Vallée-Poussin Theorem for Equations with Caputo Fractional Derivative

Abstract

ABSTRACT In this paper, the functional differential equation ( D C a + α x ) ( t ) + ∑ i = 0 m ( T i x ( i ) ) ( t ) = f ( t ) , t ∈ [ a , b ] , with Caputo fractional derivative D C a + α is studied. The operators Ti act from the space of continuous to the space of essentially bounded functions. They can be operators with deviations (delayed and advanced), integral operators and their various linear combinations and superpositions. Such equations could appear in various applications and in the study of systems of, for example, two fractional differential equations, when one of the components can be presented from the first equation and substituted then to another. For two-point problems with this equation, assertions about negativity of Green’s functions and their derivatives with respect to t are obtained. Our technique is based on an analog of the Vallée-Poussin theorem for differential inequalities, which is proven in our paper and gives necessary and sufficient conditions of negativity of Green’s functions and their derivatives for two-point problems: there exists a positive function v satisfying corresponding boundary conditions and the inequality ( D C a + α ν ) ( t ) + ∑ i = 0 m ( T i v ( i ) ) ( t ) < 0 ,   t ∈ [ a ,   b ] . Choosing the function v, we obtain explicit sufficient tests of sign-constancy of Green’s functions and its derivatives. It is demonstrated that these tests cannot be improved in a general case. Influences of delays on these sufficient conditions are analyzed. It is demonstrated that the tests can be essentially improved for “small” deviations.