On Hilfer generalized proportional fractional derivative

Advances in Difference Equations

article

Ahmed, Idris¹  Kumam, Poom¹  Jarad, Fahd⁴  Borisut, Piyachat¹  Jirakitpuwapat, Wachirapong¹ 

[1] KMUTTFixed Point Research Laboratory, KMUTT-Fixed Point Theory and Applications Research Group (KMUTT-FPTA), Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT);Department of Mathematics and Computer Science, Sule Lamido University;Department of Medical Research, China Medical University Hospital, China Medical University;Department of Mathematics, Çankaya University

关键词: Existence;    Proportional fractional derivative;    Fixed point theorems;    Nonlocal condition;    Volterra integral equation;   

DOI  :  10.1186/s13662-020-02792-w

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

Motivated by the Hilfer and the Hilfer–Katugampola fractional derivative, we introduce in this paper a new Hilfer generalized proportional fractional derivative, which unifies the Riemann–Liouville and Caputo generalized proportional fractional derivative. Some important properties of the proposed derivative are presented. Based on the proposed derivative, we consider a nonlinear fractional differential equation with nonlocal initial condition and show that this equation is equivalent to the Volterra integral equation. In addition, the existence and uniqueness of solutions are proven using fixed point theorems. Furthermore, we offer two examples to clarify the results.

【 授权许可】

CC BY