The aim of this paper is to show that the following difference equation：Xn＋1=α＋（xn-k/xn-m）^p, n=0,1,2,…,where α 〉 -1, p 〉 O, k,m ∈ N are fixed, 0 ≤ m 〈 k, x-k, x-k＋1,…,x-m,…,X-1, x0 are positive, has p...The aim of this paper is to show that the following difference equation：Xn＋1=α＋（xn-k/xn-m）^p, n=0,1,2,…,where α 〉 -1, p 〉 O, k,m ∈ N are fixed, 0 ≤ m 〈 k, x-k, x-k＋1,…,x-m,…,X-1, x0 are positive, has positive nonoscillatory solutions which converge to the positive equilibrium x=α＋1. It is interesting that the method described in the paper, in some cases can also be applied when the parameter α is variable.展开更多
In this paper, we investigate the solutions of the system of difference equations xn+1=xn-1/ynxn-1-1,yn+1=yn-1/xnyn-1-1,zn+1=xn/ynzn-1 where x0,x-1,y0,y-1,z0,z-1∈R.
The aim of this paper is to establish the existence of at least two non-zero homoclinic solutions for a nonlinear Laplacian difference equation without using AmbrosettiRabinowitz type-conditions. The main tools are mo...The aim of this paper is to establish the existence of at least two non-zero homoclinic solutions for a nonlinear Laplacian difference equation without using AmbrosettiRabinowitz type-conditions. The main tools are mountain pass theorem and Palais-Smale compactness condition involving suitable functionals.展开更多
In this article, the existence of finite order entire solutions of nonlinear difference equations fn＋ Pd（z, f） = p1 eα1 z＋ p2 eα2 z are studied, where n ≥ 2 is an integer, Pd（z, f） is a difference polynomial ...In this article, the existence of finite order entire solutions of nonlinear difference equations fn＋ Pd（z, f） = p1 eα1 z＋ p2 eα2 z are studied, where n ≥ 2 is an integer, Pd（z, f） is a difference polynomial in f of degree d（≤ n-2）, p1, p2 are small meromorphic functions of ez, and α1, α2 are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.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
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
In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex d...In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.展开更多
This paper is concerned with the nonlinear neutral functional difference equations△x(n) =-a(n)x(n) +h(n)f(n,x(n-T(n)),△x(n-δ(n))),where a,h and f are nonnegative sequences.Sufficient conditions for the existence of...This paper is concerned with the nonlinear neutral functional difference equations△x(n) =-a(n)x(n) +h(n)f(n,x(n-T(n)),△x(n-δ(n))),where a,h and f are nonnegative sequences.Sufficient conditions for the existence of at least three positive T-periodic solutions are established by using a fixed point theorem due to Avery and Peterson.展开更多
In this paper, we investigate the growth of the meromorphic solutions of the following nonlinear difference equations F（Z）N＋pN-1（F）=0,where n ≥ 2 and small functions as proposed by Yang than 1. Pn-1（f） is a di...In this paper, we investigate the growth of the meromorphic solutions of the following nonlinear difference equations F（Z）N＋pN-1（F）=0,where n ≥ 2 and small functions as proposed by Yang than 1. Pn-1（f） is a difference polynomial of degree at most n - 1 in f with coefficients. Moreover, we give two examples to show that one conjecture and Laine [2] does not hold in general if the hyper-order of f（z） is no less展开更多
In this work we discuss the existence of ψ-bounded solutions for linear difference equations. We present a necessary and sufficient condition for the existence of ψ--bounded solutions for the linear nonhomogeneous d...In this work we discuss the existence of ψ-bounded solutions for linear difference equations. We present a necessary and sufficient condition for the existence of ψ--bounded solutions for the linear nonhomogeneous difference equation x（n＋1）=A（n）x（n）＋f（n） for every ψ-bounded sequence f（n）.展开更多
This paper studies the nonautonomous nonlinear system of difference equations △x（n） A（n）x（n） ＋ f（n,x（n））, n∈Z ,（＊）where x（n）∈R^N,A（n） = （aij（n））N×N is an N×N matrix, with aij∈C（R,R...This paper studies the nonautonomous nonlinear system of difference equations △x（n） A（n）x（n） ＋ f（n,x（n））, n∈Z ,（＊）where x（n）∈R^N,A（n） = （aij（n））N×N is an N×N matrix, with aij∈C（R,R） for i,j = 1,2,3 ,N, and f = （f1,f2,... ,fN）^T ∈C（R×R^N,R^N）, satisfying A（t＋） = A（t）, f（t＋ω,z） = f（t, z） for any t∈R, （t, z） ∈R× RN and ∈is a positive integer. Sufficient conditions for the existence of ω-periodic solutions to equations （＊） are obtained.展开更多
This paper is concerned with a class of partial difference equations with variable coefficients. Explicit growth bounds are found for their solutions. These bounds provide information on the existence of exponentially...This paper is concerned with a class of partial difference equations with variable coefficients. Explicit growth bounds are found for their solutions. These bounds provide information on the existence of exponentially bounded solutions.展开更多
