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

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

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

laguerre expansions of tempered distributions and generalized functions abstract let ln t n be for 1 the sequence of generalized laguerre polynomials then we consider the orthonormal system n n 1 1 2 l n t t 2 e t 2 n in l2 0 we study the expansions of certain spaces of generalized functions with respect to this orthonormal system for 0 this space of generalized functions is the space of tempered distributions with positive support we characterize the sequences of fourier laguerre coefficients which appear in these expansions finally we give some applications

on the difference equation xn 1 α xn 1xn in this paper we study the behavior of the solutions of the difference equation x n 1 a x n 1x n n 0 1 where a is a negative number included are results which considerably improve and correct those in the recently published paper a e hamza on the recursive sequence x n 1 a x n 1x n j math anal appl 322 2006 668 674 we also refute conjecture 2 in g ladas open problems and conjectures j difference equ appl 7 2 2001 477 482

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

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

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

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

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

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

application of difference equations in divisibility abstract we present an easy applicable method which for a large class of integer sequences a n shows when all their terms are divisible by a natural number m 1

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

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

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

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 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

length two representations of quantum affine superalgebras and baxter operators associated to quantum affine general linear lie superalgebras are two families of short exact sequences of representations whose first and third terms are irreducible the baxter tq relations involving infinite dimensional representations the extended t systems of kirillov reshetikhin modules we make use of these representations over the full quantum affine superalgebra to define baxter operators as transfer matrices for the quantum integrable model and to deduce bethe ansatz equations under genericity conditions

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

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

on 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

oscillator versus prefundamental representations we find the l weights and the corresponding l weight vectors for the finite and infinite dimensional representations of the quantum loop algebras uq x1d530 x1d529 2 and uq x1d530 x1d529 3 obtained from the verma representations of the quantum groups uq x1d524 x1d529 2 and uq x1d524 x1d529 3 via the jimbo s homomorphism then we find the l weights and the l weight vectors for the q oscillator representations of the positive borel subalgebras of the same quantum loop algebras this allows in particular to relate the q oscillator and prefundamental representations

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

solutions of a max type system of difference equations we show in an elegant way that the general solution to the following max type system of difference equations xn 1 maxaxn ynxn yn 1 maxayn xnyn n n0 for the case y0 x0 a 0 y0 x0 max a 1 a is given by xn afk n 1 n x0fk n y0fk n 1 n n n y1 x0 y0 and yn y0fk n 1 1afk n 1 n 1x0fk n 1 1 1 n n 2 where k n n for n even k n n 1 for n odd n n0 n n mod2 for n n0 and where the sequence fn n n is the fibonacci sequence correcting and giving a proof of a wrong result in the paper d simsek b demir c cinar on the solutions of the system of difference equations xn 1 max a xn yn xn yn 1 max a yn xn yn discrete dyn nat soc 2009 2009 11 pp article id 325296

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

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 the system xn 1 ynx n k yn k 1 an bnynxn k yn 1 xnyn k xn k 1 cn dnxnyn k it is shown that the system of difference equationsx n 1 y nx n ky n k 1 a n b ny nx n k y n 1 x ny n kx n k 1 c n d nx ny n k email protected n 0 where email protected n the sequences a n b n c n d n email protected n 0 and the initial values x i y i i 0 email protected are real numbers can be solved in closed form asymptotic behavior of well defined solutions of the system for the case when all the sequences a n b n c n and d n are constant is studied in detail

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

oscillator versus prefundamental representations ii arbitrary higher ranks we find the l weights and the l weight vectors for the highest l weight q oscillator representations of the positive borel subalgebra of the quantum loop algebra uq l sll 1 for arbitrary values of l having this we establish the explicit relationship between the q oscillator and prefundamental representations our consideration allows us to conclude that the prefundamental representations can be obtained by tensoring q oscillator representations

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

difference equations and nonexistence of solutions of an integral equation abstract by using difference equations we give some easily applicable sufficient conditions for the non existence of nonnegative solutions satisfying some additional conditions of the integral equation f x x a x t b x t f t p d t where p 1 and x is a measurable space with probability measure

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

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

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