Existence and uniqueness of positive solutions of semilinear elliptic equations

This paper is devoted to the study of existence,uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations.A necessary and sufficient condition for the existence of positive solutions to problems is given.We prove that if the uniqueness and non-degeneracy results are valid for positive solutions of a class of semi-linear elliptic equations,then they are still valid when one perturbs the differential operator a little bit.As consequences,some uniqueness results of positive solutions under the domain perturbation are also obtained.     

NONDEGENERACY OF POSITIVE SOLUTIONS TO HOMOGENEOUS SECOND-ORDER DIFFERENTIAL SYSTEMS AND ITS APPLICATIONS

This article considers the Dirichlet problem of homogeneous and inhomogeneous second-order ordinary differential systems. A nondegeneracy result is proven for positive solutions of homogeneous systems. Sufficient and necessary conditions for the existence of multiple positive solutions for inhomogeneous systems are obtained by making use of the nondegeneracy and uniqueness results of homogeneous systems.     

Unicity of meromorphic solutions of partial differential equations

In this survey, results on the existence, growth, uniqueness, and value distribution of meromorphic (or entire) solutions of linear partial differential equations of the second order with polynomial coefficients that are similar or different from that of meromorphic solutions of linear ordinary differential equations have been obtained. We have characterized those entire solutions of a special partial differential equation that relate to Jacobian polynomials. We prove a uniqueness theorem of meromorphic functions of several complex variables sharing three values taking into account multiplicity such that one of the meromorphic functions satisfies a nonlinear partial differential equations of the first order with meromorphic coefficients, which extends the Brosch??s uniqueness theorem related to meromorphic solutions of nonlinear ordinary differential equations of the first order.     

Strong solutions of the Navier–Stokes equations based on the maximal Lorentz regularity theorem in Besov spaces

We show existence and uniqueness theorem of local strong solutions to the Navier–Stokes equations with arbitrary initial data and external forces in the homogeneous Besov space with both negative and positive differential orders which is an invariant space under the change of scaling. If the initial data and external forces are small, then the local solutions can be extended globally in time. Our solutions also belong to the Serrin class in the usual Lebesgue space. The method is based on the maximal Lorentz regularity theorem of the Stokes equations in the homogeneous Besov spaces. As an application, we may handle such singular data as the Dirac measure and the single layer potential supported on the sphere.     

非线性随机分数阶微分方程Euler方法的弱收敛性

论文首先证明了非线性随机分数阶微分方程解的存在唯一性,然后构造了数值求解该方程的Euler方法,并证明了当方程满足一定约束条件时,该方法是弱收敛的.特别地,当分数阶α=0时,该方程退化为非线性随机微分方程,所获结论与现有文献中的相关结论是一致的;当α≠0,且初值条件为齐次时,所获结论可视为现有文献中线性随机分数阶微分方...     

Existence and uniqueness of positive solutions of boundary value problems for coupled systems of singular second-order three-point non-linear differential and difference equations

Using the mixed monotone method we establish the existence and uniqueness of positive solutions for coupled systems of second-order three-point non-linear differential and difference equations. The theorems obtained are very general and complement previously known results.     

STRUCTURE OF POSITIVE SOLUTIONS OF A CLASS OF QUASILINEAR ORDINARY DIFFERENTIAL EQUATIONS

The existence and uniqueness of positive solutions for a class of quasilinearordinary differential equations with a large parameter are obtained. It is shownthat the flat core of positive solutions can exist. So some results of [1-3] aresharpened.     

Existence and Uniqueness of Positive Solutions for a System of Multi-order Fractional Differential Equations

In this paper, we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations. The system is used to represent constitutive relation for viscoelastic model of fractional differential equa-tions. Our results are based on the fixed point theorems of increasing operator and the cone theory, some illustrative examples are also presented.     

Transmission problems in the theory of elastic hemitropic materials

The purpose of this article is to investigate mixed transmission-boundary value problems for the differential equations of the theory of hemitropic (chiral) elastic materials. We consider the differential equations corresponding to the time harmonic dependent case, the so called pseudo-oscillation equations. Applying the potential method and the theory of pseudodifferential equations we prove uniqueness and existence theorems of solutions to the Dirichlet, Neumann and mixed transmission-boundary value problems for piecewise homogeneous hemitropic composite bodies and analyze their regularity properties. We investigate also interface crack problems and establish almost best regularity results.     

