This paper is concerned with the existence of nontrivial homoclinic solutions for a class of second order Hamiltonian systems with external forc-ing perturbations q+Aq+Vq(t,q)=f(t),where q=(q1,q2,..qN)∈R^(N),A is an ...This paper is concerned with the existence of nontrivial homoclinic solutions for a class of second order Hamiltonian systems with external forc-ing perturbations q+Aq+Vq(t,q)=f(t),where q=(q1,q2,..qN)∈R^(N),A is an antisymmetric constant N×N matrix,V(t,q)=-K(t,q)+W(t,q)with K,W ∈C^(1)(R,R^(N))and satisfying b1|q|^(2)≤K(t,q)≤b_(2)|q|^(2)for some positive constants b_(2)≥b_(1)>0 and external forcing term f∈C(R,R^(N))being small enough.Under some new weak superquadratic conditions for W,by using the mountain pass theorem,we obtain the existence of at least one nontrivial homoclinic solution.展开更多
By means of an extension of Mawhin＇s continuation theorem and some analysis methods, the existence of a set with 2kT-periodic solutions for a class of second order neutral functional differential systems is studied, ...By means of an extension of Mawhin＇s continuation theorem and some analysis methods, the existence of a set with 2kT-periodic solutions for a class of second order neutral functional differential systems is studied, and then the homoclinic solutions are obtained as the limit points of a certain subsequence of the above set.展开更多
This paper mainly discusses the existence of nontrivial homoclinic solutions for nonperiodic semilinear fourth-order ordinary differential equation u^（4）＋pu″＋a（x）u-b（x）u^2=c（x）u^3=3arising in the study of p...This paper mainly discusses the existence of nontrivial homoclinic solutions for nonperiodic semilinear fourth-order ordinary differential equation u^（4）＋pu″＋a（x）u-b（x）u^2=c（x）u^3=3arising in the study of pattern formation by means of Mountain Pass Lemma.展开更多
The existence of homoclinic solutions for the second-order p-Laplacian differential system( ρ( t) Φp( u'( t))) '-s( t) Φp( u( t))+ λf( t,u( t)) = 0 with impulsive effects Δ( ρ( tj) Φp( u'( tj))) = I...The existence of homoclinic solutions for the second-order p-Laplacian differential system( ρ( t) Φp( u'( t))) '-s( t) Φp( u( t))+ λf( t,u( t)) = 0 with impulsive effects Δ( ρ( tj) Φp( u'( tj))) = Ij( u( tj)) is studied. By using three critical points theorem and variational methods, the sufficient condition is established to guarantee that this p-Laplacian differential system with impulsive effects has at least one nontrivial homoclinic solution. Besides,an example is presented to illustrate the main result in the end of this paper.展开更多
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 paper,we study the existence of infinitely many homoclinic solutions for a class of first order Hamiltonian systems ż=J H_(z)(t,z),where the Hamiltonian function H possesses the form H(t,z)=1/2L(t)z⋅z+G(t,z),a...In this paper,we study the existence of infinitely many homoclinic solutions for a class of first order Hamiltonian systems ż=J H_(z)(t,z),where the Hamiltonian function H possesses the form H(t,z)=1/2L(t)z⋅z+G(t,z),and G(t,z)is only locally defined near the origin with respect to z.Under some mild conditions on L and G,we show that the existence of a sequence of homoclinic solutions is actually a local phenomenon in some sense,which is essentially forced by the subquadraticity of G near the origin with respect to z.展开更多
A generalized hyperbolic perturbation method is presented for homoclinic solutions of strongly nonlinear autonomous oscillators, in which the perturbation proce- dure is improved for those systems whose exact homoclin...A generalized hyperbolic perturbation method is presented for homoclinic solutions of strongly nonlinear autonomous oscillators, in which the perturbation proce- dure is improved for those systems whose exact homoclinic generating solutions cannot be explicitly derived. The generalized hyperbolic functions are employed as the basis functions in the present procedure to extend the validity of the hyperbolic perturbation method. Several strongly nonlinear oscillators with quadratic, cubic, and quartic nonlinearity are studied in detail to illustrate the efficiency and accuracy of the present method.展开更多
Homoclinic solutions were introduced for a prescribed mean curvature Lienard p-Laplacian equation with a deviating argument.It was divided into three parts to discuss the existence of homoclinic solutions.By using an ...Homoclinic solutions were introduced for a prescribed mean curvature Lienard p-Laplacian equation with a deviating argument.It was divided into three parts to discuss the existence of homoclinic solutions.By using an extension of Mawhin's continuation theorem,the existence of a set with 2kT-periodic for a prescribed mean curvature Lienard p-Laplacian equation with a deviating argument was studied.According to a limit on a certain subsequence of 2kT-periodic set,homoclinic solutions were obtained.A numerical example demonstrates the validity of the main results.展开更多
