the yang lee edge singularity in spin models on connected and non connected rings renormalization group arguments based on a 3 field theory lead us to expect a certain universal behavior for the density of partition function zeros in spin models with short range interaction such universality has been tested analytically and numerically in different d 1 and higher dimensional spin models in d 1 one finds usually the critical exponent 1 2 recently we have shown in the d 1 blume emery griffiths beg model on a periodic static lattice one ring that a new critical behavior with 2 3 can arise if we have a triple degeneracy of the transfer matrix eigenvalues here we define the d 1 beg model on a dynamic lattice consisting of connected and non connected rings non periodic lattice and check numerically that also in this case we have mostly 1 2 while the new value 2 3 can arise under the same conditions of the static lattice triple degeneracy which is a strong check of universality of the new value of we also show that although such conditions are necessary they are not sufficient to guarantee the new critical behavior

h2 boundedness of the pullback attractor for the non autonomous sir equations with diffusion abstract we prove some regularity results for the pullback attractor of a non autonomous sir model with diffusion in a bounded domain of r d where d 1 we show a regularity result for the unique solution of the problem we establish a general result about h 2 3 boundedness of invariant sets for the associate evolution process then as a consequence we deduce that the pullback attractor of the non autonomous system of sir equations with diffusion is bounded in h 2 3

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

pullback attractors in v for non autonomous 2d navier stokes equations and their tempered behaviour abstract in this paper the asymptotic behaviour of the solutions to a non autonomous 2d navier stokes model is analyzed when the initial datum belongs to v for two frameworks the universe of fixed bounded sets and also for another universe given by a tempered condition the existence of pullback attractors in these different universes is established and thanks to regularity properties the relation between these several families of attractors and the corresponding in h is successfully studied finally two results about the tempered behaviour in v and h 2 2 of the pullback attractors when time goes to are obtained

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 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

pullback attractors for asymptotically compact non autonomous dynamical systems abstract first we introduce the concept of pullback asymptotically compact non autonomous dynamical system as an extension of the similar concept in the autonomous framework our definition is different from that of asymptotic compactness already used in the theory of random and non autonomous dynamical systems as developed by crauel flandoli kloeden schmalfuss amongst others which means the existence of a random or time dependent family of compact attracting sets next we prove a result ensuring the existence of a pullback attractor for a non autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets this attractor is minimal and in most practical applications it is unique finally we illustrate the theory with a 2d navier stokes model in an unbounded domain

the partition function zeros in the one dimensional q state potts model the zeros of the partition function in the one dimensional q state potts model with arbitrary and continuous q or 0 have been studied using a transfer matrix the location of zeros and the yang lee edge singularity have been analysed and two different regimes corresponding to q 1 and q 1 have been observed a duality relation has also been derived which relates the zeros in the complex field plane to those in the complex temperature plane

global stability of a rational difference equation abstract in this paper we study the global stability of the difference equation x n 1 x n a a 0 x n a k x n k n 0 1 where the parameters a a i 0 for i 0 k x k x 1 0 and x 0 0 we prove that the unique positive equilibrium is globally asymptotically stable if and only if it is locally asymptotically also we provide sufficient condition for it to be globally asymptotically stable and our results solve the open problem proposed by kulenovic and ladas dynamics of second order rational difference equations with open problems and conjectures chapman hall crc boca raton 2002

density of yang lee zeros for the ising ferromagnet the densities of yang lee zeros for the ising ferromagnet on the l x l square lattice are evaluated from the exact grand partition functions l 3 approximately 16 the properties of the density of yang lee zeros are discussed as a function of temperature t and system size l the three different classes of phase transitions for the ising ferromagnet first order phase transition second order phase transition and yang lee edge singularity are clearly distinguished by estimating the magnetic scaling exponent yh from the densities of zeros for finite size systems the divergence of the density of zeros at yang lee edge in high temperatures yang lee edge singularity which has been detected only by the series expansion until now for the square lattice ising ferromagnet is obtained from the finite size data the identification of the orders of phase transitions in small systems is also discussed using the density of yang lee zeros

dynamics of a higher order rational difference equation abstract our aim in this paper is to investigate the dynamics of difference equation y n 1 p qy n 1 y n k 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 the global stability the boundedness of positive solutions and the character of semicycles of the above mentioned difference equation in particular our results solve the open problem introduced by kulenovic and ladas in their monograph m r s kulenovic g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc boca raton 2002

