Consider the second order nonlinear neutral difference equationThe sufficient conditions are established for the oscillation and asymptotic behavior of the solutions of this equation.
The objective of this paper is to study the oscillatory and asymptotic properties of the mixed type third order neutral difference equation of the form△（an△^2（xn＋bnxn-τ1＋cnxn＋τ2））＋qnx^βn＋1-σ1＋pnx^^βn...The objective of this paper is to study the oscillatory and asymptotic properties of the mixed type third order neutral difference equation of the form△（an△^2（xn＋bnxn-τ1＋cnxn＋τ2））＋qnx^βn＋1-σ1＋pnx^^βn＋1＋σ2=0,where （an）, （bn}, （cn}, （qn} and （pn} are positive real sequences, β is a ratio of odd positive integers, τ1, τ2, and σ2 are positive integers. We establish some sufficient conditions which ensure that all solutions are either oscillatory or converges to zero. Some examples are presented to illustrate the main results.展开更多
In this paper,we investigate oscillation of solutions to a class of second order neutral delay difference equations with continuous arguments. We then obtain some sufficient conditions for bounded oscillation of the s...In this paper,we investigate oscillation of solutions to a class of second order neutral delay difference equations with continuous arguments. We then obtain some sufficient conditions for bounded oscillation of the solutions.展开更多
The aim of this paper is to study the oscillation of second order neutral difference equations.Our results are based on the new comparison theorems,that reduce the problem of the oscillation of the second order equati...The aim of this paper is to study the oscillation of second order neutral difference equations.Our results are based on the new comparison theorems,that reduce the problem of the oscillation of the second order equation to that of the first order equation.The comparison principles obtained essentially simplify the examination of the equations.展开更多
In this paper, new oscillation criteria are obtained for all solution of the firstorder neutral difference equation △(x(n) - P(n)x(n-m)) + Q(n)x(n-k) =0, n ≥n0 where m > 0, k≥ 0 are integers. p(n), Q(n)∈ C({n0...In this paper, new oscillation criteria are obtained for all solution of the firstorder neutral difference equation △(x(n) - P(n)x(n-m)) + Q(n)x(n-k) =0, n ≥n0 where m > 0, k≥ 0 are integers. p(n), Q(n)∈ C({n0, n0 + 1,...}, R+). Our results do not need the usual hypothesis ∑Q(s) = ∞. Some example are given s=N to demonstrate the advantage of our results than those in the literature.展开更多
Consider the second Order nonlinear neutral difference equation for n≥n0 The sufficient conditions are obtained for the oscillatory and asymptotic behavior of the solutions of this equation.
A class of higher order neutral difference equations is considered and some sufficient conditions are obtained for all solutions to oscillate or tend to zero.
In this paper, we are concerned with the second order neutral difference equation with continuous variable. Using the integral transformation and generalized Riccati transformation, we obtain some oscillation criteria.
