Some Existence Theorems Of Solutions For Nonlinear Operator Equations

In this paper, we mainly study the existence theorems of solutions for nonlinear operator equation. The paper is divided into three chapters.Chapter 1: Some people have studied the eigenvalue problem of complete continuous mapping and k-set contraction mapping and obtained many new results. Such as H. Amann[1]. DA Jun Gwo[2-4], I. Massobo, C.A. Stuart. But there are few people to study the existence of eigenvectors and eigenvalues for 1-set-contraction mapping. In this chapter, the author uses the fixed point index of 1-set contraction mapping to study the problem of their eigenvectors and eigenvalues and gets some new eigenvalues and eigenvectors existence theorems.Chapter 2: In this chapter, the author obtains the multiple fixed point theorem of increasing operator and the coupled quasi-fixed point theorem of mixed monotone operator by using the relative sigma-complete set. Both of them satisfy the counter upper-down fixed point condition. The results also are new.Chapter 3: In this chapter, the author is to study the existence theorems of random solutions for random semi-closed 1-set contraction operators equation A(co,x)=ux (u>l) in different border condition and obtains some new results.