In this paper, by using Mawhin＇s continuation theorem and some analysis methods,the existence of a set with 2kT-periodic solutions for a kind of prescribed mean curvature equation with a deviating argument is studied...In this paper, by using Mawhin＇s continuation theorem and some analysis methods,the existence of a set with 2kT-periodic solutions for a kind of prescribed mean curvature equation with a deviating argument is studied, and then a homoclinic solution is obtained as a limit of a certain subsequence of the above set.展开更多
In this paper,we study the existence and multiplicity of homoclinic solutions for a class of second-order Hamiltonian system:u″(t)−L(t)u(t)+▽V(t,u)=0,where L(t)and V(t,u)are not periodic in t.First,we introduce the ...In this paper,we study the existence and multiplicity of homoclinic solutions for a class of second-order Hamiltonian system:u″(t)−L(t)u(t)+▽V(t,u)=0,where L(t)and V(t,u)are not periodic in t.First,we introduce the definition of index and establish the corresponding index theory.Then,by using the index theory and critical point theory,we prove our main results under the asymptotic quadratic conditions of the potential function.展开更多
In this paper,we investigate a class of reversible dynamical systems in four dimensions.The spectrums of their linear operators at the equilibria are assumed to have a pair of positive and negative real eigenvalues an...In this paper,we investigate a class of reversible dynamical systems in four dimensions.The spectrums of their linear operators at the equilibria are assumed to have a pair of positive and negative real eigenvalues and a pair of purely imaginary eigenvalues for the small parameterμ>0,where these two real eigenvalues bifurcate from a double eigenvalue 0 forμ=0.It has been shown that this class of systems owns a generalized homoclinic solution with one hump at the center(a homoclinic solution exponentially approaching a periodic solution with a small amplitude).This paper gives a rigorous existence proof of two-hump solutions.These two humps are far away and are glued by the small oscillations in the middle if some appropriate free constants are activated.The obtained results are also applied to some classical systems.The ideas here may be used to study the existence of 2^(k)-hump solutions.展开更多
In this paper,we consider the following perturbed fractional Hamiltonian systems{tD_(∞)^(α)(_(-∞)D_(t)^(α)u(t))+L(t)u(t)=■_(u)W(t,u(t))+■(u)G(t,u(t)),t∈R,u∈H^(α)(R,R^(N)),whereα∈(1/2,1],L∈C(R,R^(N×N))...In this paper,we consider the following perturbed fractional Hamiltonian systems{tD_(∞)^(α)(_(-∞)D_(t)^(α)u(t))+L(t)u(t)=■_(u)W(t,u(t))+■(u)G(t,u(t)),t∈R,u∈H^(α)(R,R^(N)),whereα∈(1/2,1],L∈C(R,R^(N×N))is symmetric and not necessarily required to be positive definite,W∈C1(R×R^(N,R))is locally subquadratic and locally even near the origin,and perturbed term G∈C1(R×R^(N,R))maybe has no parity in u.Utilizing the perturbed method improved by the authors,a sequence of nontrivial homo clinic solutions is obtained,which generalizes previous results.展开更多
In this paper, we consider the existence of homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity. The classical Ambrosetti–Rabinowitz superlinear condition is improved by a ge...In this paper, we consider the existence of homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity. The classical Ambrosetti–Rabinowitz superlinear condition is improved by a general superlinear one. The proof is based on the critical point theory in combination with periodic approximations of solutions.展开更多
In this paper, a periodic difference equation with saturable nonlinearity is considered. Using the linking theorem in combination with periodic approximations, we establish sufficient conditions on the nonexistence an...In this paper, a periodic difference equation with saturable nonlinearity is considered. Using the linking theorem in combination with periodic approximations, we establish sufficient conditions on the nonexistence and on the existence of homoclinic solutions. Our results not only solve an open problem proposed by Pankov, but also greatly improve some existing ones even for some special cases.展开更多
By means of Mawhin＇s continuation theorem and some analysis methods, the existence of 2kT-periodic solutions is studied for a class of neutral functional differential equations, and then a homoclinic solution is obta...By means of Mawhin＇s continuation theorem and some analysis methods, the existence of 2kT-periodic solutions is studied for a class of neutral functional differential equations, and then a homoclinic solution is obtained as a limit of a certain subsequence of the above periodic solutions set.展开更多
In this paper, we investigate a non-autonomous second-order p-Laplacian system. Based on critical point theory, we discuss the existence of homoclinic orbits of the system.
