Vallée-Poussin Theorem For Equations With Caputo Fractional Derivative

Author

Martin Bohner, Missouri University of Science and TechnologyFollow
Alexander Domoshnitsky
Seshadev Padhi
Satyam Narayan Srivastava

Abstract

In this paper, the functional differential equation (^CD_a^α+x)(t) + ^mΣ_i=0 (T_ix(ⁱ))(t) = f(t); t 2 [a; b]; with Caputo fractional derivative ^CD_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 Vallee'-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 (^CD_a^α+v)(t) + ^mΣ_i=0 (T_iv(ⁱ))(t) < 0, t 2 [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.

Recommended Citation

M. Bohner et al., "Vallée-Poussin Theorem For Equations With Caputo Fractional Derivative," Mathematica Slovaca, vol. 73, no. 3, pp. 713 - 728, De Gruyter, Jun 2023.

The definitive version is available at https://doi.org/10.1515/ms-2023-0052

Department(s)

Mathematics and Statistics

Publication Status

Open Access

Keywords and Phrases

boundary value problems; Caputo derivative; differential inequality; fractional differential equations; positive solutions; sign constancy of Green's function; Vallée-Poussin theorem

International Standard Serial Number (ISSN)

1337-2211; 0139-9918

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2023 De Gruyter, All rights reserved.

Publication Date

01 June, 22023