Boundary Value Problem for Impulsive Delay Fractional Differential Equations with Several Generalized Proportional Caputo Fractional Derivatives

Fractal and Fractional

article

Ravi P. Agarwal¹  Snezhana Hristova² 

[1] Department of Mathematics, Texas A&M University-Kingsville;Faculty of Mathematics and Informatics, Plovdiv University “P. Hilendarski”

关键词: generalized proportional Caputo fractional derivatives;    delays;    impulses;    boundary value problem;    integral presentation of the solution;    existence;    uniqueness34A34;    34K45;    34A08;   

DOI  :  10.3390/fractalfract7050396

学科分类：社会科学、人文和艺术（综合）

来源: mdpi

【 摘 要 】

A scalar nonlinear impulsive differential equation with a delay and generalized proportional Caputo fractional derivatives (IDGFDE) is investigated. The linear boundary value problem (BVP) for the given fractional differential equation is set up. The explicit form of the unique solution of BVP in the special linear case is obtained. This formula is a generalization of the explicit solution of the case without any delay as well as the case of Caputo fractional derivatives. Furthermore, this integral form of the solution is used to define a special proportional fractional integral operator applied to the determination of a mild solution of the studied BVP for IDGFDE. The relation between the defined mild solution and the solution of the BVP for the IDGFDE is discussed. The existence and uniqueness results for BVP for IDGFDE are proven. The obtained results in this paper are a generalization of several known results.

【 授权许可】

CC BY