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

free vibration of axially loaded rotating timoshenko shaft systems by the wave train closure principle a systematic approach for the free vibration analysis of a rotating timoshenko shaft system subjected to axial forces is presented in this paper the system has multiple point discontinuities such as elastic supports rotor masses and cross sectional changes wave reflection and transmission matrices are employed to characterize the wave motions between the sub spans of the shaft system these matrices are combined with the field transfer matrices expressed in wave forms to obtain the characteristic equation in a straightforward manner the solutions are exact since effects of attenuating wave components are included in the formulation the wave propagation based matrix algebra leads to recursive algorithms which are suitable for computer coding three examples are presented to illustrate the numerical procedure

global asymptotic stability in a class of generalized putnam equations abstract it was conjectured that for every integer m 3 the unique equilibrium c 1 of the generalized putnam equation x n 1 i 0 m 2 x n i x n m 1 x n m x n x n 1 i 2 m x n i n 0 1 2 with positive initial conditions is globally asymptotically stable in this paper we prove this conjecture

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 behavior for a fourth order rational difference equation in this paper we use a method different from the known literature to investigate the global behavior of the following fourth order rational difference equation xn 1 xn 1xn 2xn 3 xn 1 xn 2 xn 3 a xn 1xn 2 xn 1xn 3 xn 2xn 3 1 a n 0 1 2

on two rational difference equations abstract in this paper we study the oscillating property of positive solutions and the global asymptotic stability of the unique equilibrium of the two rational difference equations x n 1 x n x n 1 x n 2 a x n x n 1 x n 2 a n 0 1 2 and x n 1 x n x n 1 x n 2 a x n 1 x n x n 2 a n 0 1 2 where a is a nonnegative constant

asymptotic weights of syzygies of toric varieties abstract the purpose of the paper is to give a sharp asymptotic description of the weights that appear in the syzygies of a smooth toric variety we prove that as the positivity of the embedding increases in any strand of syzygies torus weights after normalization stabilize to the same fixed shape that we explicitly specify

on a max type and a min type difference equation this note shows that every positive solution to the following third order non autonomous max type difference equationx n 1 maxa nx n x n 2 email protected n 0 when a n n n 0 is a three periodic sequence of positive numbers is periodic with period three the same result was proved for the following min type difference equationx n 1 mina nx n x n 2 email protected n 0

free vibrations of a stepped spinning timoshenko beam the objective of the present note is to employ polynomial based generalized force mode functions with the method of galerkin

two rational recursive sequences two sufficient conditions are obtained for the global asymptotic stability of the following two rational difference equations x n 1 x nx n 2 ax n x n 2 n 0 1 2 and x n 1 x n 1x n 2 ax n 1 x n 2 n 0 1 2 where a 0 and the initial values x 2 x 1 x 0 0

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

asymptotic syzygies of algebraic varieties we study the asymptotic behavior of the syzygies of a smooth projective variety as the positivity of the embedding line bundle grows the main result asserts that the syzygy modules are non zero in almost all degrees allowed by castelnuovo mumford regularity we also give an effective statement for veronese varieties that we conjecture to be optimal

the difference equation xn 1 α xn k i 0k 1cixn i has solutions converging to zero abstract the aim of this note is to show that the following difference equation x n 1 x n k i 0 k 1 c i x n i n 0 1 where k n c i 0 i 0 k 1 i 0 k 1 c i 1 and 1 has solutions which monotonically converge to zero this result shows the existence of such solutions which was not shown in the recently accepted paper a e hamza on the recursive sequence x n 1 x n 1 x n j math anal appl in press

free vibration analysis of planar curved beams by wave propagation in this paper a systematic approach for the free vibration analysis of a planar circular curved beam system is presented the system considered includes multiple point discontinuities such as elastic supports attached masses and curvature changes neglecting transverse shear and rotary inertia harmonic wave solutions are found for both extensional and inextensional curved beam models dispersion equations are obtained and cut off frequencies are determined wave reflection and transmission matrices are formulated accounting for general support conditions these matrices are combined with the aid of field transfer matrices to provide a concise and efficient method for the free vibration problem of multi span planar circular curved beams with general boundary conditions and supports the solutions are exact since the effects of attenuating wave components are included in the formulation several examples are presented and compared with other methods

global stability of some classes of higher order nonlinear difference equations we give elegant and short proofs of some recent results on global stability of some classes of higher order nonlinear difference equations

