Existence and Uniqueness of Mild Solutions for Neutral Stochastic Fractional Sum-Difference Equations with Impulses Driven by a Rosenblatt Process

Yogesh Hanmant Shirole; Suryakant Muralidhar Jogdand

doi:doi:10.11648/j.ajam.20261405.11

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Existence and Uniqueness of Mild Solutions for Neutral Stochastic Fractional Sum-Difference Equations with Impulses Driven by a Rosenblatt Process

Yogesh Hanmant Shirole^*, Suryakant Muralidhar Jogdand

Published in American Journal of Applied Mathematics (Volume 14, Issue 5)

Received: 7 June 2026 Accepted: 14 July 2026 Published: 9 September 2026

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Abstract

This paper investigates the existence and uniqueness of mild solutions for a class of neutral stochastic fractional difference equations with impulsive effects in a Hilbert space framework, driven by a Rosenblatt process. The considered system is formulated using the Caputo fractional difference operator and incorporates delay terms, impulsive effects, and stochastic perturbations exhibiting long-range dependence. The primary objective is to establish sufficient conditions ensuring the existence and uniqueness of mild solutions for the proposed system. To achieve this objective, the theory of resolvent operators is combined with stochastic analysis techniques and the Banach fixed point theorem. In particular, an appropriate operator is constructed from the mild solution formulation, and suitable conditions are imposed to guarantee its contractive property. Consequently, the existence and uniqueness of a mild solution are established. The obtained results extend and generalize several existing results for fractional differential and stochastic systems to the discrete fractional setting involving Rosenblatt stochastic processes. The proposed framework effectively accounts for the combined influence of fractional memory, delays, impulsive effects, and long-range-dependent stochastic disturbances. Furthermore, an application to a class of impulsive stochastic partial fractional difference equations is presented to demonstrate the applicability and effectiveness of the theoretical results. The application verifies that the established assumptions can be satisfied in a relevant stochastic fractional model. Thus, the results contribute to the qualitative theory of neutral stochastic fractional difference equations and provide a useful framework for the analysis of discrete-time stochastic systems with memory, delay, impulsive phenomena, and long-range dependence. These findings may also serve as a basis for further investigations of more general stochastic fractional difference systems.

Published in	American Journal of Applied Mathematics (Volume 14, Issue 5)
DOI	10.11648/j.ajam.20261405.11
Page(s)	277-286
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Fractional Difference Equations, Neutral Stochastic Systems, Impulsive Effects, Rosenblatt Process, Mild Solution, Existence and Uniqueness, Banach Fixed Point Theorem, Resolvent Pperator

