In this paper, we obtain a Lyapunov-type and a Hartman–Wintner-type inequalities for a nonlinear fractional hybrid equation with left Riemann–Liouville and right Caputo fractional derivatives of order $$1/2<\alpha \le 1,$$ subject to Dirichlet boundary conditions. It is also shown that failure of the Lyapunov-type and Hartman–Wintner-type inequalities, corresponding nonlinear boundary value problem has only trivial solutions. In the case $$\alpha =1$$ , our results coincide with the classical Lyapunov and Hartman–Wintner inequalities, respectively.

中文翻译：

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非线性分数边值问题的李雅普诺夫型不等式

在本文中，我们获得了具有左黎曼-刘维尔和右卡普托分数阶导数$$1/2<\alpha \le 1,$$subject 的非线性分数式混合方程的Lyapunov 型和Hartman-Wintner 型不等式Dirichlet 边界条件。还表明，Lyapunov 型和 Hartman-Wintner 型不等式的失效，对应的非线性边值问题只有平凡解。在 $$\alpha =1$$ 的情况下，我们的结果分别与经典的 Lyapunov 和 Hartman-Wintner 不等式一致。