Existence and uniqueness results for double jump fractional uncertain differential equations

Authors

  • Philip Ajibola Bankole
    Department of Mathematics Education, Lagos State University of Education, Oto/Ijanikin, Lagos State, Nigeria
  • Mabel Eruore Adeosun
    Department of Mathematics and Statistics, Osun State College of Technology, Esa-Oke, Osun State, Nigeria
  • Sunday Emmanuel Fadugba
    Department of Mathematics, Ekiti State University, Ado-Ekiti, 360001, Ekiti State, Nigeria
  • Tolulope Fadina
    Department of Mathematics, University of Illinois Urbana - Champaign, Urbana Illnois 61801, USA
  • Christopher Thron
    Department of Science and Mathematics, Texas A&M University - Central Texas, Killeen, USA

Keywords:

Double V-jump, Banach fixed point, Existence, Uniqueness, Fractional uncertain differential equation

Abstract

Fractional uncertain differential equations have been used to model random processes in economics and other fields that exhibit jumps, dependency, and nonlinearities, and which possess uncertainties due to limited data and inadequate models. In this paper, a double V-jump fractional uncertain differential equation (DV-FUDE) is presented as a $\theta^{th}$-order Riemann-Liouville or Caputo fractional uncertain differential equation with the addition of two V-jump independent processes on different filtrations. The equation models systems possessing two sources of uncertain shocks attributed to internal and external factors, respectively. Exact solutions in the case of time-dependent coefficients are given in terms of the Mittag-Leffler function. Sample continuity, existence, and uniqueness for the general Riemann-Liouville and Caputo DV-FUDE are established using the Banach Fixed Point Theorem, under  global Lipschitz and linear growth conditions on the coefficients. Some extensions and possible areas of application are highlighted for future research.    

Dimensions

[1] A. Einstein, “Über die von molekularkinetischen theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen”, Annalen der Physik 17 (1905) 549. https://doi.org/10.1002/andp.19053220806.

[2] F. Black & M. Scholes, “The pricing of options and corporate liabilities”, Journal of political economy 81 (1973) 637. http://dx.doi.org/10.1086/260062.

[3] D. Filipovic, Term structure models: a graduate course, Springer-Verlag, Berlin-Heidelberg-New York, 2009. https://doi.org/10.1007/978-3-540-68015-4.

[4] R. Cont & P. Tankov, Financial modelling with jump processes, Chapman & Hall/CRC, 2004. https://doi.org/10.1201/9780203485217.

[5] M. E. Adeosun & O. O. Ugbebor, “An empirical assessment of symmetric and asymmetric jump-diffusion models for the Nigerian stock market indices”, Scientific African 12 (2021) e00733. https://doi.org/10.1016/j.sciaf.2021.e00733.

[6] T. Bollerslev, V. Todorov & S. Li, “Jump tails, extreme dependencies, and the distribution of stock returns”, Journal of Econometrics 172 (2013) 307. https://doi.org/10.1016/j.jeconom.2012.08.014.

[7] A. A. Kilbas, H. M. Srivastava & J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies Vol. 204, Elsevier Science B.V., Amsterdam, 2006. https://doi.org/10.1016/S0304-0208(06)80001-0.

[8] V. Lakshmikantham & A. S. Vatsala, “Basic theory of fractional differential equations”, Nonlinear Analysis: Theory, Methods & Applications 69 (2008) 2677. https://doi.org/10.1016/j.na.2007.08.042.

[9] V. Lakshmikantham, S. Leela & J. Vasundhara Devi, Theory of fractional dynamic systems, Cambridge Scientific Publishers Ltd., Cambridge, 2009. https://cambridgescientificpublishers.com/product/theory-of-fractional-dynamic-systems.

[10] B. Ahmad & J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions”, Computers & Mathematics with Applications 58 (2009) 1838. https://doi.org/10.1016/j.camwa.2009.07.091.

[11] F. Mainardi, “Fractional calculus: theory and applications”, Mathematics 6 (2018) 145. https://doi.org/10.3390/math6090145.

[12] L. Deng & Y. Zhu, “An uncertain optimal control model with n jumps and application”, Computer Science and Information Systems 9 (2012) 1453. https://doi.org/10.2298/CSIS120225049D.

[13] Y. Shu & B. Li, “Existence and uniqueness of solutions to uncertain fractional switched systems with an uncertain stock model”, Chaos, Solitons & Fractals 155 (2022) 111746. https://doi.org/10.1016/j.chaos.2021.111746.

