Existence and Uniqueness of Solutions for FDEs with Nonlocal Discrete, Integral Delay and Nonlinear Terminal Boundary Conditions

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-303
Published: Sep 21, 2026
Cite as:
Selvakumar Viswanathan, Jeyachandhiran Ramaprabu, Gunaseelan Mani, Abdulkareem Saleh Hamarsheh, Rajagopalan Ramaswamy, S. Padmaja, Existence and Uniqueness of Solutions for FDEs with Nonlocal Discrete, Integral Delay and Nonlinear Terminal Boundary Conditions, Int. J. Anal. Appl., 24 (2026), 303.

Main Article Content

Selvakumar Viswanathan, Jeyachandhiran Ramaprabu, Gunaseelan Mani, Abdulkareem Saleh Hamarsheh, Rajagopalan Ramaswamy, S. Padmaja

Abstract

This paper investigates a nonlinear FDEs involving the Caputo derivative of order \(\kappa\in(0,1)\). The boundary condition is novel: it includes a finite sum of interior point values, an integral with a time-delay term and a nonlinear function of the terminal state. By applying Krasnoselskii’s FPT, the existence of at least one solution is established under appropriate growth assumptions. and using Banach’s contraction principle we prove uniqueness under a Lipschitz condition. The problem is transformed into an equivalent integral equation via the Green’s function approach. A complete preliminaries section covers Caputo fractional calculus, function spaces and key compactness results including the Arzelà-Ascoli theorem. An illustrative example confirms the theoretical results.

