This article studies one dimensional viscous Camassa-Holm equation with a periodic boundary condition. The existence of the almost periodic solution is investigated by using the Galerkin method.
In this paper, the existence and uniqueness of almost periodic solutions for some infinite delay integral equations are discussed. By using Krasnoselskii fixed point theorem,some new results are obtained.
This paper deals with the problems on the existence and uniqueness and stability of almost periodic solutions for functional differential equations with infinite delays.The author obtains some sufficient conditions wh...This paper deals with the problems on the existence and uniqueness and stability of almost periodic solutions for functional differential equations with infinite delays.The author obtains some sufficient conditions which ganrantee the existence and uniqueness and stability of almost periodic solutions with module containment.The results extend all the results of the paper and solve the two open problems proposed in under much weaker conditions than that proposed in.展开更多
This paper deals with the uniqueness and existence of periodic solutions of neutral Volterraintegrodifferential equations (1) and (2). Some new unique existence criteria are obtained.
In this paper, the concept of generalized ω-periodic solution is given for Riccati's equationy'=a(t)y^2+b(t)y+c(t)with perriodic coefficients, the relation between generalized ω-periodicsolutions and the cha...In this paper, the concept of generalized ω-periodic solution is given for Riccati's equationy'=a(t)y^2+b(t)y+c(t)with perriodic coefficients, the relation between generalized ω-periodicsolutions and the characteristic numbers of system x'_1=c(t)x_2, x'_2=-a(t)x_1-b(t)x_2 is indicated, andseveral necessary and sufficient conditions are given using the coefficients. Moreover, in the case of a(t)without zero, the relation between the number of continuous ω-periodic solutions of y'=a(t)y^2+b(t)y+c(t)+δand the parameter δ is given; thus the problem on the existence of continuous ω-periodic.solutions is basically solved.展开更多
In this paper, using Fourier series, we study the problem of the existence of periodic solutionsof a type of periodic neutral differential difference system. Some necessary and sufficient conditionsfor the existence o...In this paper, using Fourier series, we study the problem of the existence of periodic solutionsof a type of periodic neutral differential difference system. Some necessary and sufficient conditionsfor the existence of periodic solutions of a type of neutral functional equation system are obtained,and at the same time, we present a method with formula shows how to find the periodicsolutions.展开更多
This paper deals with the existence of positive periodic solutions for a kind of nonautonomous Volterra intergo-differential equations by employing the Krasnoselskii fixed point theorem. Applying the general theorems ...This paper deals with the existence of positive periodic solutions for a kind of nonautonomous Volterra intergo-differential equations by employing the Krasnoselskii fixed point theorem. Applying the general theorems established to several biomathematical models, the paper improves some previous results and obtains some new results.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
In this paper, we consider a certain fourth-order nonlinear ordinary differential equation. Some sufficient conditions which guarantee the existence of at least one ω-periodic solution to the system are obtain.
