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

on continuous solutions of a class of systems of nonlinear functional difference equations with deviating argument abstract bounded continuous solutions on r of a class of systems of nonlinear functional difference equations with a complicated nonlinear deviation of the argument that depends on unknown functions are studied here

on families of bounded continuous solutions of some systems of functional difference equations on half intervals some sufficient conditions for the existence of families of bounded continuous solutions of some classes of systems of functional difference equations on the intervals 0 and 0 are given

solutions of a system of functional differential equations in a right neighborhood of zero abstract we give some sufficient conditions for the system of nonlinear functional differential equations x qt x t f t x t x 1 qt x 2 qt x k qt x qt x g t where q 0 1 and functions f 0 t r n r n r n r n j 0 t r n 0 t j 1 k and g 0 t 0 t are continuous for the unique existence of continuously differentiable solutions x t on the interval 0 t 0 t satisfying the condition x 0 0

the fundamental conjecture for homogeneous kahler manifolds toute variete de kahler homogene est un fibre de fibres holomorphe sur un domaine borne homogene dans lequel la fibre est avec la metrique de kahler induite le produit d une variete de kahler homogene plate et d une variete de kahler homogene compacte simplement connexe

bounded solutions of nonlinear discrete equations abstract the goal of this paper is to give sufficient conditions for the existence of bounded solutions of a system of discrete equations u k 1 f k u k k a a 1 a 2 with a n n 0 1 and f n a r n r n a principle giving sufficient conditions for the existence of at least one initial problem generating solution with its graph staying in a previously prescribed domain for every k a a 1 a 2 is formulated illustrative examples are considered as well

asymptotic behaviour of a two dimensional differential system with delay under the conditions of instability abstract the asymptotic behaviour of the solutions of a real two dimensional system x a t x t b t x t r h t x t x t r where r 0 is a constant delay is studied under the assumption of instability here a b and h are matrix functions and a vector function respectively the conditions for the existence of bounded solutions or solutions tending to the origin as t are given the method of investigation is based on the transformation of the considered real system to one equation with complex valued coefficients asymptotic properties of this equation are studied by means of a suitable lyapunov krasovskii functional and by virtue of the wazewski topological principle the results supplement those of kalas and barakova j math anal appl 269 1 2002 278 300 where the stability and asymptotic behaviour were investigated for the stable case

permanence and global attractivity of a discrete multispecies lotka volterra competition predator prey systems in this paper we propose a discrete multispecies lotka volterra competition predator prey systems for general non autonomous case sufficient conditions which ensure the permanence and the global stability of the system are obtained for periodic case sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system are obtained

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

on periodic and solutions converging to zero of some systems of differential difference equations sufficient conditions for the existence of a periodic c 1 solution of a system of differential difference equations are given we also give some sufficient conditions for the existence of an n parameter family of solutions of a system of differential difference equations converging to zero along with their first derivatives

linear dynamical systems coherent state manifolds flows and matrix riccati equation the classical motion and the quantum evolution determined by linear hamiltonians on coherent state manifolds of compact and noncompact hermitian symmetric space structure are considered a matrix riccati equation with opposite signs of the quadratic terms in the compact and noncompact cases determines the classical motion and the quantum evolution the geometric meaning of equations of motion as flow on a complex grassmann manifold and his noncompact dual is emphasized possibilities of generalizing the results to flag manifolds are pointed out

existence of periodic solutions in predator prey with watt type functional response and impulsive effects abstract in this paper the existence of a periodic solution and the permanence of a periodic watt type predator prey system where the prey disperses in patchy environment with two patches are investigated with the continuation theorem of coincidence degree theory we obtain the existence of a positive periodic solution for such a system we assume the watt type functional response within patch dynamics and provide a sufficient condition to guarantee the predator and prey species to be permanent furthermore we give numerical analysis to confirm our theoretical results it will be useful to ecosystem control

on some solvable systems of difference equations abstract we show that the following systems of difference equations x n 1 u n 1 v n y n 1 w n 1 s n n n 0 where u n v n w n s n are some of the sequences x n or y n with real initial values x 0 and y 0 are solvable in fourteen out of sixteen possible cases two cases are left unsolved probably the most interesting is the result in the case u n x n v n x n w n x n s n y n where a fascinating formula is obtained in an elegant way by using some ad hoc ideas

