developments in the stability analysis of difference approximations is examined for initial boundary value problems of the hyperbolic type for the numerical experiments the one dimensional inviscid gasdynamic equations in conservation law form are selected a class of implicit schemes based on linear multistep methods for ordinary differential equations is chosen and the use of space or space time extrapolations as implicit or explicit boundary schemes is emphasized some numerical examples with various inflow outflow conditions highlight the commonly discussed issues explicit vs implicit boundary schemes and unconditionally stable schemes hen finite difference schemes are used to solve initial boundary value problems for the equations of fluid dynamics it is well known that most methods require more boundary conditions than those required by the governing partial differential equations these additional boundary conditions for the finite difference equations are often called numerical boundary conditions the numerical boundary conditions cannot be imposed arbitrarily but are determined in general using interior information for example by extrapolation or uncentered approximations in this paper any numerical procedure used to provide a numerical boundary condition will be called a boundary scheme whatever schemes are used for the numerical boundary conditions it is a common practice to assume that the boundary scheme has a local effect and will not affect the solution globally during the early 1970s kreiss 1 2 osher 3 gustafsson et al 4 varah 5 and gustafsson6 published a series of papers establishing methods for checking the stability and accuracy of difference approximations with boundary schemes included since then further progress has been made in the theory of linear difference approximations for initial boundary value problems of the hyperbolic and parabolic type 7 12 because improper treatment of the boundary conditions can lead to instability and inaccuracy even though we start with a stable interior scheme i e scheme for the interior points it is appropriate to adopt an approach that includes the stability and accuracy of the combined interior and boundary schemes surveys of recent developments and extensive bibliographies are included in papers by coughran13 and yee 14 the purpose of this paper is to examine the applicability to practical calculations for nonlinear gasdynamic problems of recent theoretical stability analyses of implicit difference approximations for initial boundary value problems of the hyperbolic type as numerical computations have progressed the use of the conservative form of the gasdynamic equations has gained popularity for physical reasons it is sometimes desirable to specify boundary conditions in the nonconservative variables and to compute with conservative variables in the interior we will consider the additional complications introduced by this procedure

on almost periodic processes in uncertain impulsive delay models of price fluctuations in commodity markets we present an impulsive price model for a single commodity market with delays and uncertain terms impulsive perturbations are realized at fixed moments of time and are proposed to model price shocks in the case of continuous time representation to do so the paper resorts to the theory of impulsive differential equations uncertain terms are due to modeling errors measurement inaccuracy mutations in the fluctuation processes and so on we investigate conditions under which the extended model is capable of generating a stable almost periodic process

invariance and stability of global attractors for multi valued impulsive dynamical systems abstract in this paper we consider infinite dimensional impulsive dynamical systems generated by evolution problems with impulses at the moments of intersection of trajectories with given subset of the phase space uniqueness of the corresponding cauchy problem is not assumed so we use multi valued semiflows to describe the long time dynamics of such systems considering global attractor as a compact minimal uniformly attracting set of the corresponding multi valued semiflow we propose a criterion of existence of such a set under additional mild assumptions we prove the invariance of the global attractor despite the fact that our approach does not require the invariance property of the set in the definition of the global attractor moreover we prove that the obtained global attractor is stable under perturbation of the right hand part of the evolution problem we also provide examples which show that in general the global attractor is not preserved under perturbation of impulsive parameters

partial persistence and extinction in n dimensional competitive systems abstract in this paper we first consider an n dimensional system slightly more general than the lotka volterra system and give conditions that imply strong persistence of all species using this result we then give computable conditions on the coefficients which imply the ultimate extinction of certain species in an n dimensional lotka volterra system

two classes of positive solutions of first order functional differential equations of delayed type abstract the equation y t f t y t is considered the existence of two classes of positive solutions asymptotically different for t is proved using the retract method with the aid of solutions of two auxiliary linear equations with several delays constructed using upper and lower linear functional estimates of the right hand side of the equation considered inequalities for both types of positive solutions are given as well

