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
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 authors consider the following second order neutral difference equation with maxima △（αn△（yn＋pnyn-k））-qn max [n-l,n]ys=0,n=0,1,2,…,（＊）where {αn}, {pn} and （qn} are sequences of real numbers, and k an...The authors consider the following second order neutral difference equation with maxima △（αn△（yn＋pnyn-k））-qn max [n-l,n]ys=0,n=0,1,2,…,（＊）where {αn}, {pn} and （qn} are sequences of real numbers, and k and l are integers with k ≥ 1 and l 〉 0. And the asymptotic behavior of nonoscillatory solutions of （＊）. An example is given to show the difference between the equations with and without ＂maxima＂ is studied.展开更多
A neutral difference equation with positive and negative coefficientsΔ(x n-c nx n-k )+p nx n-l -q nx n-r =0, n=0,1,2,...,is considered and a sufficient condition for the global attractivity of the ze...A neutral difference equation with positive and negative coefficientsΔ(x n-c nx n-k )+p nx n-l -q nx n-r =0, n=0,1,2,...,is considered and a sufficient condition for the global attractivity of the zero solution of this equation is obtained, which improves and extends the all known results in the literature.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
In this paper the sufficient conditions for the existence of positive solutions of the neutral difference equations with positive and negative coefficients are established. The results improve some known conclusions i...In this paper the sufficient conditions for the existence of positive solutions of the neutral difference equations with positive and negative coefficients are established. The results improve some known conclusions in the literature.展开更多
Consider the second-order semi-linear neutral difference equation △[αn｜△（xn＋pnxn-τ）｜^α-1△（xn＋pnxn-τ）]＋∑i=1^k qi（n）｜xn-σi｜^α-σi=0,（1）The sufficient conditions are established for oscillation o...Consider the second-order semi-linear neutral difference equation △[αn｜△（xn＋pnxn-τ）｜^α-1△（xn＋pnxn-τ）]＋∑i=1^k qi（n）｜xn-σi｜^α-σi=0,（1）The sufficient conditions are established for oscillation of the solutions of （1）. These results generalize and improve some known results about both neutral and delay difference equation.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
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, 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.
In this paper, we consider a certain forced higher-order nonlinear neutral difference equation. By using Krasnoselskii’s fixed point theorem and some new techniques, we obtain necessary and sufficient conditions for ...In this paper, we consider a certain forced higher-order nonlinear neutral difference equation. By using Krasnoselskii’s fixed point theorem and some new techniques, we obtain necessary and sufficient conditions for the existence of non-oscillatory solutions for the system. Our results extend the results of Zhang [5].展开更多
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
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
Aim To obtain new criteria for asymptotic behavior and nonexistence of positive solutions of nonlinear neutral delay difference equations. Methods By means of Hlder inequality and a method of direct analysis, some i...Aim To obtain new criteria for asymptotic behavior and nonexistence of positive solutions of nonlinear neutral delay difference equations. Methods By means of Hlder inequality and a method of direct analysis, some interesting Lemmas were offered. Results and Conclusion New criteria for asymptotic behavior and nonexistence of positive solutions of nonlinear neutral delay difference equations are established, which extend and improve the results obtained in the literature. Some interesting examples illustrating the importance of our results are also included.展开更多
Consider the first order neutral delay differential equation with positive and negative coefficients:[x(t)-c(t)x(t-γ)]+p(t)x(t-τ)-Q(t)x(t-δ)=0,t≥t 0,(1)where c,p,Q∈C((t 0,∞),R +),R +=(0,∞),γ】0,t】δ≥0. W...Consider the first order neutral delay differential equation with positive and negative coefficients:[x(t)-c(t)x(t-γ)]+p(t)x(t-τ)-Q(t)x(t-δ)=0,t≥t 0,(1)where c,p,Q∈C((t 0,∞),R +),R +=(0,∞),γ】0,t】δ≥0. We obtain the sufficient condition for the existence of positive solutions of Eq.(1). As a corollary, we improve the correspondent result in by removing the condition∫ ∞ c 0 (t) d t=∞,where (t)=p(t)-Q(t-τ+δ)≥0.展开更多
In this paper, we consider the neutral difference equation△(x n-cx n-m )+p nx n-k =0, n=N, N+1, N+2, …,where c and p n are real numbers, k, m are positive integers with m<k, and △ den...In this paper, we consider the neutral difference equation△(x n-cx n-m )+p nx n-k =0, n=N, N+1, N+2, …,where c and p n are real numbers, k, m are positive integers with m<k, and △ denotes the forward difference operator: △ u n=u n+1 -u n. By using the Krasnoselskii fixed theorem, we obtain some sufficient conditions under which such an equation has a bounded and eventually positive solution which tends to zero as n→∞.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.
