Leray-Schauder度的一种推广及不动点和固有值

A GENERALIZATION OF LERAY-SCHAUDER DEGREE, FIXED POINTS AND EIGENVALUES

设Χ是实Banach空间,dimΧ=∞,Ω(?)Χ是有界开集,F:(?)→Χ全连续,f=I-F,p∈Χ\kf((?)Ω),k∈R且k>O.我们定义d_L(kf,Ω,p)=d_(LS)(f,Ω,(1/k)p).于是d_L具有Leray-Schauder度的基本性质.应用这个拓扑度可以推广Schauder不动点定理和Rothe不动点定理.并且我们得到固有值存在定理:设F:(?)→Χ全连续,O∈(?),F(0)=0.假设S(?)(?)是非空闭集,使得inf{||x-y|| |x∈X\S,y∈(?)F(S)}>0,则F有无穷多个固有值.

Let X be real Banach space with dimX = , open bounded, F: X completely continuous, f = I-F, pX \ kf, kR and k>0. The author of thispaper defines . Then d has the basic properties of Leray-Schauder degree. This topdlogical degree may be applied to obtaining of some extension for both of Schauder's and Rothc's fixed point theorem. Moreover, some theorems for existence of eigenvalue can be obtained.Theorem Let F: Q -X be completely continuous, 0 and F(0)=0- Let S be nonempty closed, such that inf . Then, there are infinite eigenvalues of F.