References

[1]	Xianmin Zhang, Praveen Agarwal, Zuohua Liu, Hui Peng, Fang You and Yajun Zhu; Existence and uniqueness of solutions for stochastic differential equations of fractional-order q > 1 with finite delays, (2017). https://doi.org/10.1186/s13662-017-1169-3
[2]	B. P. Moghaddam, Lei Zhang, A. M. Lopes, J. A. Tenreiro Machado and Z. S. Mostaghim; Sufficient conditions for existence and uniqueness of fractional stochastic delay differential equations, Stochastics, 92(3), 379-396, 2020. https://doi.org/10.1080/17442508.2019.1625903
[3]	B. P. Moghaddam, L. Zhang, A. M. Lopes, J. A. T. Machado and Z. Mostaghim, Sufficient conditions for existence and uniqueness of fractional stochastic delay differential equations, Stochastics, 92(3), 379-396, 2020. https://doi.org/10.1080/17442508.2019.1625903
[4]	Alka Chadha and Swaroop Nandan Bora; Existence and exponential stability for neutral stochastic fractional differential equations with impulses driven by Poisson jumps, Stochastics, 90(5), 663-681, 2018. https://doi.org/10.1080/17442508.2017.1402899
[5]	I. Podlubny; Fractional Differential equations, Academic Press USA, (1999).
[6]	Shantanu Das; Functional Fractional Calculus, Springer-Verlag, Berlin Heidelberg, 2011. https://doi.org/10.1007/978-3-642-20545-3
[7]	Mahamat hamit mahamat hassan, mamadou abdoul diop, ramkumar kasinathan, ravikumar kasinathan; existence, global attracting sets and exponential decay of solution to stochastic functional integro-differential equations driven by rosenblatt process, Vol. (2020).
[8]	B. Boufoussi, E. Lakhel and A. Tlidi; Neutral stochastic functional differential evolution equations with varying-time delays driven by Rosenblatt process in Hilbert spaces, Gulf Journal of Mathematics, Vol. 6, No. 1, pp. 89-103, 2018. https://doi.org/10.56947/gjom.v6i1.127
[9]	El Hassan Lakhel; Exponential stability for stochastic neutral functional differential equations driven by Rosenblatt process with delay and Poisson jumps, Random Operators and Stochastic Equations, 24(2), 113-127, 2016. https://doi.org/10.1515/rose-2016-0008
[10]	Ravikumar Kasinathan, Ramkumar Kasinathan, Mahamat Hassan Mahamat Hamit, and Mamadou Abdoul Diop; Exponential behavior of neutral impulsive stochastic integro-differential equations driven by Poisson jumps and Rosenblatt process, Nonautonomous Dynamical Systems, 7(1), 1-21, 2020. https://doi.org/10.1515/msds-2020-0001
[11]	I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[12]	S. Das, Functional Fractional Calculus, 2nd ed., Springer, Berlin, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-20545-3
[13]	F. M. Atici and P. W. Eloe, A transform method in discrete fractional calculus, International Journal of Difference Equations, 2(2) (2007), 165-176.
[14]	F. M. Atici and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electronic Journal of Qualitative Theory of Differential Equations, Special Edition I (2009), Paper No. 3, pp. 1-12. https://doi.org/10.14232/ejqtde.2009.4.3
[15]	A. Chadha and S. N. Bora, Existence and exponential stability for neutral stochastic fractional differential equations with impulses driven by Poisson jumps, Stochastics, 89(8), 1102-1121, 2018. https://doi.org/10.1080/17442508.2017.1402899
[16]	D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, 1993.
[17]	C. A. Tudor, Analysis of Variations for Self-Similar Processes, Springer, Cham, 2013. https://doi.org/10.1007/978-3-319-00642-9
[18]	M. S. Taqqu, Weak convergence to fractional Brownian motion and to the Rosenblatt process, Probability Theory and Related Fields, 31, 287-302, 1975. https://doi.org/10.1007/BF00532868
[19]	E. H. Lakhel, Exponential stability for stochastic neutral functional differential equations driven by Rosenblatt process with delay and Poisson jumps, Statistics and Probability Letters, 140, 53-61, 2018. https://doi.org/10.1016/j.spl.2018.04.003 newpage
[20]	M. H. M. Hassan, M. A. Diop, R. Kasinathan and R. Kasinathan, Existence, global attracting sets and exponential decay of solutions to stochastic functional integrodifferential equations driven by Rosenblatt process, Journal of Computational and Applied Mathematics, 369, 112586, 2020. https://doi.org/10.1016/j.cam.2019.112586
[21]	R. Kasinathan, R. Kasinathan, M. H. M. Hassan and M. A. Diop, Exponential behavior of neutral impulsive stochastic integrodifferential equations driven by Poisson jumps and Rosenblatt process, Mathematics, 8(11), 2015, 2020. https://doi.org/10.3390/math8112015
[22]	X. Zhang, P. Agarwal, Z. Liu, H. Peng, F. You and Y. Zhu, Existence and uniqueness of solutions for stochastic differential equations of fractional order, Stochastics and Dynamics, 17(5), 1750036, 2017. https://doi.org/10.1142/S0219493717500368
[23]	C. Goodrich and A. Peterson, Discrete Fractional Calculus, Springer, Cham, 2015. https://doi.org/10.1007/978-3-319-25562-6
[24]	T. Abdeljawad, On Riemann and Caputo fractional differences, Computers & Mathematics with Applications, 62(3), 1602-1611, 2011. https://doi.org/10.1016/j.camwa.2011.07.006

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APA Style

Shirole, Y. H., Jogdand, S. M. (2026). Existence and Uniqueness of Mild Solutions for Neutral Stochastic Fractional Sum-Difference Equations with Impulses Driven by a Rosenblatt Process. American Journal of Applied Mathematics, 14(5), 277-286. https://doi.org/10.11648/j.ajam.20261405.11

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ACS Style

Shirole, Y. H.; Jogdand, S. M. Existence and Uniqueness of Mild Solutions for Neutral Stochastic Fractional Sum-Difference Equations with Impulses Driven by a Rosenblatt Process. Am. J. Appl. Math. 2026, 14(5), 277-286. doi: 10.11648/j.ajam.20261405.11

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AMA Style

Shirole YH, Jogdand SM. Existence and Uniqueness of Mild Solutions for Neutral Stochastic Fractional Sum-Difference Equations with Impulses Driven by a Rosenblatt Process. Am J Appl Math. 2026;14(5):277-286. doi: 10.11648/j.ajam.20261405.11

