半序空间中广义混合单调算子方程的可解性与应用

SOLVABILITY OF GENERALIZED MIXED MONOTONE OPERATOR EQUATIONS IN PARTIALLY ORDERED SPACES AND APPLICATIONS

引进了广义混合单调算子N∶X×X×X→X与算子方程Lx=N(x,y,z)的三元拟解的新定义.利用半序方法和锥理论,首先在完备度量空间和Banach空间中讨论了广义混合单调算子方程Lx=N(x,y,z)在正向和反向上下解条件下的三元拟解的存在性,其次,在Banach空间中研究了广义混合单调算子方程Lx=N(x,x,x)解的存在唯一性.所得结果改进和推广了相关文献中的的结果.最后,给出了主要结果的一个应用.

Abstract In this paper, the new concepts of generalized mixed monotone operators N∶X×X×X→X and tripled quasi-solutions for operator equations Lx = N（x, y, z） are introduced. By using partial order method and cone theory, the existence of tripled quasi-solutions for generalized mixed monotone operator equations Lx = N（x, y, z） under the condition of forward and counter upper-down solutions are studied in corn- plete metric spaces and Banach spaces. raixed monotone operator equations Lx Then the existence of solutions for generalized = N（x, x, x） is studied in Banach spaces. The results extend and improve some existing results. Finally, an example is given to il- lustrate the validity of the main results.