asymptotic behavior of nonlinear difference systems in this paper we consider two dimensional nonlinear difference systems of the email protected n a nf y n dy n 1 b ng x n we classify their solutions according to asymptotic behavior and give some necessary and sufficient conditions for the existence of solutions of such classes

oscillation and nonoscillation theorems for second order nonlinear difference equations abstract some new oscillation and nonoscillation theorems are obtained for second order nonlinear difference equations a p n a x n q n a x n f n x n b x n which generalize and improve all results proved by popenda in j math anal appl 123 1987 34 48

oscillation and nonoscillation of two dimensional difference systems this paper is concerned with the oscillation and nonoscillation behavior of solution of the nonlinear two dimensional difference system xn ang yn yn 1 f n xn n n n0 n0 n0 1 some necessary and sufficient conditions are given for the system to admit the existence of oscillatory and nonoscillatory solutions with special asymptotic properties some important results in the literatures are generalized examples to illustrate the result are included

fixed sign solutions of second order difference equations with neumann boundary conditions in this paper we study the existence of solution of the nonlinear second order difference problem with neumann boundary conditions assuming the existence of a pair of ordered lower and upper solutions and we obtain optimal existence results for the case and even for to this end we do an exhaustive study of the values of the real parameters and for which the associated green s function to the linear operator l u u 2 2 u 1 u with neumann boundary conditions has fixed sign

higher order pathwise numerical approximations of spdes with additive noise in this article the pathwise numerical approximation of semilinear parabolic stochastic partial differential equations spdes driven by additive noise is considered a new numerical scheme for the time and space discretization of such spdes is proposed and this scheme is shown to converge for such spdes faster than standard numerical schemes such as the linear implicit euler scheme or the linear implicit crank nicholson scheme the suggested scheme takes advantage of the smoothing effect of the dominant linear operator and of two linear functionals of the noise process of the spde the abstract setting under which the scheme is analyzed includes spdes driven by fractional brownian motions too

invariant and attracting sets of impulsive delay difference equations with continuous variables the aim of this paper is to study the invariant and attracting sets of impulsive delay difference equations with continuous variables some criteria for the invariant and attracting sets are obtained by using the decomposition approach and delay difference inequalities with impulsive initial conditions

an existence theorem for a nonlinear difference equation where p 1 and s p is a real sequence we establish conditions under which 1 1 has a positive nondecreasing solution here a solution of 1 1 is a real sequence y y which satisfies 1 1 since 1 1 is a recurrence relation given real initial values y and y it is clear that we can inductively obtain y y since p is a real number greater than 1 the sequence yk may not be real that is not a solution note however that any constant multiple of a solution is again a solution the existence question originated in l where it was shown that hardy s inequality for a series 2 can be viewed as a necessary condition for the existence of a positive nondecreasing solution of 1 1 our existence theorem reduces to a nonoscillation criterion in the case p 2 since then 1 1 changes to a second order linear difference equation see e g 3 a brief outline of the paper is as follows section 2 introduces a riccati type transformation and then develops various necessary conditions for the existence of a positive nondecreasing solution section 3 discusses some of the functional analytic background which is needed in order to apply schauder s theorem section 4 contains the proof of our existence theorem and section 5 describes a comparison theorem which allows us to drop the assumption that the coefficients sk in 1 1 are nonnegative

oscillation of second order nonlinear difference equation with continuous variable in this paper we are mainly concerned with the second order nonlinear difference equation with continuous variable here by using the iterated integral transformations generalized riccati transformations and integrating factors we give some oscillatory criteria for this equation

oscillation criteria for two dimensional dynamic systems on time scales we establish some oscillation criteria for the two dimensional dynamic system x d t b t g y s t y d t a t f x t on a time scale t our results not only unify the oscillation of two dimensional differential systems and difference systems but include the oscillation results for differential systems and provide new oscillation criteria for difference systems several examples are considered to illustrate the main results

polynomial time quantum algorithms for pell s equation and the principal ideal problem we give polynomial time quantum algorithms for three problems from computational algebraic number theory the first is pell s equation given a positive nonsquare integer d pell s equation is x2 dy2 e 1 and the goal is to find its integer solutions factoring integers reduces to finding integer solutions of pell s equation but a reduction in the other direction is not known and appears more difficult the second problem we solve is the principal ideal problem in real quadratic number fields this problem which is at least as hard as solving pell s equation is the one way function underlying the buchmann williams key exchange system which is therefore broken by our quantum algorithm finally assuming the generalized riemann hypothesis this algorithm can be used to compute the class group of a real quadratic number field

