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Peer-reviewedCC BYFractal and Fractional

Existence, Uniqueness, and Continuous Dependence on Initial/Final Values for Liouville–Caputo Fractional Difference Equations

  • Xiaomin Li1,2
  • Huaigu Tian1,2
  • Peijun Zhang1,2
  • Liu X1,2
  1. 1Shaanxi International Joint Research Center for Applied Technology of Controllable Neutron Source, Xijing University, Xi’an 710123, China
  2. 2Xi’an Key Laboratory of Human-Machine Integration and Control Technology for Intelligent Rehabilitation, Xijing University, Xi’an 710123, China
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Abstract

This paper develops a unified qualitative framework for four classes of Liouville–Caputo fractional difference equations arising from different combinations of fractional sums and integer-order differences. Based on the equivalence between initial/final value problems and Volterra-type summation equations, sufficient conditions for the existence and uniqueness of solutions are established by applying the Banach contraction mapping principle together with refined combinatorial estimates. Furthermore, the continuous dependence of solutions on prescribed initial or final data is investigated. By deriving explicit error estimates through a discrete fractional Gronwall-type inequality, we prove that Lipschitz solutions depend continuously on perturbations of boundary data. Numerical experiments for a representative case are presented to verify the theoretical results, including the influence of the fractional order and the sensitivity with respect to boundary data, while additional examples illustrate the applicability of the framework. The obtained results extend the unified discrete fractional calculus framework by providing a rigorous well-posedness analysis and offering a theoretical foundation for further applications of discrete fractional models with memory effects and diverse boundary conditions.

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Cite

BibTeXRIS
BibTeX
@article{li2026existence,
  title = {{Existence, Uniqueness, and Continuous Dependence on Initial/Final Values for Liouville–Caputo Fractional Difference Equations}},
  author = {Xiaomin Li and Huaigu Tian and Peijun Zhang and Liu X},
  journal = {Fractal and Fractional},
  year = {2026},
  month = jul,
  doi = {10.3390/fractalfract10080504},
  url = {https://doi.org/10.3390/fractalfract10080504},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Existence, Uniqueness, and Continuous Dependence on Initial/Final Values for Liouville–Caputo Fractional Difference Equations
AU  - Xiaomin Li
AU  - Huaigu Tian
AU  - Peijun Zhang
AU  - Liu X
PY  - 2026
DA  - 2026/07/26
JO  - Fractal and Fractional
DO  - 10.3390/fractalfract10080504
UR  - https://doi.org/10.3390/fractalfract10080504
AB  - This paper develops a unified qualitative framework for four classes of Liouville–Caputo fractional difference equations arising from different combinations of fractional sums and integer-order differences. Based on the equivalence between initial/final value problems and Volterra-type summation equations, sufficient conditions for the existence and uniqueness of solutions are established by applying the Banach contraction mapping principle together with refined combinatorial estimates. Furthermore, the continuous dependence of solutions on prescribed initial or final data is investigated. By deriving explicit error estimates through a discrete fractional Gronwall-type inequality, we prove that Lipschitz solutions depend continuously on perturbations of boundary data. Numerical experiments for a representative case are presented to verify the theoretical results, including the influence of the fractional order and the sensitivity with respect to boundary data, while additional examples illustrate the applicability of the framework. The obtained results extend the unified discrete fractional calculus framework by providing a rigorous well-posedness analysis and offering a theoretical foundation for further applications of discrete fractional models with memory effects and diverse boundary conditions.
SN  - 2504-3110
ER  -