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.     

On the existence of periodic solutions for the equation\ddot x + f(x)\dot x + g(x) = 0

We establish in this work sufficient conditions for the existence of periodic solutions for the Liénard equation .     

Nonlinear integrodifferential equations anda priori bounds on periodic solutions

Summary This paper studies the existence of aperiodic solution of a nonlinear integrodifferential system of the form , for each continuous periodic function p and under suitable assumptions on f, k and g. A topological transversality method is employed to obtain the existence of periodic solutions. This method relies ona priori bounds on periodic solutions. Several examples are provided where a variant of Liapunov's direct method is employed to obtaina priori bounds on periodic solutions.     

Solutions of the equation \documentclass{article}\pagestyle{empty}\begin{document}$$ \partial _t^n f(x,t) = \hat L(x,t)f(x,t) + S(x,t) $$\end{document}. Exact solutions of some partial differential equations in mechanics and physics

The solutions of the equation $ \partial _t^n f(x,t) = \hat L(x,t)f(x,t) + S(x,t) $, for L? a linear operator are derived. Different forms for L? whether it is time independent or time dependent and self-commutative (or not) at different times are considered separately. By using the results obtained, exact solutions of some partial differential equations are found for the first time.     

Multiple equilibria,periodic solutions and a priori bounds for solutions in superlinear parabolic problems

Consider the Dirichlet problem for the parabolic equation in , where $\Omega$ is a bounded domain in and f has superlinear subcritical growth in u. If f is independent of t and satisfies some additional conditions then using a dynamical method we find multiple (three, six or infinitely many) nontrivial stationary solutions. If f has the form where m is periodic, positive and m,g satisfy some technical conditions then we prove the existence of a positive periodic solution and we provide a locally uniform bound for all global solutions.     

Chaotic behavior of a class of nonlinear differential delay equations

Summary For the differential delay equation the existence of infinitely many periodic as well as infinitely many aperiodic solutions («choatic behavior in the sense of Li and Yorke») is proved.     

The stability of RDDE\ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) + q_2 (t)x(t - r(t)) = O

In this paper, the author considered the stability of zero solution of linear RDDE $$\begin{gathered} \ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) + q_2 (t)x(t - r(t)) = O, \hfill \\ \ddot x(t) + p_1 (t)\dot x(t) + q_1 (t)x(t) + p_2 (t)\dot x(t - r(t)) = O \hfill \\ \end{gathered} $$ using Liapunov-Razumikhin functional and transformations and obtained some sufficient conditions for the stability of Eqs.(1) and (2). These results are suitable both for boundedp _i (t),q _i (t) andr(t).i=1,2.     

On controllability of systems of the form\dot x=A(t)x+g(t,u)

In an earlier paper, the author established a sufficient condition for controllability of systems of the form =A(t)x+g(t, u). This condition is a growth condition which generalizes the concept of an asymptotically proper system introduced by LaSalle for linear systems. The purpose of this paper is examine and apply this growth condition. We first show that the condition is also necessary for controllability. Then, we use these results to consider the controllability of perturbations of the above system. The main result of the paper is a class of systems which in many applications can be assumed to be controllable.During the writing of this paper, the author held a Junior Faculty Summer Fellowship from the Research Council of the University of Nebraska.     

On solutions of quasilinear wave equations with nonlinear damping terms

In this paper we consider the existence and asymptotic behavior of solutions of the following problem:

Chaos in almost periodic systems

Let be an almost periodic differential equation with a hyperbolic almost periodic solutionu(t) and another hyperbolic solutionv(t) satisfying ¦v(t)–u(t)¦0 as ¦t¦. It is shown that the solutions of such an equation exhibit chaotic behavior.

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Zusammenfassung Sei eine fastperiodische Differentialgleichung mit einer fastperiodischen hyperbolischen Lösungu(t) und seiv(t) eine weitere hyperbolische Lösung für welche ¦v(t)–u(t)¦0 für ¦t¦. Es wird gezeigt, daß die Differentialgleichung unter diesen Bedingungen chaotisches Verhalten aufweist.

     

Subharmonics of a Nonconvex Noncoercive Hamiltonian System

We study the problem of the existence of multiple periodic solutions of the Hamiltonian system

When all trajectories in the Liénard plane cross the vertical isocline?

In this paper we study the problem whether all trajectories of the system =y–F(x) and =–g(x) cross the vertical isocline which is very important for the existence of periodic solutions and oscillation theory. The problem has not been solved for the critical case:

Asymptotic behavior of solutions of a nonlinear difference equation with continuous argument

We consider the difference equation with continuous argument

Das asymptotische Verhalten der Peanokerne einiger interpolatorischer Quadraturverfahren

Summary Interpolatory quadrature formulae consist in replacing by wherep _f denotes the interpolating polynomial off with respect to a certain knot setX. The remainder may in many cases be written as wherem=n resp. (n+1) forn even and odd, respectively. We determine the asymptotic behaviour of the Peano kernelP _X (t) forn for the quadrature formulae of Filippi, Polya and Clenshaw-Curtis.

     

Note on irregularities of distribution

Letf be a periodic function on with period 1, piecewise continuously differentiable, satisfying . For an arbitrary sequence = ( _i ) in [0,1) put and . If then _n (f,) >c· logn holds for some positive constantc (depending onf only) and almost alln. In a certain sense the converse is also true: there is a class of functionsf with such that _n (f,) =o (logn).Support has been received from Netherlands Organization for the Advancement of Pure Research (Z. W. O.).     

Critical Exponents for the Heat-Conduction Equation with Nonlocal,Nonlinear Perturbations

We study the conditions for the existence and nonexistence of global in time (t > 0), nonnegative solutions of the problem

Multiple solutions for three-point boundary value problem with nonlinear terms depending on the first order derivative

In this paper, we consider the following second-order three-point boundary value problem

Interior points of the convex hull of few points in\mathbb{E}^d

We show that ifP , |P|=d+k,dk1 andO int convP, then there exists a simplexS of dimension with vertices inP, satisfyingO rel intS, the bound being sharp. We give an upper bound for the minimal number of vertices of facets of a (j-1)-neighbourly convex polytope in withv vertices.Research (partially) supported by Hung. Nat. Found. for Sci. Research, grant no. 1817Research (partially) supported by Hung. Nat. Found. for Sci. Research, grant no. 326-0213     

On the mixed problem for hyperbolic partial differential-functional equations of the first order

We consider the mixed problem for the hyperbolic partial differential-functional equation of the first order where is a function defined by z _(x,y)(t, s) = z(x + t, y + s), (t, s) [–, 0] × [0, h]. Using the method of bicharacteristics and the method of successive approximations for a certain integral-functional system we prove, under suitable assumptions, a theorem of the local existence of generalized solutions of this problem.     

Asymptotic properties of solutions of the viscous Hamilton-Jacobi equation

The purpose of the paper is to study properties of solutions of the Cauchy problem for the equation under the assumption . General selfsimilar solutions are constructed. Moreover, for initial data with some decay at infinity, we determine the leading term of the asymptotics of solutions in which is described by either solutions of the linear heat equation or by particular selfsimilar solutions of the original equation.