In this paper, we prove some fixed point theorems under a convex combination of generalized (\(\epsilon -\delta \)) type rational contractions in which the fixed point may or may not be a point of discontinuity. As a by-product we explore some new answers to the open question posed by Rhoades (Contemp Math 72:233–245, 1988). Furthermore, we consider geometric properties of the fixed point set of a self-mapping on a metric space. We define a new kind of contractive mapping and prove that the fixed point set of this kind of contraction contains a circle (resp. a disc). Several non-trivial examples are given to illustrate our results. Apart from these, an application of discontinuous activation functions, frequently used in neural networks is also given.

在本文中，我们证明了广义（\（\ epsilon-\ delta \））型有理收缩的凸组合下的一些不动点定理，其中不动点可能是也可能不是不连续点。作为副产品，我们探索了对Rhoades提出的开放性问题的一些新答案（Contemp Math 72：233–245，1988）。此外，我们考虑度量空间上自映射的不动点集的几何性质。我们定义了一种新的收缩映射，并证明了这种收缩的不动点集包含一个圆（分别是一个圆盘）。给出了几个非平凡的例子来说明我们的结果。除此之外，还给出了在神经网络中经常使用的不连续激活函数的应用。