In the current manuscript, two fixed-point theorems for Dass-Gupta and Gupta-Saxena rational interpolative-type operators are studied in the setting of metric spaces. For the authenticity of the presented work, examples and applications to the existence of a solution to the Caputo-Fabrizio fractional derivative and Caputo-Fabrizio fractal-fractional derivative are also discussed.

中文翻译：

---

有理插值型算子的不动点定理及其应用

在当前的手稿中，在度量空间的设置中研究了Dass-Gupta和Gupta-Saxena有理插值型算子的两个不动点定理。为了保证所提出的工作的真实性，还讨论了Caputo-Fabrizio分形导数和Caputo-Fabrizio分形-分形导数解的存在性的示例和应用。