By using Banach contraction principle, we obtain the global results （with respect to ｜｜A｜｜≠1） on the sufficient conditions for the existence of nonoscillatory solutions to a system of nonlinear neutral delay di...By using Banach contraction principle, we obtain the global results （with respect to ｜｜A｜｜≠1） on the sufficient conditions for the existence of nonoscillatory solutions to a system of nonlinear neutral delay difference equations with matrix coefficients.展开更多
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.
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.
基金the Natural Science Foundation of Hebei Province (103141)Key Science Foundation of Hebei Normal University (1301808)
文摘The authors consider the following second order neutral difference equation with maxima △（αn△（yn＋pnyn-k））-qn max [n-l,n]ys=0,n=0,1,2,…,（＊）where {αn}, {pn} and （qn} are sequences of real numbers, and k and l are integers with k ≥ 1 and l 〉 0. And the asymptotic behavior of nonoscillatory solutions of （＊）. An example is given to show the difference between the equations with and without ＂maxima＂ is studied.
基金the Science Foundation of Educational Committee of Hunan Provinc
文摘A neutral difference equation with positive and negative coefficientsΔ(x n-c nx n-k )+p nx n-l -q nx n-r =0, n=0,1,2,...,is considered and a sufficient condition for the global attractivity of the zero solution of this equation is obtained, which improves and extends the all known results in the literature.
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.
基金the National Natural Science Foundation of China(1 0 0 71 0 1 8)
文摘In this paper the sufficient conditions for the existence of positive solutions of the neutral difference equations with positive and negative coefficients are established. The results improve some known conclusions in the literature.
基金the National Natural Science Foundation of China (60274021)
文摘Consider the second-order semi-linear neutral difference equation △[αn｜△（xn＋pnxn-τ）｜^α-1△（xn＋pnxn-τ）]＋∑i=1^k qi（n）｜xn-σi｜^α-σi=0,（1）The sufficient conditions are established for oscillation of the solutions of （1）. These results generalize and improve some known results about both neutral and delay difference equation.
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.
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, 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 the KEY Project of Chinese Ministry of Education.
文摘In this paper, we consider a certain forced higher-order nonlinear neutral difference equation. By using Krasnoselskii’s fixed point theorem and some new techniques, we obtain necessary and sufficient conditions for the existence of non-oscillatory solutions for the system. Our results extend the results of Zhang [5].
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.
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.
文摘Aim To obtain new criteria for asymptotic behavior and nonexistence of positive solutions of nonlinear neutral delay difference equations. Methods By means of Hlder inequality and a method of direct analysis, some interesting Lemmas were offered. Results and Conclusion New criteria for asymptotic behavior and nonexistence of positive solutions of nonlinear neutral delay difference equations are established, which extend and improve the results obtained in the literature. Some interesting examples illustrating the importance of our results are also included.
文摘Consider the first order neutral delay differential equation with positive and negative coefficients:[x(t)-c(t)x(t-γ)]+p(t)x(t-τ)-Q(t)x(t-δ)=0,t≥t 0,(1)where c,p,Q∈C((t 0,∞),R +),R +=(0,∞),γ】0,t】δ≥0. We obtain the sufficient condition for the existence of positive solutions of Eq.(1). As a corollary, we improve the correspondent result in by removing the condition∫ ∞ c 0 (t) d t=∞,where (t)=p(t)-Q(t-τ+δ)≥0.
文摘In this paper, we consider the neutral difference equation△(x n-cx n-m )+p nx n-k =0, n=N, N+1, N+2, …,where c and p n are real numbers, k, m are positive integers with m<k, and △ denotes the forward difference operator: △ u n=u n+1 -u n. By using the Krasnoselskii fixed theorem, we obtain some sufficient conditions under which such an equation has a bounded and eventually positive solution which tends to zero as n→∞.
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This work is supported by the National Natural Sciences Foundation of China under Grant 10361006the Natural Sciences Foundation of Yunnan Province under Grant 2003A0001MYouth Natural Sciences Foundation of Yunnan University under Grant 2003Q032C and Sciences Foundation of Yunnan Educational Community under Grant 04Y239A.
文摘By using Banach contraction principle, we obtain the global results （with respect to ｜｜A｜｜≠1） on the sufficient conditions for the existence of nonoscillatory solutions to a system of nonlinear neutral delay difference equations with matrix coefficients.