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

location of zeros in the complex temperature plane absence of lee yang theorem in the yang lee theory 1952 the occurrence of a phase transition at real physical values of the magnetic field is related to the behaviour of the density of zeros of the partition function in the complex magnetic field plane as shown by lee and yang these zeros fall on lines for a wide class of ferromagnetic models in extensions of the yang lee theory to the complex temperature plane it has often been assumed that the zeros of the partition function would again fall on lines rather than in areas though this happens to be true for the isotropic ising model the authors point out that there is in general no basis for this assumption they discuss the location of the zeros of the partition function of the anisotropic ising model in the complex temperature plane and show that they indeed always fall in areas

yang lee zeros of the potts model the yang lee zeros of the three component ferromagnetic potts model in one dimension in the complex plane of an applied field are determined the phase diagram consists of a triple point where three phases coexist emerging from the triple point are three lines on which two phases coexist and which terminate at critical points yang lee edge singularity the zeros do not all lie on the imaginary axis but along the three two phase lines the model can be generalized to give rise to a tricritical point which is a new type of yang lee edge singularity gibbs phase rule is generalized to apply to coexisting phases in the complex plane

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

sufficient number of integrals for the pth order lyness equation in this communication we present a sufficient number of first integrals for the lyness equation of arbitrary order we first use the staircase method quispel et al 1991 physica a 173 243 66 to construct integrals of a derivative equation of the lyness equation closed form expressions for the integrals are given based on a non commutative vieta expansion the integrals of the lyness equation then follow directly from these integrals previously found integrals for the lyness equation arise as special cases of our new set of integrals

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

on the global periodicity of discrete dynamical systems and application to rational difference equations a new necessary condition for global periodicity of discrete dynamical systems and of difference equations is obtained here this condition will be applied to contribute to solving the problem of global periodicity for second order rational difference equations

on the periodic cycle of xn 1 an bnxn cnxn 1 the behaviour of solutions of the difference equation x n 1 a n b nx n c nx n 1 n 0 1 2 are investigated where the sequence a n b n and c n are periodic

partition function zeros and magnetization plateaus of the spin 1 ising heisenberg diamond chain abstract we study the properties of the generalized spin 1 ising heisenberg model on a diamond chain which can be considered as a theoretical model for the homometallic magnetic complex ni 3 c 4 h 2 o 4 2 3 oh 2 h 2 o 4 n 2 h 2 o n the model possesses a large variety of ground state phases due to the presence of biquadratic and single ion anisotropy parameters magnetization and quadrupole moment plateaus are observed at one and two thirds of the saturation value the distributions of yang lee and fisher zeros are studied numerically for a variety of values of the model parameters the usual value 1 2 alongside an unusual value 2 3 is determined for the yang lee edge singularity exponents

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

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 relation between two different concepts of pullback attractors for non autonomous dynamical systems abstract for an abstract dynamical system we establish under minimal assumptions the existence of d attractor i e a pullback attractor for a given class d of families of time varying subsets of the phase space we relate this concept with the usual attractor of fixed bounded sets pointing out its usefulness in order to ensure the existence of this last attractor in particular situations moreover we prove that under a simple assumption these two notions of attractors generate in fact the same object this is then applied to a navier stokes model improving some previous results on attractor theory

h2 boundedness of the pullback attractor for a non autonomous reaction diffusion equation abstract we prove some regularity results for the pullback attractor of a reaction diffusion model first we establish a general result about h 2 boundedness of invariant sets for an evolution process then as a consequence we deduce that the pullback attractor of a non autonomous reaction diffusion equation is bounded not only in l 2 h 0 1 but also in h 2

partition function zeros of the one dimensional blume capel model in transfer matrix formalism zeros of the partition function of the one dimensional ferromagnetic and antiferromagnetic blume capel models have been studied by using the transfer matrix method in the thermodynamic limit and for finite size chains the equation for the distribution of zeros of the partition function in the thermodynamic limit is derived the distribution of the yang lee and fisher zeros are studied for a variety of values of the parameters of the model densities of the yang lee and fisher zeros are investigated and a singular behavior of the corresponding densities of zeros at the edge points is shown the edge singularity exponents are calculated analytically for the densities of the yang lee and fisher zeros it is found that for both cases edge singularity exponents are universal and equal to sigma 1 2

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

the yang lee edge singularity in one dimensional ising and n vector models the distribution of zeros of the partition function in the complex magnetic field plane is studied for linear chains ofn vector spins and finite width strips of ising spins with nearest neighbor interactions by means of transfer matrix operator techniques the exponent characterizing the behavior of the density of zeros near the yang lee edge is shown to have the exact valuea 1 2 i analytically forn vector chains in the high temperature limit and for ising strips in the low temperature limit and ii numerically for intermediate temperatures the crossover of from itsn vector value to its spherical model value 1 2 asn as well as fromd 1 tod 2 ising as the width of the strips increases seems to proceed by an accumulation of branch points in the spectrum of the transfer operator for then vector models the position of the gap edge and the free energy at the edge approach their spherical model values with corrections of order l n with 3 4

