Antiperiodic Boundary Value Problem for a Semilinear Differential Equation of Fractional Order

• Main Author: G. G. Petrosyan
• Format: Article
• Language: English
• Published: Irkutsk State University 2020-12-01
• Series: Известия Иркутского государственного университета: Серия "Математика"
• Subjects: caputo fractional derivative; semilinear differential equation; boundary value problem; fixed point; condensing mapping; measure of noncompactness
• Online Access: http://mathizv.isu.ru/en/article/file?id=1359

The present paper is concerned with an antiperiodic boundary value problem for a semilinear differential equation with Caputo fractional derivative of order $q\in(1,2)$ considered in a separable Banach space. To prove the existence of a solution to our problem, we construct the Green’s function corresponding to the problem employing the theory of fractional analysis and properties of the Mittag-Leffler function . Then, we reduce the original problem to the problem on existence of fixed points of a resolving integral operator. To prove the existence of fixed points of this operator we investigate its properties based on topological degree theory for condensing mappings and use a generalized B.N. Sadovskii-type fixed point theorem.