In this paper, hy using successive approximation method and fixed-point theorem, we discuss a class of infinite delay integral equation and obtain some sufficient conditions which guarantee the existence and uniquenes...In this paper, hy using successive approximation method and fixed-point theorem, we discuss a class of infinite delay integral equation and obtain some sufficient conditions which guarantee the existence and uniqueness of the periodic and almost periodic solutions of the system.展开更多
A fully nonlinear generalization of the Camassa-Holm equation is investigated. Using the 3 pseudoparabolic regularization technique, its local well-posedness in Sobolev space Hs（R） with s 〉 is established via a li...A fully nonlinear generalization of the Camassa-Holm equation is investigated. Using the 3 pseudoparabolic regularization technique, its local well-posedness in Sobolev space Hs（R） with s 〉 is established via a limiting procedure. Provided that the initial momentum （1 - 02）u0 satisfies the sign condition,the existence and uniqueness of global solutionsfor the equation are shown to be true in the space C.展开更多
We consider the solution of the good Boussinesq equation Utt -Uxx ＋ Uxxxx = （U2）xx, -∞ 〈 x 〈 ∞, t ≥ 0, with periodic initial value U（x, 0） = ε（μ ＋ φ（x））, Ut（x, 0） = εψ（x）, -∞ 〈 x 〈 ∞, where...We consider the solution of the good Boussinesq equation Utt -Uxx ＋ Uxxxx = （U2）xx, -∞ 〈 x 〈 ∞, t ≥ 0, with periodic initial value U（x, 0） = ε（μ ＋ φ（x））, Ut（x, 0） = εψ（x）, -∞ 〈 x 〈 ∞, where μ = 0, φ（x） and ψ（x） are 2π-periodic functions with 0-average value in [0, 2π], and ε is small. A two parameter Bcklund transformation is found and provide infinite conservation laws for the good Boussinesq equation. The periodic solution is then shown to be uniformly bounded for all small ε, and the H1-norm is uniformly bounded and thus guarantees the global existence. In the case when the initial data is in the simplest form φ（x） = μ＋a sin kx, ψ（x） = b cos kx, an approximation to the solution containing two terms is constructed via the method of multiple scales. By using the energy method, we show that for any given number T 〉 0, the difference between the true solution u（x, t; ε） and the N-th partial sum of the asymptotic series is bounded by εN＋1 multiplied by a constant depending on T and N, for all -∞ 〈 x 〈 ∞, 0 ≤ ｜ε｜t ≤ T and 0 ≤ ｜ε｜≤ε0.展开更多
The anti-periodic traveling wave solutions to a forced two-dimensional generalized KdV-Burgers equation are studied. Some theorems concerning the boundness, existence and uniqueness of the solution to this equation ar...The anti-periodic traveling wave solutions to a forced two-dimensional generalized KdV-Burgers equation are studied. Some theorems concerning the boundness, existence and uniqueness of the solution to this equation are proved.展开更多
In this article, we are concerned with the global weak solutions to the 1D com- pressible viscous hydrodynamic equations with dispersion correction δ2ρ（（φ（ρ））xxφ′（ρ））x with φ（ρ） = ρα. The model co...In this article, we are concerned with the global weak solutions to the 1D com- pressible viscous hydrodynamic equations with dispersion correction δ2ρ（（φ（ρ））xxφ′（ρ））x with φ（ρ） = ρα. The model consists of viscous stabilizations because of quantum Fokker-Planck operator in the Wigner equation and is supplemented with periodic boundary and initial con- ditions. The diffusion term εuxx in the momentum equation may be interpreted as a classical conservative friction term because of particle interactions. We extend the existence result in [1] （α=1/2） to 0 〈 α ≤ 1. In addition, we perform the limit ε→0 with respect to 0 〈 α ≤1/2.展开更多
This paper deals with even order nonlinear differential equations. Applying the Schauder fixed point theorem and the Schwartz inequality technique, we give a criterion to ensure the existence and uniqueness of periodi...This paper deals with even order nonlinear differential equations. Applying the Schauder fixed point theorem and the Schwartz inequality technique, we give a criterion to ensure the existence and uniqueness of periodic solutions. This criterion is a complement of the well known Lazer type results.展开更多
In the present paper we are concerned with the following differential equation with pcriodic coefficientsy’ = f(t,y) = A(t)ym + B(t)y + C(t) (m 2,m ∈N) (1)and obtain the theorems for the existence and the number of ...In the present paper we are concerned with the following differential equation with pcriodic coefficientsy’ = f(t,y) = A(t)ym + B(t)y + C(t) (m 2,m ∈N) (1)and obtain the theorems for the existence and the number of equation (1). Our results extend and improve the corresponding ones in [1,4].展开更多
In this paper we study a periodic two-component Camassa-Holm equation with generalized weakly dissipation. The local well-posedness of Cauchy problem is investigated by utilizing Kato’s theorem. The blow-up criteria ...In this paper we study a periodic two-component Camassa-Holm equation with generalized weakly dissipation. The local well-posedness of Cauchy problem is investigated by utilizing Kato’s theorem. The blow-up criteria and the blow-up rate are established by applying monotonicity. Finally, the global existence results for solutions to the Cauchy problem of equation are proved by structuring functions.展开更多
In this paper, we investigate periodic solutions of neutral linear functional differential equations of arbitrary order with constant coefficients by Fourier serieses. We obtain the necessary and sufficient conditions...In this paper, we investigate periodic solutions of neutral linear functional differential equations of arbitrary order with constant coefficients by Fourier serieses. We obtain the necessary and sufficient conditions for the existence and uniqueness of periodic solutions by finite systems of linear algebraic equations except for one special case. For the special case, we also give some applicable results.展开更多
In the present paper we investigate the number of periodic solutions of the followingdifferential equationwhich is discussed in paper [1,2], and the theorem is obtained by the method of crossratio of the solutions of ...In the present paper we investigate the number of periodic solutions of the followingdifferential equationwhich is discussed in paper [1,2], and the theorem is obtained by the method of crossratio of the solutions of ( ) without the traditional condition assumption that the functions Ai(t), aj(t) (i=1, 2, 3; j = 0, 1, 2) are differential.展开更多
基金Supported by Natural Science Foundation of China (10471047)Natural Science Foundation of Guangdong Province (05300162)
文摘This article studies one dimensional viscous Camassa-Holm equation with a periodic boundary condition. The existence of the almost periodic solution is investigated by using the Galerkin method.
