The Well-Posedness of Fractional Systems with Affine-Periodic Boundary Conditions

• Published in: Differential equations and dynamical systems Vol. 28; no. 4; pp. 1015 - 1031
• Main Authors: Xu, Fei, Li, Yong, Gao, Yixian, Xu, Xu
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01.10.2020; Springer Nature B.V
• Subjects: Boundary conditions; Computer Science; Derivatives; Engineering; Integrals; Kinetic equations; Mathematics; Mathematics and Statistics; Nonlinearity; Original Research; Uniqueness; Nonlinear fractional dynamical systems; Affine-periodic boundary values; Existence; Uniqueness
• ISSN: 0971-3514; 0974-6870
• DOI: 10.1007/s12591-017-0360-z

This paper is devoted to study the existence and uniqueness of solutions for a class of nonlinear fractional dynamical systems with affine-periodic boundary conditions. We can show that there exists a solution for an α -fractional system via the homotopy invariance of Brouwer degree, where 0 < α ≤ 1 . Furthermore, using Gronwall–Bellman inequality, we can prove the uniqueness of the solution if the nonlinearity satisfies the Lipschitz continuity. We apply the main theorem to the fractional kinetic equation and fractional oscillator with constant coefficients subject to affine-periodic boundary conditions. And in appendix, we give the proof of the nonexistence of affine-periodic solution to a given ( α , Q , T ) -affine-periodic system in the sense of Riemann–Liouville fractional integral and Caputo derivative for 0 < α < 1 .