oscillation theorems for certain second order nonlinear difference equations in this paper discrete inequalities are used to offer sufficient conditions for the oscillation of all solutions of the difference equation delta bigl a n delta y n sigma bigr q n 1 f y n 1 r n qquad n ge 0 where 0 p qwithp qodd integers orpeven andqodd integers several examples which dwell upon the importance of our results are also included

positive solutions of higher order nonlinear delay difference equations abstract classification schemes for positive solutions of a class of higher order nonlinear delay difference equations are given in terms of their asymptotic behavior and necessary as well as sufficient conditions for the existence of these solutions are also provided

oscillation of certain second order nonlinear difference equations abstract in this paper we consider the second order nonlinear difference equation formula where y n y n 1 y n is the forward difference operator f c r r f is nondecreasing and uf u 0 as u 0 q n is a real sequence some new sufficient conditions for the oscillation of all solutions of 1 are obtained

oscillation for linear non autonomous systems of difference equations with continuous arguments abstract in this paper we establish sufficient conditions for the oscillation of the linear non autonomous systems of difference equations with continuous arguments n x t p x t j 1 n q j t x t j 0 where x t x t x t n 1 p r m m q j c r r m m j 1 2 n and j j 1 2 n are non negative constants furthermore assume q j t q j l k t where q j l k t l k 1 2 m j 1 2 n are continuous bounded and do not change signs on r

quantum computation and the evaluation of tensor networks we present a quantum algorithm that additively approximates the value of a tensor network to a certain scale when combined with existing results this provides a complete problem for quantum computation the result is a simple new way of looking at quantum computation in which unitary gates are replaced by tensors and time is replaced by the order in which the tensor network is swallowed we use this result to derive new quantum algorithms that approximate the partition function of a variety of classical statistical mechanical models including the potts model

oscillatory and asymptotic behavior of second order nonlinear difference equations in this paper we are dealing with the oscillatory and asymptotic behaviour of solutions of second order nonlinear difference equations of the form rn xn f n xn 0 n n n0 1 we obtain the following results a if k n0 l rk any nonoscillatory solution of 1 must belong to one of the following four types k k k 0 k 0 b if k n0 l rk any nonoscillatory solution of 1 must belong to one of the following three types k0 k k0 c necessary and sufficient conditions for 1 to have a nonoscillatory solution which belongs to k k k 0 k0 or k are given depending on whether f is a superlinear or sublinear function all these results include and improve b szmanda s results in bull polish acad sci math 34 nos 3 4 1986 133 141

on the stability in periodic and almost periodic difference systems abstract in this paper we consider the system of difference equations x n 1 f n x n f n 0 0 n 0 1 2 this system has the trivial zero solution x n 0 sufficient conditions of its asymptotic stability are obtained in the cases when functions f n x are periodic and almost periodic in n

on unbounded positive solutions of second order difference equations with a singular nonlinear term abstract in this paper several necessary and or sufficient conditions are given for the existence of unbounded positive solutions of nonlinear difference equations of the form r n x n f n x n 0 n n n 0 where n n0 n0 n0 1 n0 is a fixed nonnegative integer rn 0 for n n0 such that n n0 1 rn and f n n0 r r is a real valued continuous function such that f n x 0 for all n x n n0 0

oscillation of the emden fowler difference systems abstract in this paper we are concerned with some new criteria for the oscillation of the discrete emden fowler system

oscillation and nonoscillation criteria for linear dynamic systems on time scales in this paper we establish oscillation and nonoscillation criteria for the linear dynamic system u d pv v d qu s here we assume that p and q are nonnegative rd continuous functions on a time scale t such that supt indeed we extend some known oscillation results for differential systems and difference systems to the so called dynamic systems

a second order nonlinear difference equation oscillation and asymptotic behavior discrete analogues are investigated for well known results on oscillation growth and asymptotic behavior of solutions of y q t y 0 for q t 0 and for q t 0 the analogue of atkinson s oscillation criterion is shown to be true for 2yn 1 qn yn 0 but the analogue for atkinson s nonoscillation criterion is shown to be false

oscillation criteria for two dimensional difference systems of first order linear difference equations sufficient conditions are established for the oscillation of the linear two dimensional difference system dx n p ny n dy n 1 q nx n n n n 0 n 0 n 0 1 where p n q n are nonnegative real sequences our results extend the results in the literature

oscillation of certain two dimensional nonlinear difference systems abstract several new oscillation criteria for two dimensional nonlinear difference systems are established examples which dwell upon the importance of our results are also included

