INVARIANT MEASURES AND UNIFORM POSITIVE ENTROPY PROPERTY FOR INVERSE LIMITS

In this paper, the interconnection of som e ergodic properties betw een a continuous selfm ap and its inverse lim itis studied. Ithas been proved that(1) theirinvariantBorelproba- bility m easures are identicalup to hom eom orphism and (2) they preserve uniform positive en- tropy property sim ultaneously. As applications, it is also proved that the upper sem i-continu- ous properties of their entropy m aps are restricted each other, and the entropy m ap of the asym ptotically h-expansivecontinuous m ap isuppersem i-continuous, atthe sam e tim e acontin- uous m ap having u.p.e. is topologicalweak-m ixing.

THE RELATION BETWEEN SIGN-CHANGING SOLUTION AND POSITIVE-NEGATIVE SOLUTIONS FOR NONLINEAR OPERATOR EQUATIONS AND ITS APPLICATIONS

By fixed point index theory and a result obtained by Amann, existence of the solution for a class of nonlinear operator equations x = Ax is discussed. Under suitable conditions, a couple of positive and negative solutions are obtained. Finally, the abstract result is applied to nonlinear Sturm-Liouville boundary value problem, and at least four distinct solutions are obtained.

ON THE NUMBER OF FIXED POINTS OF NONLINEAR OPERATORS AND APPLICATIONS

In this paper, the famous Amann three-solution theorem is generalized. Multiplicity question of fixed points for nonlinear operators via two coupled parallel sub-super solutions is studied. Under suitable conditions, the existence of at least six distinct fixed points of nonlinear operators is proved. The theoretical results are then applied to nonlinear system of Hammerstein integral equations.

A convergence analysis of the inexact Rayleigh quotient iteration and simplified Jacobi-Davidson method for the large Hermitian matrix eigenproblem

The inexact Rayleigh quotient iteration (RQI) is used for computing the smallest eigenpair of a large Hermitian matrix. Under certain condition, the method was proved to converge quadratically in literature. However, it is shown in this paper that under the original given condition the inexact RQI may not quadratically converge to the desired eigenpair and even may misconverge to some other undesired eigenpair. A new condition, called the uniform positiveness condition, is given that can fix misconvergence problem and ensure the quadratic convergence of the inexact RQI. An alternative to the inexact RQI is the Jacobi-Davidson (JD) method without subspace acceleration. A new proof of its linear convergence is presented and a sharper bound is established in the paper. All the results are verified and analyzed by numerical experiments.

A New H^2 Regularity Condition of the Solution to Dirichlet Problem of the Poisson Equation and Its Applications

We study the regularity of the solution of Dirichlet problem of Poisson equations over a bounded domain.A new sufficient condition,uniformly positive reach is introduced.Under the assumption that the closure of the underlying domain of interest has a uniformly positive reach,the H^2 regularity of the solution of the Poisson equation is established.In particular,this includes all star-shaped domains whose closures are of positive reach,regardless if they are Lipschitz domains or non-Lipschitz domains.Application to the strong solution to the second order elliptic PDE in non-divergence form and the regularity of Helmholtz equations will be presented to demonstrate the usefulness of the new regularity condition.

Pointwise Pseudo-Orbit Tracing Property and Chaos

In this article, we discuss the relationship between pointwise pseudo-orbit tracing property and chaotic properties such as topological mixing. When f has pointwise pseudo-orbit tracing property, we give some equal conditions of uniform positive entropy and completely positive entropy.