Zhang Lihong, Liu Xuehui, Wang Guotao
School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan, Shanxi 030031, China.
Heliyon. 2024 May 17;10(10):e31285. doi: 10.1016/j.heliyon.2024.e31285. eCollection 2024 May 30.
This article combines -Hilfer fractional calculus with glucose molecular graph, defines fractional differential and inclusion systems on each edge of a glucose molecular graph by the assumption that 0 or 1 marks the vertices, and studies the single-valued and multi-valued -Hilfer type fractional boundary value problems on the glucose molecular graph. On the one hand, the existence and uniqueness of solutions in the single-valued case are proved by using several fixed point theorems. On the other hand, in the multi-valued case, we consider that the right side of the inclusion has convex valued and non-convex value. By applying Leray-Schauder nonlinear alternative method of multi-valued maps as well as Covitz-Nadler fixed point theorem of multi-valued contractions, two existence results are obtained respectively. On this basis, we also get the topological structure of the solution set, which is a pioneering work for ()-Hilfer fractional differential inclusion on the glucose graph. Finally, several examples are provided to verify the reliability of our proposed results.
本文将-Hilfer分数阶微积分与葡萄糖分子图相结合，通过假设0或1标记顶点，在葡萄糖分子图的每条边上定义分数阶微分和包含系统，并研究葡萄糖分子图上的单值和多值-Hilfer型分数阶边值问题。一方面，利用几个不动点定理证明了单值情况下解的存在唯一性。另一方面，在多值情况下，考虑包含关系右侧具有凸值和非凸值。通过应用多值映射的Leray-Schauder非线性替代方法以及多值压缩映射的Covitz-Nadler不动点定理，分别得到了两个存在性结果。在此基础上，我们还得到了解集的拓扑结构，这对于葡萄糖图上的（）-Hilfer分数阶微分包含关系来说是一项开创性的工作。最后，给出了几个例子来验证我们所提出结果的可靠性。
