Existence of exactlyn+1 simple 4-periodic solutions of the differential delay equation\dot x(t) = - f(x(t - 1))

By discussing the zeros of periodic solutions we give in this paper a criterion for the existence of exactlyn+1 simple 4-periodic solutions of the differential delay equation Supported by the Chinese National Foundation for Natural Sciences.     

Boundedness of solutions for sublinear reversible systems

We are concerned with the boundedness of all the solutions for second order differential equation

On the structure of the spectrum of a linear time periodic wave equation

In this paper we study the structure of the spectrum of the evolution operator generated by the time periodic wave equation with 2π periodicity inx as boundary condition. We show that such structure depends heavily on the rationality or irrationality of the rotation numbers of the characteristics deiffential equations as well as on the phase space.     

Existence of Minimizers for Nonconvex Variational Problems with Slow Growth

Consider the minimization problem

On the controllability of convex processes with delay

We study the controllability of differential-difference inclusions , where the graph of the multifunctionA(t, ·) is a closed convex cone.     

Existence,uniqueness, and global attractivity of positive solutions and MLE of the parameters to the logistic equation with random perturbation

This paper discusses a randomized Logistic equation N(t)=(r αB(t))N(t)[1-(N(t)/K)] with an initial value N(0)=N_0,and N_0 is a random variable satisfying 0＜N_0＜K.The existence, uniqueness and global attractivity of positive solutions and maximum likelihood estimate(MLE)of the parameters of the equation are studied.     

Convergence of solutions of nonlinear differential equations

Summary Solutions of are said to converge if every pair of solutions x(t), y(t) satisfy x(t) − y(t) →0 as t → ∞. An invariance principle of LaSalle is used to determine conditions under which the solutions of converge. In certain cases the approach used does not require boundedness of solutions as has been required in most previous results on convergence of solutions. The results of this investigation are applied to a number of nonlinear second order differential equations. Sufficient conditions are also found for the convergence of solutions of certain functional differential equations. Entrata in Redazione il 10 febbraio 1976.     

Global dynamics of delay differential equations

In this survey paper the delay differential equation (t) = −μx(t) + g(x(t − 1)) is considered with μ ≥ 0 and a smooth real function g satisfying g(0) = 0. It is shown that the dynamics generated by this simple-looking equation can be very rich. The dynamics is completely understood only for a small class of nonlinearities. Open problems are formulated. Supported in part by the Hungarian NFSR, Grant No. T049516.     

声波导中的局部正交变换

In this paper, a local orthogonal transformation is created to transform the Helmholtz waveguide with curved interface to the one with a flat interface within the two-layer medium, and the Helmholtz equation uxx uzz k^2(x,z)u=0 is transformed to Vxx αVzz βVz γV=0. Numerical results demonstrate that the transformation is more feasible. This transformation is particularly useful for the research on wave propagation in acoustic waveguide.     

A problem of adaptive control with standard model

We consider the problem of adaptive control by a linear stochastic system with lag

Smoluchowski-Kramers approximation for a general class of SPDEs

We prove that the so-called Smoluchowski-Kramers approximation holds for a class of partial differential equations perturbed by a non-Gaussian noisy term. Namely, we show that the solution of the one-dimensional semi-linear stochastic damped wave equations , u(0) = u₀, u_t (0) = v₀, endowed with Dirichlet boundary conditions, converges as the parameter μ goes to zero to the solution of the semi-linear stochastic heat equation , u(0) = u₀, endowed with Dirichlet boundary conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Positive solutions for some 1-dimensional boundary value problems of Laplace-type

This paper deals with the existence of triple positive solutions for the 1-dimensional equation of Laplace-type(φ(x'(t)))' q(t)f(t,x(t),x'(t)) = 0, t ∈ (0, 1),subject to the following boundary condition:a1φ(x(0)) - a2φ(x'(0)) = 0, a3φ(x(1)) a4φ(x'(1)) = 0,where φ is an odd increasing homogeneous homeomorphism. By using a new fixed point theorem,sufficient conditions are obtained that guarantee the existence of at least three positive solutions. The emphasis here is that the nonlinear term f is involved with the first order derivative explicitly.     

Criteria for the coincidence of the kernel of a function with the kernels of its Riesz and Abel integral means

We indicate criteria for the coincidence of the Knopp kernels K(f) K(A _f), and K (R _f) of bounded functions f(t); here,

Propagation of oscillations to 2D incompressible Euler equations

The asymptotic expansions are studied for the vorticity to 2D incompressible Euler equations with-initial vorticity , where ϕ₀(x) satisfies |d ϕ₀(x)|≠0 on the support of and is sufficiently smooth and with compact support in ℝ² (resp. ℝ²×T) The limit,v(t,x), of the corresponding velocity fields {v ^ɛ(t,x)} is obtained, which is the unique solution of (E) with initial vorticity ω₀(x). Moreover, (ℤ²)) for all 1≽p∞, where and ϕ(t,x) satisfy some modulation equation and eikonal equation, respectively.     

On the existence of periodic solutions for the third-order nonlinear ordinary differential equations

In this paper,the existence of periodic solution for the third-order nonlinear ordinarydifferential equation of the form -^x f(-^x) g(-^x) h(x)=p(t) is consldered,where f,g,h andp are the continuous functlons, and p (t T) =p (t). By using the Leray-Schauder degreemethod,some sufficient conditions to guarantee the existence of T periodic solution for the equation are obtained. Some previous results are extended and improved.     

Some properties of essential spectra of a positive operator, II

Let T be a positive operator on a Banach lattice E. Some properties of Weyl essential spectrum σ_ew(T), in particular, the equality , where is the set of all compact operators on E, are established. If r(T) does not belong to Fredholm essential spectrum σ_ef(T), then for every a ≠ 0, where T₋₁ is a residue of the resolvent R(., T) at r(T). The new conditions for which implies , are derived. The question when the relation holds, where is Lozanovsky’s essential spectrum, will be considered. Lozanovsky’s order essential spectrum is introduced. A number of auxiliary results are proved. Among them the following generalization of Nikol’sky’s theorem: if T is an operator of index zero, then T = R + K, where R is invertible, K ≥ 0 is of finite rank. Under the natural assumptions (one of them is ) a theorem about the Frobenius normal form is proved: there exist T-invariant bands such that if , where , then an operator on D_i is band irreducible.      

The existence of multiple positive solutions of p-Laplacian boundary value problems

In this paper, we establish sufficient conditions to guarantee the existence of at least three or 2n − 1 positive solutions of nonlocal boundary value problems consisting of the second-order differential equation with p-Laplacian

Hamilton-Jacobi equations having only action functions as solutions

Let L(x, v) be a Lagrangian which is convex and superlinear in the velocity variable v, and let H(x, p) be the associated Hamiltonian. Conditions are obtained under which every viscosity solution of the Hamilton-Jacobi equation

A spectral characterization of the uniform continuity of strongly continuous groups

Let X be a Banach space and a strongly continuous group of linear operators on X. Set and where is the unit circle and denotes the spectrum of T(t). The main result of this paper is: is uniformly continuous if and only if is non-meager. Similar characterizations in terms of the approximate point spectrum and essential spectra are also derived. Received: 14 June 2006, Revised: 27 September 2007