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

finite size scaling of the density of zeros of the partition function in first and second order phase transitions the finite size scaling form for the density of zeros of the partition function in first and second order phase transitions is derived using the finite size scaling of the density of zeros the order of a phase transition can be easily determined and the order parameter calculated from finite size data we illustrate the scaling theory using exact values for the zeros of the partition function of the two dimensional ising model in the complex magnetic field plane

finite lattice extrapolation algorithms two algorithms for sequence extrapolation due to van den broeck and schwartz 1979 and bulirsch and stoer 1964 are reviewed and critically compared applications to three state and six state quantum chains and to the 2 1 d ising model show that the algorithm of bulirsch and stoer is superior in particular if only very few finite lattice data are available

complex temperature properties of the two dimensional ising model for nonzero magnetic field we study the complex temperature phase diagram of the square lattice ising model for nonzero external magnetic field h i e for 0 le mu le infty where mu e 2 beta h we also carry out a similar analysis for infty le mu le 0 the results for the interval 1 le mu le 1 provide a new way of continuously connecting the two known exact solutions of this model viz at mu 1 onsager yang and mu 1 lee and yang our methods include calculations of complex temperature zeros of the partition function and analysis of low temperature series expansions for real nonzero h the inner branch of a lima c c on bounding the fm phase breaks and forms two complex conjugate arcs we study the singularities and associated exponents of thermodynamic functions at the endpoints of these arcs for mu 0 there are two line segments of singularities on the negative and positive u axis and we carry out a similar study of the behavior at the inner endpoints of these arcs which constitute the nearest singularities to the origin in this case finally we also determine the exact complex temperature phase diagrams at mu 1 on the honeycomb and triangular lattices and discuss the relation between these and the corresponding zero field phase diagrams

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 asymptotic stability of a second order rational difference equation abstract the main goal of this paper is to investigate the global asymptotic stability of the difference equation y n 1 p n y n p n y n 1 n 0 1 2 where p n if n is even if n is odd and 0 0 and the initial conditions y 1 y 0 0 we show that the unique equilibrium y 1 is globally asymptotically stable under the certain conditions

critical behavior at edge singularities in one dimensional spin models in 1 yang and lee have initiated a method to study phase transitions based on partition function zeros since then several applications have appeared in equilibrium and more recently in nonequilibrium statistical mechanics see 2 for a review and references it is expected that with the increase of computer power new applications will appear es pecially where analytic methods can hardly be used in many statistical models the so called yang lee zeros ylzs lie on arcs on the complex u plane u exph where h is an external magnetic field and 1 kbt the zeros at end points of the arcs tend to pinch the positive real axis reu 0 at the critical value of the magnetic field for t tcr while for t tcr they accumulate at complex values ue exphe which are called yang lee edge singularities yless the linear density of zeros has a universal power like divergence u uue in the neighborhood of the yless the critical properties of the yless are described by a nonunitary o 3 field theory 3 which corresponds in two dimensions 4 to a p q 2 1 nonunitary conformal minimal model leading to the prediction 1 6 this is in agreement with numerical results for the ferromagnetic ising model see e g 5 6 and experimental magnetization data 7 even in the antiferromagnetic ising model where ue rnumerical results 8 also lead to 1 6 analo gously for the ferromagnetic and antiferromagnetic s 1 2 ising model in d 1 one derives exactly 1 2 in the fer romagnetic case the value 1 2 has been found in other d 1 models such as the n vector chain 9 and the q state potts model with q 1 and 0 q 1 10 furthermore in 11 by assuming that the transfer matrix eigenvalues are real in the gap of zeros and that the magnetic field is pure imaginary one obtains 1 2 for the s 1 3 2 ising model and also for the blume emery griffiths beg model investigated here the same assumptions of 11 have been made in the proof of 1 2 given in 12 9 more recently 13 another proof which holds without those assumptions has appeared in the blume capel model we show here that the latter proof contains also some hypotheses which are not satisfied in part of the parameter space of the beg model where a critical behavior appears

global stability two conjectures and maple abstract consider the second order difference equation u 1 0 u 0 0 and u n 1 f u n 1 u n for n 0 where either a f u v u p v u q v or b f u v p q v 1 u if 0 q p in case a or p 0 and q 0 in case b it has been conjectured see m r s kulenovic g ladas dynamics of second order rational difference equations chapman and hall crc press 2001 that lim n u n exists and equals l where l 0 and l f l l we prove this conjecture in case a and significantly extend the range of p and q for which it is known in case b in cases a and b these questions are equivalent to global stability of the fixed point l l of the planar map u v v f u v for as in case a we consider natural four dimensional extensions t of 3 and s of 2 for 0 q p we prove that l l l l is a global stable fixed point of t but we also describe precisely a range of parameters 0 q p for which s has at least three distinct fixed points in the positive orthant we describe section 3 some general principles underlying our arguments symbolic calculations using maple play a crucial role in our arguments in section 4