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

global behavior of a higher order nonlinear difference equation abstract in this paper we consider a higher order difference equation of the form x n 1 f x n x n k n 1 2 3 under some certain assumptions we obtain that the length of its finite semicycle is less than or equal to k moreover we concluded that the equation is permanent under some special conditions and a sufficient condition for its global asymptotical stability is given when these results are applied to the difference equation x n 1 x n k x n n 1 2 3 some sufficient and necessary conditions of global asymptotical stability of the equation are obtained we should state out that the main theorem in j math anal appl 233 1999 790 798 is included in one of our results

on a system of difference equations of odd order solvable in closed form it is shown that the next system of difference equationsx n c ny n 2 k 1 a n b ny n 2 k 1 i 1 k 1y n 2 i 1 x n 2 i y email protected nx n 2 k 1 a email protected nx n 2 k 1 i 1 k 1x n 2 i 1 y n 2 email protected n 0 where a n b n c n a n b n c n email protected n 0 and initial values x i y i email protected 1 2 2k 1 are real numbers is solved in closed form and the behavior of its well defined solutions when all the coefficients in the system are constant is described

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

dynamics of two classes of recursive sequences we investigate the dynamics of two classes of recursive sequences x n email protected email protected i 0 i 1 i 2 j a 2 jx n i 0x n i 1 x n i 2 email protected email protected i 0 i 1 i 2 j 1 a 2 j 1x n i 0x n i 1 x n i 2 j 1 1 b and x n email protected email protected i 0 i 1 i 2 j 1 a 2 j 1x n i 0x n i 1 x n i 2 j email protected email protected i 0 i 1 i 2 j a 2 jx n i 0x n i 1 x n i 2 j b we prove that their unique equilibrium email protected 1 is globally asymptotically stable in addition a new approach is presented to research the theory of recursive sequences

unbounded increasing solutions of a system of difference equations with delays we consider a homogeneous system of difference equations with deviating arguments in the form y n k 1 q k n y n p k y n r k where n n 0 n 0 z p k r k are integers r k p k 0 q is a positive integer y y 1 y s t y n 0 r n 0 r 1 r s is an unknown discrete vector function s 1 is an integer r max r 1 r q y n y n 1 y n and k n ij k n i j 1 s are real matrices such that ij k n 0 n 0 1 0 and k 1 q j 1 s ij k n 0 for each admissible i and all n n 0 the behavior of solutions of this system is discussed for n the existence of unbounded increasing solutions in an exponential form is proved and estimates of solutions are given the scalar case is discussed as well

on the asymptotic stability of xn 1 a xnxn k xn xn k we prove that the equilibrium solution of the rational difference equation x n 1 a x nx n kx n x n k n 0 1 2 where k is a nonnegative integer a 0 and x k x 0 0 is globally asymptotically stable

general form of some rational recursive sequences in this note we study the general form of some rational recursive sequences by some modification of the methods and ideas as well as the transformation from the paper k s berenhaut j d foley and s stevic the global attractivity of the rational difference equation y n y n k y n m1 y n ky n m appl math lett 20 2007 54 58 we give a new proof for the conjectures posed therein

global asymptotic stability in a class of putnam type equations abstract in this paper we initiate the study of a class of putnam type equation of the form x n 1 a 1 x n a 2 x n 1 a 3 x n 2 x n 3 a 4 b 1 x n x n 1 b 2 x n 2 b 3 x n 3 b 4 n 0 1 2 where a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4 are positive constants with a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4 x 3 x 2 x 1 x 0 are positive numbers a sufficient condition is given for the global asymptotic stability of the equilibrium point c 1 of such equations

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

global behavior of solutions of the generalized lyness difference equations under quadratic perturbations we study the global asymptotic stability of solutions of the following two difference equations x n 2 x n a bx n 1 1 c x n 1 2 cx n 2 n 0 1 2 and x n 2 x n a bx n 1 d 1 c x n 1 2 d x n 1 cx n 2 n 0 1 2 where a 0 d 0 c 0 1 and the initial values x 0 x 1 0 bastien and rogalski 2004 showed if c 0 then there exist all the possible periods for the solutions of the above equations using an extension of the quasi lyapunov method we prove that the sequences generated by the first difference equation are globally asymptotically stable where 0 c 1 0 b 2 a 1 c and the initial values x 0 x 1 0 the global convergence property of the second difference equation has also been obtained for b 0 and 0 c 1 conversely the sequences generated by the above equations are increasing to for c 1 and b 0

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

qualitative properties for a fourth order rational difference equation abstract in this paper we use a method different from the known literature to investigate the qualitative properties of the following fourth order rational difference equation x n 1 x n x n 1 x n 3 x n x n 1 x n 3 a x n x n 1 x n x n 3 x n 1 x n 3 1 a n 0 1 2 where a 0 and the initial values x 3 x 2 x 1 x 0 0 the successive lengths of positive and negative semicycles of nontrivial solutions of the above equation is found to periodically occur that is 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 or 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 3 2 1 1 3 by using the rule the positive equilibrium point of the equation is verified to be globally asymptotically stable

