We introduce the notions of a generalized -contraction, a generalized -weak contraction, a -weak JS-contraction, an integral-type -weak contraction, and an integral-type -weak JS-contraction to establish the fixed point, fixed ellipse, and fixed elliptic disc theorems. Further, we verify these by illustrative examples with geometric interpretations to demonstrate the authenticity of the postulates. The motivation of this work is the fact that the set of nonunique fixed points may include a geometric figure like a circle, an ellipse, a disc, or an elliptic disc. Towards the end, we provide an application of -contraction to chemical sciences.

中文翻译：

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度量空间中的唯一性和非唯一性不动点及其在化学科学中的应用

我们引入一个广义的概念-收缩，广义-弱收缩，一-弱JS-收缩，整体型-弱收缩，整体型-弱JS-收缩，建立定点，定椭圆形，和固定的椭圆盘定理。此外，我们通过带有几何解释的说明性例子来验证这些假设，以证明假设的真实性。这项工作的动机是这样的事实，即一组非唯一的固定点可能包括一个几何图形，例如圆形，椭圆形，圆盘形或椭圆形圆盘。最后，我们提供-收缩在化学科学中的应用。