In this paper, we investigate the asymptotic behavior of the following quasilinear difference equations (E) where , . We classified the solutions into six types by means of their asymptotic behavior. We establish the ...In this paper, we investigate the asymptotic behavior of the following quasilinear difference equations (E) where , . We classified the solutions into six types by means of their asymptotic behavior. We establish the necessary and/or sufficient conditions for such equations to possess a solution of each of these six types.展开更多
This paper is concerned with the oscillatory (and nonoscillatory) behavior of solutions of second oder quasilinear difference equations of the type Some necessary and sufficient conditions are given for the equation t...This paper is concerned with the oscillatory (and nonoscillatory) behavior of solutions of second oder quasilinear difference equations of the type Some necessary and sufficient conditions are given for the equation to admit oscillatory and nonoscillatory solutions with special asymptotic properties. These results generalize and improve some known results.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 study the oscillatory and asymptotic behavior of second order neutral delay difference equation with “maxima” of the form? Examples are given to illustrate the main result.
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
For functional difference equations with unbounded delay,we characterized the existence of totally stable and asymptotically almost periodic solution by using stability properties of a bounded solution in a certain li...For functional difference equations with unbounded delay,we characterized the existence of totally stable and asymptotically almost periodic solution by using stability properties of a bounded solution in a certain limiting equation.展开更多
The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obta...The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the asymptotic stability of global solutions by means of a difference inequality.展开更多
We study the convergence of the positive solutions of the system of the following two difference equations: where K is a positive integer, the parameters?A,B,?α, β? and the initial conditions are positive real numbe...We study the convergence of the positive solutions of the system of the following two difference equations: where K is a positive integer, the parameters?A,B,?α, β? and the initial conditions are positive real numbers. Our results generalize well known results in [1,2].展开更多
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
Consider the retarded difference equation x<sub>n</sub>-x<sub>n-1</sub>=F（-f（x<sub>n</sub>）+g（x<sub>n</sub>-k））, （*） where k is a positive integer, F,f,g:R→R a...Consider the retarded difference equation x<sub>n</sub>-x<sub>n-1</sub>=F（-f（x<sub>n</sub>）+g（x<sub>n</sub>-k））, （*） where k is a positive integer, F,f,g:R→R are continuous, F and f are increasing on R, and uF（u）】0 for all u≠0. We show that when f（y）≥g（y）（resp. f（y）≤g（y）） for y∈R, every solution of （*） tends to either a constant or -∞ （resp. ∞） as n→∞. Furthermore, if f（y）≡g（y） for y∈R, then every solution of （*） tends to a constant as n→∞.展开更多
We propose a class of delay difference equation with piecewise constant nonlinearity. Such a delay difference equation can be regarded as the discrete analog of a differential equation. The convergence of solutions an...We propose a class of delay difference equation with piecewise constant nonlinearity. Such a delay difference equation can be regarded as the discrete analog of a differential equation. The convergence of solutions and the existence of asymptotically stable periodic solutions are investigated for such a class of difference equation.展开更多
This paper is concerned with the study of asymptotic behavior of nonoscillatory solutions of second order neutral nonlinear difference equations of theformwhere λ∈ {-1,1},△ is the forword difference operator define...This paper is concerned with the study of asymptotic behavior of nonoscillatory solutions of second order neutral nonlinear difference equations of theformwhere λ∈ {-1,1},△ is the forword difference operator defined by △x_n=x_n+1 x+n.展开更多
In this paper, we investigate the asymptotic behavior of the extremal solutions of a difference equation and their character and prove the existence of the non-extremal solutions.