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

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

asymptotic behavior of the positive solutions of an exponential type system of difference equations in this paper we study the existence of a unique positive equilibrium the boundedness persistence and global attractivity of the positive solutions of a system of the following two difference equations x n 1 ax n by n 1 e x n y n 1 cy n dx n 1 e y n n 0 1 where a b c d are positive constants and the initial values x 1 x 0 y 1 y 0 are positive numbers

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

zero entropy for some birational maps of c2 abstract in this study we consider a special case of the family of birational maps f c 2 c 2 which were dynamically classified by 13 we identify the zero entropy subfamilies of f and explicitly give the associated invariant fibrations in particular we highlight all of the integrable and periodic mappings

the periodic character of positive solutions of the difference equation x n 1 x n x n k in this note we consider the nonlinear difference equation x n 1 f x n x n k n 0 1 where k 1 2 and the initial values gk g k 1 g 0 0 we give sufficient conditions under which every positive solution of this equation converges to a not necessarily prime period two solution if k is odd and the unique positive equilibrium if k is even

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

modal analysis of rotating shafts a body fixed axis formulation approach abstract a spinning timoshenko beam subjected to a constant moving load is analyzed using a modal expansion technique the formulation is based on a body fixed axis reference system and thus should be applicable to handle any general cross sections using this alternative approach dynamic quantities such as natural frequencies mode shapes and system responses are easily computable it is shown that simply supported spinning timoshenko beams posses two pairs of natural frequencies corresponding to each mode shape through appropriate simplifications the coupled eighth order differential equations governing the problem are degenerated into a set of uncoupled fourth order equations with this simplified theory closed form expressions for natural frequencies and system transient response are derived however one pair of the natural frequencies is lost in the process but this is acceptable as the frequencies of this pair are much higher than those of the retained pair in addition a linearized approximation of the natural frequency similar to the one given for euler bernoulli beams is also proposed for a timoshenko shaft numerical simulations are performed to demonstrate the characteristics of the response

on the global asymptotic stability of the difference equation xn xn 1xn 2 xn 3 a xn 1 xn 2xn 3 a li and zhu applied mathematics and computation 150 2004 481 studied the difference equationx n x n 1x n 2 x n 3 ax n 1 x n 2x n 3 a n 1 2 where a 0 they claimed that the unique positive equilibrium of the equation is globally asymptotically stable unfortunately there is a subtle mistake in their proof in this paper we present a correct proof for this claim

on the system of difference equations x n x n 1 y n 2 a y n 2 b y n 1 y n y n 1 x n 2 c x n 2 d x n 1 we show that the following system of difference equations x n x n 1 y n 2 a y n 2 b y n 1 y n y n 1 x n 2 c x n 2 d x n 1 n n 0 where parameters a b c d as well as initial values x 2 x 1 y 2 y 1 are real numbers is solvable in closed form and by using obtained formulas we give some results on the long term behavior of their solutions we also give natural explanations and considerably extend some recent results in the literature

a presic type contractive condition and its applications abstract we study discrete dynamic systems in a complete metric space m d defined by mappings which satisfy presic type contractive conditions their counterparts in an ordered banach space are investigated and applied to solve the global asymptotic stability of the equilibriums of some nonlinear difference equations

global stability of a rational symmetric difference equation motivated by ideas from papers c cinar i yalcinkaya s stevic a note on global asymptotic stability of a family of rational equations rostock math kolloq 59 2004 41 49 and s stevic global stability and asymptotics of some classes of rational difference equations j math anal appl 316 2006 60 68 here we confirm a conjecture on a rational symmetric difference equation from the paper k berenhaut j foley s stevic the global attractivity of the rational difference equation y n y n k y n m1 y n ky n m appl math lett 20 2007 54 58

global asymptotic stability of a second order nonlinear difference equation in this paper the global asymptotic stability of the nonlinear difference equationx n 1 a bx na bx n cx n 1 n 0 1 is investigated where a b a b c 0 are real numbers and the initial conditions x 1 is nonnegative real numbers and x 0 is a positive real number we show that the unique positive equilibrium of the equation is globally asymptotically stable in particular our results solve two conjectures proposed 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 hall crc boca raton 2002 and by kulenovic et al in their paper m r s kulenovic g ladas l f martins i w rodrigues the dynamics of x n 1 a bx na bx n cx n 1 facts and conjectures comput math appl 45 2003 1087 1099 099

on the recursive sequence xn 1 a i lkxn 2i 1b c i lk 1xn 2i abstract the aim of this work is to investigate the global stability periodic nature oscillation and the boundedness of solutions of the difference equation x n 1 a i l k x n 2 i 1 b c i l k 1 x n 2 i n 0 1 where a b c are nonnegative real numbers and l k are nonnegative integers l k we discuss the existence of unbounded solutions under certain conditions when l 0

the global attractivity of a higher order rational difference equation abstract this paper studies global asymptotic stability for positive solutions to the equation y n y n k y n l y n m y n k y n l y n m 1 y n k y n l y n k y n m y n l y n m n 0 1 with y m y m 1 y 1 0 and 1 k l m the paper also includes a listing of possible semi cycle structures for various k l m the results generalize several others in the recent literature