an linear programming based lower bound for the simple assembly line balancing problem the simple assembly line balancing problem is a classical integer programming problem in operations research a set of tasks each one being an indivisible amount of work requiring a number of time units must be assigned to workstations without exceeding the cycle time we present a new lower bound namely the lp relaxation of an integer programming formulation based on dantzig wolfe decomposition we propose a column generation algorithm to solve the formulation therefore we develop a branch and bound algorithm to exactly solve the pricing problem we assess the quality of the lower bound by comparing it with other lower bounds and the best known solution of the various instances from the literature computational results show that the lower bound is equal to the best known objective function value for the majority of the instances moreover the new lp based lower bound is able to prove optimality for an open problem c 2004 elsevier b v all rights reserved

singular limit of a competition diffusion system with large interspecific interaction abstract we consider a competition diffusion system for two competing species the density of the first species satisfies a parabolic equation together with an inhomogeneous dirichlet boundary condition whereas the second one either satisfies a parabolic equation with a homogeneous neumann boundary condition or an ordinary differential equation under the situation where the two species spatially segregate as the interspecific competition rate becomes large we show that the resulting limit problem turns out to be a free boundary problem we focus on the singular limit of the interspecific reaction term which involves a measure located on the free boundary

a variational principle for impulsive semiflows abstract we deduce a variational principle for impulsive semiflows defined on compact metric spaces in particular we generalize the classical notion of topological entropy to our setting of discontinuous semiflows

on existence of a globally attractive periodic solution of impulsive delay logarithmic population model in this paper it is shown that a logarithmic population model which is governed by impulsive delay differential equation has a globally attractive periodic solution

new modelling approach concerning integrated disease control and cost effectivity two new models for controlling diseases incorporating the best features of different control measures are proposed and analyzed these models would draw from poultry livestock and government expertise to quickly cooperatively and cost effectively stop disease outbreaks the combination strategy of pulse vaccination and treatment or isolation is implemented in both models if the number of infectives reaches the risk level rl firstly for one time impulsive effect we compare three different control strategies for both models in terms of cost the theoretical and numerical results show that there is an optimal vaccination and treatment proportion such that integrated pulse vaccination and treatment or isolation reaches its minimum in terms of cost moreover this minimum cost of integrated strategy is less than any cost of single pulse vaccination or single treatment secondly a more realistic case for the second model is investigated based on periodic impulsive control strategies the existence and stability of periodic solution with the maximum value of the infectives no larger than rl is obtained further the period t of the periodic solution is calculated which can be used to estimate how long the infectious population will take to return back to its pre control level rl once integrated control tactics cease this implies that we can control the disease if we implement the integrated disease control tactics every period t for periodic control strategy if we aim to control the disease such that the maximum number of infectives is relatively small our results show that the periodic pulse vaccination is optimal in terms of cost

some new results of nonautonomous lotka volterra competitive systems with delays abstract in this paper we improve and generalize the results of montes de oca and zeeman j math anal appl 192 1995 360 370 proc amer math soc 124 1996 3677 3687 to general n species nonautonomous lotka volterra type competitive systems with finite or infinite delays we establish a series of criteria under which a part of the n species of the systems is driven to extinction while the remaining part of the n species is persistent or uniformly persistent or coexists globally asymptotically stable

lasalle s theorems in impulsive semidynamical systems abstract we consider a semidynamical system subject to variable impulses and we obtain the lasalle invariance principle and the asymptotic stability theorem for this semidynamical system

on impulsive semidynamical systems it is now realized that the theory of impulsive differential equations is an important area of investigation moreover the theory of such equations is much richer than the corresponding theory of differential equations 1 21 in this paper we isolate the ideas involved in the theory of impulsive differential equations and initiate the study of impulsive semidynamical systems for allied results on discontinuous dynamical systems see 3 4