dynamic behaviors of the impulsive periodic multi species predator prey system the dynamic behaviors of an impulsive periodic predator prey model with n preys and m predators are studied in this paper by constructing a suitable lyapunov function and using the comparison theorem of impulsive differential equation sufficient conditions which ensure the permanence and global attractivity of the system are obtained at the same time a set of criteria which guarantee that some species in the system are permanent and globally attractive while the remaining species are driven to extinction is obtained our results show that for the multi species predator prey community impulsivity is one of the important reasons that can change the long time behaviors of species

stability and asymptotic properties of a system of functional differential equations with nonconstant delays the asymptotic properties of a real two dimensional differential system x t a t x t k 1 mb k t x q k t h t x t x q 1 t x q m t with unbounded nonconstant delays t q k t 0 satisfying lim t q k t are studied here a b k and h are supposed to be matrix functions and a vector function the conditions for the stability and asymptotic stability of solutions and the conditions under which all solutions tend to zero are given the methods are based on the transformation of the considered real system to one equation with complex valued coefficients asymptotic properties are studied by means of a lyapunov krasovskii functional the results generalize some previous ones where the asymptotic properties for two dimensional systems with one or more constant delays or one nonconstant delay were studied

bounded solutions of some systems of nonlinear functional differential equations with iterated deviating argument abstract for a class of systems of nonlinear functional differential equations with iterated deviating arguments dependent of unknown functions are given some conditions for the existence of bounded solutions on the real line satisfying some additional conditions

asymptotically convergent solutions of a system of nonlinear functional differential equations of neutral type with iterated deviating arguments some conditions that guarantee the existence of a family of solutions converging to zero of a system of nonlinear functional differential equations with iterated deviating arguments are given

retract principle for neutral functional differential equations abstract the investigation of asymptotic behaviour of solutions of ordinary differential equations is often based on the application of the retract principle initially developed for ordinary differential equations this technique was extended to other classes of equations not answered remains a problem concerning the possibility of extending this principle to neutral differential equations the goal of the present paper is to partially fill this gap and develop a corresponding technique for the application of this principle the applicability of the main result is illustrated on a nonlinear equation and sufficient conditions for existence of a positive solution are derived

permanence in nonautonomous lotka volterra system with predator prey the nonautonomous lotka volterra system with m predators and n preys is considered in this paper average conditions are obtained for permanence and global attractivity in the system it is shown that our results are generalization of those of the periodic system

unique existence of bounded continuous solutions on the real line of a class of nonlinear functional equations with complicated deviations abstract the existence and uniqueness of bounded continuous solutions on the real line of a general class of nonlinear functional equations with quite complicated deviations dependent of unknown functions are established in this paper

an existence criterion of positive solutions of p type retarded functional differential equations the conditions of existence of a positive solution i e a solntion with positive coordinates on a considered interval of systems of retarded functional equations in the case of unbounded delay with finite memory are discussed a general criterion for nonlinear case is given as well as its application to a linear system illustrative special cases are considered too

on solutions of a class of systems of nonlinear functional equations in a neighborhood of zero satisfying lipschitz condition unique existence of solutions of a system of nonlinear functional equations with iterated deviations in a neighborhood of zero and satisfying the lipschitz condition is studied here

nonoscillatory bounded solutions of neutral differential systems abstract the article contains some sufficient conditions for the existence of a nonoscillatory bounded solution of the nonlinear neutral differential systems the main results are presented in several theorems

on a generalized nonlinear functional equation abstract the paper deals with the functional equation f x f f u x f v x under some special assumptions concerning the given functions u v and f our main result extends some results in the literature

on some systems of difference equations abstract in this note we show that the following systems of difference equations u n 1 v n 1 v n v n 1 u n 1 u n u n 1 v n 1 u n v n 1 u n 1 v n u n 1 u n 1 v n v n 1 v n 1 u n with n n 0 and complex initial values u 0 and v 0 are solvable explicitly by means of riccati equations and based on the formulae for the general solutions we present the behaviour of their solutions as n while the result is natural for the first system it is a bit surprising for the other two systems

a topological principle for retarded functional differential equations of caratheodory type abstract the paper presents a razumikhin type version of wazewski s principle for rfdes of caratheodory type thus generalizing the author s previous results j differential equations 36 117 138 1980 a few facts from the theory of rfdes are reviewed and the concept of a regular polyfacial set with respect to an rfde of caratheodory type is introduced and the results are state and proved most of the examples given in the above work can easily be adapted to the more general setting of caratheodory systems an indication how to do it is given in only one case throughout the paper standard notation is used

