on a solvable system of difference equations of kth order it is shown that the next system of k difference equationsx n j a n j x n k j b n j i 1 kx n i s j i 1 c n j n n 0 j 1 k where a n j b n j c n j

coherence of certain overconvergent isocrystals without frobenius structures on curves in this paper we will prove the coherence as fancyscript d dagger modules of overconvergent isocrystals on curves which do not contain tame parts at each singular points of the given isocrystal even without frobenius structures in the proof we will construct finite presentations concretely using special differential operators defined by garnier and the decomposition theorem of christol mebkhout

cut by curves criterion for the log extendability of overconvergent isocrystals in this paper we prove a cut by curves criterion for an overconvergent isocrystal on a smooth variety over a field of characteristic p 0 to extend logarithmically to its smooth compactification whose complement is a simple normal crossing divisor under certain assumption this is a p adic analogue of a version of cut by curves criterion for regular singularity of an integrable connection on a smooth variety over a field of characteristic 0 in the course of the proof we also prove a kind of cut by curves criteria on solvability highest ramification break and exponent of modules

on a generalized max type difference equation from automatic control theory abstract this paper studies the boundedness character of positive solutions of the following generalization of a max type difference equation from automatic control theory see mishkis 1977 18 and popov 1966 21 x n 1 max a x n p x n 1 q x n 2 r n n 0 where the parameters a p q and r are positive numbers in the study of the equation we also introduce a new method called oachkatzlschwoif

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

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

periodic solutions of large norm of hamiltonian systems abstract hamiltonian systems of ordinary differential equations are studied the only assumption made on the hamiltonian is appropriately rapid growth at infinity it is proved that for any given period there is an unbounded sequence of periodic solutions of the system having the given period

global attractivity of a family of nonautonomous max type difference equations abstract this paper studies a family of nonautonomous max type difference equations with several delays using some transformations we prove global attractivity of positive solutions to these equations under some conditions some of our results considerably extend related ones in the literature two examples are given to illustrate the main results

homotopy invariants of time reversible periodic orbits ii towards applications abstract for a reversible periodic orbit we apply the sequence of homotopy invariants deg n n 1 2 3 defined in fiedler heinze 1996 we use this sequence to prove a global bifurcation result for reversible periodic orbits with prescribed minimal period this result will be applied to second order systems with neumann boundary conditions a discussion and remarks on the sequence of degrees concludes the paper

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

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

domains of undefinable solutions of some equations and systems of difference equations for some classes of difference equations and systems of difference equations we find the sets of initial values for which solutions of these equations and systems are not well defined so called domains of undefinable solutions

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

asymptotically periodic solutions of second order difference equations abstract in this paper we obtain sufficient conditions for the existence of asymptotically periodic solutions of the second order difference equations of the form r n x n a n f x n b n moreover we give sufficient conditions under which every bounded solution of the above equation is asymptotically periodic our approach allows us to control the degree of approximation

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

global behavior of the max type difference equation x n max 1 x n m a n x n r in this paper we study the global behavior of the following max type difference equation x n max 1 x n m a n x n r n 0 1 2 where a n n 0 is a sequence of positive numbers with a n 0 1 for every n 0 and sup a n 1 and m r 1 2 3 and the initial values x d x d 1 x 1 0 with d max m r we show that 1 if x n n d is a positive solution of this equation then for every 0 k 2 m 1 x 2 mn k n 0 is eventually monotone 2 if a n n 0 is a periodic sequence then every positive solution of this equation is eventually periodic with period 2 m

more on a rational second order difference equation abstract here we give natural explanations for some recent relations for the solutions of the following difference equation z n 2 z n 1 3 z n 2 2 z n 1 2 n n 0 and extend them for the case of complex initial values we also present some other properties related to solutions of the equation as well as some properties of the solutions of an associate difference equation of first order

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 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

on the eventual periodicity of xn 1 max 1xn anxn 1 with a period five parameter in this paper we will study the max difference equation x n 1 max 1x n a nx n 1 when a n is a period five sequence of real numbers greater than one we will prove that every solution is eventually periodic with period twenty

