Two homoclinic solutions for a nonperiodic fourth‐order differential equation without coercive condition

In this paper, we investigate the existence of homoclinic solutions for a class of fourth‐order nonautonomous differential equations where w is a constant, and . By using variational methods and the mountain pass theorem, some new results on the existence of homoclinic solutions are obtained under some suitable assumptions. The interesting is that a(x) and f(x,u) are nonperiodic in x,a does not fulfil the coercive condition, and f does not satisfy the well‐known (AR)‐condition. Furthermore, the main result partly answers the open problem proposed by Zhang and Yuan in the paper titled with Homoclinic solutions for a nonperiodic fourth‐order differential equations without coercive conditions (see Appl. Math. Comput. 2015; 250:280–286). Copyright © 2016 John Wiley & Sons, Ltd.     

On the existence of free-surface azimuthal equatorial flows

This paper is concerned with the governing equations (together with the free boundary and the bottom boundary conditions) for an inviscid, incompressible fluid, written in spherical coordinates which are fixed at a point on the rotating Earth. For a special type of a steady flow moving only in the azimuthal direction, with no variation in this direction, we provide a functional-analytic study of the link between the pressure at the free surface of the ocean and the distortion of that surface.     

Further study of alternating iterations for rectangular matrices

Theory of matrix splittings is a useful tool in the analysis of iterative methods for solving systems of linear equations. When two splittings are given, it is of interest to compare the spectral radii of the corresponding iteration matrices. This helps to arrive at the conclusion that which splitting should one choose so that one can reach the desired solution of accuracy or the exact solution in a faster way. In the case of many splittings are provided, the comparison of the spectral radii is time-consuming. Such a situation can be overcome by introducing another iteration scheme which converges to the same solution of interest in a much faster way. In this direction, the theory of alternating iterations for real rectangular matrices is recently proposed. In this note, some more results to the theory of alternating iterations are added. A comparison result of two different alternating iteration schemes is then presented which will help us to choose the iteration scheme that will guarantee the faster convergence of the alternating iteration scheme. In addition to these results, a comparison result for proper weak regular splittings is also obtained.     

Continuity in law with respect to the Hurst index of some additive functionals of sub-fractional Brownian motion

In this article, first, we prove some properties of the sub-fractional Brownian motion introduced by Bojdecki et al. [Statist. Probab. Lett. 69(2004):405–419]. Second, we prove the continuity in law, with respect to small perturbations of the Hurst index, in some anisotropic Besov spaces, of some continuous additive functionals of the sub-fractional Brownian motion. We prove that our result can be obtained easily, by using the decomposition in law of the sub-fractional Brownian motion given by Bardina and Bascompte [Collect. Math. 61(2010):191–204] and Ruiz de Chavez and Tudor [Math. Rep. 11(2009):67–74], without using the result of Wu and Xiao [Stoch. Proc. Appl. 119(2009):1823–1844] by connecting the sub-fractional Brownian motion to its stationary Gaussian process through Lamperti’s transform. This decomposition in law leads to a better understanding and simple proof of our result.     

Existence of multiplicity harmonic and subharmonic solutions for second‐order quasilinear equation via Poincaré‐Birkhoff twist theorem

In this paper, we investigate the existence and multiplicity of harmonic and subharmonic solutions for second‐order quasilinear equation where , g satisfies the superlinear condition at infinity. We prove that the given equation possesses harmonic and subharmonic solutions by using the phase‐plane analysis methods and a generalized version of the Poincaré‐Birkhoff twist theorem.     

Exact separation theorem for closed sets in Asplund spaces

In this paper, in terms of the Fréchet normal cone, we establish exact separation results for finitely many disjoint closed sets in an Asplund space, which supplement the extremal principle and some fuzzy separation theorems. As an application, we provide a new optimality condition for a constraint optimization problem in terms of Fréchet subdifferential and Fréchet normal cone.     

Bolzano's Theorems for Holomorphic Mappings

The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano's ones for real functions on an interval is deduced in a very simple way from Cauchy's theorem for holomorphic functions.A more complicated proof,using Cauchy's argument principle,provides uniqueness of the zero,when the sign conditions on the boundary are strict.Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions.Extensions to holomorphic mappings from Cn to Cn are obtained using Brouwer degree.     

一类1 < α < 2非局部条件下的脉冲分数阶微分方程mild解的存在性

本文研究了一类1 < α < 2非局部条件下的脉冲分数阶偏微分方程mild解的存在性问题.利用解算子的相关性质及Krasnoselskii不动点理论的方法,获得了这类方程的mild解并予以证明,且得到了解的存在性结果.     

Banach空间中一类二阶具阻尼项的积分-微分包含的可控性

本文讨论了一类二阶具阻尼项的积分-微分包含问题.利用集值不动点定理将问题转化为求不动点问题,进而得到系统可控性的充分条件.     

一类非线性椭圆方程的刘维尔型定理

设（Mⁿ,g）是一个n维非紧的完备黎曼流行.本文考虑有正解的非线性椭圆方程△_fu+au log u=0的刘维尔型定理,其中a是一个非零常数.利用Bochner公式和极大值原理,获得了以上方程在Bakry-Emery里奇曲率有下界时正解的Li-Yau型梯度估计和某些有关的刘维尔理论,推广了文献[7]的结果.