Lie-Mei symmetry and conserved quantities of the Rosenberg problem

The Rosenberg problem is a typical but not too complicated problem of nonholonomic mechanical systems.The Lie-Mei symmetry and the conserved quantities of the Rosenberg problem are studied.For the Rosenberg problem,the Lie and the Mei symmetries for the equation are obtained,the conserved quantities are deduced from them and then the definition and the criterion for the Lie-Mei symmetry of the Rosenberg problem are derived.Finally,the Hojman conserved quantity and the Mei conserved quantity are deduced from the Lie-Mei symmetry.     

The solution of the global relation for the derivative nonlinear Schrödinger equation on the half-line

We consider initial-boundary value problems for the derivative nonlinear Schrödinger (DNLS) equation on the half-line x>0. In a previous work, we showed that the solution q(x,t) can be expressed in terms of the solution of a Riemann-Hilbert problem with jump condition specified by the initial and boundary values of q(x,t). However, for a well-posed problem, only part of the boundary values can be prescribed; the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. In general, an effective solution of the problem therefore requires solving the global relation. Here, we present the solution of the global relation in terms of the solution of a system of nonlinear integral equations. This also provides a construction of the Dirichlet-to-Neumann map for the DNLS equation on the half-line.     

Positive solution of singularly perturbed Dirichlet problem with singularities

A class of singularly perturbed boundary value problem with singularities is considered. Introducing the stretched variables, the boundary layer corrective terms near x = 0 and x = 1 are constructed. Under suitable conditions, by using the theory of differential inequalities the existence and asymptotic behavior of solution for boundary value problem are proved, uniformly valid asymptotic expansion of solution with boundary layers are obtained,     

A Remark on the Cauchy Problem of 1D Compressible Navier-Stokes Equations with Density-dependent Viscosity Coefcients

This paper is concerned with the Cauchy problems of one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefcients.By assumingρ0∈L1(R),we will prove the existence of weak solutions to the Cauchy problems forθ0.This will improve results in Jiu and Xin’s paper(Kinet.Relat.Models,1(2):313–330(2008))in whichθ12is required.In addition,We will study the large time asymptotic behavior of such weak solutions.     

Spatial‐skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses

A spectral problem for the Laplace operator in a thick cascade junction with concentrated masses is considered. This cascade junction consists of the junction's body and a great number of ?‐alternating thin rods belonging to two classes. One class consists of rods of finite length, and the second one consists of rods of small length of order . The density of the junction is of order on the rods from the second class and outside of them. The asymptotic behavior of eigenvalues and eigenfunctions of this problem is studied as ? → 0. There exist five qualitatively different cases in the asymptotic behavior of eigenmagnitudes as ? → 0, namely the case of ‘light’ concentrated (α ∈ (0,1)), ‘middle’ concentrated (α = 1), and ‘heavy’ concentrated masses (α ∈ (1, + ∞ )) that we divide into ‘slightly heavy’ concentrated (α ∈ (1,2)), ‘intermediate heavy’ concentrated (α = 2), and ‘very heavy’ concentrated masses (α > 2). In the paper, we study in detail the influence of the concentrated masses on the asymptotic behavior if α ∈ (1,2). We construct the leading terms of asymptotic expansions both for the eigenvalues and eigenfunctions and prove the corresponding asymptotic estimates. Copyright © 2013 John Wiley & Sons, Ltd.     

Fixed point theorems in piecewise continuous function spaces and applications to some nonlinear problems

The piecewise continuous function space is an important state space in various nonlinear problems. In this paper, we establish some new fixed point theorems for the perturbed contraction operators in piecewise continuous function spaces. Our results can be applied to various integral operators. Some previous results are improved in this literature. As applications, the existence and uniqueness of solutions of impulsive anti‐periodic boundary value problems and the Lasota–Wazewska models with delays are exhibited at last two sections. Copyright © 2013 John Wiley & Sons, Ltd.     

Spectral asymptotics and basis properties of fourth order differential operators with regular boundary conditions

In this paper, we consider the problem where λ is a spectral parameter; q(x) ∈ L₁(0,1) is complex‐valued function; α_s, s = 1,2,3, are arbitrary complex constants that satisfy α₂ = α₁ + α₃ and σ = 0,1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established. It is proved that all the eigenvalues, except for finite number, are simple and the system of root functions of this spectral problem forms a basis in the space L_p(0,1), 1 < p < ∞ , when ; moreover, this basis is unconditional for p = 2. We note that the considered problem was previously investigated in the condition of α₂ ≠ α₁ + α₃. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Well-Posedness of Determination of Two Coefficients in a Fractional Integrodifferential Equation

The authors study an inverse problem for a fractional integrodifferential equation, which aims to determine simultaneously two time varying coefficients, a kernel function and a source function, from the additional integral overdetermination condition. By using the fixed point theorem in suitable Sobolev space, the global existence and uniqueness results of this inverse problem are obtained.