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

on 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

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 stability of a difference equation with maximum we prove that every positive solution to the difference equationx n maxa 1x n 1 a 1 a 2x n 2 a 2 a kx n k a k email protected n 0 where email protected n a i 0 a email protected 0 1 i 1 k converges to the following quantity maxa 1 1 a 1 1 a k 1 a k 1 confirming a quite recent conjecture of interest we also prove another result on global convergence which concerns some cases when not all a i i 1 k belong to the interval 0 1

eventual periodicity of some systems of max type difference equations the periodicity of the next system of max type difference equations xn 1 max1 i1 m1f1i1xn ki1 1 1 1 xn ki1 2 1 2 xn ki1 l 1 l n xn t1s 1 xn 2 max1 i2 m2f2i2xn ki2 1 2 1 xn ki2 2 2 2 xn ki2 l 2 l n xn t2s 2 xn l max1 il mlflilxn kil 1 l 1 xn kil 2 l 2 xn kil l l l n xn tls l n n0n n0 where s l mj tj kij h j n j h 1 l j h 1 l 1 2 l 1 2 l is a permutation of 1 2 l 1 2 l and fjij 0 l n0 0 fjij 0 l n0 0 j 1 l j 1 l ij 1 mj ij 1 mj is studied it is shown that under some conditions posed on functions fjijfjij all positive solutions of the system are eventually periodic with period ts for some t nt n

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

on the periodic nature of the solutions of the reciprocal difference equation with maximum abstract we prove that every positive solution of the third order difference equation x n max a x n 1 b x n 2 c x n 3 n 0 1 is eventually periodic of not necessarily prime period t which is explicitly determined by the coefficients a b and c

periodic solutions with prescribed minimal period for convex autonomous hamiltonian systems clarke has shown that the problem of findingt periodic solutions for a convex hamiltonian system is equivalent to the problem of finding critical points to a certain functional dual to the classical action functional in this paper we relate the morse index of the critical point to the minimal period of the correspondingt periodic solution in particular we show that if the critical point is obtained by the ambrosetti rabinowitz mountain pass theorem the corresponding solution has minimal periodt that is it cannot bet k periodic withk integer k 2 as a consequence we prove that if the hamiltonian is flat near an equilibrium and superquadratic near infinity then for anyt 0 the corresponding hamiltonian system has a periodic solution with minimal periodt

on a solvable system of difference equations of fourth order it is shown that the next system of k difference equationsx n j a n j x n k j b n j i 1 kx n i s j i 1 c n j n n 0 j 1 k where a n j b n j c n j n n 0 j 1 k and initial values x i j i j 1 k are real numbers and where s n 1 k is defined by s km j j 1 j 1 k 1 s km k 1 m n 0 can be solved in closed form in an elegant way some applications of obtained formulas for the case when the sequences a n j b n j and c n j are k periodic are also given

dynamics of a rational difference equation abstract in this note we investigate the solution of the difference equation x n 1 x n 1 a x n 1 x n n 0 1 2 where x 1 x 0 r and a 0 moreover we discuss the stability properties and semi cycle behavior of this solution

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

an index for global hopf bifurcation in parabolic systems on considere le systeme non lineaire d t u a u f u 0 et on cherche des solutions periodiques en temps u t qui bifurquent a partir d une solution stationnaire

on a system of difference equations with maximum in this paper we study the periodic character of the solutions of the system of difference equations with maximum x n 1 max a y n x n 1 y n 1 max b x n y n 1 where a b are positive constants and the initial values x 1 x 0 y 1 y 0 are positive numbers we prove that every solution is eventually periodic

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

on a difference equation with maximum abstract this paper studies the asymptotic behavior of positive solutions to the difference equation x n max 1 x n 1 a x n 2 n 0 1 where 0 0 we prove that every positive solution of this equation approaches x 1 or is eventually periodic with period 4 according as a 1 or a 1

on a symmetric system of max type difference equations we study behavior of positive solutions of the following max type system of difference equationsx n 1 maxc y n px n 1 p y n 1 maxc x n py n 1 p email protected n 0 where p email protected 0 extending some results in the literature among other results we prove that if p email protected 0 1 then every positive solution of the system converges to 1 1

on some periodic systems of max type difference equations we show that all positive solutions to the system of max type difference equations xn 1 max1 i m1f1ixn ki 1 1 1 xn ki 2 1 2 xn ki l 1 l n xn s 1 xn 2 max1 i m2f2ixn ki 1 2 1 xn ki 2 2 2 xn ki l 2 l n xn s 2 xn l max1 i mlflixn ki 1 l 1 xn ki 2 l 2 xn ki l l l n xn s l n n0 where s l mj ki t j n j t 1 l and for a fixed j i 1 mj and where the functions fji 0 l n0 0 j 1 l i 1 mj satisfy some conditions are eventually periodic with not necessarily prime period s a related result for the corresponding system of min type difference equations is also proved

snakes oriented families of periodic orbits their sources sinks and continuation abstract poincare observed that for a differential equation x x depending on a parameter each periodic orbit generally lies in a connected family of orbits in x space in order to investigate certain large connected sets denoted q of orbits containing a given orbit we introduce two indices an orbit index and a center index defined at certain stationary points we show that genetically there are two types of hopf bifurcation those we call sources 1 and sinks 1 generically if the set q is bounded in x space and if there is an upper bound for periods of the orbits in q then q must have as many source hopf bifurcations as sink hopf bifurcations and each source is connected to a sink by an oriented one parameter snake of orbits a snake is a maximal path of orbits that contains no orbits whose orbit index is 0 see fig 1 1

