The Well-Posedness of Fractional Systems with Affine-Periodic Boundary Conditions
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The Well-Posedness of Fractional Systems with Affine-Periodic Boundary Conditions Fei Xu1 · Yong Li2 · Yixian Gao2 · Xu Xu3
© Foundation for Scientific Research and Technological Innovation 2017
Abstract 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 )-affineperiodic system in the sense of Riemann–Liouville fractional integral and Caputo derivative for 0 < α < 1. Keywords Affine-periodic boundary values · Nonlinear fractional dynamical systems · Existence · Uniqueness
The research of YL was supported in part by NSFC Grant: 11571065, 11171132 and National Research Program of China Grant 2013CB834100. The research of YG was supported in part by NSFC Grant: 11671071 and JLSTDP20160520094JH.
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Yixian Gao [email protected] Fei Xu [email protected] Yong Li [email protected] Xu Xu [email protected]
1
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, People’s Republic of China
2
School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, Jilin, People’s Republic of China
3
College of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
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Differ Equ Dyn Syst
Introduction The structure of the solution of an differential equation plays an important role in the theory of dynamical systems, such as periodicity, anti-periodicity, harmonic-periodicity, quasiperiodicity and so on. In 2103 [46], Li et al. introduced the concept “affine-periodic” which is a kind of symmetry rather than periodicity, which is the general version of periodicity, antiperiodicity, harmonic-periodicity and quasi-periodicity. There are some natural phenomena presenting affine-periodicity [9,34], such as, spiral wave (or affine-periodic wave), spiral line in geometry, and the orbit of the earth goes round the sun: the orbit is a circle or ellipse exactly in a plane, but in fact, the circle in the plane is only a projection in the space along the time axis. The orbit of the earth when it goes round the sun is cubic rather than just in a plane and the space position is rotating as the time walks a periodic, that is to say, the orbit of the earth goes round the sun presents the “affine-symmetry”. The subject of affine-periodic problem is essential and more and more researchers pay attention to “affine-symmetry” and consider the existence of aff
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