global asymptotic stability of a higher order rational difference equation abstract in this note we consider the following rational difference equation x n 1 f x n r 1 x n r k g x n m 1 x n m l 1 f x n r 1 x n r k g x n m 1 x n m l n 0 1 where f c 0 k 0 and g c 0 l 0 with k l 1 2 0 r 1 r k and 0 m 1 m l and the initial values are positive real numbers we give sufficient conditions under which the unique equilibrium x 1 of this equation is globally asymptotically stable which extends and includes corresponding results obtained in the recent literature

the periodicity and solutions of the rational difference equation with periodic coefficients abstract in this paper we give necessary and sufficient conditions for generalized solution and periodicity of the difference equation x n 1 p n x n k x n k 1 q n x n k 1 with k 2 periodic coefficients where k n x k 1 x k x 0 r also we obtain that the generalized solution is periodic with k 1 period

global attractivity in the recursive sequence xn 1 α βxn γ xn 1 our aim in this paper is to investigate the global attractivity of the recursive sequencex n email protected bx email protected n 1 n 0 1 where a 0 b c 0 we show that one positive equilibrium point of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients

global attractivity in a second order nonlinear difference equation abstract consider the difference equation x n 1 x n x n 1 n 0 1 2 1 where the function satisfies the following conditions 0 0 and is nonincreasing in the equation 1 has a unique positive solution if denotes the unique positive solution of 1 then x x x x x 0 for x x then x is a global attractor of all positive solutions of eq 1

global attractivity in a rational recursive sequence our aim in this paper is to investigate the periodic character invariant intervals and the global attractivity of all positive solutions of the nonlinear difference equationx n email protected email protected email protected n 1 n 0 1 where a 0 b c 0 we show that the positive equilibrium of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients

on the higher order rational recursive sequence xn axn k bx n 3k the main purpose of the current paper is to prove that every positive solution of the delay difference equationx n ax n k bx n 3 k where a b 0 the initial conditions x 3 k 1 x 3 k 2 x 0 0 and k 1 2 3 converges eventually to a period k solution we also give the results of computational examples to support our theoretical discussion

on the difference equation yn 1 α βe yn γ yn 1 in this paper we investigate the convergence the boundedness and the periodic character of the positive solutions of the difference equationy n email protected email protected y email protected y n 1 n 0 1 2 where the parameters a b c are positive numbers and the initial conditions y 1 y 0 are arbitrary non negative numbers

on the global behavior of higher order recursive sequences our basic purpose in this paper is to establish some conditions for the global attractivity and convergence of the kth order recursive sequencex n f x n 1 x n 2 x n k n 1 2 3 we will show that if f is a real valued function with a convex basin w in r k which is differentiable with bounded differential then f has an equilibrium point our final comments are about informative examples

on the efficiency and accuracy of the boundary element method and the finite element method the efficiency and computational accuracy of the boundary element and finite element methods are compared in this paper this comparison is carried out by employing different degrees of mesh refinement to solve a specific illustrative problem by the two methods

a rational difference equation abstract we investigate the boundedness character the periodic nature and the global asymptotic stability of all positive solutions of the equation in the title with positive parameters and nonnegative initial conditions

application of the complex variable boundary element method to solving potential problems in doubly connected domains the complex variable boundary element method cvbem developed by hromadka for the solution of potential problems in simply connected domains is extended to the solution of heat conduction problems in doubly connected domains a cut is made in the doubly connected domain and it was found that the complex potentials along the cut do not cancel out but result in a complex stream function that plays the role of perturbation in the nodal equations cauchy riemann conditions are used to derive additional equations which relate the stream functions and the boundary heat fluxes and potentials when neumann and robin conditions are imposed on the boundaries the resulting nodal equations are expressed in matrix form and coding rules and methods for checking the matrix elements are developed three solution methods implicit explicit and hybrid are described and by means of examples the efficacy of these methods is discussed and compared

some dynamically trivial mappings with applications to the improvement of simple iteration abstract we consider the iteration algorithm defined by x n k g x n x n 1 x n k 1 n 0 1 2 where is a k dimensional means we show that under certain conditions on the function g the algorithm will converge globally to a fixed point of g and that under less stringent conditions it will converge locally even when the corresponding algorithm for simple iteration x n 1 g x n n 0 1 2 does not a corollary of our main theorem answers a related question concerning the dynamic triviality of a class of mappings of r k into itself

