引进了广义混合单调算子$N:X\times X\times X\to X$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.

In this paper, the new concepts of generalized mixed monotone operators $N:X\times X\times X\to X$ 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 complete  metric spaces and Banach spaces. Then the existence of solutions for   generalized mixed monotone operator equations $Lx=N(x,x,x)$ is studied in Banach spaces. The results extend and improve some  existing  results. Finally, an example is given to illustrate  the validity of the main results.