Multiplicity results for asymptotically linear elliptic problems at resonance

In this paper several new multiplicity results for asymptotically linear elliptic problem at resonance are obtained via Morse theory and minimax methods. Some new observations on the critical groups of a local linking-type critical point are used to deal with the resonance case at 0.     

Nontrivial periodic solutions for asymptotically linear resonant difference problem

In this paper the existence of nontrivial periodic solution for second order asymptotically linear difference equation at resonance is obtained. The methods used here are based on combining the minimax methods and the Morse theory, especially the observation on the critical groups.     

Nontrivial solutions of semilinear equations at resonance

In this paper we prove new existence results concerning nontrivial solutions to semilinear elliptic problem at resonance. The methods used here are based on combining the minimax methods and the Morse theory.     

Stability of the quadratic functional equation in Lipschitz spaces

Let G be an Abelian group with a metric d and E a normed space. For any f:G→E we define the quadratic difference of the function f by the formula

Qf(x,y):=2f(x)+2f(y)−f(x+y)−f(x−y)     

含临界位势与临界参数的超线性椭圆型方程

本文考虑一类含临界位势与临界参数的超线性椭圆型方程解的存在性.本文应用Morse理论,考虑非线性项f(x,s)在零点附近以及无穷远处的性质,给出了方程在某个新的Sobolev-Hardy空间中解的存在性.     

A convergence proof for an affine-scaling algorithm for convex quadratic programming without nondegeneracy assumptions

This paper presents a theoretical result on convergence of a primal affine-scaling method for convex quadratic programs. It is shown that, as long as the stepsize is less than a threshold value which depends on the input data only, Ye and Tse's interior ellipsoid algorithm for convex quadratic programming is globally convergent without nondegeneracy assumptions. In addition, its local convergence rate is at least linear and the dual iterates have an ergodically convergent property.Research supported in part by the NSF under grant DDM-8721709.     

Computations of critical groups and applications to asymptotically linear wave equation and beam equation

In this paper, by using the Morse index theory for strongly indefinite functionals developed in [Nonlinear Anal. TMA, in press], we compute precisely the critical groups at the origin and at infinity, respectively. The abstract theorems are used to study the existence (multiplicity) of nontrivial periodical solutions for asymptotically wave equation and beam equation with resonance both at infinity and at zero.     

Regularity and Morse Index of the Solutions to Critical Quasilinear Elliptic Systems

Regularity results and critical group estimates are studied for critical (p, r)-systems. Multiplicity results of solutions for a critical potential quasilinear system are also proved using Morse theory.     

Existence results for a nonlinear system with a parameter

In this paper, the monotone operator principle, the critical point theory and the morse theory are employed to discuss the system of nonlinear equations of the form Au=λf(u), some existence results are obtained.     

Existence and multiple solutions for a nonlinear system with a parameter

In this paper, the critical point theory and the Morse theory are employed to discuss the system of nonlinear equations of the form Au=λf(u)$Au=\lambda f\left(u\right)$, where the matrix A$A$ is not necessarily positive definite. Some existence and multiple solutions are obtained.     

Generalized saddle point theorem and asymptotically linear problems with periodic potential

We prove a critical point theorem, which is an infinite dimensional generalization of the classical saddle point theorem of P. H. Rabinowitz. As an application we obtain solution of asymptotically linear Schrödinger equations with periodic potential.     

Common fixed point theorem for noncommuting mappings satisfying a generalized asymptotically nonexpansive condition

We present a common fixed point theorem for generalized asymptotically nonexpansive and noncommuting mappings in normed linear spaces.      

Stability for quadratic functional equation in the spaces of generalized functions

In this paper, we consider the general solution of quadratic functional equation

f(ax+y)+f(ax−y)=f(x+y)+f(x−y)+2(a²−1)f(x)     

The stability theorem for smooth pseudoisotopies

The stability theorem states that the suspension map C(M) C(M X I) defined on the pseudoisotopy space C(M)=Diff(M X I rel M X O U M X I) of a compact smooth n-manifold M is n/3-connected. This implies that C(M) has the R~ n/3-homotopy type of the stable pseudoisotopy space P(M) which is related to Waldhausen's algebraic K-theory of spaces by Waldhausen's formula A(X) S(X₊) X B2P(X). This paper gives a detailed proof of the smooth stability theorem following ideas by Hatcher for the proof of a PL stability theorem.Supported by NSF Grant No. MCS-85-02317.     

Nontrivial solutions for discrete boundary value problems with multiple resonance via computations of the critical groups

In this paper, we study the existence of nontrivial solutions for a class of second-order difference equations with multiple resonance at both infinity and the origin by applying the critical point theory and Morse theory.     

Bifurcation for strongly indefinite functional and applications to Hamiltonian system and noncooperative elliptic system

In this paper, we discuss the bifurcation problems for strongly indefinite functional via Morse theory. The generalized topological degree for a class of vector fields is defined. As applications, we study the bifurcation problems for Hamiltonian system and noncooperative elliptic system.     

Multiple solutions of some boundary value problems with parameters

In this paper, we study the existence and multiplicity of nontrivial solutions for the following second-order Dirichlet nonlinear boundary value problem with odd order derivative: −u^″(t)+au^′(t)+bu(t)=f(t,u(t)) for all t∈[0,1] with u(0)=u(1)=0, where a,b∈R¹, f∈C¹([0,1]×R¹,R¹). By using the Morse theory, we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution, two nontrivial solutions and infinitely many solutions, separately.     

Asymptotic aspect of the quadratic functional equation in multi-normed spaces

We investigate the Hyers-Ulam stability of the quadratic functional equation for mappings from abelian groups into multi-normed spaces. We also study the stability on a restricted domain and present an asymptotic behavior of the quadratic equation in the framework of multi-normed spaces.