on the positive solutions of the difference equation xn 1 xn 1 1 x nxn 1 the positive solutions of the difference equation x n 1 x n 1 1 x nx n 1 n 0 1 2 are investigated where x 1 and x 0 are the positive real numbers

the caratheodory fejer problem and h sub spl infin l sub 1 identification a time domain approach in this paper we study a worse case robust control oriented identification problem this problem is in the framework of h sub spl infin identification but the formulation here is more general the available a priori information in our problem consists of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant and an upper bound on the noise level the plant to be identified is assumed to lie in a certain subset in the space of h sub spl infin characterized by the assumed a priori information the available experimental information consists of a corrupt finite output time series obtained in response to a known nonzero but otherwise arbitrary input the proposed algorithm is in the class of interpolatory algorithms which are known to possess desirable optimality properties in reducing the identification error this algorithm is obtained by solving an extended caratheodory fejer problem via standard convex programming methods both the algorithm and error bounds ran be applied to l sub 1 identification problems as well

on a nonlinear second order system of difference equations we prove that all positive solutions of the next system of difference equationsx n 1 a y n px n 1 q y n 1 a x n py n 1 q email protected n 0 where parameters a p and q are positive numbers are bounded if one of the following conditions is satisfied p 2 4 or p 1 q q 1 completely describing the boundedness character of the system in this case

a note on the periodic cycle of xn 2 1 xn 1 xn the behaviour of solutions of the difference equation x n 1 a n b nx nx n 1 n 0 1 are investigated where the sequences a n and b n are periodic

periodicity and attractivity of a nonlinear higher order difference equation in this paper a nonlinear higher order difference equation is considered the existence and nonexistence of two period positive solution and the attractivity are investigated

the dynamics of χn 1 α βχna bχn cxn 1 facts and conjectures abstract we investigate the global character of solutions of the equation in the title with non negative parameters and nonnegative initial conditions

paper bounded error parameter estimation noise models and recursive algorithms this paper deals with some issues involving a parameter estimation approach that yields estimates consistent with the data and the given a priori information the first part of the paper deals with the relationships between various noise models and the size of the resulting membership set the set of parameter estimates consistent with the data and the a priori information when there is some flexibility about the choice of the noise model this analysis can be helpful for noise model selection so that the resulting membership set yields a better estimate of the unknown parameter the second part of the paper presents algorithms for various commonly encountered noise models that have the following properties a they are recursive and easy to implement and b after a finite learning period the estimates provided by these algorithms are guaranteed to be in or very close to the membership set in general the interpolatory algorithms that produce an estimate in the membership set do not possess nice statistical and worst case properties similar to those of classical approaches such as least mean squares lms and least squares ls algorithms in the third part of the paper we propose an algorithm that is optimal in a certain worst case sense but gives an estimate that is in or is close to the membership set

on the recursive sequence x n 1 α βx n k g x n x n 1 x n k 1 this paper is devoted to investigating the asymptotic behavior of the recursive sequence x n email protected bx n kg x n x n 1 x n k 1 n 0 1 where a 0 and b 0 and g is continuous on r k we show that under certain conditions this equation has a unique positive negative equilibrium point which is a global attractor with some basin email protected r k 1 also we establish the oscillation of all solutions with initial conditions x i i 0 k such that x 0 x 1 x k s we apply these results to the recursive sequence x n email protected bx n email protected email protected k 1i 0 a ix n i b ix n i 2 n 0 1 where a c a i b i 0 i 0 k 1 and b 0

model validation a connection between robust control and identification the gap between the models used in control synthesis and those obtained from identification experiments is considered by investigating the connection between uncertain models and data the model validation problem addressed is given experimental data and a model with both additive noise and norm bounded perturbations is it possible that the model could produce the observed input output data this problem is studied for the standard h sub infinity mu framework models a necessary condition for such a model to describe an experimental datum is obtained for a large class of models in the robust control framework this condition is computable as the solution of a quadratic optimization problem

on the nonlinear difference equation system xn 1 a yn m xn yn 1 a xn m yn in this paper we study the boundedness persistence and global asymptotic stability of positive solutions of the system of two difference equations x n 1 a y n mx n y n 1 a x n my n n 0 1 where a 0

dynamics of a non linear difference equation abstract our aim in this paper is to investigate the dynamics of difference equation y n 1 y n py n k y n q n 0 1 2 where the initial conditions y k y 1 y0 are non negative k 1 2 3 and the parameters p and q are non negative we study characteristics such as periodicity invariant interval the character of semicycles the global stability and the boundedness of positive solutions of the above mentioned difference equation in particular our results solve the open problem introduced by kulenovic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002