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

yang lee edge singularity of a one dimensional ising ferromagnet with arbitrary spin it is shown that for a one dimensional lattice system in a purely imaginary magnetic field if the interaction is finite range the nature of the yang lee edge singularity is universal independent of the spin and interaction strengths the edge singularity corresponds to the twofold degeneracy of the largest eigenvalues of the transfer matrix for the blume emery griffiths ferromagnet the tricritical point and the edge pseudosingularity may exist the tricritical point corresponds to the triple degeneracy of the eigenvalues the edge pseudosingularity corresponds to the twofold degeneracy of the nonlargest eigenvalues

generalized partition function zeros of 1d spin models and their critical behavior at edge singularities here we study the partition function zeros of the one dimensional blume emery griffiths model close to their edge singularities the model contains four couplings h j k including the magnetic field h and the ising coupling j we assume that only one of the three couplings j k is complex and the magnetic field is real the generalized zeros zi tend to form continuous curves on the complex z plane in the thermodynamic limit the linear density at the edges ze diverges usually with z z ze and 1 2 however as in the case of complex magnetic fields yang lee edge singularity if we have a triple degeneracy of the transfer matrix eigenvalues a new critical behavior with 2 3 can appear as we prove here explicitly for the cases where either or k is complex our proof applies for a general three state spin model with short range interactions the fisher zeros complex j are more involved in practice we have not been able to find an explicit example with 2 3 as far as the other couplings h k are kept as real numbers our results are supported by numerical computations of zeros we show that it is absolutely necessary to have a non vanishing magnetic field for a new critical behavior the appearance of 2 3 at the edge closest to the positive real axis indicates its possible relevance for tricritical phenomena in higher dimensional spin models

the yang lee zeros of the 1d blume capel model on connected and non connected rings we carry out a numerical and analytic analysis of the yang lee zeros of the 1d blume capel model with periodic boundary conditions and its generalization on feynman diagrams for which we include sums over all connected and non connected rings for a given number of spins in both cases for a specific range of the parameters the zeros originally on the unit circle are shown to depart from it as we increase the temperature beyond some limit the curve of zeros can bifurcate and become two disjoint arcs as in the 2d case we also show that in the thermodynamic limit the zeros of both blume capel models on the static connected ring and on the dynamical feynman diagrams lattice tend to overlap in the special case of the 1d ising model on feynman diagrams we can prove for arbitrary number of spins that the yang lee zeros must be on the unit circle the proof is based on a property of the zeros of legendre polynomials

normal forms for rational difference equations with applications to the global periodicity problem we propose a classification and derive the associated normal forms for rational difference equations with complex coefficients as an application we study the global periodicity problem for second order rational difference equations with complex coefficients we find new necessary conditions as well as some new examples of globally periodic equations

properties of a certain lyness equation we investigate oscillation cycle length and extreme values for the difference equationxn 1 a k 1i 0 bixn i xn k whereaandbiare nonnegative numbers and a k 1i 0 bi 0 ifa 0 it is known from theorem 2 2 1 of v l kocic g ladas and i w rodrigues j math anal appl 173 1993 127 157 that every cycle has no more than 2k 1 terms ifa 0 we show this is not necessarily true a general statement concerning cycle length ifa 0 remains an open question

global attractivity in a nonlinear second order difference equation in this paper we obtain a global attractivity result for the positive equilibrium of a nonlinear second order difference equation of the form xn 1 f xn xn 1 n 0 1 the result applies to the difference equation xn 1 a bxn a n 1 n 0 1 where a b a 0 1996 john wiley sons inc

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 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

h2 boundedness of the pullback attractors for non autonomous 2d navier stokes equations in bounded domains abstract we prove some regularity results for the pullback attractors of a non autonomous 2d navier stokes model in a bounded domain of r 2 we establish a general result about h 2 2 v boundedness of invariant sets for the associate evolution process then as a consequence we deduce that under adequate assumptions the pullback attractors of the non autonomous 2d navier stokes equations are bounded not only in v but also in h 2 2

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

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

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

global asymptotic stability for a higher order nonlinear rational difference equations abstract in this paper we investigate the boundedness periodic character invariant intervals and the global asymptotic stability of all nonnegative solutions of the difference equation x n 1 ax n bx n k a bx n n 0 1 where a b a b are positive real numbers k 1 is a positive integer and the initial conditions x k x 1 x 0 are nonnegative real numbers it is shown that the zero equilibrium of this equation is globally asymptotically stable if b a a and the unique positive equilibrium is globally asymptotically stable if a a b a a the results obtained solve a open problem proposed by kulenovic and ladas m r s kulenovi g ladas dynamics of second order rational difference equations chapman hall crc boca raton fl 2002 p 129

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