On the existence of positive solutions of nonlinear second order differential equations

Under suitable conditions on , the boundary value problem

has at least one positive solution. Moreover, we also apply this main result to establish several existence theorems of multiple positive solutions for some nonlinear (elliptic) differential equations.

     

Arbitrarily large solutions of the conformal scalar curvature problem at an isolated singularity

We study the conformal scalar curvature problem

where is a continuous function. We show that a necessary and sufficient condition on for this problem to have positive solutions which are arbitrarily large at is that be less than 1 on a sequence of points in which tends to .

     

On the strong maximum principle

This paper presents a necessary and sufficient condition on the convex function in order that continuous solutions to

satisfy a Strong Maximum Principle on any open connected .

     

Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations

In this paper, the authors consider the boundary value problem

and give sufficient conditions for the existence of any number of symmetric positive solutions of (E)-(B). The relationships between the results in this paper and some recent work by Henderson and Thompson (Proc. Amer. Math. Soc. 128 (2000), 2373-2379) are discussed.

     

Unprovability of sharp versions of Friedman's sine-principle

For every and every function of one argument, we introduce the statement : ``for all , there is such that for any set of rational numbers, there is of size such that for any two -element subsets and in , we have

We prove that for and any function eventually dominated by , the principle is not provable in . In particular, the statement is not provable in Peano Arithmetic. In dimension 2, the result is: does not prove , where and is the inverse of the Ackermann function.

     

一类n-阶m-点奇异边值问题的正解

研究n-阶m-点奇异边值问题其中h(t)允许在t=0,t=1处奇异,f(t,v_0,v_1,…,v_(n-2))允许在v_i=0(i=0,1,…,n-2)处奇异.利用锥拉伸与压缩不动点定理得到了上述奇异边值问题正解的存在性.     

A Hodge decomposition interpretation for the coefficients of the chromatic polynomial

Let be a simple graph with nodes. The coloring complex of , as defined by Steingrimsson, has -faces consisting of all ordered set partitions, in which at least one contains an edge of . Jonsson proved that the homology of the coloring complex is concentrated in the top degree. In addition, Jonsson showed that the dimension of the top homology is one less than the number of acyclic orientations of .

In this paper, we show that the Eulerian idempotents give a decomposition of the top homology of into components . We go on to prove that the dimensions of the Hodge pieces of the homology are equal to the absolute values of the coefficients of the chromatic polynomial of . Specifically, if we write , then .

     

A two-parameter ``Bergman space' inequality

For , define , where is a tensor product of one-parameter Haar functions. Let and . We prove a sufficient condition, which is close to necessary, on double sequences of weights and non-negative , which ensures that the inequality

holds for all . We extend our result to an inequality concerning two-parameter wavelet families.

     

On a Liouville-type theorem and the Fujita blow-up phenomenon

The main purpose of this paper is to obtain the well-known results of H.Fujita and K.Hayakawa on the nonexistence of nontrivial nonnegative global solutions for the Cauchy problem for the equation

with on the half-space as a consequence of a new Liouville theorem of elliptic type for solutions of () on . This new result is in turn a consequence of other new phenomena established for nonlinear evolution problems. In particular, we prove that the inequality

has no nontrivial solutions on when We also show that the inequality

has no nontrivial nonnegative solutions for , and it has no solutions on bounded below by a positive constant for 1.$">

     

Global existence for the critical generalized KdV equation

We discuss results regarding global existence of solutions for the critical generalized Korteweg-de Vries equation,

The theory established shows the existence of global solutions in Sobolev spaces with order below the one given by the energy space , i.e. solutions corresponding to data , 3/4$">, with , where is the solitary wave solution of the equation.

     

Inequalities based on a generalization of concavity

The concept of concavity is generalized to functions, , satisfying order differential inequalities, , and homogeneous two-point boundary conditions, , for some . A piecewise polynomial, which bounds the function, , below, is constructed, and then is employed to obtain that , where max and denotes the supremum norm. An analogous inequality for a related Green's function is also obtained. These inequalities are useful in applications of certain cone theoretic fixed point theorems.

     

Large time behavior of small solutions to subcritical derivative nonlinear Schrödinger equations

We consider the derivative nonlinear Schrödinger equations

where the coefficient satisfies the time growth condition

is a sufficiently small constant and the nonlinear interaction term consists of cubic nonlinearities of derivative type

where and . We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate , for all , where . Furthermore we show that for there exist the usual scattering states, when and the modified scattering states, when

     

On the stability of the standard Riemann semigroup

We consider the dependence of the entropic solution of a hyperbolic system of conservation laws

on the flux function . We prove that the solution is Lipschitz continuous w.r.t. the norm of the derivative of the perturbation of . We apply this result to prove the convergence of the solution of the relativistic Euler equation to the classical limit.

     

The best bounds in Wallis' inequality

For all natural numbers , let denote a double factorial. Then

The constants and are the best possible. From this, the well-known Wallis' inequality is improved.

     

Fixed points and stability of a nonconvolution equation

In this note we consider an equation of the form

and give conditions on and to ensure that the zero solution is asymptotically stable. When applied to the classical case of , these conditions do not require that , nor do they involve the sign of or the sign of any derivative of .

     

Existence and uniqueness theorems for singular anisotropic quasilinear elliptic boundary value problems

On bounded domains we consider the anisotropic problems in with 1$"> and on and in with and on . Moreover, we generalize these boundary value problems to space-dimensions 2$">. Under geometric conditions on and monotonicity assumption on we prove existence and uniqueness of positive solutions.     

On random algebraic polynomials

This paper provides asymptotic estimates for the expected number of real zeros and -level crossings of a random algebraic polynomial of the form , where are independent standard normal random variables and is a constant independent of . It is shown that these asymptotic estimates are much greater than those for algebraic polynomials of the form .

     

Almost automorphic solutions of semilinear evolution equations

We are concerned with the semilinear differential equation in a Banach space ,

where generates an exponentially stable -semigroup and is a function of the form . Under appropriate conditions on and , and using the Schauder fixed point theorem, we prove the existence of an almost automorphic mild solution to the above equation.

     

Mean value of mixed exponential sums

For integers , , , with , and Dirichlet character , we define a mixed exponential sum

where , and denotes the summation over all with . The main purpose of this paper is to study the mean value of

and to give a related identity on the mean value of the general Kloosterman sum

where .

     

Monoid of self-equivalences and free loop spaces

Let be a simply-connected closed oriented -dimensional manifold. We prove that for any field of coefficients there exists a natural homomorphism of commutative graded algebras where is the loop algebra defined by Chas and Sullivan. As usual denotes the monoid of self-equivalences homotopic to the identity, and the space of based loops. When is of characteristic zero, yields isomorphisms where denotes the Hodge decomposition on .