optimal algorithms theory for robust estimation and prediction this paper deals with the theory of optimal algorithms for problems which cannot be solved exactly the theory developed allows for the derivation of new and interesting results in parameter estimation and in time series prediction in situations where no reliable statistical hypothesis can be made on the functions and modeling errors involved but only a bound on them is known in particular the derivation of computationally simple optimal algorithms for these two problems is investigated the practical effectiveness of the algorithms obtained is illustrated by several numerical examples

global attractivity of the recursive sequence xn 1 α βxn 1 γ xn our aim in this paper is to investigate the global attractivity of the recursive sequence x n 1 a bx n 1 c x n under specified conditions we show that the positive equilibrium point of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients

on the recursive sequence yn 1 p yn 1 qyn yn 1 abstract we investigate the periodic character and the global stability of solutions of the equation y n 1 p y n 1 qy n y n 1 with positive parameters and positive initial conditions

recursive system identification in the presence of noise and model uncertainties the main contribution of this paper is a recursive algorithm for parametric system identification in the presence of both noise and model uncertainties the estimates provided by this algorithm are not invalidated after a learning period by the observed input output data and the assumed system and uncertainty structures a complementary off line algorithm derived from the on line algorithm is also presented

global attractivity of the difference equation xn 1 xn αxn 1 β xn our aim in this paper is to investigate the global attractivity of nonnegative solution of the difference equationx n 1 x n ax n 1 b x n n 0 1 where a 0 b 0 are real numbers the initial conditions x 1 x 0 are nonnegative real numbers and with b and x 1 not both zero we show that the positive equilibrium point of the equation is a global attractor

on the difference equation xn 1 pxn s xn tqxn s xn t abstract in this paper we address the difference equation x n 1 px n s x n t qx n s x n t n 0 1 with positive initial conditions we establish a sufficient condition for the global asymptotic stability of the positive equilibrium of this equation

on the rational difference equation y n 1 a y n y n k we find conditions for the global asymptotic stability of the unique negative equilibrium y 1 a of the equation yn 1 a yn yn k where y k y k 1 y0 a r and k 1 2 3 4

oscillation and asymptotic behavior of a class of higher order nonlinear recursive sequences abstract we investigate the global attractivity the periodic behavior and the invariant interval with the semicycles character of solutions of the equation x n 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 2 for all admissible non negative values of the parameters b c and the initial conditions x 2 k 1 x 1 x 0 we show that the solutions exhibit a periodic attitude depending on the parameters of the family of the above mentioned difference equation in comparison with each other

the complex variable boundary element method applications the complex variable boundary element method cvbem is used to approximate several potential problems where analytical solutions are known a modelling result produced from the cvbem is a measure of relative error in matching the known boundary condition values of the problem a cvbem error reduction algorithm is used to reduce the relative error of the approximation by adding nodal points in boundary regions where error is large from the test problems overall error is reduced significantly by utilizing the adaptive integration algorithm

global attractivity in a nonlinear difference equation abstract we consider the nonlinear difference equation x n a k 1 m b k x n k n 0 1 2 where a and bk k 1 2 m are nonnegative numbers with b mk 1bk 0 and we are interested in whether all positive solutions are attracted by the positive equilibrium l a 2 a 2 2 b

on the recursive sequence xn axn 1 bxn 2c dxn 1xn 2 in this paper we study the invariant intervals the global attractivity of the equilibrium points and the asymptotic behavior of the solutions of the difference equationx n ax n 1 bx n 2c dx n 1x n 2 n 1 2 where a 0 b c d 0

global asymptotic stability in a rational recursive sequence in this paper we study the global stability of the difference equationx n a bx n 1 cx n 1 2d x n 2 n 1 2 where a b 0 and c d 0 we show that one nonnegative equilibrium point of the equation is a global attractor with a basin that is determined by the parameters and every positive solution of the equation in the basin exponentially converges to the attractor

qualitative analysis of a discrete logistic equation with several delays in this paper we study the qualitative behavior of solutions of the discrete delay logistic equation with several delaysx n 1 x email protected i 1 mr i1 x n k ik n 0 1 2 where r email protected 0 for i 1 m k i i 1 2 m are positive integers and email protected 0 we obtain sufficient conditions for the global attractivity of all positive solutions about the positive equilibrium furthermore we show that such model is uniformly persistent and that all its eventually positive solutions are bounded the oscillation about the positive equilibrium is also discussed

