This paper introduces two new spectral properties (H) and (gH), and investigates the two properties in connection with Weyl type theorems. Also the preservation of the two properties are studied under commuting nilpotent, quasi-nilpotent, finite rank or Riesz perturbation.

In this paper, the expression under the action of fractional derivative and fractional integral for a common function on the unit ball of several complex variables is improved. At the same time, the equivalent norms of the fractional differential on two holomorphic function spaces are improved, and the constraint conditions β=s+N for the fractional differential R^{s, t} and R^{β, t} in the equivalent norms are removed, where N is a positive integer.

This paper is concerned with the generalized Riemann problem for the nonlinear chromatography equations, where the delta shock wave occurs in the corresponding Riemann solution. It is quite different from the previous generalized Riemann problems which focus on classical elementary waves. We constructively solve the generalized Riemann problem in a neighborhood of the origin on the x-t plane. In solutions, we find that the generalized Riemann solutions have a structure similar to the solution of the corresponding Riemann problem for most of cases. However, a delta shock wave in the corresponding Riemann solution may turn into a shock wave followed by a contact discontinuity, which provides us with a detailed method for analyzing the internal mechanism of a delta shock wave.

In this paper, we study the existence of minimum of a constrained variational problem in the Sobolev space W^{1, 4}($\mathbb{S}$). If ∫$_\mathbb{S}$g(θ)dθ>0, the minimum is a positive solution to the related Euler-Lagrange equation

${u^{\prime \prime }} + u = \frac{{g(\theta )}}{{u\left( {{u^2} + {u^{\prime 2}}} \right)}}, \theta \in {\rm{\mathbb{S}}}$

Based on this, we prove the solvability of the dual Minkowski problem in $\mathbb{R}$² posed by Huang-Lutwak-Yang-Zhang[Acta Math, 2016, 216(2):325-338].

In this paper, we are considered with a constrained variational problem for certain type of Kirchhoff equation with trapping potential and the bottom of the potential is an ellipsoid. We are interested in the asymptotic behavior of solutions of variational problem and we prove that the minimizers of the minimization problem blows up at one of the endpoints of the major axis of the ellipsoid as the related parameter approaches a critical value.

In this paper, the authors studied the existence results of the mild solutions for a class of fractional semilinear integro-differential equation of mixed type by using the measure of noncompactness, k-set contraction and β-resolvent family. It is well known that the k-set contraction requires additional condition to ensure the contraction coefficient 0 < k < 1. We don't require additional condition to ensure the contraction coefficient 0 < k < 1. An example is introduced to illustrate the main results of this paper.

In this paper, we study the generalized hyperelastic-rod wave equation. We changed the generalized hyperelastic-rod wave equation into a complex differential equation by using traveling wave transform and show that meromorphic solutions of the complex differential equation belong to the class W by the weak $ \left\langle {h, k} \right\rangle $ condition and the Fuchs index. Furthermore, we find out all meromorphic solutions of the complex differential equation, then we obtain the traveling wave solutions of the generalized hyperelastic-rod wave equation. We can apply the idea of this study to some related mathematical physics equations.