半序空间中二元算子方程的可解性及应用

SOLVABILITY OF BINARY OPERATOR EQUATIONS IN PARTIAL ORDERED SPACES AND APPLICATIONS

首先在Banach空间中,利用半序方法和锥理论,研究了混合单调算子方程Lx=N(x,y)在反向上下解条件下的耦合解的存在性.然后在完备度量空间和Banach空间中,利用半序方法和锥理论,研究算子方程Lx=N(x,x)解的存在性唯一性问题,得到了一些新结论.所得的部分结论改进了最近一些文献中的重要结果.最后,将所得的部分结果应用于非线性算子固有值与固有元的存在性的研究中.

In this paper, the existence of the coupled solution for the operator equation Lx = N（x, x） is proved by the techniques of partial order and the theory of cone in Banach spaces under the condition of counter upper-down solutions. Then, by means of the techniques of partial order and the theory of cone, the existence and uniqueness of solution for the mixed monotone operator equation Lx = N（x, x） is discussed in complete metric spaces and Banach spaces, respectively. Some of the main results presented in this work improve the corresponding results in the recent works. Finally, part of the given results is applied to the studies of the existence of eigenvalues and eigenvectors for nonlinear operators.