基金supported by the National Natural Science Foundation of China（11371027） the Projects of Outstanding Young Talents of Universities in Anhui Province（gxyq2018116）+2 种基金 the Teaching Groups in Anhui Province（2016jxtd080,2015jxtd048） the NSF of Educational Bureau of Anhui Province（KJ2017A702,KJ2017A704） the NSF of Bozhou University（BZSZKYXM201302,BSKY201539）
文摘In this paper, the existence and uniqueness of almost periodic solutions for some infinite delay integral equations are discussed. By using Krasnoselskii fixed point theorem,some new results are obtained.
文摘This paper deals with the problems on the existence and uniqueness and stability of almost periodic solutions for functional differential equations with infinite delays.The author obtains some sufficient conditions which ganrantee the existence and uniqueness and stability of almost periodic solutions with module containment.The results extend all the results of the paper and solve the two open problems proposed in under much weaker conditions than that proposed in.
基金project is supported by National Natural Science Foundation of China
文摘This paper deals with the uniqueness and existence of periodic solutions of neutral Volterraintegrodifferential equations (1) and (2). Some new unique existence criteria are obtained.
文摘In this paper, the concept of generalized ω-periodic solution is given for Riccati's equationy'=a(t)y^2+b(t)y+c(t)with perriodic coefficients, the relation between generalized ω-periodicsolutions and the characteristic numbers of system x'_1=c(t)x_2, x'_2=-a(t)x_1-b(t)x_2 is indicated, andseveral necessary and sufficient conditions are given using the coefficients. Moreover, in the case of a(t)without zero, the relation between the number of continuous ω-periodic solutions of y'=a(t)y^2+b(t)y+c(t)+δand the parameter δ is given; thus the problem on the existence of continuous ω-periodic.solutions is basically solved.
文摘In this paper, using Fourier series, we study the problem of the existence of periodic solutionsof a type of periodic neutral differential difference system. Some necessary and sufficient conditionsfor the existence of periodic solutions of a type of neutral functional equation system are obtained,and at the same time, we present a method with formula shows how to find the periodicsolutions.
基金The research supported by the National Natural Science Foundation of China.
文摘This paper deals with the existence of positive periodic solutions for a kind of nonautonomous Volterra intergo-differential equations by employing the Krasnoselskii fixed point theorem. Applying the general theorems established to several biomathematical models, the paper improves some previous results and obtains some new results.
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In this paper, we consider a certain fourth-order nonlinear ordinary differential equation. Some sufficient conditions which guarantee the existence of at least one ω-periodic solution to the system are obtain.
文摘In this paper, hy using successive approximation method and fixed-point theorem, we discuss a class of infinite delay integral equation and obtain some sufficient conditions which guarantee the existence and uniqueness of the periodic and almost periodic solutions of the system.