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

existence of solutions bounded together with their first derivatives of some systems of nonlinear functional differential equations we give sufficient conditions for the system of nonlinear functional differential equations of the formx t 1 ax t f t x t x v 1 t x v k t x g 1 t x g l t where a is a matrix deta 0 v j t email protected j 1 t x f j 2 x f j m j t x t j 1 email protected and functions f r x r n k l 1 r n g j r r j 1 email protected f j i r xr n r j 1 email protected i 1 m email protected are continuous for the unique existence of continuously differentiable solutions x t which are bounded together with their first derivatives on r 0 and satisfy the condition lim t x t 1 ax t 0

existence and global attractivity of an almost periodic ecological model the almost periodic lotka volterra model with m predators and n preys is considered in this paper by constructing suitable lyapunov function some sufficient conditions are obtained for the existence and global attractivity of a unique positive almost periodic solution of this model examples show that our criteria are new general and easily verifiable

on a periodic multi species ecological model a periodic predator prey model with m predators and n preys is proposed in this paper which can be seen as the modification of the traditional lotka volterra model by using comparison theorem the ultimately bounded region of the system is obtained by using comparison theorem and brouwer fixed point theorem sufficient conditions with guarantee the existence of a positive periodic solution of the system is obtained finally by constructing a suitable lyapunov function some sufficient conditions are obtained for the existence of a unique globally attractive periodic solution of system the results obtained here generalized the main results of j d zhao w c chen global asymptotic stability of a periodic ecological model applied mathematics and computation 147 3 2004 881 892

on bounded continuously differentiable solutions with lipschitz first derivatives of a system of functional differential equations sufficient conditions for the unique existence of a continuously differentiable solution bounded together with its first derivative and whose first derivative satisfies the lipschitz condition of a boundary value problem of a system of functional differential equations with iterated deviating arguments dependent of unknown function are given

on reproducing kernels and quantization of states quantization of a mechanical system with the phase space a kahler manifold is studied it is shown that the calculation of the feynman path integral for such a system is equivalent to finding the reproducing kernel function the proposed approach is applied to a scalar massive conformal particle interacting with an external field which is described by deformation of a hermitian line bundle structure

on the difference equation xn xn 2 bn cnxn 1xn 2 abstract we prove that the difference equation x n x n 2 b n c n x n 1 x n 2 n n 0 where b n n n 0 and c n n n 0 are sequences periodic with period two and the initial values x 2 x 1 are real numbers can be solved explicitly some applications of obtained formulae are given

permanence in nonautonomous multi species predator prey system with feedback controls a nonautonomous multi species predator prey system with feedback controls is proposed in this paper where the competition among the predator species and among the prey species is considered also some important factors such as the effect of toxins and the age structure are also considered average conditions are obtained for permanence and global attractivity in the system the results obtained here generalized the main results of zhao and jiang j d zhao j f jiang permanence in nonautonomous lotka volterra system with predator prey appl math computat 152 2004 99 109

global stability and asymptotics of some classes of rational difference equations in this note we prove that the positive solutions of some classes of rational difference equations are globally asymptotically stable using a berg s result we also find asymptotics of some solutions of these equations

on solutions of a class of systems of nonlinear functional differential equations of neutral type with complicated deviations of an argument abstract sufficient conditions for the existence of a unique bounded solution together with its first derivative on r of a system of nonlinear functional differential equations with an iterated deviating argument dependent of unknown function and satisfying the condition lim t x t 1 x t 0 are given

global attractivity of the periodic lotka volterra system a lotka volterra periodic model withm predators andn preys is studied in this paper a set of easily verifiable sufficient conditions that guarantee the existence uniqueness and global attractivity of the positive periodic solutions is obtained finally a suitable example is given to illustrate that the conditions of the main theorem are feasible