In this paper,we study the asymptotic behavior of solutions to a class of higher order difference equations.With the aid of the discrete inequality,we obtain some sufficient conditions which ensure that all the soluti...In this paper,we study the asymptotic behavior of solutions to a class of higher order difference equations.With the aid of the discrete inequality,we obtain some sufficient conditions which ensure that all the solutions to the equation are some high order of infinities,and also that some conditions which guarantee that every oscillatory solution to the equation has the property that the i order L operator of it tends to infinity when its independent variable tends to zero.展开更多
The asymptotic behavior of the nonoscillatory solutions of the difference equations △[r(n)△x(n)]+f(n,x(n),x(r(n,x(n))))=0 is considered. In the case when f is a strongly sublinear (superlinear) function, conditions ...The asymptotic behavior of the nonoscillatory solutions of the difference equations △[r(n)△x(n)]+f(n,x(n),x(r(n,x(n))))=0 is considered. In the case when f is a strongly sublinear (superlinear) function, conditions for oscillations of (1) are also found.展开更多
In this paper we prove the solution of explicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the relevant nonlinear stationary problem as t→∞. For nonlinea...In this paper we prove the solution of explicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the relevant nonlinear stationary problem as t→∞. For nonlinear parabolic problem, we obtain the long time asymptotic behavior of its discrete solution which is analogous to that of its continuous solution. For simplicity, we discuss one-dimensional problem.展开更多
In this paper, the solution of back-Euler implicit difference scheme for a semi-linear parabolic equation is proved to converge to the solution of difference scheme for the corresponding semi-linear elliptic equation ...In this paper, the solution of back-Euler implicit difference scheme for a semi-linear parabolic equation is proved to converge to the solution of difference scheme for the corresponding semi-linear elliptic equation as t tends to infinity. The long asymptotic behavior of its discrete solution is obtained which is analogous to that of its continuous solution. At last, a few results are also presented for Crank-Nicolson scheme.展开更多
In this paper we prove that the solution of implicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the corresponding nonlinear stationary problem as t -&g...In this paper we prove that the solution of implicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the corresponding nonlinear stationary problem as t -> infinity. For the discrete solution of nonlinear parabolic problem, we get its long time asymptotic behavior which is similar to that of the continuous solution. For simplicity, we consider one-dimensional problem.展开更多
Let B^(H) be a fractional Brownian motion with Hurst index 1/2≤H<1.In this paper,we consider the equation(called the Ornstein-Uhlenbeck process with a linear self-repelling drift)dX_(t)^(H)=dB_(t)^(H)+σ X_(t)^(H)...Let B^(H) be a fractional Brownian motion with Hurst index 1/2≤H<1.In this paper,we consider the equation(called the Ornstein-Uhlenbeck process with a linear self-repelling drift)dX_(t)^(H)=dB_(t)^(H)+σ X_(t)^(H)dt+vdt-θ(∫_(0)^(t)(X_(t)^(H)-X_(s)^(H))ds)dt,whereθ<0,σ,v∈ℝ.The process is an analogue of self-attracting diffusion(Cranston,Le Jan.Math Ann,1995,303:87–93).Our main aim is to study the large time behaviors of the process.We show that the solution X^(H)diverges to infinity as t tends to infinity,and obtain the speed at which the process X^(H)diverges to infinity.展开更多
文摘In this paper, we investigate the asymptotic behavior of the following quasilinear difference equations (E) where , . We classified the solutions into six types by means of their asymptotic behavior. We establish the necessary and/or sufficient conditions for such equations to possess a solution of each of these six types.
基金The Science Foundation (00C029) of Hunan Educational Committee.
文摘This paper is concerned with the oscillatory (and nonoscillatory) behavior of solutions of second oder quasilinear difference equations of the type Some necessary and sufficient conditions are given for the equation to admit oscillatory and nonoscillatory solutions with special asymptotic properties. These results generalize and improve some known results.
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 study the oscillatory and asymptotic behavior of second order neutral delay difference equation with “maxima” of the form? Examples are given to illustrate the main result.
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For functional difference equations with unbounded delay,we characterized the existence of totally stable and asymptotically almost periodic solution by using stability properties of a bounded solution in a certain limiting equation.
基金supported by National Natural Science Foundation of China(61273016)The Natural Science Foundation of Zhejiang Province(Y6100016)The Public Welfare Technology Application Research Project of Zhejiang Province Science and Technology Department(2015C33088)
文摘The initial-boundary value problem for a class of nonlinear hyperbolic equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the asymptotic stability of global solutions by means of a difference inequality.
文摘We study the convergence of the positive solutions of the system of the following two difference equations: where K is a positive integer, the parameters?A,B,?α, β? and the initial conditions are positive real numbers. Our results generalize well known results in [1,2].
