A curve tracing algorithm for computing the pseudospectrum

The boundary curve of the pseudospectrum of a matrix is defined as a contour line of its resolvent norm. A rather simple and efficient continuation method is presented, which determines the implicitly given curve by a prediction-correction scheme, where the correction step is accomplished by one single Newton step. Besides its efficiency the algorithm turns out to be very accurate as long as the boundary of the pseudospectrum is a smooth curve. Problems may arise at bifurcation points where the resolvent norm is not differentiable.This work was partially supported by Deutsche Forschungsgemeinschaft.     

The problem of continuation with least coanalytic deviation

The problem of continuing a function from the unit circle to the unit disk so that the continuation has the least deviation from the Sobolev subspace of analytic functions is considered. A mathematical model of this problem is constructed. It is proved that the problem is well-posed.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 45–57, July, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00621.     

Unique continuation property for a higher order nonlinear Schrödinger equation

We prove that, if a sufficiently smooth solution u to the initial value problem associated with the equation

     

Unique continuation for the system of elasticity in the plane

We prove the strong unique continuation property for the Lamé system of elastostatics in the plane, , with variable Lamé coefficients , , when is Lipschitz and is measurable.

     

Periodic solutions for a neutral nonlinear dynamical equation on a time scale

Let T be a periodic time scale. We use a fixed point theorem due to Krasnosel'ski? to show that the nonlinear neutral dynamic system with delay

     

Study on a kind of $p$-Laplacian neutral differential equation with multiple variable coefficients

In this paper, we first discuss some properties of the neutral operator with multiple variable coefficients $(Ax)(t):=x(t)-\sum\limits_{i=1}^{n}c_i(t)x(t-\delta_i)$. Afterwards, by using an extension of Mawhin''s continuation theorem, a kind of second order $p$-Laplacian neutral differential equation with multiple variable coefficients as follows $$\left(\phi_p\left(x(t)-\sum\limits_{i=1}^{n}c_i(t)x(t-\delta_i)\right)''\right)''=\tilde{f}(t,x(t),x''(t))$$ is studied. Finally, we consider the existence of periodic solutions for two kinds of second-order $p$-Laplacian neutral Rayleigh equations with singularity and without singularity. Some new results on the existence of periodic solutions are obtained. It is worth noting that $c_i$ ($i=1,\cdots,n$) are no longer constants which are different from the corresponding ones of past work.     

Multiplicity of periodic solutions for a delayed ratio-dependent predator-prey model with Holling type III functional response and harvesting terms

By using the generalized continuation theorem, the existence of four positive periodic solutions for a delayed ratio-dependent predator-prey model with Holling type III functional response