In this paper,the multiplicity of homoclinic solutions for second order non-autonomous Hamiltonian systemsü(t)-L(t)u(t)+▽uW(t,u(t))=0 is obtained via a new Symmetric Mountain Pass Lemma established by Kajikiya,w...In this paper,the multiplicity of homoclinic solutions for second order non-autonomous Hamiltonian systemsü(t)-L(t)u(t)+▽uW(t,u(t))=0 is obtained via a new Symmetric Mountain Pass Lemma established by Kajikiya,where L∈C(R,RN×N)is symmetric but non-periodic,W∈C1(R×RN,R)is locally even in u and only satisfies some growth conditions near u=0,which improves some previous results.展开更多
This paper considers the steady Swift-Hohenberg equation u＇＇＇＋β2u＇＇＋u^3-u=0.Using the dynamic approach, the authors prove that it has a homoclinic solution for each β∈ （4√8-ε0,4 √8）, where ε0 is a smal...This paper considers the steady Swift-Hohenberg equation u＇＇＇＋β2u＇＇＋u^3-u=0.Using the dynamic approach, the authors prove that it has a homoclinic solution for each β∈ （4√8-ε0,4 √8）, where ε0 is a small positive constant. This slightly complements Santra and Wei＇s result [Santra, S. and Wei, J., Homoclinic solutions for fourth order traveling wave equations, SIAM J. Math. Anal., 41, 2009, 2038-2056], which stated that it admits a homoclinic solution for each β∈C （0,β0） where β0 = 0.9342 ....展开更多
By using the theory of exponential dichotomies and the method of Liapunov-Schmidt,in this paper we investigate the existence of homoclinic solutions for autonomous differential equations and obtain a Melnikov-type vec...By using the theory of exponential dichotomies and the method of Liapunov-Schmidt,in this paper we investigate the existence of homoclinic solutions for autonomous differential equations and obtain a Melnikov-type vector. We show that if the Melnikov-type vector satisfies some conditions then autonomous differential equations have homoclinic solutions. Moreover, our result holds only for autonomous differential equations.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.12171253).
文摘This paper is concerned with the existence of nontrivial homoclinic solutions for a class of second order Hamiltonian systems with external forc-ing perturbations q+Aq+Vq(t,q)=f(t),where q=(q1,q2,..qN)∈R^(N),A is an antisymmetric constant N×N matrix,V(t,q)=-K(t,q)+W(t,q)with K,W ∈C^(1)(R,R^(N))and satisfying b1|q|^(2)≤K(t,q)≤b_(2)|q|^(2)for some positive constants b_(2)≥b_(1)>0 and external forcing term f∈C(R,R^(N))being small enough.Under some new weak superquadratic conditions for W,by using the mountain pass theorem,we obtain the existence of at least one nontrivial homoclinic solution.
基金sponsored by the National Natural Science Foundation of China(11271197)the Science and Technology Foundation in Ministry of Education of China(207047)the Science Foundation of NUIST of China(20090202 and 2012r101)
文摘By means of an extension of Mawhin＇s continuation theorem and some analysis methods, the existence of a set with 2kT-periodic solutions for a class of second order neutral functional differential systems is studied, and then the homoclinic solutions are obtained as the limit points of a certain subsequence of the above set.
基金The Project sponsored by SRF for ROCS, SEM"985 Engineer" of China (CUN 985-3-3)
文摘This paper mainly discusses the existence of nontrivial homoclinic solutions for nonperiodic semilinear fourth-order ordinary differential equation u^（4）＋pu″＋a（x）u-b（x）u^2=c（x）u^3=3arising in the study of pattern formation by means of Mountain Pass Lemma.
基金National Natural Science Foundations of China(No.11271371,No.10971229)
文摘The existence of homoclinic solutions for the second-order p-Laplacian differential system( ρ( t) Φp( u'( t))) '-s( t) Φp( u( t))+ λf( t,u( t)) = 0 with impulsive effects Δ( ρ( tj) Φp( u'( tj))) = Ij( u( tj)) is studied. By using three critical points theorem and variational methods, the sufficient condition is established to guarantee that this p-Laplacian differential system with impulsive effects has at least one nontrivial homoclinic solution. Besides,an example is presented to illustrate the main result in the end of this paper.