optimal nonparametric identification from arbitrary corrupt finite time series formulates and solves a worst case system identification problem for single input single output linear shift invariant distributed parameter plants the available a priori information in this problem consists of time dependent upper and lower bounds on the plant impulse response and the additive output noise the available a posteriori information consists of a corrupt finite output time series obtained in response to a known nonzero but otherwise arbitrary input signal the authors present a novel identification method for this problem this method maps the available a priori and a posteriori information into an uncertain model of the plant which comprises a nominal plant model a bounded additive output noise and a bounded additive model uncertainty the upper bound on the model uncertainty is explicit and expressed in terms of both the l sub 1 and h sub spl infin system norms the identification method and the nominal model possess certain well defined optimality properties and are computationally simple requiring only the solution of a single linear programming problem

dynamics of a higher order rational difference equation abstract in this paper we will investigate a nonlinear rational difference equation of higher order our concentration is on invariant intervals periodic character the character of semicycles and global asymptotic stability of all positive solutions of x n 1 x n x n k bx n cx n k n 0 1 it is worth to mention that our results solve the open problem proposed by kulenvic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman hall crc boca raton 2002

locating a crack of arbitrary but known shape by the method of path independent integrals abstract the solution of a crack problem of an arbitrary but known shape inside an infinite plane isotropic elastic medium can be achieved in general by the method of complex singular integral equations and their numerical solution by using the gauss or the lobatto chebyshev methods in a few special cases like straight or circular arc shaped cracks this solution is available in closed form in this paper we will not contribute to the above methods of solution of crack problems but we will propose a method for the determination of the exact position of such a crack inside a closed contour in the elastic medium by gathering and using information along this contour only by experimental techniques and applying the method of complex path independent integrals for the location of the crack this paper constitutes a nontrivial generalization of relevant previous results by the author and it is of quite general applicability in fracture mechanics for nondestructive testing numerical results for the particular case of a straight crack are displayed for the illustration of the efficiency of the method the generalization of the present results to the determination of additional geometric and loading parameters of the crack is also suggested very briefly and related numerical results concerning the length of the crack and the pressure distribution on it in the aforementioned numerical application are also presented

set membership identification of systems with parametric and nonparametric uncertainty a method for parameter set estimation in which the system model is assumed to contain both parametric and nonparametric uncertainty is presented in the disturbance free case the parameter set estimate is guaranteed to contain the parameter set of the true plant in the presence of stochastic disturbances the parameter set estimate obtained from finite data records is shown to have the property that it contains the true plant parameter set with probability one as the data length tends to infinity

boundedness character of a class of difference equations abstract a complete picture regarding the boundedness character of positive solutions to the following difference equation x n max a x n 1 p x n k p n n 0 where k 2 and the parameters a and p are positive real numbers is given in particular for the case p k 1 0 k k k 1 k 1 we prove that all solutions to the equation are bounded we also present corresponding results concerning the following closely related difference equation x n a x n 1 p x n k p n n 0

stability properties of nonlinear difference equations and conditions for boundedness abstract in this paper we obtain sufficient conditions for the global asymptotic stability of the difference equations x n 1 f x n g x n k n 0 1 we also obtain sufficient conditions for all solutions to be bounded

global attractivity of the difference equation xn 1 α xn k xn in this paper we investigate the global stability of all negative solutions of the difference equationx n email protected x n kx n n 0 1 where a 1 is an integer and the initial conditions x k x 0 are arbitrary real numbers we show that the unique negative equilibrium of the equation is a global attractor with a basin that depends on certain conditions of the coefficient

global attractivity for a class of higher order nonlinear difference equations our aim in this paper is to investigate the global attractivity of all positive solutions of the higher order nonlinear difference equationx n 1 a bx email protected k i 0b ix n i n 0 1 where a b b email protected 0 a b 0 b k email protected 0 email protected 1 2 and the initial conditions x k x 0 are arbitrary real numbers we show that one positive equilibrium of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients

comparison of boundary element and finite element methods in the inelastic torsion of prismatic shafts the accuracy and computational efficiency of the boundary element and the finite element methods are compared in this paper for problems of time dependent inelastic torsion of prismatic shafts several cross sections and two types of twisting history are considered in the numerical examples the shaft material is assumed to obey an elastic time hardening creep constitutive model

