ABSTRACT

The main aim of this paper is to prove the Ulam–Hyers stability of solutions for a new form of nonlinear fractional Langevin differential equations involving two fractional orders in the ψ-Caputo sense. Prior to proceeding to the main results, the proposed system is converted into an equivalent integral form by the help of fractional calculus. Next, we proceed to investigate the existence and uniqueness of the solution by applying Schauder and Banach fixed point theorems. Finally, we study the Ulam–Hyers stability criteria for the main fractional system. Illustrative examples are presented to demonstrate the validity of the obtained results. The results are new and provide extensions to some known results in the literature.

摘要

本文的主要目的是证明一种新形式的非线性分数 Langevin 微分方程的解的 Ulam-Hyers 稳定性，该方程涉及ψ -Caputo 意义上的两个分数阶。在进行主要结果之前，通过分数微积分将所提出的系统转换为等效的积分形式。接下来，我们通过应用 Schauder 和 Banach 不动点定理来研究解的存在性和唯一性。最后，我们研究了主要分数系统的 Ulam-Hyers 稳定性标准。举例说明了所获得的结果的有效性。这些结果是新的，并为文献中的一些已知结果提供了扩展。