In this paper, we investigate sufficient conditions for the existence of solutions to the system

$$\begin{aligned} \left\{ \begin{array}{l} Tx = x,\\ \alpha _i(x) = 0_E,\quad i=1,2,\ldots ,r , \end{array} \right. \end{aligned}$$

where \(0_E\) is the zero vector of E, and \(\alpha _i :E\rightarrow E \; \; i=1,2,\ldots , r\) are mappings, T is a mapping satisfying the Pachpatte-contraction.

中文翻译：

---

Pachpatte算子方程组的不动点定理。

在本文中，我们研究了系统解存在的充分条件

$$ \ begin {aligned} \ left \ {\ begin {array} {l} Tx = x，\\ \ alpha _i（x）= 0_E，\ quad i = 1,2，\ ldots，r，\ end {数组} \ right。\ end {aligned} $$

其中\（0_E \）是E的零向量，而\（\ alpha _i：E \ rightarrow E \; \; i = 1,2，\ ldots，r \）是映射，T是满足Pachpatte的映射-收缩。