a complex variable boundary element method development a generalized boundary integral equation method for the solution of the laplace equation is developed based on the cauchy integral theorem for analytical complex variable functions although the approach is complicated by the utilization of complex variable theory the resulting model involves direct integration along straight line boundary segments elements rather than using quadrature formulae that are required in current real variable boundary element formulations previously published boundary integral equation methods based on the cauchy integral theorem are shown to be a subset of the generalized model theory developed in this paper

an approach to identification for robust control given measured data generated by a discrete time linear system we propose a model consisting of a linear time invariant system affected by norm bounded perturbation under mild assumptions the plants belonging to the resulting uncertain family form a convex set the approach depends on two key parameters an a priori given bound of the perturbation and the input used to generate the data it turns out that the size of the uncertain family can be reduced by intersecting the model families obtained by making use of different inputs the model validation problem in this identification scheme is analyzed for a given energy level the invalidation problem yields the family of those models which can never be invalidated for any possible input of fixed energy and any possible perturbation this leads to the intersection of all uncertain families a consequence of the invalidation problem is that for finite length measurements not all models can be invalidated using fixed energy inputs

on the difference equation yn 1 a ynyn k with a 0 abstract we find conditions for the global asymptotic stability of the unique negative equilibrium y 1 a of the equation 0 1 y n 1 a y n y n k where y k y k 1 y0 0 a

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

a time domain approach to model validation in this paper we offer a novel approach to control oriented model validation problems the problem is to decide whether a postulated nominal model with bounded uncertainty is consistent with measured input output data our approach directly uses time domain input output data to validate uncertainty models the algorithms we develop are computationally tractable and reduce to generally nondifferentiable convex feasibility problems or to linear programming problems in special cases we give analytical solutions to these problems

the asymptotic stability of xn 1 axn bxn k 0 we obtain necessary and sufficient conditions for the asymptotic stability of the difference equation xn 1 axn bxn k 0 n 0 1 2 where the coefficients a and b are real numbers and the delay k is a nonnegative integer

on the recursive sequences xn 1 αxn 1β xn our aim in this paper is to investigate the global attractivity of the recursive sequences x n 1 email protected n 1 b x n under specified conditions we show that the zero equilibrium points of these equations are global attractor with basin that depend on certain conditions posed on the coefficients

on the recursive sequence xn 1 α βxn γxn 1bxn cxn 1 we investigate the global character of solutions of the equation in the title with nonnegative parameters and positive initial conditions we give a detailed description of the semicycles of solutions and prove that the equilibrium of the equation is globally asymptotically stable

on the recursive sequence xn 1 α βxn k 1 γxn 2k 1bxn k 1 cxn 2k 1 abstract our aim in this paper is to investigate the global asymptotic stability of all positive solutions of the higher order nonlinear difference equation x n 1 x n k 1 x n 2 k 1 bx n k 1 cx n 2 k 1 n 0 1 2 where b c and are positive k 1 2 3 and the initial conditions x 2 k 1 x 1 x 0 are positive real numbers we show that the unique positive equilibrium of the equation is globally asymptotically stable and has some basins that depend on certain conditions posed on the coefficients our concentration is on invariant intervals the character of semicycles and the boundedness of the above mentioned equation our final comments are about informative examples

on asymptotic behaviour of the difference equation xn 1 α xn kxn in this paper we investigate the global stability and the periodic character of positive solutions of the difference equationx n email protected x n kx nunder specified conditions

on rational recursive sequences our goal in this chapter is to gain some understanding of the dynamics of rational recursive sequences of the form x n 1 frac a sum limits i 0 k a i x n i b sum limits i 0 k b i x n i n 0 1 3 0 1 where k is a positive integer and the coefficients a b a 0 a k b 0 b k are nonnegative real numbers

global attractivity of a higher order nonlinear difference equation the main goal of this paper is to investigate the locally asymptotically stable period two solutions invariant intervals and global attractivity of all negative solutions of the nonlinear difference equationx n 1 1 x na x n k n 0 1 where email protected 1 k is a positive integer and initial conditions x k x email protected 0 it is shown that the unique negative equilibrium of the equation is a global attractor with a basin that depends on certain conditions of the coefficient

the role of model validation for assessing the size of the unmodeled dynamics the problem of assessing the quality of a given or estimated model is a central issue in system identification various new techniques for estimating bias and variance contributions to the model error have been suggested in recent literature in this contribution classical model validation procedures are placed at the focus of our attention we discuss the principles by which we reach confidence in a model through such validation techniques and also how the distance to a true description can be estimated this way in particular we stress how the typical model validation procedure gives a direct measure of the model error of the model test without referring to its ensemble properties several model error bounds are developed for various assumptions about the disturbances entering the system