[14] Z. Jia & X. Liu, “Uncertain stochastic hybrid differential game system with V-n jumps: Saddle point equilibrium strategies and application to advertising duopoly game”, Chaos, Solitons & Fractals 171 (2023) 113490. https://doi.org/10.1016/j.chaos.2023.113490.

[15] B. Liu, Uncertainty 978-3-662-44354-5. theory, 5th ed., Springer, Berlin, 2015. https://doi.org/10.1007/

[16] K. Yao, “A type of uncertain differential equations with analytic solution”, Journal of Uncertainty Analysis and Applications 1 (2013) 8. https://doi.org/10.1186/2195-5468-1-8.

[17] B. Liu, “Some research problems in uncertainty theory”, Journal of Uncertain Systems 3 (2009) 3. https://www.researchgate.net/profile/Baoding-Liu/publication/228666965_Some_research_problems_in_uncertainy_theory/links/02e7e52836f150c344000000/Some-research-problems-in-uncertainy-theory.pdf.

[18] J. Zhou, Y. Jiang, A. A. Pantelous & W. Dai, “A systematic review of uncertainty theory with the use of scientometrical method”, Fuzzy Optimization and Decision Making 22 (2023) 463. https://doi.org/10.1007/s10700-022-09400-4.

[19] S. Li, J. Peng & B. Zhang, “Multifactor uncertain differential equation”, Journal of uncertainty Analysis and Applications 3 (2015) 7. https://doi.org/10.1186/s40467-015-0031-y.

[20] L. Deng & Y. Zhu, “Uncertain optimal control with jump”, ICIC Express Letters, Part B: Applications 3 (2012) 419. https://www.researchgate.net/publication/275314904_Uncertain_Optimal_Control_with_Jump.

[21] Z. Jia, X. Liu & C. Li, “Fixed point theorems applied in uncertain fractional differential equation with jump”, Symmetry 12 (2020) 1. https://doi.org/10.3390/sym12050765.

[22] L. Deng & Y. Zhu, “Existence and uniqueness theorem of solution for uncertain differential equations with jump”, ICIC Express Letters, Part B: Applications 6 (2012) 2693. https://www.researchgate.net/publication/286590908_Existence_and_uniqueness_theorem_of_solution_for_uncertain_differential_equations_with_jump.

[23] B. Liu, Uncertainty theory, 2nd ed., Springer, Berlin, 2007. https://doi.org/10.1007/978-3-540-73165-8.

[24] X. Chen & B. Liu, “Existence and uniqueness theorem for uncertain differential equations”, Fuzzy OptimDecis. Mak. 9 (2010) 69. http://dx.doi.org/10.1007/s10700-010-9073-2.

[25] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Vol. 198, Academic Press, New York, 1999. https://www.sciencedirect.com/bookseries/mathematics-in-science-and-engineering/vol/198/suppl/C.

[26] X. Chen & J. Gao, “Uncertain term structure model of interest rate”, Soft Comput. 17 (2013) 597. http://dx.doi.org/10.1007/s00500-012-0927-0.

[27] T. Abdeljawad, “On Riemann and Caputo fractional differences”, Computers & Mathematics with Applications 62 (2011) 1602. https://doi.org/10.1016/j.camwa.2011.03.036.

[28] Y. Zhu, “Existence and uniqueness of the solution to uncertain fractional differential equation”, Journal of Uncertainty Analysis and Applications 3 (2015) 5. https://dx.doi.org/10.1186/s40467-015-0028-6.

[29] Z. Jia, X. Liu & C. Li, “Existence, uniqueness, and stability of uncertain delay differential equations with V -jump”, Advances in difference Equations 2020 (2020) 1. https://doi.org/10.1186/s13662-020-02895-4.

[30] Y. Zhu, “Uncertain fractional differential equations and an interest rate model”, Method Appl. Sci. 38 (2015) 3359. http://dx.doi.org/10.1002/mma.3335.

Published

2026-05-04

How to Cite

Existence and uniqueness results for double jump fractional uncertain differential equations. (2026). Journal of the Nigerian Society of Physical Sciences, 8(2), 3153. https://doi.org/10.46481/jnsps.2026.3153

Issue

Volume 8, Issue 2, May 2026

Section

Mathematics & Statistics
Copyright & Licensing

Copyright (c) 2026 Philip Ajibola Bankole, Mabel Eruore Adeosun, Sunday Emmanuel Fadugba, Tolulope Fadina, Christopher Thron (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Existence and uniqueness results for double jump fractional uncertain differential equations. (2026). Journal of the Nigerian Society of Physical Sciences, 8(2), 3153. https://doi.org/10.46481/jnsps.2026.3153

Similar Articles

121-130 of 148

You may also start an advanced similarity search for this article.

Most read articles by the same author(s)