Unique existence results and numerical solutions for fourth-order impulsive differential equations with nonlinear boundary conditions

The work is concerned with three kinds of fourth-order impulsive differential equations with nonlinear boundary conditions. We at first focused on studying the existence and uniqueness of positive solutions for these kinds of problems. By converting the problem to an equivalent integral equation, then applying the new class of fixed point theorems for the sum operator on cone, we obtain the sufficient conditions which not only guarantee the existence of a unique positive solution, but also be applied to construct two iterative sequences for approximating it. Further, we present the numerical methods for solving the fourth-order differential equations. At last, some examples are given with numerical verifications to illustrate the main results.     

Generalized Newtonian fluids in moving domains

In this paper we prove the existence of weak solutions for the equations describing the unsteady motion of an incompressible, viscous and homogeneous generalized Newtonian fluid in a non-cylindrical domain ${?}_{t\in I}\{t\}\times \mathrm{\Omega }(t)$.     

Partial regularity of weak solutions to a PDE system with cubic nonlinearity

In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the underlying biological network coupled to a diffusion equation for the conductance vector of the network. There are several different types of nonlinearities in the system. Of particular mathematical interest is a term that is a polynomial function of solutions and their partial derivatives and this polynomial function has degree three. That is, the system contains a cubic nonlinearity. Only weak solutions to the system have been shown to exist. The regularity theory for the system remains fundamentally incomplete. In particular, it is not known whether or not weak solutions develop singularities. In this paper we obtain a partial regularity theorem, which gives an estimate for the parabolic Hausdorff dimension of the set of possible singular points.     

Symmetry and uniqueness of solutions to some Liouville-type equations and systems

We prove symmetry and uniqueness results for three classes of Liouville-type problems arising in geometry and mathematical physics: asymmetric Sinh-Gordon equation, cosmic string equation and Toda system, under certain assumptions on the mass associated to these problems. The argument is in the spirit of the sphere covering inequality which for the first time is used in treating different exponential nonlinearities and systems.     

Boundary value problems for an iterative di?erential equation

This paper discusses the solutions of an iterative di?erential equation under general boundary value conditions. Using an auxiliary integral equation without the help of Green’s functions usually being constructed in higher order equations, we prove the existence and uniqueness of solutions by the ?xed point theorems of Schauder and Banach, respectively. Our theorems generalize and revise the related results.     

Sobolev orthogonal Legendre rational spectral methods for problems on the half line

Modified Legendre rational spectral methods for solving second-order differential equations on the half line are proposed. Some Sobolev orthogonal Legendre rational basis functions are constructed, which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier series. Numerical results demonstrate the effectiveness and the spectral accuracy of this approach.     

Lp~(p>1) Solutions of BSDEs with Generators Satisfying Some Non-uniform Conditions in t and \omeg

This paper is devoted to the $L^p$ ($p>1$) solutions of one-dimensional backward stochastic differential equations (BSDEs for short) with general time intervals and generators satisfying some non-uniform conditions in $t$ and $\omega$. An existence and uniqueness result, a comparison theorem and an existence result for the minimal solutions are respectively obtained, which considerably improve some known works. Some classical techniques used to deal with the existence and uniqueness of $L^p$ ($p>1$) solutions of BSDEs with Lipschitz or linear-growth generators are also developed in this paper.     

Existence and approximated results of solutions for a class of nonlocal elliptic variational-hemivariational inequalities

In this paper, we consider the regularization of a class of elliptic variational-hemivariational inequalities driven by the fractional Laplace operator. First, we demonstrate characterizations of nonlocal elliptic variational-hemivariational inequalities. Next, we provide coercivity conditions that guarantee the existence and uniqueness of solution. Finally, by virtue of the so-called Browder–Tikhonov regularization method, we introduce a strongly convergent approximation procedure for the considered nonlocal problems considered without any coercivity condition.     

RAMAN SPECTROSCOPIC STUDY ON THE CHAIN CONFORMATION OF ELECTROSYNTHESIZED POLY(p-PHENYLENE) FILMS

Poly(p-phenylene) (PPP) films have been electrochemically deposited on mirrorlike stainless steel (SS) electrode surfaces by direct oxidation of benzene in boron trifluoride diethyl etherate (BFEE) solutions at a constant applied potential of 1.7 V (versus Ag/AgCl). The chain conformations of as-grown PPP films have been studied by Raman spectroscopy. The results demonstrated that high synthesis temperature favors the formation of planar PPP, and the conformation of planar PPP can be changed into torsional PPP by cooling treatment.     

一类三阶常微分方程非线性三点边值问题解的存在性

In this paper, existence of solutions of third-order differential equation
y′″（t）=f（t,y（t）,y′（t）,y″（t））
with nonlinear three-point boundary condition
{g（y（a）,y′（a）,y″（a））=0,
h（y（b）,y′（b））=0,
I（y（c）,y′（c）,y″（c））=0
is obtained by embedding Leray-Schauder degree theory in upper and lower solutions method,where a, b, c∈ R,a〈 b〈 c; f ： [a,c]×R^3→R,g：R^3→R,h：R^2→R and I：R^3→R are continuous functions. The existence result is obtained by defining the suitable upper and lower solutions and introducing an appropriate auxiliary boundary value problem. As an application, an example with an explicit solution is given to demonstrate the validity of the results in this paper.