Consider the second order nonlinear neutral difference equationThe sufficient conditions are established for the oscillation and asymptotic behavior of the solutions of this equation.
In this paper, we obtain some new Razumikhin type theorems of stability andboundedness for functional differential equations with infinite delay. Under thecondition of V((ξ)) V(t), we substitute the requirement sati...In this paper, we obtain some new Razumikhin type theorems of stability andboundedness for functional differential equations with infinite delay. Under thecondition of V((ξ)) V(t), we substitute the requirement satisfying V 0 insome sets of points {(t, x)|V(t, x) = αi or βi, i = 1, 2,...} for the requirementV 0 in classical theorems of stability and boundedness (for reference, see[1]-[3]).展开更多
This paper is a brief survey of our recent study on the connection between two parts of Hilbert′s 16th problem and equivariant bifurcation problem. We hope to understand the following questions: can we use the period...This paper is a brief survey of our recent study on the connection between two parts of Hilbert′s 16th problem and equivariant bifurcation problem. We hope to understand the following questions: can we use the periodic solution family of ( m-1) degree planar Hamiltonian systems with Z q equivariant (or D q equivariant) symmetry to realize some schemes of ovals for planar algebraic curves? On the contrary, if an algebraic curve of degree m has maximal number of branches of ovals (it is called M -curve), can we make his all ovals become limit cycles of a planar polynomial system? What schemes of distribution of limit cycles can be realized by polynomial system.展开更多
In this paper, we investigate the complex oscillation of the higher order differential equation where B0, ...,Bk-1,,F 0 are transcendental meromorpic functions having only finitely many poles. We obtain some precise e...In this paper, we investigate the complex oscillation of the higher order differential equation where B0, ...,Bk-1,,F 0 are transcendental meromorpic functions having only finitely many poles. We obtain some precise estimates of the exponent of convergence of the zero sequence of meromorphic solutions for the above equation.展开更多
We study the boundary value problem for the vector system which is equivalent to a singular singularly-perturbed boundary value problem involving a slow variable. Under appropriate assumptions, we obtain that the asym...We study the boundary value problem for the vector system which is equivalent to a singular singularly-perturbed boundary value problem involving a slow variable. Under appropriate assumptions, we obtain that the asymptotic expansion of a solution is uniformly valid on a finit interval. Meanwhile, we find an intrinsic relation between a solution of Riccati equations in the technique of diagnolization and an invariant manifold a boundary layer.MSC: 34E15展开更多
In this paper, Lyapunov-like exponential stability and unstability of differentialalgebraic equation are considered from the viewpoint of stability of system motion, and the criteria of exponential stability and unsta...In this paper, Lyapunov-like exponential stability and unstability of differentialalgebraic equation are considered from the viewpoint of stability of system motion, and the criteria of exponential stability and unstability of nonlinear nonautonomous differential-algebraic equation are given by using Lyapunov-like function similar to ordinary differential equation.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
In this paper, we establish the existence of positive radially symmetric solutions of div(|Du|p-2Du) + λf(r,u(r) ) = 0 in domain R1 < r < R0 or 0 < r < ∞ with a variety of Dirichlet boundary conditions. ...In this paper, we establish the existence of positive radially symmetric solutions of div(|Du|p-2Du) + λf(r,u(r) ) = 0 in domain R1 < r < R0 or 0 < r < ∞ with a variety of Dirichlet boundary conditions. The function f is allowed to be singular when u = 0.展开更多
In this paper, we are concerned with the maximum number of linearly independent transcendental solutions with finite exponent of convergence of the zeros for a higher order homogeneous linear differential equation whe...In this paper, we are concerned with the maximum number of linearly independent transcendental solutions with finite exponent of convergence of the zeros for a higher order homogeneous linear differential equation where its coefficients are entire functions with order less than 1/2 and one dominant. The result obtained here is an extension and a complement of J. K. Langley's.展开更多
This paper considers absolute stability of control systems with superposition of nonlinear elements in the case of infinite sector. Necessary and sufficient conditions are obtained for the existence of Lyapunov funct...This paper considers absolute stability of control systems with superposition of nonlinear elements in the case of infinite sector. Necessary and sufficient conditions are obtained for the existence of Lyapunov function in the class of quadratic forms augmented by integrals of nonlinear functions.展开更多
This paper outlines some results on the degenerate differential systems with delays and illustrates the importance of studying such systems. We aslo present some problems to bring about people′s interest.
