This work intends to treat the existence of mild solutions for the Hilfer fractional hybrid differential equation (HFHDE) with linear perturbation of first and second type in partially ordered Banach spaces. First, we establish the results concerning the actuality of fixed point of sum and product of operators via the concepts of measure of noncompactness and simulation functions in partially ordered spaces. Then combining these fixed point theorems with the concepts in fractional calculus, new existence results for mild solutions of HFHDE’s are established. Furthermore, the presented fixed point results and existence results improve and extend the present state-of-art in the literature. Competent examples in support of theory are illustrated for better understanding.

中文翻译：

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通过新的 Krasnoselskii 型不动点定理解决 Hilfer 分数式混合问题的存在性

这项工作旨在处理 Hilfer 分数式混合微分方程 (HFHDE) 在偏序 Banach 空间中具有第一类和第二类线性扰动的温和解的存在。首先，我们通过偏序空间中的非紧性测度和模拟函数的概念，建立了关于算子的和和乘积不动点的现实性的结果。然后将这些不动点定理与分数阶微积分中的概念相结合，建立了HFHDE 的温和解的新存在结果。此外，所提出的不动点结果和存在结果改进和扩展了文献中的现有技术。为了更好地理解，展示了支持理论的有效例子。