Let,. We study the existence and multiple positive solutions of n-th nonlinear discrete fractional boundary value problem of the form By using a fixed-point theorem on cone, the parameter intervals of problem is estab...Let,. We study the existence and multiple positive solutions of n-th nonlinear discrete fractional boundary value problem of the form By using a fixed-point theorem on cone, the parameter intervals of problem is established.展开更多
In this paper, we investigate the existence of positive solutions for a class of nonlinear q-fractional boundary value problem. By using some fixed point theorems on cone, some existence results of positive solutions ...In this paper, we investigate the existence of positive solutions for a class of nonlinear q-fractional boundary value problem. By using some fixed point theorems on cone, some existence results of positive solutions are obtained.展开更多
In this article, we consider the non-linear difference equation(f(z + 1)f(z)-1)(f(z)f(z-1)-1) =P(z, f(z))/Q(z, f(z)),where P(z, f(z)) and Q(z, f(z)) are relatively prime polynomials in f(z) with rational coefficients....In this article, we consider the non-linear difference equation(f(z + 1)f(z)-1)(f(z)f(z-1)-1) =P(z, f(z))/Q(z, f(z)),where P(z, f(z)) and Q(z, f(z)) are relatively prime polynomials in f(z) with rational coefficients. For the above equation, the order of growth, the exponents of convergence of zeros and poles of its transcendental meromorphic solution f(z), and the exponents of convergence of poles of difference △f(z) and divided difference △f(z)/f(z)are estimated. Furthermore, we study the forms of rational solutions of the above equation.展开更多
In this paper,we investigate the growth of meromorphic solutions of some kind of non-homogeneous linear difference equations with special meromorphic coefficients.When there are more than one coefficient having the sa...In this paper,we investigate the growth of meromorphic solutions of some kind of non-homogeneous linear difference equations with special meromorphic coefficients.When there are more than one coefficient having the same maximal order and the same maximal type,the estimates on the lower bound of the order of meromorphic solutions of the involved equations are obtained.Meanwhile,the above estimates are sharpened by combining the relative results of the corresponding homogeneous linear difference equations.展开更多
The aim of this paper is to prove the existence of one or multiple solutions of nonlinear difference equations coupled to a general set of boundary conditions.Before to do this,we construct a discrete operator whose f...The aim of this paper is to prove the existence of one or multiple solutions of nonlinear difference equations coupled to a general set of boundary conditions.Before to do this,we construct a discrete operator whose fixed points coincide with the solutions of the problem we are looking for.Moreover,we introduce a strong positiveness condition on the related Green's function that allows us to construct suitable cones where to apply adequate fixed point theorems.Once we have the general existence result,we deduce,as a particular case,the existence of solutions of a second order difference equation with nonlocal perturbed Dirichlet conditions.展开更多
An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations i...An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.展开更多
文摘The aim of this paper is to show that the following difference equation：Xn＋1=α＋（xn-k/xn-m）^p, n=0,1,2,…,where α 〉 -1, p 〉 O, k,m ∈ N are fixed, 0 ≤ m 〈 k, x-k, x-k＋1,…,x-m,…,X-1, x0 are positive, has positive nonoscillatory solutions which converge to the positive equilibrium x=α＋1. It is interesting that the method described in the paper, in some cases can also be applied when the parameter α is variable.
文摘In this paper, we investigate the solutions of the system of difference equations xn+1=xn-1/ynxn-1-1,yn+1=yn-1/xnyn-1-1,zn+1=xn/ynzn-1 where x0,x-1,y0,y-1,z0,z-1∈R.
基金supported by the Bulgarian National Science Fund under Project DN 12/4 “Advanced analytical and numerical methods for nonlinear differential equations with applications in finance and environmental pollution”, 2017。
文摘The aim of this paper is to establish the existence of at least two non-zero homoclinic solutions for a nonlinear Laplacian difference equation without using AmbrosettiRabinowitz type-conditions. The main tools are mountain pass theorem and Palais-Smale compactness condition involving suitable functionals.
基金supported by the National Natural Science Foundation of China(11661044)
文摘In this article, the existence of finite order entire solutions of nonlinear difference equations fn＋ Pd（z, f） = p1 eα1 z＋ p2 eα2 z are studied, where n ≥ 2 is an integer, Pd（z, f） is a difference polynomial in f of degree d（≤ n-2）, p1, p2 are small meromorphic functions of ez, and α1, α2 are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.
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.
文摘In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.