anti lyapunov method for systems of discrete equations abstract in the present paper problems concerning the qualitative behaviour of solutions of discrete equations are considered a general principle which gives a guarantee that the graph of at least one solution stays in a prescribed domain is formulated as a consequence the auxiliary tool namely a method of regular polyfacial sets is developed and an application of the general principle is given this application is based with exceptions on the fact that points of the boundary of a domain considered are strict egress points such a situation is often in the theory of ordinary differential equations termed the anti lyapunov case illustrative applications and examples are given and open problems are formulated

study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form in this paper we study the boundedness the persistence and the asymptotic behavior of the positive solutions of the following systems of two difference equations of exponential form xn 1 a be yn c yn 1 y n 1 d e xn f xn 1

on the difference equation xn xn kb cxn 1 xn k abstract we describe the behavior of well defined solutions of the difference equation x n x n k b cx n 1 x n k n n 0 where k n parameters b c and the initial values x k x 1 are real numbers these results among others improve and extend our results in s stevic more on a rational recurrence relation appl math e notes 4 2004 80 85

existence of bounded solutions of a class of neutral systems of functional differential equations some results on the existence of bounded solutions together with their first derivatives of a class of neutral systems of functional differential equations with complicated deviations which extend and unify numerous results in the literature are proved

on difference equations with powers as solutions and their connection with invariant curves abstract we look for nonlinear systems of difference equations which have power functions as solutions and we find a connection to the theory of invariant curves and to self reversed polynomials in particular we construct algebraic solutions of a special equation of invariant curves

unique existence of periodic solutions of a class of systems of nonlinear difference equations with continuous argument some sufficient conditions which guarantee the unique existence of continuous periodic solutions of a class of systems of nonlinear difference equations with continuous argument are given here more important part in the proofs of presented results are descriptions of the methods for constructing these solutions which extend some methods in the literature

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

note on the existence of periodic solutions of a class of systems of differential difference equations abstract we prove a result on the existence of periodic solutions of a class of systems of differential difference equations partially solved with respect to the first order derivative by constructing it in a form of a series

stability of the zero solution of a family of functional differential equations abstract we give some sufficient conditions under which the zero solution of the next functional differential equation x t ax t bx 0 t j 1 k c j x j t f x t x k 1 t x k l t where a b c j j 1 k are real numbers f r l 1 r is a continuous function such that f 0 0 0 and j t c 2 0 j 0 k l is asymptotically stable

local existence of lipschitz continuous solutions of systems of nonlinear functional equations with iterated deviations abstract some conditions which guarantee the local existence of lipschitz continuous solutions of some systems of nonlinear functional equations with complicated deviations dependent of unknown functions are presented

dynamics of a nonautonomous predator prey system with the beddington deangelis functional response abstract in this paper we systematically study the dynamics of a nonautonomous predator prey system with the beddington deangelis functional response the explorations involve the permanence extinction global asymptotic stability general nonautonomous case the existence uniqueness and stability of a positive almost periodic solution and a boundary almost periodic solution for the periodic almost periodic case the paper ends with some interesting numerical simulations that complement our analytical findings

on the dynamics of the difference equation xn 1 1bnxn xn 1 in this note we study the following non autonomous equation x n 1 1b nx n x n 1 n 0 1 where b n 0 is a period 2 sequence and the initial values x 1 x 0 0 we show that the unique prime period 2 solution of this equation is globally asymptotically stable which answer open problem 5 proposed by camouzis and ladas 6

the periodic solutions for general periodic impulsive population systems of functional differential equations and its applications in this paper the general periodic impulsive population systems of functional differential equations are investigated by using the method of poincare map and horn s fixed point theorem we prove that the ultimate boundedness of all solutions implies the existence of periodic solutions as applications of this result the existence of positive periodic solutions for the general periodic impulsive kolmogorov type population dynamical systems are discussed we further prove that as long as the system is permanent there must exist at least one positive periodic solution in addition the permanence and existence of positive periodic solutions are discussed for the periodic impulsive single species logistic models and the periodic impulsive n species lotka volterra competitive models with delays

