On a Periodic Boundary Value Problem for Fractional Quasilinear Differential Equations with a Self-Adjoint Positive Operator in Hilbert Spaces

Mikhail Kamenskii, Garik Petrosyan, Paul Raynaud de Fitte and Jen-Chih Yao
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Mikhail Kamenskii: Faculty of Mathematics, Voronezh State University, 394018 Voronezh, Russia
Garik Petrosyan: Faculty of Physics and Mathematics, Voronezh State Pedagogical University, 394043 Voronezh, Russia
Paul Raynaud de Fitte: Raphael Salem Mathematics Laboratory, University of Rouen Normandy, 76821 Rouen, France
Jen-Chih Yao: Research Center for Interneural Computing, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

Mathematics, 2022, vol. 10, issue 2, 1-12

Abstract: In this paper we study the existence of a mild solution of a periodic boundary value problem for fractional quasilinear differential equations in a Hilbert spaces. We assume that a linear part in equations is a self-adjoint positive operator with dense domain in Hilbert space and a nonlinear part is a map obeying Carathéodory type conditions. We find the mild solution of this problem in the form of a series in a Hilbert space. In the space of continuous functions, we construct the corresponding resolving operator, and for it, by using Schauder theorem, we prove the existence of a fixed point. At the end of the paper, we give an example for a boundary value problem for a diffusion type equation.

Keywords: fractional differential equation; Caputo fractional derivative; self-adjoint positive operator; fixed point; Mittag–Leffler function; Green function; completely continuous map (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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