A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations

This paper is concerned with an operator equation Ax+Bx+Cx=x on ordered Banach spaces, where A is an increasing α-concave operator, B is an increasing sub-homogeneous operator and C is a homogeneous operator. The existence and uniqueness of its positive solutions is obtained by using the properties of cones and a fixed point theorem for increasing general β-concave operators. As applications, we utilize the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for two classes nonlinear problems which include fourth-order two-point boundary value problems for elastic beam equations and elliptic value problems for Lane-Emden-Fowler equations.     

一类超椭圆方程有正整数解的必要条件的问题

研究一类超椭圆方程(n d)(n 2d)…(n kd)=y3有正整数解的必要条件,利用数论的一些基本方法,给出了一个新的结果,推进了Erd(o)s和Selfridge的猜想的证明.     

Positive solutions for some Schrödinger equations having partially periodic potentials

This paper is concerned with the problem of finding positive solutions of the equation −Δu+(a_∞+a(x))u=|u|^q−2u, where q is subcritical, Ω is either RN or an unbounded domain which is periodic in the first p coordinates and whose complement is contained in a cylinder , a_∞>0, a∈C(RN,R) is periodic in the first p coordinates, infx∈RN(a_∞+a(x))>0 and a(x^′,x^″)→0 as |x^″|→∞ uniformly in x^′. The cases a?0 and a?0 are considered and it is shown that, under appropriate assumptions on a, the problem has one solution in the first case and p+1 solutions in the second case when p?N−2.     

On the solutions of a class of difference equations

In this note we investigate the solutions of a class of difference equations and prove that Conjectures 4.8.2, 4.8.3, 5.4.6 and 6.10.3 proposed by M. Kulenovic and G. Ladas in [M. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures, Chapman & Hall/CRC Press, 2002] are true.     

On reduction of some classes of partial differential equations to equations with fewer variables and exact solutions

We establish a connection between the fundamental solutions to some classes of linear nonstationary partial differential equations and the fundamental solutions to other nonstationary equations with fewer variables. In particular, reduction enables us to obtain exact formulas for the fundamental solutions of some spatial nonstationary equations of mathematical physics (for example, the Kadomtsev-Petviashvili equation, the Kelvin-Voigt equation, etc.) from the available fundamental solutions to one-dimensional stationary equations.     

A note on blow-up solutions to some nonlinear Volterra integral equations

In this paper we investigate certain nonlinear Volterra integral equations with the power nonlinearity. The basic results provide a criterion involving the kernel and the nonlinearity for the existence of the nontrivial and blow-up solution.     

Sharp estimates of bounded solutions to some semilinear second order dissipative equations

Let H, V be two real Hilbert spaces such that VH with continuous and dense imbedding, and let FC¹(V) be convex. By using differential inequalities, a close-to-optimal ultimate bound of the energy is obtained for solutions in to u^″+cu^′+bu+F(u)=f(t) whenever .     

Positive solutions to a class of second order differential equations with singular nonlinearities

A class of second order quasilinear differential equations with singular nonlinearities is considered. The set of all possible solutions defined on a positive half-line [a,∞) is classified into six types according to their aymptotic behavior as t→∞, and sharp conditions are established for the existence of solutions belonging to each of the classified types.     

对一类随机算子方程的随机解的注记

给出在Banach空间中一类随机算子方程的随机解的某些新结果,它推广文[4]与文[6]中几个结果.     

Twin positive solutions of boundary value problems for finite difference equations with p-Laplacian operator

In this paper, a new twin fixed point theorem is applied to obtain the existence of at least two positive solutions for the boundary value problem

     

On the existence of positive solutions of second order differential equations

Using the fixed point method we prove an existence result for positive solutions of nonlinear second order ordinary differential equations. An application to semilinear Schrödinger equations in exterior domains is also presented. Mathematics Subject Classification (2000) 34C10     

Nonexistence of multi-bubble solutions to some elliptic equations on convex domains

We prove the nonexistence of multi-bubble solutions for several types of problems on smooth bounded convex domains. Problems we study include the Liouville equation

     

Convergence rates of regularized solutions of nonlinear ill-posed operator equations involving monotone operators

The convergence rates of regularized solutions of nonlinear ill-posed operator equations involving monotone operators are investigated, and conditions that guarantee convergence rates like and are given, where δ denotes the noise level of the data perturbation. Project supported by the National Natural Science Foundation of China (Grant No. 19671029).     

偶数阶中立型泛函微分方程最终正解的存在性

获得了偶数阶中立型泛函微分方程[a(t)x(t)-b(t)x(t-r)](n) q(t)f(x(t-σ))=0 (t≥0)存在最终正解的充分条件．     

Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients

We investigate linear and quasilinear evolutionary partial integro-differential equations of second order which include time fractional evolution equations of time order less than one. By means of suitable energy estimates and De Giorgi's iteration technique we establish results asserting the global boundedness of appropriately defined weak solutions of these problems. We also show that a maximum principle holds for such equations.     

非线性微分方程的正周期解

§ 1　 IntroductionIn[1 ] ,Saker and Agarwal studied the existence and uniqueness of positive periodicsolutions of the nonlinear differential equationN′(t) =-δ(t) N(t) + p(t) N(t) e- a N(t) ,(1 )whereδ(t) and p(t) are positive T-periodic functions.They proved that if p* >δ* ,then(1 ) has a unique T-periodic positive solution,wherep* =min0≤ t≤ Tp(t) ,δ* =max0≤ t≤ Tδ(t) .　　In view ofthe papermentioned above,whatcan be said aboutequation(1 ) when p* ≤δ* ?In this paper,we conside…     

Existence of periodic solutions for neutral functional differential equations with nonlinear difference operator

In this paper, the authors consider the problem of existence of periodic solutions for a second order neutral functional differential system with nonlinear difference D-operator. For such a system, since the possible periodic solutions may not be differentiable, our method is based on topological degree theory of condensing field, not based on Leray Schauder topological degree theory associated to completely continuous field.