An analog of the classical Vallée-Poussin theorem about differential inequality in the theory of ordinary differential equations is developed for fractional functional differential equations. The main results are obtained in a form of a theorem about several equivalent assertions. Among them solvability of two-point boundary value problems with fractional functional differential equation, negativity of Green’s function, and its derivatives and existence of a function v(t) satisfying a corresponding differential inequality. Thus the Vallée-Poussin theorem presents one of the possible “entrances” to assertions on nonoscillating properties and assertions about the negativity of Green’s functions and their derivatives for various two-point problems. Choosing the function v(t) in the condition, we obtain explicit tests of sign-constancy of Green’s functions and their derivatives. It can be stressed that a choice of a corresponding function in the Vallée-Poussin theorem leads to explicit criteria in the form of algebraic inequalities, which, as we demonstrate with examples, cannot be improved. Replacing strict inequalities with non-strict ones will already lead to incorrect statements. In some cases, the well-known results obtained in the form of the Lyapunov inequalities for fractional differential equations can be improved based on ours. Another development, we propose, is a concept to consider fractional functional differential equations. The basis of this concept is to reduce boundary value problems for fractional functional differential equations to operator equations in the space of essentially bounded functions. Fractional functional differential equations can appear in various applications and in the process of construction of equations for a corresponding component of a solution-vector of a system of fractional equations.

为分数泛函微分方程开发了关于常微分方程理论中关于微分不等式的经典 Vallée-Poussin 定理的类比。主要结果以关于几个等价断言的定理的形式获得。其中包括分数泛函微分方程的两点边值问题的可解性、格林函数的负性及其导数以及满足相应微分不等式的函数v ( t ) 的存在性。因此，Vallée-Poussin 定理提出了关于非振荡性质的断言和关于格林函数及其导数对于各种两点问题的负性的断言的可能“入口”之一。选择函数v（吨) 在该条件下，我们获得了格林函数及其导数的符号恒常性的显式检验。可以强调的是，在 Vallée-Poussin 定理中选择相应的函数会导致代数不等式形式的明确标准，正如我们通过示例所证明的那样，它无法改进。用非严格的不等式代替严格的不等式已经导致错误的陈述。在某些情况下，以分数阶微分方程的李雅普诺夫不等式形式获得的众所周知的结果可以在我们的基础上得到改进。我们提出的另一个发展是考虑分数泛函微分方程的概念。这个概念的基础是将分数泛函微分方程的边值问题简化为本质有界函数空间中的算子方程。