Uniqueness in Determining a Ball with a Single Incoming Wave

We prove that a ball with the impedance boundary condition is uniquely determined by the far-field pattern corresponding to an incident plane wave at one given wavenumber and one given incident direction. In the uniqueness proof, the impedance parameter in the impedance boundary condition is unknown.     

Initial value problem for mixed-type differential equations

The initial value problem is discussed for a class of linear functional differential equations of mixed type and various criteria on existence and uniqueness are obtained.     

Fixed point theorems of operators with ppf dependence in banach spaces

In this paper the theory of fixed point theorems of nonlinear whose domain and range are different Banach spaces are considered. Also the analogues of the Contraction Mapping Principle. Krasnoselskii's fixed point theorem and a result on the convergence of quasinonexpansive mappings are dealt with. As an application, the existence and uniqueness of solutions of differential equations with retarded argument in Banach spaces are discussed     

Limit cycles in a general two-stroke oscillation

Conditions of the existence of limit cycles in a general two-stroke oscillation system are obtained. The general oscillation is a special kind of Liénard-type system with isoclines approaching both positive infinites or negative infinites as t→±∞$t\to \pm \infty$. The conditions for the uniqueness of the two-stroke oscillation are also proved. Some examples are used to illustrate our results.     

Existence and uniqueness of periodic solutions for a class of generalized Liénard systems with forcing term

By using topological degree theory and some analysis skills, we obtain some sufficient conditions for the existence and uniqueness of periodic solutions for a class of forced generalized Liénard systems.     

Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern

The problem considered is that of determining the shape of aplane acoustically sound-soft obstacle from the knowledge ofthe far-field pattern for one time-harmonic incident field.An iterative procedure is proposed based on two boundary integralsrepresenting the incident field and the far-field pattern, respectively.Numerical examples are included which show that the proceduregives accurate numerical approximations in relatively few iterations.     

The escaping disaster: A problem related to state-dependent delays

We study a second order state-dependent delay equation with an in general unbounded delay that describes an idealized one-dimensional motion. Although governed by a negative feedback mechanism, the set of equilibria, formed by the so called constant velocity solutions, is unstable and a new kind of instability, called the escaping instability, evolves.     

On the behavior of attractors under finite difference approximation

We recall that the long-time behavior of the Kuramoto-Sivashinsky equation is the same as that of a certain finite system of ordinary differential equations. We show how a particular finite difference scheme approximating the Kuramoto-Sivashinsky may be viewed as a small C ¹ perturbation of this system for the grid spacing sufficiently small. As a consequence one may make deductions about how the global attractor and the flow on the attractor behaves under this approximation. For a sufficiently refined grid the long-time behavior of the solutions of the finite difference scheme is a function of the solutions at certain grid points, whose number and position remain fixed as the grid is refined. Though the results are worked out explicitly for the Kuramoto-Sivashinsky equation, the results extend to other infinite-dimensional dissipative systems.     

The Cauchy-Nicoletti problem with poles

The Cauchy-Nicoletti boundary value problem for a system of ordinary differential equations with pole-type singularities is investigated. The conditions of the existence, uniqueness, and nonuniqueness of a solution in the class of continuously differentiable functions are given. The classical Banach contraction principle is combined with a special transformation of the original problem.     

Multiplicity results for periodic solutions to delay differential equations via critical point theory

By the critical point theory, we study the existence and multiplicity of periodic solutions to the following system of delay differential equations:

(*)     

Exponentially asymptotically stable dynamical systems

In this paper, we give some sufficient conditions which guarantee that the zero solution of the differential equation with two delayed terms x(t) = a(t)x(t-p(t)) - b(t)x(t-r(t))is uniformly stable     

The existence of periodic solutions of non-autonomous second-order Hamiltonian systems

The purpose of this paper is to study the existence of periodic solutions for a class of non-autonomous second-order Hamiltonian systems. Some new existence theorems are obtained by using the least action principle and the saddle point theorem.     

Autonomous and non-autonomous attractors for differential equations with delays

The asymptotic behaviour of some types of retarded differential equations, with both variable and distributed delays, is analysed. In fact, the existence of global attractors is established for different situations: with and without uniqueness, and for both autonomous and non-autonomous cases, using the classical notion of attractor and the recently new concept of pullback one, respectively.     

Weighted pseudo almost periodic solutions for some partial functional differential equations

In this work, we give sufficient conditions for the existence and uniqueness of a weighted pseudo almost periodic solution for some partial functional differential equations. To illustrate our main result, we study the existence of a weighted pseudo almost periodic solution for some diffusion equation with delay.     

Global attractivity in monotone and subhomogeneous almost periodic systems

A global attractivity theorem is first proved for a class of skew-product semiflows. Then this result is applied to monotone and subhomogeneous almost periodic reaction-diffusion equations, ordinary differential systems and delay differential equations for their global dynamics.     

Monotone iterative technique for functional differential equations with retardation and anticipation

The method of the monotone iterative technique together with coupled lower and upper solutions is employed to prove existence and uniqueness results for functional differential equations involving retardation and anticipation.     

The Korteweg-de Vries equation and beyond

We review a new method for linearizing the initial-boundary value problem of the KdV on the semi-infinite line for decaying initial and boundary data. We also present a novel class of physically important integrable equations. These equations, which include generalizations of the KdV, of the modified KdV, of the nonlinear Schrödinger and of theN-wave interactions, are as generic as their celebrated counterparts and, furthermore it appears that they describe certain physical situations more accurately.     

A second-order nonlinear problem with two-point and integral boundary conditions

The paper gives sufficient conditions for the existence and nonuniqueness of monotone solutions of a nonlinear ordinary differential equation of the second order subject to two nonlinear boundary conditions one of which is two-point and the other is integral. The proof is based on an existence result for a problem with functional boundary conditions obtained by the author in [6].     

The Far-Field Equations in Linear Elasticity for Disconnected Rigid Bodies and Cavities

In this paper the far-field equations in linear elasticity for scattering from disjoint rigid bodies and cavities are considered. The direct scattering problem is formulated in differential and integral form. The boundary integral equations are constructed using a combination of single- and double-layer potentials. Using a Fredholm type theory it is proved that these boundary integral equations are uniquely solvable. Assuming that the incident field is produced by a superposition of plane incident waves in all directions of propagation and polarization it is established that the scattered field is also expressed as the superposition of the corresponding scattered fields. A pair of integral equations of the first kind which hold independently of the boundary conditions are constructed for the far-field region. The properties of the Herglotz functions are used to derive solvability conditions for the far-field equations. It is also proved that the far-field operators, in terms of which we can express the far-field equations, are injective and have dense range. An analytical example for spheres illuminates the theoretical results.