the rule of semicycle and global asymptotic stability for a fourth order rational difference equation in this paper the rule for the lengths of positive and negative semicycles of nontrivial solutions of the following fourth order rational difference equation x n 1 x n bx n 2 x n 3 b ax n b x n 2x n 3 b a n 0 1 2 where a b e 0 and the initial values x 3 x 2 x 1 x 0 e 0 to successively occur is found to be 4 3 1 2 2 1 1 1 4 3 1 2 2 1 1 1 4 3 1 2 2 1 1 1 by which the positive equilibrium point of the equation is verified to be globally asymptotically stable

dynamics of a family of two dimensional difference systems the boundedness and global asymptotic behavior of positive solutions of the nonlinear two dimensional difference systems x n f x n t 1 y n s 1 y n j y n s 2 x n t 2 where t 1 s 1 s 2 t 2 are all positive integers are investigated some sufficient conditions are established such that the positive solutions converge to the unique equilibrium finally some examples are given to illustrate our results

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

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

dynamic behavior of a recursive sequence in this paper the global asymptotic stability the global attractivity the boundedness character the periodic nature and chaotic behavior of solutions of the difference equationx n 1 a x n 1x n n 0 1 are investigated where a r is a real number and the initial conditions x 1 x 0 are arbitrary real numbers as might be expected the six cases a 3 give rise to substantially different dynamic behavior

stability of the recursive sequence xn 1 α βxn γ xn 1 abstract in this paper we investigate the global asymptotic stability of the recursive sequence xn 1 xn xn 1 n 0 1 where 0 we show that the unique positive equilibrium point of the equation is a global attractor with a basin that depends on the conditions posed on the coefficients

on the dynamics of the recursive sequence xn 1 xn 1β γxn 22xn 4 γxn 2xn 42 in this paper we investigate the global behavior of the difference equation x n 1 x n email protected email protected n 2 2x n email protected n 2x n 4 2 n 0 1 with positive parameters and non negative initial conditions

on the recursive sequence yn 1 α yn 1β yn yn 1yn abstract we study the global stability the boundedness character and the periodic nature of the positive solutions of the difference equation y n 1 y n 1 y n y n 1 y n n 0 1 2 where 0 2 and the initial conditions y 1 y0 are arbitrary positive real numbers

dynamics of a rational difference equation in this paper we investigate the periodic character invariant intervals oscillation and global stability of all positive solutions of the equationx n 1 px n x n kq x n k n 0 1 where the parameters p and q and the initial conditions x k x 0 are nonnegative real numbers as might be expected the two cases p q give rise to different dynamic behaviors in particular our results solve the open problem proposed by kulenvic and ladas in their monograph dynamics of second order rational difference equations with open problems and conjectures chapman hall crc boca raton 2001

on the difference equation xn p xn sqxn t xn s abstract we investigate the persistence and global attractivity of the difference equation x n p x n s qx n t x n s n 1 2 3 with positive parameters and positive initial conditions our results not only include some previously known results but apply to some difference equations that have not been investigated so far

stability of solutions for the recursive sequence x n 1 α β x n γ g x n k in this paper we investigate the global behavior of the recursive sequence xn 1 x xn g xn k for n 0 1 where 0 and g x is a continuous function defined on which satisfies some additional conditions

global stability of a higher order rational recursive sequence abstract in this paper we study the qualitative behavior of solutions of the class of delay difference equation x n 1 x n k 1 x n 2 k 1 a bx n k 1 n 0 1 2 where the initial conditions x 2 k 1 x 1 x 0 are positive k 1 2 and the parameters a b are positive our concentration is on invariant intervals and the global stability of the above mentioned equation we obtain sufficient conditions for the global attractivity of all positive solutions about the zero and positive equilibrium points with basins that depend on specific conditions posed on the coefficients furthermore the oscillation and the character of semicycles about the positive equilibrium are thoroughly discussed it is also illustrated that for some special case of parameters the solution will be either a period 2 solution or will converge to the equilibrium point or will have unbounded solutions

global attractivity in a recursive sequence in this paper we study the invariant intervals the globally attractivity of the two equilibrium points and the oscillatory behavior of the solutions of the difference equationx n ax n 1 bx n 2c x n 2 n 1 2 where a b c 0