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Project supported by NNSF (19601016) of China NSF (97-37-42) of Hunan
文摘Consider the retarded difference equation x<sub>n</sub>-x<sub>n-1</sub>=F（-f（x<sub>n</sub>）+g（x<sub>n</sub>-k））, （*） where k is a positive integer, F,f,g:R→R are continuous, F and f are increasing on R, and uF（u）】0 for all u≠0. We show that when f（y）≥g（y）（resp. f（y）≤g（y）） for y∈R, every solution of （*） tends to either a constant or -∞ （resp. ∞） as n→∞. Furthermore, if f（y）≡g（y） for y∈R, then every solution of （*） tends to a constant as n→∞.
基金Research supported by National Natural Science Foundation of China (10071016)the Key Research Program of Science and Technology of the Ministry of Education of China, the Doctor Program Foundation of the Ministry of Education of China (20010532002)Foundation for University Excellent Teacher by the Ministry of Education.
文摘We propose a class of delay difference equation with piecewise constant nonlinearity. Such a delay difference equation can be regarded as the discrete analog of a differential equation. The convergence of solutions and the existence of asymptotically stable periodic solutions are investigated for such a class of difference equation.
文摘This paper is concerned with the study of asymptotic behavior of nonoscillatory solutions of second order neutral nonlinear difference equations of theformwhere λ∈ {-1,1},△ is the forword difference operator defined by △x_n=x_n+1 x+n.
基金Research supported by Distinguished Expert Science Foundation of Naval Aeronautical Engineering Institute.
文摘In this paper, we investigate the asymptotic behavior of the extremal solutions of a difference equation and their character and prove the existence of the non-extremal solutions.
文摘In this paper,we study the asymptotic behavior of solutions to a class of higher order difference equations.With the aid of the discrete inequality,we obtain some sufficient conditions which ensure that all the solutions to the equation are some high order of infinities,and also that some conditions which guarantee that every oscillatory solution to the equation has the property that the i order L operator of it tends to infinity when its independent variable tends to zero.
基金the National Natural Science Foundation of China (No.69982002) and theNationa1 Key Basic Research Special Found (No.G199802030
文摘The asymptotic behavior of the nonoscillatory solutions of the difference equations △[r(n)△x(n)]+f(n,x(n),x(r(n,x(n))))=0 is considered. In the case when f is a strongly sublinear (superlinear) function, conditions for oscillations of (1) are also found.
文摘In this paper we prove the solution of explicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the relevant nonlinear stationary problem as t→∞. For nonlinear parabolic problem, we obtain the long time asymptotic behavior of its discrete solution which is analogous to that of its continuous solution. For simplicity, we discuss one-dimensional problem.
基金The work was supported by Jiangsu Province's Natural Science Foundation (BK97004)National Natural Science Foundation (19801007) of CHINA.
文摘In this paper, the solution of back-Euler implicit difference scheme for a semi-linear parabolic equation is proved to converge to the solution of difference scheme for the corresponding semi-linear elliptic equation as t tends to infinity. The long asymptotic behavior of its discrete solution is obtained which is analogous to that of its continuous solution. At last, a few results are also presented for Crank-Nicolson scheme.
文摘In this paper we prove that the solution of implicit difference scheme for a semilinear parabolic equation converges to the solution of difference scheme for the corresponding nonlinear stationary problem as t -> infinity. For the discrete solution of nonlinear parabolic problem, we get its long time asymptotic behavior which is similar to that of the continuous solution. For simplicity, we consider one-dimensional problem.
文摘Let B^(H) be a fractional Brownian motion with Hurst index 1/2≤H<1.In this paper,we consider the equation(called the Ornstein-Uhlenbeck process with a linear self-repelling drift)dX_(t)^(H)=dB_(t)^(H)+σ X_(t)^(H)dt+vdt-θ(∫_(0)^(t)(X_(t)^(H)-X_(s)^(H))ds)dt,whereθ<0,σ,v∈ℝ.The process is an analogue of self-attracting diffusion(Cranston,Le Jan.Math Ann,1995,303:87–93).Our main aim is to study the large time behaviors of the process.We show that the solution X^(H)diverges to infinity as t tends to infinity,and obtain the speed at which the process X^(H)diverges to infinity.