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

on a system of difference equations which can be solved in closed form we present an elegant method for solving the system of difference equationsx n 1 1x email protected n y n email protected n 1y n email protected n 0 where parameters a and b and the initial values x 0 and y 0 are positive numbers and by using obtained formulae we show that every solution of the system converges to a two periodic solution

on the asymptotic behavior and periodic nature of a difference equation with maximum in this paper we investigate the asymptotic behavior and periodic nature of positive solutions of the difference equation x n max ax n 1 1xn 2 a n 0 where a 0 and 0 a 1 we prove that every positive solution of this difference equation approaches x 1 or is eventually periodic with a period 2 3 or 4

dynamics of a kth order rational difference equation in this article we investigate the dynamics of the difference equation x n 1 x n ka x n kx n where x k x 0 0 and a 0 k is any positive integer in some particular cases we allow for zero initial conditions

eventually constant solutions of a rational difference equation we describe all the solutions of a rational difference equation from putnam s mathematical competition which are eventually equal to its positive equilibrium email protected 1 as a consequence we give a new elegant and short proof of the fact that the equation has a positive solution which is not eventually equal to one moreover we show that almost all solutions of the equation are not eventually equal to one

periodicity of a second order switched difference system over integers abstract in this paper a second order switched difference system which consists of two linear difference equations with a switching rule is proposed to study specifically the periodicity of a particular case is addressed deriving the appropriate rational values for parameter r which possess periodic integer solutions by the transformation method the particular second order difference system is transformed into a first order switched system and we prove that 1 this system possesses periodic integer solutions of prime period two if and only if r 1 2 2 any rational r except for the integers arises periodic integer solutions of prime period three 3 periodic integer solutions of prime period four exist if and only if r 1 2 4 this system possesses no periodic solutions of prime period five we also prove that if r 0 and the system has periodic integer solutions of prime period k 6 then the only possible values of r are reciprocals of integers

on positive solutions of some classes of max type systems of difference equations global attractivity results on positive solutions of some special cases of the next max type system of difference equations xn i max1 xn l1 i i p1 i j1 1m1 i xn j1 i1 sj1 i1 qj1 i1 xn lri i i pri i jri 1mri i xn jri iri sjri iri qjri iri n n0 where ri n i 1 k k n sjt it 1 k qjt it 0 i 1 k t 1 ri and pj i 0 1 mj i n i 1 k j 1 ri are presented we also present some results on the behavior of positive solutions of the difference equation xn max p1xn 1 pkxn k xn max p1xn 1 pkxn k where pj j 1 k are nonnegative numbers

on the system of difference equations xn cnyn 3 an bnyn 1xn 2y n 3 yn γnxn 3 αn βnxn 1yn 2 xn 3 we show that the system of difference equationsx n c ny n 3a n b ny n 1x n 2y n 3 y email protected nx n email protected email protected nx n 1y n 2x n 3 email protected n 0 where the sequences 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 3 are real numbers such that c n 0 c n 0 email protected n 0 can be solved in closed form and for the case when all the sequences a n b n c n a n b n and c n are constant we describe the asymptotic behavior of well defined solutions of the system

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

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

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

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

on the max type equation xn 1 max axn xn 2 abstract we show that every well defined solution of the difference equation x n 1 max a x n x n 2 n n 0 where a r is eventually periodic with period three

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

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

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

on the solutions of a system of difference equations with maximum in this paper we study the following max type system of difference equations x n max 1 x n m min 1 a y n r y n max 1 y n m min 1 b x n t n n 0 where a b 0 m r t 1 2 with r m and t m we show that every solution of this system with the initial values x d y d x d 1 y d 1 x 1 y 1 0 is eventually periodic with period 2m where d max m r t

long term behavior of positive solutions of a system of max type difference equations abstract some sufficient conditions which guaranty the attractivity of all positive solutions of the next system of difference equations x n max a y n 1 p j 2 k x n j q j y n max a x n 1 p j 2 k y n j q j n n 0 where a p 0 and q j j 2 k are nonnegative numbers are given the boundedness character of positive solutions of the system is also studied