An oscillation criterion is obtained for even order neutral type difference equations of the following formwhere m > 2 is even, no is a nonnegative integer, A is the forward difference operator defined by zn =xn+1 ...An oscillation criterion is obtained for even order neutral type difference equations of the following formwhere m > 2 is even, no is a nonnegative integer, A is the forward difference operator defined by zn =xn+1 - xn,and for i >1, i is the ith-order forward difference operator defined by ixn = A(i-1xn), r and o are positive integers.展开更多
By using the Riccati transformation and the Cauchy mean inequality, we shall derive some oscillatory criteria for the second order neutral delay difference equationΔ [a nΔ (x n+p nx g(n) )]+q nf(x σ(n) ...By using the Riccati transformation and the Cauchy mean inequality, we shall derive some oscillatory criteria for the second order neutral delay difference equationΔ [a nΔ (x n+p nx g(n) )]+q nf(x σ(n) )=0. These results generalize and improve some known results about both neutral and delay difference equations.展开更多
Consider the nonlinear neutral difference equation Δ [x n+p ng(x n-k )]+q nh(x n-l )=0 (1.1) we prove that, under appropriate hypotheses, if an associated equation with Eq. (1.1) oscillates, then Eq. ...Consider the nonlinear neutral difference equation Δ [x n+p ng(x n-k )]+q nh(x n-l )=0 (1.1) we prove that, under appropriate hypotheses, if an associated equation with Eq. (1.1) oscillates, then Eq. (1.1) oscillates. We also give an example to illustrate the main result.展开更多
Consider the n-th order neutral differential equation where n > 1 is an odd integer, p(t) ∈ C([t0, ∞ ), R), Q(t), R(t)∈ C([t0, ∞), R+), τ > 0, σ, r ∈ R+. In this paper we will survey some recent results o...Consider the n-th order neutral differential equation where n > 1 is an odd integer, p(t) ∈ C([t0, ∞ ), R), Q(t), R(t)∈ C([t0, ∞), R+), τ > 0, σ, r ∈ R+. In this paper we will survey some recent results on the oscillation of Eq.() without the usual requirement that We also give some interesting open problems on this topic.展开更多
The objective of this paper is to study the oscillatory and asymptotic properties of the general mixed type third order neutral difference equation of the form △（aαn△2（xn＋ bnxn-τ1＋ cnxn＋τ2）） ＋ qnx^a n＋1...The objective of this paper is to study the oscillatory and asymptotic properties of the general mixed type third order neutral difference equation of the form △（aαn△2（xn＋ bnxn-τ1＋ cnxn＋τ2）） ＋ qnx^a n＋1-σ＋ pnx^βn＋1＋=σ2 0,where {an}, {bn}, {cn}, {qn} and {pn} are positive real sequences, both α and β are ratios of odd positive integers, τ1, τ2, σ1 and σ2 are positive integers. We establish some sufficient conditions which ensure all solutions are either oscillatory or converge to zero.展开更多
Our aim in this paper is to obtain a necessary and sufficient condition characterized by coefficients and delays under which every solution of higher-order neutral equation will be oscillating, where p ≠ 0, τ > ...Our aim in this paper is to obtain a necessary and sufficient condition characterized by coefficients and delays under which every solution of higher-order neutral equation will be oscillating, where p ≠ 0, τ > 0, σ> 0 and q are all constants.展开更多
A class of neutral type higher order difference equations is considered. Some sufficient conditions of oscillation and asymptotic behavior of solutions is given.
In this paper, a class of second order nonlinear neutral difference equations with variable delays are studied. The criteria for existence of bounded eventually positive solution is obtained by using Banach contractio...In this paper, a class of second order nonlinear neutral difference equations with variable delays are studied. The criteria for existence of bounded eventually positive solution is obtained by using Banach contraction mapping principle and some necessary techniques. Moreover, some sufficient conditions for oscillation of the equations are given. Some results available in documents are extended in this paper. Illustrative examples are given.展开更多
文摘Consider the second order nonlinear neutral difference equationThe sufficient conditions are established for the oscillation and asymptotic behavior of the solutions of this equation.
文摘The objective of this paper is to study the oscillatory and asymptotic properties of the mixed type third order neutral difference equation of the form△（an△^2（xn＋bnxn-τ1＋cnxn＋τ2））＋qnx^βn＋1-σ1＋pnx^^βn＋1＋σ2=0,where （an）, （bn}, （cn}, （qn} and （pn} are positive real sequences, β is a ratio of odd positive integers, τ1, τ2, and σ2 are positive integers. We establish some sufficient conditions which ensure that all solutions are either oscillatory or converges to zero. Some examples are presented to illustrate the main results.
基金This work are supported by the NNSF of China (No.10571050)the Science and Research Fund for Higher Colleges of Hunan (No.06C054).
文摘In this paper,we investigate oscillation of solutions to a class of second order neutral delay difference equations with continuous arguments. We then obtain some sufficient conditions for bounded oscillation of the solutions.