基金Supported by the Natural Science Foundation of Hunan Province（12JJ6006） Supported by the Science Foundation of Department of Science and Technology of Hunan Province（2012FJ3107）
文摘This paper is concerned with the nonlinear neutral functional difference equations△x(n) =-a(n)x(n) +h(n)f(n,x(n-T(n)),△x(n-δ(n))),where a,h and f are nonnegative sequences.Sufficient conditions for the existence of at least three positive T-periodic solutions are established by using a fixed point theorem due to Avery and Peterson.
基金supported by the NNSF of China(11171013,11371225,11201014)the YWF-14-SXXY-008 of Beihang Universitythe Fundamental Research Funds for the Central University
文摘In this paper, we investigate the growth of the meromorphic solutions of the following nonlinear difference equations F（Z）N＋pN-1（F）=0,where n ≥ 2 and small functions as proposed by Yang than 1. Pn-1（f） is a difference polynomial of degree at most n - 1 in f with coefficients. Moreover, we give two examples to show that one conjecture and Laine [2] does not hold in general if the hyper-order of f（z） is no less
基金The NSF(Y2008A30,ZR2010AL011)of Shandong Province
文摘In this work we discuss the existence of ψ-bounded solutions for linear difference equations. We present a necessary and sufficient condition for the existence of ψ--bounded solutions for the linear nonhomogeneous difference equation x（n＋1）=A（n）x（n）＋f（n） for every ψ-bounded sequence f（n）.
基金Supported by the NNSF of China(10571050),the EYTP of China and the Science Foundation of the Education Committee of Hunan Province(04C457).
文摘This paper studies the nonautonomous nonlinear system of difference equations △x（n） A（n）x（n） ＋ f（n,x（n））, n∈Z ,（＊）where x（n）∈R^N,A（n） = （aij（n））N×N is an N×N matrix, with aij∈C（R,R） for i,j = 1,2,3 ,N, and f = （f1,f2,... ,fN）^T ∈C（R×R^N,R^N）, satisfying A（t＋） = A（t）, f（t＋ω,z） = f（t, z） for any t∈R, （t, z） ∈R× RN and ∈is a positive integer. Sufficient conditions for the existence of ω-periodic solutions to equations （＊） are obtained.
文摘This paper is concerned with a class of partial difference equations with variable coefficients. Explicit growth bounds are found for their solutions. These bounds provide information on the existence of exponentially bounded solutions.
文摘Let,. We study the existence and multiple positive solutions of n-th nonlinear discrete fractional boundary value problem of the form By using a fixed-point theorem on cone, the parameter intervals of problem is established.
文摘In this paper, we investigate the existence of positive solutions for a class of nonlinear q-fractional boundary value problem. By using some fixed point theorems on cone, some existence results of positive solutions are obtained.
基金supported by the National Natural Science Foundation of China(11371225)National Natural Science Foundation of Guangdong Province(2016A030313686)
文摘In this article, we consider the non-linear difference equation(f(z + 1)f(z)-1)(f(z)f(z-1)-1) =P(z, f(z))/Q(z, f(z)),where P(z, f(z)) and Q(z, f(z)) are relatively prime polynomials in f(z) with rational coefficients. For the above equation, the order of growth, the exponents of convergence of zeros and poles of its transcendental meromorphic solution f(z), and the exponents of convergence of poles of difference △f(z) and divided difference △f(z)/f(z)are estimated. Furthermore, we study the forms of rational solutions of the above equation.
基金Supported by the National Natural Science Foundation of China(No.11761035)the Natural Science Foundation of Jiangxi Province in China(No.20171BAB201002)
文摘In this paper,we investigate the growth of meromorphic solutions of some kind of non-homogeneous linear difference equations with special meromorphic coefficients.When there are more than one coefficient having the same maximal order and the same maximal type,the estimates on the lower bound of the order of meromorphic solutions of the involved equations are obtained.Meanwhile,the above estimates are sharpened by combining the relative results of the corresponding homogeneous linear difference equations.
基金partially supported by Xunta de Galicia(Spain),project EM2014/032 and AIE,Spain and FEDER,grant MTM2016-75140-Psupported by the Bulgarian National Science Fundation under Project DN 12/4“Advanced Analytical and Numerical Methods for Nonlinear Differential Equations with Applications in Finance and Environmental Pollution”,2017。
文摘The aim of this paper is to prove the existence of one or multiple solutions of nonlinear difference equations coupled to a general set of boundary conditions.Before to do this,we construct a discrete operator whose fixed points coincide with the solutions of the problem we are looking for.Moreover,we introduce a strong positiveness condition on the related Green's function that allows us to construct suitable cones where to apply adequate fixed point theorems.Once we have the general existence result,we deduce,as a particular case,the existence of solutions of a second order difference equation with nonlocal perturbed Dirichlet conditions.
文摘An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.