stability in mean of partial variables for stochastic reaction diffusion systems abstract it is with the aid of the lyapunov method that stochastic ordinary differential equations and stochastic functional differential equations have been investigated intensively in the past decades however for stochastic reaction diffusion equations this useful technique seems to feel helpless on account of the emptiness of its own ito s formula to get over this difficulty we will manage to regard the integral of the considered trajectory with respect to spatial variables as the solution of the corresponding stochastic ordinary differential equations and then employ the ito formula to study stability in mean of partial variables for ito stochastic reaction diffusion equations some sufficient conditions for uniform stability in mean asymptotic stability in mean uniformly asymptotic stability in mean exponential stability in mean of partial variables are presented

hidden translation and translating coset in quantum computing we give efficient quantum algorithms for the problems of hidden translation and hidden subgroup in a large class of nonabelian solvable groups including solvable groups of constant exponent and of constant length derived series our algorithms are recursive for the base case we solve efficiently hidden translation in mathbb z p n whenever p is a fixed prime for the induction step we introduce the problem translating coset generalizing both hidden translation and hidden subgroup and prove a powerful self reducibility result translating coset in a finite solvable group g is reducible to instances of translating coset in g n and n for appropriate normal subgroups n of g our self reducibility framework combined with kuperberg s subexponential quantum algorithm for solving hidden translation in any abelian group leads to subexponential quantum algorithms for hidden translation and hidden subgroup in any solvable group

classification schemes for positive solutions of a second order nonlinear difference equation classification schemes for eventually positive solutions of a class of second order nonlinear difference equations are given in terms of their asymptotic magnitudes and necessary as well as sufficient conditions for the existence of these solutions are also provided

existence of bounded positive solutions for a system of difference equations abstract this paper deals with the second order nonlinear neutral delay system of difference equations a n x n b n x n f n x h 1 n x h kn y w 1 n y w kn c n n n 0 p n y n q n y n g n x s 1 n x s kn y t 1 n y t kn r n n n 0 by virtue of the banach fixed point theorem and some natural modifications of the techniques in the literature we prove the existence results of uncountably many bounded positive solutions for the system of difference equations seven examples are constructed to explain the main results presented in this paper

necessary and sufficient conditions for the oscillation of systems of difference equations with continuous arguments abstract in this paper the oscillation of all solutions of system of the form y t i 1 n p i y t i 0 where p i r m m i r i 1 2 n is studied necessary and sufficient conditions in terms of the characteristic equation and explicit conditions in terms of the eigenvalues or the logarithmic norms of the coefficient matrices are established

oscillation of difference systems of the neutral type the authors consider the nonlinear difference systems d x n a nx s n p ng y n dy n dq nf x t n n n n 0 where d 1 a n p n and q n are real sequences s n and t n are real non negative sequences of integers and f g r r are continuous with uf u 0 and ug u 0 for u 0 the authors obtain sufficient conditions for all solutions of the system to be oscillatory examples to illustrate the results are included

on oscillation of difference equations with bounded phi laplacian oscillation of second order difference equations with bounded phi laplacian is studied by comparing it with the discrete emden fowler difference equation analogues and discrepancies in behavior of solutions of these equations are also presented and a comparison with the continuous case is given

classification schemes for nonoscillatory solutions of two dimensional nonlinear difference systems abstract classification schemes for nonoscillatory solutions of a class of nonlinear two dimensional nonlinear difference systems are given in terms of their asymptotic magnitudes and necessary as well as sufficient conditions for the existence of these solutions are also provided

existence for nonoscillatory solutions of higher order nonlinear neutral difference equations abstract in this paper we consider the following forced higher order nonlinear neutral difference equation m x n c n x n k s 1 u p s n f s x n r s q n n n 0 where m u 1 k 0 and rs 0 are integers cn p s n s 1 2 u and qn are sequence of real numbers and f s c r r s 1 2 u by using krasnoselskii s fixed point theorem and some new techniques we obtain sufficient conditions for the existence of nonoscillatory solutions for general p s n s 1 2 u and qn which means that we allow oscillatory p s n s 1 2 u and qn

classification and existence of positive solutions of systems of volterra nonlinear difference equations a classification scheme for the eventually positive solutions of a class of two dimensional volterra nonlinear difference equations is given in terms of their asymptotic magnitudes and necessary as well as sufficient conditions for the existence of such solutions are provided

convergence of the euler scheme for stochastic functional partial differential equations the study of stochastic partial different equations is motivated by the fact that when one wants to model some evolution phenomena arising in physics biology and engineering etc however in many applications the resulting equations have so far proved intractable to direct analytical solution numerical approximations such as the euler scheme are therefore a vital tool in exploring model behaviour unfortunately current results concerning the convergence of such schemes impose conditions on the drift and diffusion coefficients of the stochastic partial differential equation namely the linear growth and global lipschitz conditions which are often not met by systems of interest in this paper these conditions are relaxed and prove that numerical solutions based on the euler scheme will converge to the true solution of a broad class of stochastic partial differential equations several examples are studied to illustrate the results