solutions converging to zero of some systems of nonlinear functional differential equations with iterated deviating argument abstract some sufficient conditions for the existence of a p parameter family of solutions converging to zero of a system of nonlinear functional differential equations with an iterated deviating argument dependent of unknown function are given

existence of bounded solutions of some systems of nonlinear functional differential equations with complicated deviating argument abstract sufficient conditions for the existence of bounded solutions on the real line satisfying lipschitz s condition of a class of systems of nonlinear functional differential equations with complicated deviating argument dependent of unknown functions are given in this paper

hermitian symmetric spaces and their unitary highest weight modules abstract the purpose of this article is to determine the set of unitarizable highest weight modules corresponding to hermitian symmetric spaces of the noncompact type the major step is that of proving unitarity at the last possible place with this established the description of the full set of unitarizable highest weight modules follows by a straightforward tensor product argument combined with the main ingredients of the proof of the key theorem bernstein gelfand gelfand and a diagramatic representation of the set of positive noncompact roots

on a nonlinear nonautonomous predator prey model with diffusion and distributed delay in this paper a nonlinear nonautonomous predator prey model with diffusion and continuous distributed delay is studied where all the parameters are time dependent the system which is composed of two patches has two species the prey can diffuse between two patches but the predator is confined to one patch we first discuss the uniform persistence and global asymptotic stability of the model after that by constructing a suitable lyapunov functional some sufficient conditions for the existence of a unique almost periodic solution of the system are obtained an example shows the feasibility of our main results

on q difference asymptotic solutions of a system of nonlinear functional differential equations the asymptotic behavior and the existence of a continuously differentiable solution bounded together with its first derivative of a system of nonlinear functional differential equations is studied here

behaviour of solutions of some linear functional equations at infinity relationship between the boundedness and the existence of a finite limit at the infinity of solutions of the linear functional equationx f t email protected t f t where email protected c f r r is a continuous increasing function satisfying the condition f t t email protected r and f is a continuous function such that there is a finite limit lim t f t is studied here by using obtained results we study behaviour of bounded solutions of the functional equationx f k t email protected email protected ix f i t f t where a i i 1 email protected are real numbers such that i 0 k email protected i 1 functions f and f satisfy above mentioned conditions and lim t f t 0

global attractivity in an almost periodic multi species nonlinear ecological model abstract a nonlinear almost periodic predator prey model with n preys and m predators is studied in this paper which can be seen as the modification of the traditional multi species lotka volterra predator prey model for general nonautonomous case by using the differential inequality theory we obtain the sufficient conditions which guarantee the uniform persistence and nonpersistence of the system after that by constructing a suitable lyapunov function some sufficient conditions are obtained which ensure the global attractivity of the system for almost periodic case by constructing a suitable lyapunov function sufficient conditions which guarantee the existence of an unique globally attractive positive almost periodic solution of the system are obtained examples together with their numeric simulations show the feasibility of our main results

extinction in nonautonomous t periodic competitive lotka volterra system a nonautonomous t periodic competitive lotka volterra system of n species is considered it is shown that if the coefficients are t periodic continuous and satisfy certain inequalities then any solution with strictly positive initial conditions has the property that all but one of its components vanish while the remaining component approaches the canonical solution of a certain logistic differential equation

global asymptotic stability of a periodic ecological model the periodic lotka volterra model with m predators and n preys is considered in this paper under the suppose that the intrinsic growth rate of the prey species may be negative while the total intrinsic growth rate in a period are positive sufficient conditions are obtained for existence and global asymptotic stability of the periodic solution of this model

dynamical properties of a class of higher order nonlinear difference equations abstract in this note we consider the dynamical properties of the following higher order nonlinear difference equation x n i 1 v x n i i 1 i 1 v x n i i 1 i 1 v x n i i 1 i 1 v x n i i 1 n 0 1 with x v x v 1 x 1 0 1 0 1 i 0 i 2 3 v our result extends and includes corresponding results in the recent literatures

globally bounded solutions of a system of nonlinear functional differential equations with iterated deviating argument abstract some sufficient conditions which guarantee the existence of a bounded solution together with its first derivative on the whole real line of a system of nonlinear functional differential equations with an iterated deviating argument dependent of unknown function are given