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The first author was supported by the National Natural Science Foundation of China(11761036,11201196)the Natural Science Foundation of Jiangxi Province(20171BAB211002)+3 种基金The second author was supported by the National Natural Science Foundation of China(11790271,12171108)the Guangdong Basic and Applied basic Research Foundation(2020A1515011019)the Innovation and Development Project of Guangzhou Universitythe Nankai Zhide Foundation。
文摘In this paper,we study the existence of infinitely many homoclinic solutions for a class of first order Hamiltonian systems ż=J H_(z)(t,z),where the Hamiltonian function H possesses the form H(t,z)=1/2L(t)z⋅z+G(t,z),and G(t,z)is only locally defined near the origin with respect to z.Under some mild conditions on L and G,we show that the existence of a sequence of homoclinic solutions is actually a local phenomenon in some sense,which is essentially forced by the subquadraticity of G near the origin with respect to z.
基金Project supported by the National Natural Science Foundation of China (Nos. 10972240 and 11102045)the Natural Science Foundation of Guangdong Province of China (No. S20110400040)+2 种基金the Foundation of Guangdong Education Department of China (No. LYM10108)the Foundation of Guangzhou Education Bureau of China (No. 10A024)the Research Grant Council of Hong Kong of China (No. GRF-HKU-7173-09E)
文摘A generalized hyperbolic perturbation method is presented for homoclinic solutions of strongly nonlinear autonomous oscillators, in which the perturbation proce- dure is improved for those systems whose exact homoclinic generating solutions cannot be explicitly derived. The generalized hyperbolic functions are employed as the basis functions in the present procedure to extend the validity of the hyperbolic perturbation method. Several strongly nonlinear oscillators with quadratic, cubic, and quartic nonlinearity are studied in detail to illustrate the efficiency and accuracy of the present method.
基金Natural Foundation of Wuyi University,China(No.XQ201305)Young-Middle-Aged Teachers Education Scientific Research Project in Fujian Province,China(No.JA15524)
文摘Homoclinic solutions were introduced for a prescribed mean curvature Lienard p-Laplacian equation with a deviating argument.It was divided into three parts to discuss the existence of homoclinic solutions.By using an extension of Mawhin's continuation theorem,the existence of a set with 2kT-periodic for a prescribed mean curvature Lienard p-Laplacian equation with a deviating argument was studied.According to a limit on a certain subsequence of 2kT-periodic set,homoclinic solutions were obtained.A numerical example demonstrates the validity of the main results.
基金Supported by the National Natural Science Foundation of China(Grant No.11271197)
文摘In this paper, by using Mawhin＇s continuation theorem and some analysis methods,the existence of a set with 2kT-periodic solutions for a kind of prescribed mean curvature equation with a deviating argument is studied, and then a homoclinic solution is obtained as a limit of a certain subsequence of the above set.
基金Supported by the Fundamental Research Funds for the Central Universities(Grant No.2019XKQYMS91)。
文摘In this paper,we study the existence and multiplicity of homoclinic solutions for a class of second-order Hamiltonian system:u″(t)−L(t)u(t)+▽V(t,u)=0,where L(t)and V(t,u)are not periodic in t.First,we introduce the definition of index and establish the corresponding index theory.Then,by using the index theory and critical point theory,we prove our main results under the asymptotic quadratic conditions of the potential function.
基金supported by National Natural Science Foundation of China(Grant No.12171171)Natural Science Foundation of Fujian Province of China(Grant Nos.2022J01303 and 2023J01121)the Scientific Research Funds of Huaqiao University。
文摘In this paper,we investigate a class of reversible dynamical systems in four dimensions.The spectrums of their linear operators at the equilibria are assumed to have a pair of positive and negative real eigenvalues and a pair of purely imaginary eigenvalues for the small parameterμ>0,where these two real eigenvalues bifurcate from a double eigenvalue 0 forμ=0.It has been shown that this class of systems owns a generalized homoclinic solution with one hump at the center(a homoclinic solution exponentially approaching a periodic solution with a small amplitude).This paper gives a rigorous existence proof of two-hump solutions.These two humps are far away and are glued by the small oscillations in the middle if some appropriate free constants are activated.The obtained results are also applied to some classical systems.The ideas here may be used to study the existence of 2^(k)-hump solutions.