基金the NSF of Shanxi Province(No.2008011002-1)the Development Foundation of Higher Education Department of Shanxi Province(No.20111117)the Foundation of Datong University 2010-B-01
文摘The aim of this paper is to study the oscillation of second order neutral difference equations.Our results are based on the new comparison theorems,that reduce the problem of the oscillation of the second order equation to that of the first order equation.The comparison principles obtained essentially simplify the examination of the equations.
文摘In this paper, new oscillation criteria are obtained for all solution of the firstorder neutral difference equation △(x(n) - P(n)x(n-m)) + Q(n)x(n-k) =0, n ≥n0 where m > 0, k≥ 0 are integers. p(n), Q(n)∈ C({n0, n0 + 1,...}, R+). Our results do not need the usual hypothesis ∑Q(s) = ∞. Some example are given s=N to demonstrate the advantage of our results than those in the literature.
文摘Consider the second Order nonlinear neutral difference equation for n≥n0 The sufficient conditions are obtained for the oscillatory and asymptotic behavior of the solutions of this equation.
文摘A class of higher order neutral difference equations is considered and some sufficient conditions are obtained for all solutions to oscillate or tend to zero.
文摘In this paper, we are concerned with the second order neutral difference equation with continuous variable. Using the integral transformation and generalized Riccati transformation, we obtain some oscillation criteria.
基金This work is supported by National Natural Sciences Foundation of People's Republic of China under Grant 10061004 and Natural Sciences Foundation of Yunnan Province
文摘An oscillation criterion is obtained for even order neutral type difference equations of the following formwhere m > 2 is even, no is a nonnegative integer, A is the forward difference operator defined by zn =xn+1 - xn,and for i >1, i is the ith-order forward difference operator defined by ixn = A(i-1xn), r and o are positive integers.
文摘By using the Riccati transformation and the Cauchy mean inequality, we shall derive some oscillatory criteria for the second order neutral delay difference equationΔ [a nΔ (x n+p nx g(n) )]+q nf(x σ(n) )=0. These results generalize and improve some known results about both neutral and delay difference equations.
文摘Consider the nonlinear neutral difference equation Δ [x n+p ng(x n-k )]+q nh(x n-l )=0 (1.1) we prove that, under appropriate hypotheses, if an associated equation with Eq. (1.1) oscillates, then Eq. (1.1) oscillates. We also give an example to illustrate the main result.
文摘Consider the n-th order neutral differential equation where n > 1 is an odd integer, p(t) ∈ C([t0, ∞ ), R), Q(t), R(t)∈ C([t0, ∞), R+), τ > 0, σ, r ∈ R+. In this paper we will survey some recent results on the oscillation of Eq.() without the usual requirement that We also give some interesting open problems on this topic.
基金Supported by Science Research Foundation of Guangxi Education Board under grant YB2014117
文摘The objective of this paper is to study the oscillatory and asymptotic properties of the general mixed type third order neutral difference equation of the form △（aαn△2（xn＋ bnxn-τ1＋ cnxn＋τ2）） ＋ qnx^a n＋1-σ＋ pnx^βn＋1＋=σ2 0,where {an}, {bn}, {cn}, {qn} and {pn} are positive real sequences, both α and β are ratios of odd positive integers, τ1, τ2, σ1 and σ2 are positive integers. We establish some sufficient conditions which ensure all solutions are either oscillatory or converge to zero.
文摘Our aim in this paper is to obtain a necessary and sufficient condition characterized by coefficients and delays under which every solution of higher-order neutral equation will be oscillating, where p ≠ 0, τ > 0, σ> 0 and q are all constants.
文摘A class of neutral type higher order difference equations is considered. Some sufficient conditions of oscillation and asymptotic behavior of solutions is given.
基金Supported by the Scientific Research Fund of Education Department of Hunan Province(07C680)
文摘In this paper, a class of second order nonlinear neutral difference equations with variable delays are studied. The criteria for existence of bounded eventually positive solution is obtained by using Banach contraction mapping principle and some necessary techniques. Moreover, some sufficient conditions for oscillation of the equations are given. Some results available in documents are extended in this paper. Illustrative examples are given.