on a nonlinear generalized max type difference equation abstract the boundedness character of positive solutions of the following max type difference equation x n max a x n 1 p x n k r n n 0 where k n 1 the parameters a and r are positive and p is a nonnegative real number is studied in this paper our main results considerably improve results appearing in the literature

a fixed point approach to the stability of functional equations in non archimedean metric spaces abstract in this note we prove a simple fixed point theorem for a special class of complete metric spaces namely complete non archimedean metric spaces which are connected with some problems coming from quantum physics p adic strings and superstrings we also show that this theorem is a very efficient and convenient tool for proving the hyers ulam stability of a quite wide class of functional equations in a single variable

on a third order system of difference equations abstract we show that the system of difference equations x n 1 a 1 x n 2 b 1 y n z n 1 x n 2 c 1 y n 1 a 2 y n 2 b 2 z n x n 1 y n 2 c 2 z n 1 a 3 z n 2 b 3 x n y n 1 z n 2 c 3 n n 0 where the parameters a i b i c i i 1 2 3 and initial values x j y j z j j 0 1 2 are real numbers can be solved developing further the results in the literature

global stability of a max type difference equation we show that every positive solution to the difference equationx n maxa 1x n p 1 a 1 a 2x n p 2 a 2 a kx n p k a k n n 0 where p i i 1 k are natural numbers such that 1 0 a i 1 1 i 1 k converges to max 1 i ka i 1 a i 1 this result improves and complements the main result in our recent note s stevic global stability of a difference equation with maximum appl math comput 210 2009 525 529 since it also considers the case when a i 1 0 i 1 k

on the recursive sequencexn 1 α xn 1 xn abstract we study the global stability the boundedness character and the periodic nature of the positive solutions of the difference equationxn 1 xn 1 xn where 0 and where the initial conditionsx 1andx0are arbitrary positive real numbers

the boundedness character of two stevic type fourth order difference equations abstract the boundedness character of positive solutions of two nonlinear fourth order difference equations which are particular cases of two large classes of difference equations by stevic are studied in this paper

asymptotic behavior of solutions of a system of q functional difference equations abstract the asymptotic behavior and the existence of a bounded lipschitz continuous solution of a system of nonlinear q functional difference equations on a half interval of the real line are studied here the main results compare the behavior of such a solution of the system with bounded solutions of a linear q difference equation

periodicity of a class of nonautonomous max type difference equations abstract in this note we prove several general and easily applicable results regarding the periodicity of positive solutions of the following difference equations x n max f 1 x n k 1 n f m x n k m n x n s n n 0 and x n min f 1 x n k 1 n f m x n k m n x n s n n 0 where 1 k 1 k m m s n and where the functions f i 0 n 0 0 1 i m satisfy some additional conditions these results considerably extend and unify numerous results on max type difference equations existing in the literature

on a system of difference equations abstract we show that the system of difference equations x n 1 ax n 1 by n x n 1 c y n 1 y n 1 x n y n 1 n n 0 where the parameters a b c and initial values x 1 x 0 y 1 y 0 are real numbers can be solved considerably improving the results in the literature

on the nonautonomous difference equation xn 1 an xn 1pxnq abstract in this paper we study the asymptotic behavior and the periodicity of the positive solutions of the nonautonomous difference equation x n 1 a n x n 1 p x n q n 0 1 where an is a positive bounded sequence p q 0 and x 1 x0 are positive numbers

on some linear difference equations and inequalities with continuous argument abstract we study some properties of solutions of linear difference equations and inequalities with continuous argument in terms of the location of the zeroes of the characteristic polynomial of the associated homogeneous linear difference equation with constant coefficients

existence uniqueness and stability of positive periodic solution for a nonlinear prey competition model with delays a nonlinear periodic predator prey model with m preys and n m predators and delays is proposed in this paper which can be seen as the modification of the traditional lotka volterra prey competition model sufficient conditions which guarantee the existence of a unique globally attractive positive periodic solution of the system are obtained

on boundedness of the difference equation xn 1 pn xn 3s 1xn s 1 with period k coefficients in this note we study the following non autonomous equation xn 1 pn xn 3s 1xn s 1 n 0 1 where pn 0 is a period k sequence and s 1 2 with gcd 3s k 1 and the initial conditions x 3s 1 x0 0 we show that every positive solution of this equation is bounded