Boundary value problems for semilinear differential inclusions of fractional order in a Banach space

Abstract

In the present paper, we show that the solution set of a fractional order semilinear differential inclusion in a Banach space has the topological structure of an -set. This result allows to apply a fixed point result for condensing multimaps to the translation multioperator along the trajectories of such inclusion and to prove the existence of solutions satisfying periodic and anti-periodic boundary value conditions. An example concerning with a fractional order feedback control system is presented.

Keywords:

• Differential inclusion
• fractional derivative
• solution set
• -set
• translation multioperator
• measure of noncompactness
• condensing multimap
• fixed point
• periodic problem
• anti-periodic problem

AMS Subject Classifications:

• Primary: 34G25
• 34A08
• Secondary: 34A60
• 34B15
• 34C25
• 47H04
• 47H08
• 47H10

Acknowledgements

The work on the paper was carried out during Prof. V. Obukhovskii’s and Prof. M. Kamenskii’s visit to the Center for Fundamental Science, Kaohsiung Medical University and the Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan in 2016. They would like to express their gratitude to the members of the Center and the Department for their kind hospitality. The authors are grateful to anonymous referees for their valuable remarks. The work of the first, second and the third author is supported by the Ministry of Education and Science of the Russian Federation in the frameworks of the project part of the state work quota (Project No 1.3464.2017).

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by the joint Taiwan MOST - Russia RFBR [grant number 17-51-52002]; the RFBR [grant number 17-01-00365], [grant number 16-01-00370], [grant number 16-01-00386], as well as the RSF [grant number 14-21-00066] (in the Voronezh State University). The work of Prof. V. Obukhovskii was supported by the Ministry of Education and Science of the Russian Federation (the [Agreement number 02.a03.21.0008] of 24 June 2016).