Abstract
The purpose of this study is to establish the Freidlin–Wentzell-type large deviation principle (LDP) for the solution of a stochastic fractional pantograph differential equation. With the Picard iterative approach, the existence of the solution is demonstrated. The indistinguishability between any two solutions of the system asserts the uniqueness. The Laplace principle, equivalent to the LDP under a Polish space, is illustrated by taking up the variational representation developed by Budhiraja and Dupuis using the weak convergence approach. The corresponding controlled deterministic system is considered to establish the compactness criterion, and validated using the sequential compactness. In accordance with Yamada–Watanabe theorem, there exists a Borel measurable function with which the weak convergence criterion is done. An example is provided to illustrate the theory developed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and material
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
References
Balachandran K, Park JY, Trujillo JJ (2012) Controllability of nonlinear fractional dynamical systems. Nonlinear Anal: Theory Methods Appl 75(4):1919–1926
Fang XQ, Ma HW, Zhu CS (2022). Non-local multi-fields coupling response of a piezoelectric semiconductor nanofiber under shear force. Mech Adv Mater Struct 1–8
Magin R, Ortigueira MD, Podlubny I, Trujillo J (2011) On the fractional signals and systems. Signal Process 91(3):350–371
Tang TQ, Shah Z, Jan R, Alzahrani E (2022) Modeling the dynamics of tumor-immune cells interactions via fractional calculus. Eur Phys J Plus 137(3):1–18
Wen C, Yang J (2019) Complexity evolution of chaotic financial systems based on fractional calculus. Chaos Solitons Fractals 128:242–251
Zhu CS, Fang XQ, Liu JX, Li HY (2017) Surface energy effect on nonlinear free vibration behavior of orthotropic piezoelectric cylindrical nano-shells. Eur J Mech-A/Solids 66:423–432
Raghavan D, Gómez-Aguilar JF, Sukavanam N (2022) Analytical approach of Hilfer fractional order differential equations using iterative Laplace transform method. J Math Chem 61:1–23
Ramesh P, Sambath M, Mohd MH, Balachandran K (2021) Stability analysis of the fractional-order prey–predator model with infection. Int J Model Simul 41(6):434–450
Chaudhary R, Singh V, Pandey DN (2020) Controllability of multi-term time-fractional differential systems with state-dependent delay. J Appl Anal 26(2):241–255
Khan H, Gómez-Aguilar JF, Khan A, Khan TS (2019) Stability analysis for fractional order advection-reaction diffusion system. Physica A 521:737–751
Kumar S, Sukavanam N (2012) Approximate controllability of fractional order semilinear systems with bounded delay. J Differ Equ 252(11):6163–6174
Cui J, Yan L (2011) Existence result for fractional neutral stochastic integro-differential equations with infinite delay. J Phys A: Math Theor 44(33):335201
Ahmed HM (2009) On some fractional stochastic integrodifferential equations in Hilbert space. Int J Math Math Sci 2009:1–8
El-Borai MM, El-Nadi KES, Mostafa OL, Ahmed HM (2004) Volterra equations with fractional stochastic integrals. Math Probl Eng 2004(5):453–468
Da Prato G, Zabczyk J (1992) Stochastic evolution equations in infinite dimensions. Cambridge University Press, Cambridge
Dupuis P, Ellis RS (2011) A weak convergence approach to the theory of large deviations. Wiley, New York
Mo C, Luo J (2013) Large deviations for stochastic differential delay equations. Nonlinear Anal 80:202–210
Siva Ranjani A, Suvinthra M, Balachandran K, Ma YK (2022) Analysis of stochastic neutral fractional functional differential equations. Boundary Value Prob 2022(1):49
Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations (vol 204). Elsevier, Amsterdam
Mao X (1997) Stochastic differential equations and applications. Horwood Publishing Limited, Chichester
Dembo A, Zeitouni O (2010) Large deviations techniques and applications. Springer, New York
Freidlin MI, Wentzell AD (1984) Random perturbations of dynamical systems. Springer, New York
Budhiraja A, Dupuis P (2000) A variational representation for positive functionals of infinite dimensional Brownian motion. Probab Math Stat-Wroclaw Univ 20(1):39–61
Acknowledgements
We thank Dr. K. Balachandran, Mentor Professor, Department of Applied Mathematics, Bharathiar University, for his expertise and assistance throughout all aspects of our study.
Funding
The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
Author information
Authors and Affiliations
Contributions
All the authors have contributed equally to this paper. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ranjani, A.S., Suvinthra, M. Large deviations for stochastic fractional pantograph differential equation. Int. J. Dynam. Control 12, 136–147 (2024). https://doi.org/10.1007/s40435-023-01339-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40435-023-01339-7
Keywords
- Fractional differential equation
- Large deviation principle
- Stochastic differential equation
- Pantograph equation