Existence theory and stability analysis of neutral $ \psi $–Hilfer fractional stochastic differential system with fractional noises and non-instantaneous impulses

Yanli Ma; Hamza Khalil; Akbar Zada; Ioan-Lucian Popa; Yanli Ma; Hamza Khalil; Akbar Zada; Ioan-Lucian Popa

doi:10.3934/math.2024396

AIMS Mathematics

2024, Volume 9, Issue 4: 8148-8173. doi: 10.3934/math.2024396

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Existence theory and stability analysis of neutral $ \psi $–Hilfer fractional stochastic differential system with fractional noises and non-instantaneous impulses

1.
Department of Common Course, Anhui Xinhua University, Hefei, Anhui 230088, China
2.
Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan
3.
Department of Computing, Mathematics and Electronics, "1 Decembrie 1918" University of Alba Iulia, Alba Iulia, 510009, Romania
4.
Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania

Received: 24 December 2023 Revised: 26 January 2024 Accepted: 01 February 2024 Published: 26 February 2024
MSC : 34A37, 34N05, 35F30, 35L05

In this article, with the help of Laplace transform, the existence of solution was established in a finite dimensional setting for nonlinear $ \psi $-Hilfer fractional stochastic equation with both retarded and advanced arguments driven by multiplicative and fractional noises, with Hurst index $ H \in (\frac{1}{2}, 1) $. At first, we obtained the existence and uniqueness results by using the Banach fixed point theorem (FPT). Second, the existence result was also obtained by applying Schaefer's fixed point theorem with less conservative conditions. Furthermore, we investigated the Hyers Ulam Rasisas stability for the aforementioned system. At the end, an example was illustrated to validate the obtained theoretical results.
- neutral,
- existence and uniqueness,
- stochastic equation,
- retarded and advanced arguments,
- Ulam's stability
Citation: Yanli Ma, Hamza Khalil, Akbar Zada, Ioan-Lucian Popa. Existence theory and stability analysis of neutral $ \psi $–Hilfer fractional stochastic differential system with fractional noises and non-instantaneous impulses[J]. AIMS Mathematics, 2024, 9(4): 8148-8173. doi: 10.3934/math.2024396

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Abstract

In this article, with the help of Laplace transform, the existence of solution was established in a finite dimensional setting for nonlinear $ \psi $-Hilfer fractional stochastic equation with both retarded and advanced arguments driven by multiplicative and fractional noises, with Hurst index $ H \in (\frac{1}{2}, 1) $. At first, we obtained the existence and uniqueness results by using the Banach fixed point theorem (FPT). Second, the existence result was also obtained by applying Schaefer's fixed point theorem with less conservative conditions. Furthermore, we investigated the Hyers Ulam Rasisas stability for the aforementioned system. At the end, an example was illustrated to validate the obtained theoretical results.

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