In this paper, we use the topological degree theory (TDT) to investigate the existence and uniqueness of solution for a class of evolution fractional order differential equations (FODEs) with proportional delay using Caputo derivative under local conditions. In the same line, we will also study different kinds of Ulam stability such as Ulam–Hyers (UH) stability, generalized Ulam–Hyers (GUH) stability, Ulam–Hyers–Rassias (UHR) stability and generalized Ulam–Hyers–Rassias (GUHR) stability for the considered problem. To justify our results we provide an example.

中文翻译：

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拓扑度数理论在比例延迟演化方程中的适用性

在本文中，我们利用拓扑度理论（TDT）研究了一类具有比例延迟的进化分数阶微分方程（FODE）在局部条件下使用Caputo导数的解的存在性和唯一性。同样，我们还将研究不同种类的 Ulam 稳定性，例如 Ulam-Hyers (UH) 稳定性、广义 Ulam-Hyers (GUH) 稳定性、Ulam-Hyers-Rassias (UHR) 稳定性和广义 Ulam-Hyers-Rassias ( GUHR) 所考虑问题的稳定性。为了证明我们的结果是正确的，我们提供了一个例子。