Article Details

References

  1. S. Abbas, M. Benchohra, J.E. Lazreg, J.J. Nieto, Y. Zhou, Fractional Differential Equations and Inclusions, World Scientific, 2023. https://doi.org/10.1142/12993.
  2. B. Ahmad, A. Alsaedi, S.K. Ntouyas, J. Tariboon, Coupled Systems of Hadamard and Riemann-Liouville Fractional Differential Equations with Hadamard Type Integral Boundary Conditions, in: Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer, Cham, 2017, pp. 173–208. https://doi.org/10.1007/978-3-319-52141-1_6.
  3. S.F. Aljurbua, S. Alotaibi, Existence and Uniqueness Theorems for Solutions to Caputo Fractional Differential Equations with Nonlocal and Irregular Boundary Conditions, AIMS Math. 11 (2026), 4681–4690. https://doi.org/10.3934/math.2026190.
  4. C. Arzelà, Sulle Funzioni di Linee, Mem. Accad. Sci. Ist. Bologna 5 (1895), 55–74.
  5. G. Ascoli, Le Curve Limiti di una Varietà Data di Curve, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Nat. 18 (1884), 521–586.
  6. S. Banach, Sur les Opérations dans les Ensembles Abstraits et Leur Application aux Équations Intégrales, Fundam. Math. 3 (1922), 133–181. https://doi.org/10.4064/fm-3-1-133-181.
  7. D. O'Regan, R.P. Agarwal, S. Hristova, M.I. Abbas, Existence and Stability Results for Differential Equations with a Variable-Order Generalized Proportional Caputo Fractional Derivative, Mathematics 12 (2024), 233. https://doi.org/10.3390/math12020233.
  8. B.K. Chaurasiya, A. Kumar, Analysis of Positive Solutions for the Fractional Derivative with Delay and Integral Boundary Conditions, Gulf J. Math. 19 (2025), 315–327. https://doi.org/10.56947/gjom.v19i2.2745.
  9. C. Derbazi, H. Hammouche, Existence and Uniqueness Results for a Class of Nonlinear Fractional Differential Equations with Nonlocal Boundary Conditions, Jordan J. Math. Stat. 13 (2020), 341–361.
  10. K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, Heidelberg, 2010. https://doi.org/10.1007/978-3-642-14574-2.
  11. A. Granas, J. Dugundji, Fixed Point Theory, Springer, New York, 2003. https://doi.org/10.1007/978-0-387-21593-8.
  12. D.H. Hyers, On the Stability of the Linear Functional Equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222.
  13. M.I. Liaqat, M. Vivas-Cortez, M.A. Yousif, P.O. Mohammed, Variable-Order Fractional Delay Differential Equations with Integral Boundary Values: A Study on Existence, Uniqueness, and Stability, Eur. J. Pure Appl. Math. 19 (2026), 6855. https://doi.org/10.29020/nybg.ejpam.v19i1.6855.
  14. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006. https://doi.org/10.1016/S0304-0208(06)X8001-5.
  15. M.A. Krasnosel'skii, Two Remarks on the Method of Successive Approximations, Uspekhi Mat. Nauk 10 (1955), 123–127.
  16. V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, 2009.
  17. C. Li, Studies on Fractional Differential Equations with Functional Boundary Condition by Inverse Operators, Math. Methods Appl. Sci. 48 (2025), 11161–11170. https://doi.org/10.1002/mma.10951.
  18. W. Feng, Boundary Value Problems for Nonlinear Fractional Differential Equations: Theory, Methods, and Applications, Fractal Fract. 10 (2026), 63. https://doi.org/10.3390/fractalfract10010063.
  19. V.B. Magar, P.S. Avhale, A.V. Kawarkhe, Krasnoselskii Fixed Point Approach to Fractional Duffing Equations with Generalized ψ-Prabhakar Derivatives, Int. J. Appl. Math. 38 (2025), 1284–1314. https://doi.org/10.12732/ijam.v38i1s.745.
  20. M. Manigandan, M. Awadalla, K. Abuasbeh, S. Trabelsi, Analytical Results on Coupled Caputo Systems with Boundary Nonlocality and Ulam-Hyers Stability, Bound. Value Probl. 2026 (2026), 98. https://doi.org/10.1186/s13661-026-02284-y.
  21. A. Oumansour, H. Kadari, J.R. Graef, A. Ouahab, Existence of Solutions to a Coupled System of Implicit Fractional Differential Equations with Nonlocal Boundary Conditions, Nonlinear Stud. 32 (2025), 1165–1187.
  22. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1999. https://doi.org/10.1016/S0076-5392(99)X8001-5.
  23. T.M. Rassias, On the Stability of the Linear Mapping in Banach Spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300. https://doi.org/10.1090/S0002-9939-1978-0507327-1.
  24. M.M. Matar, M.E. Samei, S. Etemad, A. Amara, S. Rezapour, J. Alzabut, Stability Analysis and Existence Criteria with Numerical Illustrations to Fractional Jerk Differential System Involving Generalized Caputo Derivative, Qual. Theory Dyn. Syst. 23 (2024), 111. https://doi.org/10.1007/s12346-024-00970-9.
  25. A.A. Sharif, M.M. Hamood, K.P. Ghadle, Analysis of Anti-periodic Boundary Value Problems for Implicit Fractional Volterra Integro-differential Equations with Generalized Tempered Fractional Derivatives, Bound. Value Probl. 2026 (2026), 104. https://doi.org/10.1186/s13661-026-02290-0.
  26. S. Solhi, A. Kajouni, K. Hilal, Existence Results for Nonlinear Boundary Value Problems Involving Generalized ψ-Caputo Fractional Derivatives, Comput. Appl. Math. 45 (2026), 41. https://doi.org/10.1007/s40314-025-03397-3.
  27. M.S. Souid, Z. Bouazza, M. Bensaid, K.S. Mozhi, M. Mokhtar, J.K.K. Asamoah, Analytical Study of Variable-Order Fractional Differential Equations with Initial and Terminal Antiperiodic Boundary Conditions, J. Appl. Math. 2025 (2025), 8863599. https://doi.org/10.1155/jama/8863599.
  28. S.M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, 1960.
  29. W. Zhang, X. Fu, Existence, Uniqueness, and Stability Analysis for a Nonlinear Multi-term Tripled System of Fractional Differential Equations with Closed Boundary Conditions, Filomat 39 (2025), 8175–8192. https://doi.org/10.2298/fil2523175z.
  30. X. Zhang, M. Li, Existence and Ulam-Type Stability for Fractional Multi-Delay Differential Systems, Fractal Fract. 9 (2025), 288. https://doi.org/10.3390/fractalfract9050288.
  31. Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, 2014. https://doi.org/10.1142/9069.
  32. Y. Sharifov, S.A. Zamanova, R.A. Sardarova, Existence and Uniqueness of Solutions for the Nonlinear Fractional Differential Equations with Two-Point and Integral Boundary Conditions, Eur. J. Pure Appl. Math. 14 (2021), 608–617. https://doi.org/10.29020/nybg.ejpam.v14i2.3978.
  33. M. Boukedroun, S. Ayadi, F. Chita, M. Erden Ege, O. Ege, R. Ramaswamy, Solutions of Nonlinear Fractional-Order Differential Equation Systems Using a Numerical Technique, Axioms 14 (2025), 233. https://doi.org/10.3390/axioms14040233.
  34. R. Ramaswamy, Solving Fractional Differential Equations and Integral Equations via Neutrosophic Bipolar Metric Space, Eur. J. Pure Appl. Math. 18 (2025), 6251. https://doi.org/10.29020/nybg.ejpam.v18i4.6251.
  35. R. Ramaswamy, G. Mani, Application of Fixed Point Result to Solve Integral Equation in the Setting of Graphical Branciari ℵ-Metric Spaces, AIMS Math. 9 (2024), 32945–32961. https://doi.org/10.3934/math.20241576.
  36. R. Ramadan, O. Ege, R. Ramaswamy, Novel Fixed Point Results in Rectangular Gb-Metric Spaces and Some Applications on Fractional Differential Equations, Fractal Fract. 9 (2025), 527. https://doi.org/10.3390/fractalfract9080527.