This paper considers the generalized Langevin equation involving ψ-Caputo fractional derivatives in a Banach space. The fractional derivative is generalized from the Caputo derivative (ψ(t) = t), the Caputo–Katugampola (ψ(t) = ψρ(t) = (tρ − 1)/ρ, the Hadamard derivative (ψ(t) = ψH(t) =ln t). We investigate the existence of mild solutions uψ of the problem, in which the source function is assumed to satisfy some weakly singular conditions. Before proceeding to the main results, we transform the problem into an integral equation. Based on the obtained integral equation, the main results are proved via the nonlinear Leray–Schauder alternatives and Banach fixed point theorems. To prove this end, a new generalized weakly Gronwall-type inequality is established. Further, we prove that the mild solution of the problem is dependent continuously on the inputs: initial data, fractional orders and the friction constant. As a consequence, we deduce that the solution uψρ of the equation involving the Caputo–Katugampola derivative tends to the solution uψH of the equation involving the Hadamard derivative as ρ → 0+.

中文翻译：

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关于涉及 ψ-CAPUTO 分数阶导数的非线性广义 LANGEVIN 方程

本文考虑广义朗之万方程涉及ψ-Banach 空间中的 Caputo 分数导数。分数导数是从 Caputo 导数 (ψ(吨) = 吨), 卡普托-卡图甘波拉 (ψ(吨) = ψρ(吨) = (吨ρ - 1)/ρ, Hadamard 导数 (ψ(吨) = ψH(吨) =ln 吨）。我们调查温和解决方案的存在你ψ问题，其中假设源函数满足一些弱奇异条件。在进行主要结果之前，我们将问题转换为积分方程。基于获得的积分方程，主要结果通过非线性 Leray-Schauder 替代方案和 Banach 不动点定理证明。为了证明这一点，建立了一个新的广义弱 Gronwall 型不等式。此外，我们证明了问题的温和解决方案持续依赖于输入：初始数据、分数阶和摩擦常数。因此，我们推断解决方案你ψρ涉及 Caputo-Katugampola 导数的方程的你ψH涉及 Hadamard 导数的方程为ρ → 0+.