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

free vibration of elastically coupled dual span curved beams the free vibration response of dual span extensional and inextensional circular curved beams constrained by an intermediate elastic support is studied the elastic constraint consists of a transverse and rotational spring as well as a tangential spring which couples the extensional motion of one span with the other wave propagation techniques are used to formulate the eigenvalue problem which result in exact eigensolutions the effects of the support type and stiffness on the transverse and tangential modes are examined in addition the influence of the curvature on the free response of the extensional curved beam is also addressed for certain support conditions the extensional and inextensional curved beams have similar behavior in general however the behavior of the system is shown to depend strongly on the type of support the modeling of the beam as extensional or inextensional and whether the natural frequencies are above or below the cut off frequency

part metric and its applications in discrete systems this paper applies the part metric method to study some types of higher order symmetric difference equations with several different exponential parameters these difference equations are proved to have unique equilibria and some useful inequalities regarding the difference equation functions are formulated by use of the part metric and a result given by kruse and nesemann 1999 8 some sufficient conditions on the parameters are given to guarantee the global asymptotic stability of the equilibria furthermore by the part metric defined on matrices this approach is also applicable to show the global asymptotic stability of some cyclic discrete dynamic systems the results of this paper are considered a big improvement over many existing results found in the literature

existence of solutions with a single semicycle for a general second order rational difference equation by making use of inclusion theorem we show in this paper the existence of solutions with a single semicycle for a general second order rational difference equation as a corollary our results positively confirm conjectures 4 8 3 and 5 4 6 in m r s kulenovic g ladas dynamics of second order rational difference equations with open problems and conjectures chapman and hall crc 2002

global asymptotic stability for two recursive difference equations two sufficient conditions are obtained for the global asymptotic stability of the following two recursive difference equationsx n 1 x nx n 1 x n 2 ax n x n 1x n 2 a n 0 1 2 andx n 1 x n 1 x nx n 2 ax nx n 1 x n 2 a n 0 1 2 where a 0 and the initial values x 2 x 1 x 0 0 some known results are included

global asymptotic stability of a family of difference equations in this paper we study the difference equation x n f 1f 2f 3 f 1 f 2 f 3 hf 1f 2 f 1f 3 f 2f 3 g h x n 1 x n r n 1 2 x 1 r x 2 r x 0 0 where f 1 f 2 f 3 g r r r and h r r 0 are all continuous functions we present a sufficient condition for this difference equation to have a globally asymptotically stable equilibrium c 1 this condition generalizes some previous results

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

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

on the recursive sequence x n 1 aπ k i l x n 2 j 1 b c π k l i l x n 2 i the aim of this work is to investigate the global stability periodic nature oscillation and the boundedness of solutions of the difference equation x n email protected i lkx n 2 i email protected i lk 1x n 2 i n 0 1 where a b c are nonnegative real numbers and l k are nonnegative integers l

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

dichotomy of a perturbed lyness difference equation abstract we investigate in this paper the perturbed lyness difference equation bx n 2 x n x n 1 x n 2 n 0 1 2 where b are arbitrary positive real numbers and 0 and the initial values x 1 x 0 0 which is a generalization of the lyness difference equation x n 2 x n a x n 1 extensively studied it is known that for the lyness difference equation i e the perturbed lyness difference equation with 0 all its solutions are periodic or strictly oscillatory however one here finds that this perturbed lyness difference equation possesses the following dichotomy for 0 b all of its solutions are globally asymptotically stable for b all the sequences generated by it converge to we hence find that there exists the essential difference for the properties of solutions between the unperturbed lyness difference equation and the perturbed lyness difference equation

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

global asymptotic behavior and boundedness of positive solutions to an odd order rational difference equation in this note we consider the following high order rational difference equation x n 1 i 1k 1 x n i i 1kx n i n 0 1 where k 3 is odd number x k x k 1 x k 2 x 1 is positive numbers we obtain the boundedness of positive solutions for the above equation and with the perturbation of initial values we mainly use the transformation method to prove that the positive equilibrium point of this equation is globally asymptotically stable

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

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

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 recursive sequence xn 1 α βxn k 1a bxn k 1 cxn 2k 1 abstract in this paper we study the global asymptotic stability invariant intervals the character of semicycles and the boundedness of all positive solutions of the higher order nonlinear difference equation x n 1 x n k 1 a bx n k 1 cx n 2 k 1 n 0 1 where a b c and are positive k 1 2 3 and the initial conditions x 2k 1 x 1 x0 are positive real numbers we show that the unique positive equilibrium of the equation is globally asymptotically stable finally we present some informative examples

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