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@article{10.11648/j.ajam.20261405.11,
  author = {Yogesh Hanmant Shirole and Suryakant Muralidhar Jogdand},
  title = {Existence and Uniqueness of Mild Solutions for Neutral Stochastic Fractional Sum-Difference Equations with Impulses Driven by a Rosenblatt Process},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {5},
  pages = {277-286},
  doi = {10.11648/j.ajam.20261405.11},
  url = {https://doi.org/10.11648/j.ajam.20261405.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261405.11},
  abstract = {This paper investigates the existence and uniqueness of mild solutions for a class of neutral stochastic fractional difference equations with impulsive effects in a Hilbert space framework, driven by a Rosenblatt process. The considered system is formulated using the Caputo fractional difference operator and incorporates delay terms, impulsive effects, and stochastic perturbations exhibiting long-range dependence. The primary objective is to establish sufficient conditions ensuring the existence and uniqueness of mild solutions for the proposed system. To achieve this objective, the theory of resolvent operators is combined with stochastic analysis techniques and the Banach fixed point theorem. In particular, an appropriate operator is constructed from the mild solution formulation, and suitable conditions are imposed to guarantee its contractive property. Consequently, the existence and uniqueness of a mild solution are established. The obtained results extend and generalize several existing results for fractional differential and stochastic systems to the discrete fractional setting involving Rosenblatt stochastic processes. The proposed framework effectively accounts for the combined influence of fractional memory, delays, impulsive effects, and long-range-dependent stochastic disturbances. Furthermore, an application to a class of impulsive stochastic partial fractional difference equations is presented to demonstrate the applicability and effectiveness of the theoretical results. The application verifies that the established assumptions can be satisfied in a relevant stochastic fractional model. Thus, the results contribute to the qualitative theory of neutral stochastic fractional difference equations and provide a useful framework for the analysis of discrete-time stochastic systems with memory, delay, impulsive phenomena, and long-range dependence. These findings may also serve as a basis for further investigations of more general stochastic fractional difference systems.},
 year = {2026}
}

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TY  - JOUR
T1  - Existence and Uniqueness of Mild Solutions for Neutral Stochastic Fractional Sum-Difference Equations with Impulses Driven by a Rosenblatt Process
AU  - Yogesh Hanmant Shirole
AU  - Suryakant Muralidhar Jogdand
Y1  - 2026/09/09
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261405.11
DO  - 10.11648/j.ajam.20261405.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 277
EP  - 286
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261405.11
AB  - This paper investigates the existence and uniqueness of mild solutions for a class of neutral stochastic fractional difference equations with impulsive effects in a Hilbert space framework, driven by a Rosenblatt process. The considered system is formulated using the Caputo fractional difference operator and incorporates delay terms, impulsive effects, and stochastic perturbations exhibiting long-range dependence. The primary objective is to establish sufficient conditions ensuring the existence and uniqueness of mild solutions for the proposed system. To achieve this objective, the theory of resolvent operators is combined with stochastic analysis techniques and the Banach fixed point theorem. In particular, an appropriate operator is constructed from the mild solution formulation, and suitable conditions are imposed to guarantee its contractive property. Consequently, the existence and uniqueness of a mild solution are established. The obtained results extend and generalize several existing results for fractional differential and stochastic systems to the discrete fractional setting involving Rosenblatt stochastic processes. The proposed framework effectively accounts for the combined influence of fractional memory, delays, impulsive effects, and long-range-dependent stochastic disturbances. Furthermore, an application to a class of impulsive stochastic partial fractional difference equations is presented to demonstrate the applicability and effectiveness of the theoretical results. The application verifies that the established assumptions can be satisfied in a relevant stochastic fractional model. Thus, the results contribute to the qualitative theory of neutral stochastic fractional difference equations and provide a useful framework for the analysis of discrete-time stochastic systems with memory, delay, impulsive phenomena, and long-range dependence. These findings may also serve as a basis for further investigations of more general stochastic fractional difference systems.
VL  - 14
IS  - 5
ER  -

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Author Information

Yogesh Hanmant Shirole

Department of Mathematics , Shri Sant Gadge Maharaj College, Loha, India

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Suryakant Muralidhar Jogdand

Department of Mathematics, Shri Shivaji College,Kandhar, India

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APA Style

Shirole, Y. H., Jogdand, S. M. (2026). Existence and Uniqueness of Mild Solutions for Neutral Stochastic Fractional Sum-Difference Equations with Impulses Driven by a Rosenblatt Process. American Journal of Applied Mathematics, 14(5), 277-286. https://doi.org/10.11648/j.ajam.20261405.11