基金Supported by National Natural Science Foundation of China(Grant No.12171355)Elite Scholar Program in Tianjin University,P.R.China。
文摘In this paper,we consider the following perturbed fractional Hamiltonian systems{tD_(∞)^(α)(_(-∞)D_(t)^(α)u(t))+L(t)u(t)=■_(u)W(t,u(t))+■(u)G(t,u(t)),t∈R,u∈H^(α)(R,R^(N)),whereα∈(1/2,1],L∈C(R,R^(N×N))is symmetric and not necessarily required to be positive definite,W∈C1(R×R^(N,R))is locally subquadratic and locally even near the origin,and perturbed term G∈C1(R×R^(N,R))maybe has no parity in u.Utilizing the perturbed method improved by the authors,a sequence of nontrivial homo clinic solutions is obtained,which generalizes previous results.
基金Supported by Program for Changjiang Scholars and Innovative Research Team in University (Grant No.IRT1226)National Natural Science Foundation of China (Grant Nos. 11171078 and 11031002)the Specialized Fund for the Doctoral Program of Higher Education of China (Grant No. 20114410110002)
文摘In this paper, we consider the existence of homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity. The classical Ambrosetti–Rabinowitz superlinear condition is improved by a general superlinear one. The proof is based on the critical point theory in combination with periodic approximations of solutions.
基金supported partially by the Specialized Fund for the Doctoral Program of Higher Eduction (Grant No.20071078001)Key Project of National Natural Science Foundation of China (Grant No. 11031002)+1 种基金Natural Science and Engineering Research Council of Canada (NSERC)Project of Scientific Research Innovation Academic Group for the Education System of Guangzhou City
文摘In this paper, a periodic difference equation with saturable nonlinearity is considered. Using the linking theorem in combination with periodic approximations, we establish sufficient conditions on the nonexistence and on the existence of homoclinic solutions. Our results not only solve an open problem proposed by Pankov, but also greatly improve some existing ones even for some special cases.
基金supported by National Natural Science Foundation of China (Grant No. 10771215)the Scientific Research Fund of Hu’nan Provincial Education Department (Grant No. 08A053)
文摘Consider the second order discrete Hamiltonian systems
基金Supported by the key NSF of Education Ministry of China (Grant No. 207047)
文摘By means of Mawhin＇s continuation theorem and some analysis methods, the existence of 2kT-periodic solutions is studied for a class of neutral functional differential equations, and then a homoclinic solution is obtained as a limit of a certain subsequence of the above periodic solutions set.
基金supported by NNSF of China Project(No.11326124)Education Department of Henan Province Project(No.14A110002)
文摘In this paper, we investigate a non-autonomous second-order p-Laplacian system. Based on critical point theory, we discuss the existence of homoclinic orbits of the system.
基金National Natural Science Foundation of China(Grant No.10901118)Elite Scholar Program in Tianjin University,P.R.China。
文摘In this paper,the multiplicity of homoclinic solutions for second order non-autonomous Hamiltonian systemsü(t)-L(t)u(t)+▽uW(t,u(t))=0 is obtained via a new Symmetric Mountain Pass Lemma established by Kajikiya,where L∈C(R,RN×N)is symmetric but non-periodic,W∈C1(R×RN,R)is locally even in u and only satisfies some growth conditions near u=0,which improves some previous results.
基金supported by the PhD Start-up Fund of the Natural Science Foundation of Guangdong Province(No.S2011040000464)the Project of Department of Education of Guangdong Province(No.2012KJCX0074)+3 种基金the China Postdoctoral Science Foundation.Special Project(No.201104077)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry(No.(2012)940)the Natural Fund of Zhanjiang Normal University(No.LZL1101)the Doctoral Project of Zhanjiang Normal University(No.ZL1109)
文摘This paper considers the steady Swift-Hohenberg equation u＇＇＇＋β2u＇＇＋u^3-u=0.Using the dynamic approach, the authors prove that it has a homoclinic solution for each β∈ （4√8-ε0,4 √8）, where ε0 is a small positive constant. This slightly complements Santra and Wei＇s result [Santra, S. and Wei, J., Homoclinic solutions for fourth order traveling wave equations, SIAM J. Math. Anal., 41, 2009, 2038-2056], which stated that it admits a homoclinic solution for each β∈C （0,β0） where β0 = 0.9342 ....
文摘By using the theory of exponential dichotomies and the method of Liapunov-Schmidt,in this paper we investigate the existence of homoclinic solutions for autonomous differential equations and obtain a Melnikov-type vector. We show that if the Melnikov-type vector satisfies some conditions then autonomous differential equations have homoclinic solutions. Moreover, our result holds only for autonomous differential equations.