existence of uncountably many bounded positive solutions for second order nonlinear neutral delay difference equations this paper deals with the second order nonlinear neutral delay difference equation d a n d x n b n x n t c n f n x f 1 n x f 2 n x f k n d n n n 0 by using krasnoselskii s fix

summation inequalities and half linear difference equations abstract the aim of this paper is to present new summation inequalities and to show that the half linear difference equation a n x n sgn x n b n x n 1 sgn x n 1 exhibits not only similarities but also some discrepancies with the linear case especially as regards the possible coexistence of the so called extremal solutions some of our results are new also for the linear difference equation

convergence of numerical solutions for a class of stochastic age dependent capital system with random jump magnitudes in stochastic differential equations sdes there is a class of stochastic functional differential equations with random jump magnitudes which aries in many financial models in general most equations of stochastic age dependent capital system do not have explicit solutions thus numerical approximation schemes are invaluable tools for exploring their properties in this paper the numerical approximation is established for a class of stochastic age dependent capital system with random jump magnitudes the main aim of this paper is to investigate the convergence of the numerical approximation for a class of stochastic age dependent capital system with random jump magnitudes it is proved that the numerical approximate solutions converge to the analytical solutions of the equations under given conditions the numerical approximate results in zhang et al 2011 2 are improved an example is given for illustration

oscillation of two dimensional difference systems abstract the authors consider the nonlinear two dimensional difference system x n b n g y n y n 1 a n f x n n n o no no 1 where n 0 n 1 2 a n and b n n n n 0 are real sequences and f g r r are continuous with uf u 0 and ug u 0 for u 0 a solution x n y n of the system is oscillatory if both components are oscillatory the authors obtain sufficient conditions for all solutions of the system to be oscillatory some of their results allow a n to oscillate examples to illustrate the results are included

unbounded solutions of quasi linear difference equations we study positive increasing solutions of the nonlinear difference equation an p n bnf n 1 p u u p 2u p 1 where an bn are positive real sequences for n 1 fr r is continuous with uf u 0 for u 0 a full characterization of limit behavior of all these solutions in terms of an bn is established examples showing the essential role of used hypotheses are also included the tools used are the schauder fixed point theorem and a comparison method based on the reciprocity principle

oscillation criteria for a nonlinear difference equation abstract essential and nontrivial improvements and extensions of several recent oscillation criteria are obtained for the nonlinear difference equation 1 below

unbounded positive solutions of higher order difference equations with singular nonlinear term in this paper several necessary and or sufficient conditions are given for the existence of unbounded positive solutions of nonlinear difference equations of the form mxn 1 f n xn 0 nen0 where m 2 1 or 1 and f n0 r r is a real valued continuous function such that f n x 0 for all n x n0 0 where n0 0 1

an application of discrete inequality to second order nonlinear oscillation abstract by using some simple discrete inequalities oscillation criteria are provided for the second order difference equations 2 y n a n 1 y n 1 0 n n where the operator is defined by yn yn 1 yn an is a real sequence the function is such that u u 0 for u 0 and u v g u v u v for u v 0 for some nonnegative function g

oscillation for system of delay difference equations abstract consider the system of delay difference equations x i t x i t k 1 l j 1 n p ijk x j t k 0 where p ijk and k 0 i j 1 2 n k 1 2 l sufficient conditions are obtained for all solutions of the system to be oscillatory

oscillation and asymptotic behavior of two dimensional difference systems this paper is concerned with the oscillation behavior of solutions of the nonlinear two dimesional difference system dx n b ng y n dy n 1 a nf x n n n n 0 n 0 n 0 1 some necessary and sufficient conditions are given for the oscillation of all solution of the system our result improve and generalize some results in the literature examples to illustrate the result are included

oscillation criteria for a class of neutral difference equations with continuous variable in this paper we are mainly concerned with oscillatory behaviour of solutions for a class of second order nonlinear neutral difference equations with continuous variable using an integral transformation the riccati transformation and iteration some oscillation criteria are obtained

bounded and zero convergent solutions of second order difference equations abstract in this paper we study boundedness and monotonicity properties of a homogeneous second order linear difference equation under certain conditions it is shown that every solution must eventually be monotonic the question of when such monotonic solutions diverge to infinity or converge to zero and the corresponding impact this behavior has on recessive and dominant solutions is investigated a necessary and sufficient condition for all solutions to be bounded is presented comparison theorems are also discussed