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ACS Style

Shirole, Y. H.; Jogdand, S. M. Existence and Uniqueness of Mild Solutions for Neutral Stochastic Fractional Sum-Difference Equations with Impulses Driven by a Rosenblatt Process. Am. J. Appl. Math. 2026, 14(5), 277-286. doi: 10.11648/j.ajam.20261405.11

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AMA Style

Shirole YH, Jogdand SM. Existence and Uniqueness of Mild Solutions for Neutral Stochastic Fractional Sum-Difference Equations with Impulses Driven by a Rosenblatt Process. Am J Appl Math. 2026;14(5):277-286. doi: 10.11648/j.ajam.20261405.11

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@article{10.11648/j.ajam.20261405.11,
  author = {Yogesh Hanmant Shirole and Suryakant Muralidhar Jogdand},
  title = {Existence and Uniqueness of Mild Solutions for Neutral Stochastic Fractional Sum-Difference Equations with Impulses Driven by a Rosenblatt Process},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {5},
  pages = {277-286},
  doi = {10.11648/j.ajam.20261405.11},
  url = {https://doi.org/10.11648/j.ajam.20261405.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261405.11},
  abstract = {This paper investigates the existence and uniqueness of mild solutions for a class of neutral stochastic fractional difference equations with impulsive effects in a Hilbert space framework, driven by a Rosenblatt process. The considered system is formulated using the Caputo fractional difference operator and incorporates delay terms, impulsive effects, and stochastic perturbations exhibiting long-range dependence. The primary objective is to establish sufficient conditions ensuring the existence and uniqueness of mild solutions for the proposed system. To achieve this objective, the theory of resolvent operators is combined with stochastic analysis techniques and the Banach fixed point theorem. In particular, an appropriate operator is constructed from the mild solution formulation, and suitable conditions are imposed to guarantee its contractive property. Consequently, the existence and uniqueness of a mild solution are established. The obtained results extend and generalize several existing results for fractional differential and stochastic systems to the discrete fractional setting involving Rosenblatt stochastic processes. The proposed framework effectively accounts for the combined influence of fractional memory, delays, impulsive effects, and long-range-dependent stochastic disturbances. Furthermore, an application to a class of impulsive stochastic partial fractional difference equations is presented to demonstrate the applicability and effectiveness of the theoretical results. The application verifies that the established assumptions can be satisfied in a relevant stochastic fractional model. Thus, the results contribute to the qualitative theory of neutral stochastic fractional difference equations and provide a useful framework for the analysis of discrete-time stochastic systems with memory, delay, impulsive phenomena, and long-range dependence. These findings may also serve as a basis for further investigations of more general stochastic fractional difference systems.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Existence and Uniqueness of Mild Solutions for Neutral Stochastic Fractional Sum-Difference Equations with Impulses Driven by a Rosenblatt Process
AU  - Yogesh Hanmant Shirole
AU  - Suryakant Muralidhar Jogdand
Y1  - 2026/09/09
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261405.11
DO  - 10.11648/j.ajam.20261405.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 277
EP  - 286
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261405.11
AB  - This paper investigates the existence and uniqueness of mild solutions for a class of neutral stochastic fractional difference equations with impulsive effects in a Hilbert space framework, driven by a Rosenblatt process. The considered system is formulated using the Caputo fractional difference operator and incorporates delay terms, impulsive effects, and stochastic perturbations exhibiting long-range dependence. The primary objective is to establish sufficient conditions ensuring the existence and uniqueness of mild solutions for the proposed system. To achieve this objective, the theory of resolvent operators is combined with stochastic analysis techniques and the Banach fixed point theorem. In particular, an appropriate operator is constructed from the mild solution formulation, and suitable conditions are imposed to guarantee its contractive property. Consequently, the existence and uniqueness of a mild solution are established. The obtained results extend and generalize several existing results for fractional differential and stochastic systems to the discrete fractional setting involving Rosenblatt stochastic processes. The proposed framework effectively accounts for the combined influence of fractional memory, delays, impulsive effects, and long-range-dependent stochastic disturbances. Furthermore, an application to a class of impulsive stochastic partial fractional difference equations is presented to demonstrate the applicability and effectiveness of the theoretical results. The application verifies that the established assumptions can be satisfied in a relevant stochastic fractional model. Thus, the results contribute to the qualitative theory of neutral stochastic fractional difference equations and provide a useful framework for the analysis of discrete-time stochastic systems with memory, delay, impulsive phenomena, and long-range dependence. These findings may also serve as a basis for further investigations of more general stochastic fractional difference systems.
VL  - 14
IS  - 5
ER  -

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