### AIMS Mathematics

2021, Issue 7: 6749-6780. doi: 10.3934/math.2021397
Research article Special Issues

# Langevin equation with nonlocal boundary conditions involving a $\psi$-Caputo fractional operators of different orders

• Received: 13 December 2020 Accepted: 07 April 2021 Published: 21 April 2021
• MSC : 34A08, 26A33, 34A12, 34D20

• This paper studies Langevin equation with nonlocal boundary conditions involving a $\psi$-Caputo fractional operators of different orders. By the aid of fixed point techniques of Krasnoselskii and Banach, we derive new results on existence and uniqueness of the problem at hand. Further, a new $\psi$-fractional Gronwall inequality and $\psi$-fractional integration by parts are employed to prove Ulam-Hyers and Ulam-Hyers-Rassias stability for the solutions. Examples are provided to demonstrate the advantage of our major results. The proposed results here are more general than the existing results in the literature which can be obtained as particular cases.

Citation: Arjumand Seemab, Mujeeb ur Rehman, Jehad Alzabut, Yassine Adjabi, Mohammed S. Abdo. Langevin equation with nonlocal boundary conditions involving a $\psi$-Caputo fractional operators of different orders[J]. AIMS Mathematics, 2021, 6(7): 6749-6780. doi: 10.3934/math.2021397

### Related Papers:

• This paper studies Langevin equation with nonlocal boundary conditions involving a $\psi$-Caputo fractional operators of different orders. By the aid of fixed point techniques of Krasnoselskii and Banach, we derive new results on existence and uniqueness of the problem at hand. Further, a new $\psi$-fractional Gronwall inequality and $\psi$-fractional integration by parts are employed to prove Ulam-Hyers and Ulam-Hyers-Rassias stability for the solutions. Examples are provided to demonstrate the advantage of our major results. The proposed results here are more general than the existing results in the literature which can be obtained as particular cases.

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