基金Supported by the Applied and Basic Project of Sichuan Province(Grant No.2012JY0020)the Fundamental Research Funds for the Central Universities(Grant No.JBK120504)
文摘A fully nonlinear generalization of the Camassa-Holm equation is investigated. Using the 3 pseudoparabolic regularization technique, its local well-posedness in Sobolev space Hs（R） with s 〉 is established via a limiting procedure. Provided that the initial momentum （1 - 02）u0 satisfies the sign condition,the existence and uniqueness of global solutionsfor the equation are shown to be true in the space C.
基金supported by National Natural Science Foundation of China(10871199)
文摘We consider the solution of the good Boussinesq equation Utt -Uxx ＋ Uxxxx = （U2）xx, -∞ 〈 x 〈 ∞, t ≥ 0, with periodic initial value U（x, 0） = ε（μ ＋ φ（x））, Ut（x, 0） = εψ（x）, -∞ 〈 x 〈 ∞, where μ = 0, φ（x） and ψ（x） are 2π-periodic functions with 0-average value in [0, 2π], and ε is small. A two parameter Bcklund transformation is found and provide infinite conservation laws for the good Boussinesq equation. The periodic solution is then shown to be uniformly bounded for all small ε, and the H1-norm is uniformly bounded and thus guarantees the global existence. In the case when the initial data is in the simplest form φ（x） = μ＋a sin kx, ψ（x） = b cos kx, an approximation to the solution containing two terms is constructed via the method of multiple scales. By using the energy method, we show that for any given number T 〉 0, the difference between the true solution u（x, t; ε） and the N-th partial sum of the asymptotic series is bounded by εN＋1 multiplied by a constant depending on T and N, for all -∞ 〈 x 〈 ∞, 0 ≤ ｜ε｜t ≤ T and 0 ≤ ｜ε｜≤ε0.
文摘The anti-periodic traveling wave solutions to a forced two-dimensional generalized KdV-Burgers equation are studied. Some theorems concerning the boundness, existence and uniqueness of the solution to this equation are proved.
文摘In this article, we are concerned with the global weak solutions to the 1D com- pressible viscous hydrodynamic equations with dispersion correction δ2ρ（（φ（ρ））xxφ′（ρ））x with φ（ρ） = ρα. The model consists of viscous stabilizations because of quantum Fokker-Planck operator in the Wigner equation and is supplemented with periodic boundary and initial con- ditions. The diffusion term εuxx in the momentum equation may be interpreted as a classical conservative friction term because of particle interactions. We extend the existence result in [1] （α=1/2） to 0 〈 α ≤ 1. In addition, we perform the limit ε→0 with respect to 0 〈 α ≤1/2.
文摘This paper deals with even order nonlinear differential equations. Applying the Schauder fixed point theorem and the Schwartz inequality technique, we give a criterion to ensure the existence and uniqueness of periodic solutions. This criterion is a complement of the well known Lazer type results.
文摘In the present paper we are concerned with the following differential equation with pcriodic coefficientsy’ = f(t,y) = A(t)ym + B(t)y + C(t) (m 2,m ∈N) (1)and obtain the theorems for the existence and the number of equation (1). Our results extend and improve the corresponding ones in [1,4].
文摘In this paper we study a periodic two-component Camassa-Holm equation with generalized weakly dissipation. The local well-posedness of Cauchy problem is investigated by utilizing Kato’s theorem. The blow-up criteria and the blow-up rate are established by applying monotonicity. Finally, the global existence results for solutions to the Cauchy problem of equation are proved by structuring functions.
文摘In this paper, we investigate periodic solutions of neutral linear functional differential equations of arbitrary order with constant coefficients by Fourier serieses. We obtain the necessary and sufficient conditions for the existence and uniqueness of periodic solutions by finite systems of linear algebraic equations except for one special case. For the special case, we also give some applicable results.
文摘In the present paper we investigate the number of periodic solutions of the followingdifferential equationwhich is discussed in paper [1,2], and the theorem is obtained by the method of crossratio of the solutions of ( ) without the traditional condition assumption that the functions Ai(t), aj(t) (i=1, 2, 3; j = 0, 1, 2) are differential.
