Existence of the positive solutions for boundary value problems of mixed differential equations involving the Caputo and Riemann–Liouville fractional derivatives

Boundary Value Problems

Research

Weihua Jiang¹  Chenguang Yan¹  Yujing Liu¹ 

[1] School of Sciences, Hebei University of Science and Technology, 050018, Shijiazhuang, Hebei, P.R. China;

关键词: Caputo fractional derivative;    Riemann–Liouville fractional derivative;    Green’s function;    Coupled system;    Boundary value problem;   

DOI  :  10.1186/s13661-023-01696-4

 received in 2022-11-03, accepted in 2023-01-18,  发布年份 2023

来源: Springer

【 摘 要 】

We prove the existence of the solutions for the new mixed differential equations, which is characteristic of the right-sided Caputo and the left-sided Riemann–Liouville fractional derivatives. There are four major ingredients. The first is composed of some basic definitions and lemmas. The second is the Green’s function of the new mixed fractional differential equations. We calculate the corresponding Green’s functions as well as their properties. The third, which is the main new ingredient of this paper, is demonstration of the existence of the solutions for fractional equations by the fixed-point theorem in cone expansion and compression of norm type. The fourth, as applications, is the example provided to illustrate our main results.

【 授权许可】

CC BY   
© The Author(s) 2023

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