A linear homogeneous partial differential equation with entire solutions represented by Bessel polynomials

We have continued our earlier studies on entire solutions of some special type linear homogeneous partial differential equations. Specifically, we deal with entire solutions of the equations that are represented in convergent series of Bessel polynomials, and determine orders and types of the solutions, in terms of their Taylor coefficients, by establishing an analogue of Lindelöf-Pringsheim theorem as well as Wiman-Valiron type theory for such functions. Finally, by using value distribution theory of holomorphic functions, we are able to exhibit some uniqueness theorems of the entire (or meromorphic) solutions.     

Asymptotic forms of positive solutions of quasilinear ordinary differential equations with singular nonlinearities

In this paper we consider positive solutions of second order quasilinear ordinary differential equations with singular nonlinearities. We obtain asymptotic equivalence theorems for asymptotically superlinear solutions and decaying solutions. By using these theorems, exact asymptotic forms of such solutions are determined. Furthermore, we can establish the uniqueness of decaying solutions as an application of our results.     

Existence of positive solutions of nonlinear fractional delay differential equations

This paper investigates the existence and uniqueness of positive solutions for a class of nonlinear fractional delay differential equations. Using a nonlinear alternative of Leray-Schauder type, we show the existence of positive solutions for the equations in question.     

黎曼流形上一类非线性偏微分方程正解的唯一性和不存在性结果

The paper studies a class of nonlinear elliptic partial differential equations on a compact Riemannian manifold (M,g) with some curvature restriction.The authors try to prove some uniqueness and nonexistent results for the positive solutions of the equations concerned.     

A pair of difference differential equations of Euler-Cauchy type

We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory and combinatorics. They are also of use in linear control theory. Here, we study these equations in a general setting. Building on previous work going back to de Bruijn, we show how adjoint equations arise naturally in the problem of uniqueness of solutions. Exploiting the adjoint relationship in a new way leads to a significant strengthening of previous uniqueness results. Specifically, we prove here that the general Euler-Cauchy difference differential equation with advanced arguments has a unique solution (up to a multiplicative constant) in the class of functions bounded by an exponential function on the positive real line. For the closely related class of equations with retarded arguments, we focus on a corresponding class of solutions, locating and classifying the points of discontinuity. We also provide an explicit asymptotic expansion at infinity.

     

Differential equations without uniqueness and classical topological dynamics

In this paper we present a point of view which allows one to interpret the solutions of a non-autonomous differential equation as a classical dynamical system, without assuming uniqueness of solutions of the initial value problem. In addition we are able to construct a global flow without assuming the global existence of solutions of the given differential equations. This point of view seems appropriate in the sense that many applications of the results of classical topological dynamics to the study of the solutions of differential equations can now be performed, in a straight-forward manner, without the uniqueness assumption. We shall illustrate this claim with one important application concerning the existence of periodic or almost periodic solutions.     

Fundamental solutions of stochastic differential equations with drift

The paper deals with the pathwise uniqueness of solutions to one-dimensional time homogeneous stochastic differential equations with a diffusion coefficient σ satisfying the local time condition and measurable drift term b. We show that if the functions σ and b satisfy a non-degeneracy condition and fundamental solution to considered equation is unique in law, then pathwise uniqueness of solutions holds. Our result is in some sense negative, more precisely we give an example of an equation with Holder continuous diffusion coefficient and nondegenerate drift for which a fundamental solution is not unique in law and pathwise uniqueness of solutions does not hold.     

具p-Laplacian算子分数阶微分方程边值问题正解的存在唯一性

本文运用非紧性测度,给出了一类具p-Laplacian算子分数阶微分方程边值问题正解的存在性与唯一性.最后,用一个例子阐述我们的主要结果.     

On meromorphic solutions of linear partial differential equations of second order

This paper is concerned with entire and meromorphic solutions of linear partial differential equations of second order with polynomial coefficients. We will characterize entire solutions for a class of partial differential equations associated with the Jacobi differential equations, and give a uniqueness theorem for their meromorphic solutions in the sense of the value distribution theory, which also applies to general linear partial differential equations of second order. The results are complemented by various examples for completeness.     

Stabilization of functions and its application

The concepts of polynomial stabilization, strong polynomial stabilization, and strong stabilization are introduced for a fundamental system of solutions of linear differential equations. Some criteria of such kinds of stabilizations and applications to the theory of existence and uniqueness of solutions of ordinary differential equations are given. An abstract scheme of the obtained results is presented for Banach spaces.     

Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter

In this paper, by using fixed point theorems of concave operators in partial ordering Banach spaces, we establish the existence and uniqueness of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations for any given parameter. Moreover, we present some pleasant properties of positive solutions to the boundary value problem dependent on the parameter. In the end, two examples are given to illustrate our main results.