Existence and Uniqueness of Solution of Fractional Cauchy Problem

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-245
Published: Aug 17, 2026
Cite as:
Muayyad Mahmood Khalil, Hayder Swadi Hamad, Estabraq Ismael Fadhil, Existence and Uniqueness of Solution of Fractional Cauchy Problem, Int. J. Anal. Appl., 24 (2026), 245.

Main Article Content

Muayyad Mahmood Khalil, Hayder Swadi Hamad, Estabraq Ismael Fadhil

Abstract

This paper deals with initial value problems (IVPs) for nonlinear first-order modified Caputo fractional differential equations (FDEs). The innovative role of this work is that the nonlinearity f is in \(L^{p}\) spaces, where \(1 \leq p < \infty\), rather than in the conventional space of continuous functions. The equivalences between the FDEs and the corresponding integral equations are rigorously obtained. Related to the Banach contraction principle, we derive several new sets of necessary conditions for the existence and uniqueness of solutions to the considered system. These results precisely extend the existing literature by accommodating nonlinearities that exhibit jump discontinuities or integrable singularities, thereby contributing to the theoretical development of fractional calculus and providing broader analytical frameworks for further mathematical investigations.

Article Details

References

  1. K. Diethelm, N.J. Ford, Analysis of Fractional Differential Equations, J. Math. Anal. Appl. 265 (2002), 229–248. https://doi.org/10.1006/jmaa.2000.7194.
  2. K. Diethelm, N.J. Ford, A.D. Freed, A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations, Nonlinear Dyn. 29 (2002), 3–22. https://doi.org/10.1023/A:1016592219341.
  3. R. Garrappa, Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial, Mathematics 6 (2018), 16. https://doi.org/10.3390/math6020016.
  4. M.M. Khalil, S.U. Rehman, A.H. Ali, R. Nawaz, B. Batiha, New Modifications of Natural Transform Iterative Method and q-Homotopy Analysis Method Applied to Fractional Order KDV-Burger and Sawada–Kotera Equations, Partial Differ. Equ. Appl. Math. 12 (2024), 100950. https://doi.org/10.1016/j.padiff.2024.100950.
  5. K. Lan, J.R.L. Webb, A New Bihari Inequality and Initial Value Problems of First Order Fractional Differential Equations, Fract. Calc. Appl. Anal. 26 (2023), 962–988. https://doi.org/10.1007/s13540-023-00152-5.
  6. K. Li, J. Peng, Laplace Transform and Fractional Differential Equations, Appl. Math. Lett. 24 (2011), 2019–2023. https://doi.org/10.1016/j.aml.2011.05.035.
  7. A. Aghajani, E. Pourhadi, J.J. Trujillo, Application of Measure of Noncompactness to a Cauchy Problem for Fractional Differential Equations in Banach Spaces, Fract. Calc. Appl. Anal. 16 (2013), 962–977. https://doi.org/10.2478/s13540-013-0059-y.
  8. G.M. N'Guérékata, A Cauchy Problem for Some Fractional Abstract Differential Equation with Nonlocal Conditions, Nonlinear Anal. Theory Methods Appl. 70 (2009), 1873–1876. https://doi.org/10.1016/j.na.2008.02.087.
  9. J.R.L. Webb, Weakly Singular Gronwall Inequalities and Applications to Fractional Differential Equations, J. Math. Anal. Appl. 471 (2019), 692–711. https://doi.org/10.1016/j.jmaa.2018.11.004.
  10. Y. Sheng, T. Zhang, The Existence Theory of Solution in Sobolev Space for Fractional Differential Equations, Appl. Math. Lett. 149 (2024), 108896. https://doi.org/10.1016/j.aml.2023.108896.
  11. H.T. Tuan, H.D. Thai, R. Garrappa, An Analysis of Solutions to Fractional Neutral Differential Equations with Delay, Commun. Nonlinear Sci. Numer. Simul. 100 (2021), 105854. https://doi.org/10.1016/j.cnsns.2021.105854.
  12. E. Hernández, D. O'Regan, K. Balachandran, On Recent Developments in the Theory of Abstract Differential Equations with Fractional Derivatives, Nonlinear Anal. Theory Methods Appl. 73 (2010), 3462–3471. https://doi.org/10.1016/j.na.2010.07.035.
  13. J.R.L. Webb, Initial Value Problems for Caputo Fractional Equations with Singular Nonlinearities, Electron. J. Differ. Equ. 2019 (2019), 117. https://ejde.math.txstate.edu/Volumes/2019/117/abstr.html.
  14. J.R.L. Webb, A Fractional Gronwall Inequality and the Asymptotic Behaviour of Global Solutions of Caputo Fractional Problems, Electron. J. Differ. Equ. 2021 (2021), 80. https://doi.org/10.58997/ejde.2021.80.
  15. J.R.L. Webb, Compactness of Nonlinear Integral Operators with Discontinuous and with Singular Kernels, J. Math. Anal. Appl. 509 (2022), 126000. https://doi.org/10.1016/j.jmaa.2022.126000.
  16. K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer, Berlin, 2010. https://doi.org/10.1007/978-3-642-14574-2.
  17. B. Jin, Fractional Differential Equations: An Approach via Fractional Derivatives, Springer, Cham, 2021. https://doi.org/10.1007/978-3-030-76043-4.
  18. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. https://zbmath.org/438987.
  19. K. Lan, Linear First Order Riemann-Liouville Fractional Differential and Perturbed Abel's Integral Equations, J. Differ. Equ. 306 (2022), 28–59. https://doi.org/10.1016/j.jde.2021.10.025.
  20. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006. https://zbmath.org/2217537.
  21. K. Diethelm, The Mean Value Theorems and a Nagumo-Type Uniqueness Theorem for Caputo's Fractional Calculus, Fract. Calc. Appl. Anal. 15 (2012), 304–313. https://doi.org/10.2478/s13540-012-0022-3.
  22. K. Diethelm, Erratum: The Mean Value Theorems and a Nagumo-Type Uniqueness Theorem for Caputo's Fractional Calculus, Fract. Calc. Appl. Anal. 20 (2017), 1567–1570. https://doi.org/10.1515/fca-2017-0082.
  23. M. Nagumo, Eine hinreichende Bedingung für die Unität der Lösung von Differentialgleichungen erster Ordnung, Jpn. J. Math. 3 (1926), 107–112. https://zbmath.org/2586874.