This paper investigates a volterra predator-prey model with undercrowding effect:x =ax2(K-x)-bxy, y=- cy + dxy-ay2. One can transform it into u=u[u(1- u)-v], v =-γv(1-mu+nv). Supercritical Hopf bifurcation takes plac...This paper investigates a volterra predator-prey model with undercrowding effect:x =ax2(K-x)-bxy, y=- cy + dxy-ay2. One can transform it into u=u[u(1- u)-v], v =-γv(1-mu+nv). Supercritical Hopf bifurcation takes place at a certain value m, of the parameter m. If the positive equilibrium E is locally asymptotically stable,then it must be globally stable. If E is locally unstable, then there exists unique stable limit cycle around E. Throughout, the main interest is in results yielding explicit dependence on the parameters involved.展开更多
Some new integral and discrete inequalities with power nonlinearity are established. Two recent results of B.G. Pachpatte are improved by some particularcorollaries of our results. Application examples are also indica...Some new integral and discrete inequalities with power nonlinearity are established. Two recent results of B.G. Pachpatte are improved by some particularcorollaries of our results. Application examples are also indicated.展开更多
This paper is devoted to study the following dynamical system about chemical reaction with saturated output By using qualitative theory of ordinary differential equations, we have completely discussed the existence, n...This paper is devoted to study the following dynamical system about chemical reaction with saturated output By using qualitative theory of ordinary differential equations, we have completely discussed the existence, nonexistence and uniqueness of limit cycle of system (S).展开更多
In this paper, we discretize the Hénon-Heiles Hamiltonian system with Dirichlet boundary condition via spatial variable, and prove the existence of absorbing sets and global attractor of discrete system.
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.
文摘In this paper, we obtain some new Razumikhin type theorems of stability andboundedness for functional differential equations with infinite delay. Under thecondition of V((ξ)) V(t), we substitute the requirement satisfying V 0 insome sets of points {(t, x)|V(t, x) = αi or βi, i = 1, 2,...} for the requirementV 0 in classical theorems of stability and boundedness (for reference, see[1]-[3]).
文摘This paper is a brief survey of our recent study on the connection between two parts of Hilbert′s 16th problem and equivariant bifurcation problem. We hope to understand the following questions: can we use the periodic solution family of ( m-1) degree planar Hamiltonian systems with Z q equivariant (or D q equivariant) symmetry to realize some schemes of ovals for planar algebraic curves? On the contrary, if an algebraic curve of degree m has maximal number of branches of ovals (it is called M -curve), can we make his all ovals become limit cycles of a planar polynomial system? What schemes of distribution of limit cycles can be realized by polynomial system.
文摘In this paper, we investigate the complex oscillation of the higher order differential equation where B0, ...,Bk-1,,F 0 are transcendental meromorpic functions having only finitely many poles. We obtain some precise estimates of the exponent of convergence of the zero sequence of meromorphic solutions for the above equation.
文摘We study the boundary value problem for the vector system which is equivalent to a singular singularly-perturbed boundary value problem involving a slow variable. Under appropriate assumptions, we obtain that the asymptotic expansion of a solution is uniformly valid on a finit interval. Meanwhile, we find an intrinsic relation between a solution of Riccati equations in the technique of diagnolization and an invariant manifold a boundary layer.MSC: 34E15
文摘In this paper, Lyapunov-like exponential stability and unstability of differentialalgebraic equation are considered from the viewpoint of stability of system motion, and the criteria of exponential stability and unstability of nonlinear nonautonomous differential-algebraic equation are given by using Lyapunov-like function similar to ordinary differential equation.
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In this paper, we establish the existence of positive radially symmetric solutions of div(|Du|p-2Du) + λf(r,u(r) ) = 0 in domain R1 < r < R0 or 0 < r < ∞ with a variety of Dirichlet boundary conditions. The function f is allowed to be singular when u = 0.
文摘In this paper, we are concerned with the maximum number of linearly independent transcendental solutions with finite exponent of convergence of the zeros for a higher order homogeneous linear differential equation where its coefficients are entire functions with order less than 1/2 and one dominant. The result obtained here is an extension and a complement of J. K. Langley's.
文摘This paper considers absolute stability of control systems with superposition of nonlinear elements in the case of infinite sector. Necessary and sufficient conditions are obtained for the existence of Lyapunov function in the class of quadratic forms augmented by integrals of nonlinear functions.
文摘This paper outlines some results on the degenerate differential systems with delays and illustrates the importance of studying such systems. We aslo present some problems to bring about people′s interest.
文摘This paper investigates a volterra predator-prey model with undercrowding effect:x =ax2(K-x)-bxy, y=- cy + dxy-ay2. One can transform it into u=u[u(1- u)-v], v =-γv(1-mu+nv). Supercritical Hopf bifurcation takes place at a certain value m, of the parameter m. If the positive equilibrium E is locally asymptotically stable,then it must be globally stable. If E is locally unstable, then there exists unique stable limit cycle around E. Throughout, the main interest is in results yielding explicit dependence on the parameters involved.
文摘Some new integral and discrete inequalities with power nonlinearity are established. Two recent results of B.G. Pachpatte are improved by some particularcorollaries of our results. Application examples are also indicated.
文摘This paper is devoted to study the following dynamical system about chemical reaction with saturated output By using qualitative theory of ordinary differential equations, we have completely discussed the existence, nonexistence and uniqueness of limit cycle of system (S).
文摘In this paper, we discretize the Hénon-Heiles Hamiltonian system with Dirichlet boundary condition via spatial variable, and prove the existence of absorbing sets and global attractor of discrete system.
