| Abstract: |
| This paper establishes sign constancy of Green`s function criteria for a class of boundary value problems involving fractional differential equations. We study equations driven by the Riemann--Liouville fractional derivative of order
$\beta \in (n-1,n]$, coupled with linear continuous Volterra operators acting on unknown function and its derivatives.
The main development of this paper is the idea of subordination in sign properties of Green`s functions for various boundary value problems. Starting with focal problems, we can conclude about sign-constancy of others, for example, multipoint and non-local problems can be among them.
For focal problem, we demonstrate equivalence of the fact of sign constancy of Green`s function and its derivatives, and the fact that the spectral radius of corresponding compact operator is less than one. Furthermore, a Vall\`ee--Poussin--type comparison theorem is established. Several illustrative examples, including equations with integral terms and deviating arguments, are presented to demonstrate the applicability of the obtained assertions. The results extend classical Vall\`ee--Poussin theorem to fractional differential equations with functional perturbations. |
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