Upper and lower solutions for fractional integro-differential equation of higher-order and with nonlinear boundary conditions

Document Type : Research Paper

Authors

1 MISCOM, National School of Applied Sciences, Cadi Ayyad University, Marrakech, Morocco

2 LMACS, Faculty of Sciences and Technics, Sultan Moulay Slimane University, Beni Mellal, Morocco

3 Lab. LMRI, FP of Khouribga, Sultan Moulay Slimane University, Morroco

Abstract

This paper delves into the identification of upper and lower solutions for a high-order fractional integro-differential equation featuring non-linear boundary conditions. By introducing an order relation, we define these upper and lower solutions. Through a rigorous approach, we demonstrate the existence of these solutions as the limits of sequences derived from carefully selected problems, supported by the application of Arzel\`a-Ascoli's theorem. To illustrate the significance of our findings, we provide an illustrative example. This research contributes to a deeper understanding of solutions in the context of complex fractional integro-differential equations.

Keywords

Main Subjects

References

  • [1]  T. Abdeljawad1 and M. E. Samei, Applying quantum calculus for the existence of solution of q-integro-differential equations with three criteria, Discrete & Continuous Dynamical Systems Series S, 14(10) (2021), 3351–3386.
  • [2]  N. Adjimi, A. Boutiara, M. E. Samei, et al., On solutions of a hybrid generalized Caputo-type problem via the noncompactness measure in the generalized version of Darbo’s criterion, J Inequal Appl 2023, 34 (2023).
  • [3]  Y. Adjabi, M. E. Samei, M. M. Matar, and J. Alzabut, Langevin differential equation in frame of ordinary and Hadamard fractional derivatives under three point boundary conditions, AIMS Mathematics, 6(3) (2021), 2796–2843.
  • [4]  Y. Allaoui, S. Melliani, A. EL Allaoui, and K. Hilal, Coupled System of Mixed Hybrid Fractional Differential Equations: Linear Perturbations of First and Second Type, In Recent Advances in Fuzzy Sets Theory, Fractional Calculus, Dynamic Systems and Optimization, (2022), pp. 446–457.
  • [5]  P. Amiri and M. E. Samei, Existence of Urysohn and Atangana-Baleanu fractional integral inclusion systems solutions via common fixed point of multi-valued operators, Chaos, Solitons & Fractals, 165(2) (2022), 112822.
  • [6]  Z. Baitiche, C. Derbazi, J. Alzabut, M. E. Samei, M. K. A. Kaabar, and Z. Siri, Monotone Iterative Method for ψ-Caputo Fractional Differential Equation with Nonlinear Boundary Conditions, Fractal and Fractional, 5(3) (2021), 81.
  • [7]  M. Bouaouid, Mild Solutions of a Class of Conformable Fractional Differential Equations with Nonlocal Conditions. Acta Math. Appl. Sin. Engl. Ser. 39 (2023), 249–261.
  • [8]  A. Boutiara, M. Benbachir, J. Alzabut, and M. E. Samei, Monotone Iterative and Upper-Lower Solution Techniques for Solving the Nonlinear ψ-Caputo Fractional Boundary Value Problem, Fractal Fractional, 5(4) (2021), 194.
  • [9]  A. Boutiara, M. K. A. Kaabar, Z. Siri, M. E. Samei, and X. G. Yue, Investigation of a Generalized Proportional Langevin and Sturm-Liouville Fractional Differential Equations via Variable Coefficients and Anti-Periodic Boundary Conditions with a Control Theory Application arising from Complex Networks, Mathematical Problems in Engineering, (2022), 21.
  • [10]  Z. Cui and Z. Zhou, Existence of solutions for Caputo fractional delay differential equations with nonlocal and integral boundary conditions, Fixed Point Theory Algorithms Sci Eng 2023, 1 (2023).
  • [11]  C. Derbazi, Z. Baitiche, M. Benchohra, and J. R. Graef, Extremal solutions to a coupled system of nonlinear fractional differential equations with Caputo fractional derivatives, J. Math. Appl., 44 (2021), 19–34.
  • [12]  S. Etemad, I. Iqbal, M. E. Samei, et al., Some inequalities on multi-functions for applying fractional CaputoHadamard jerk inclusion system, Journal of Inequalities and Applications, 84 (2022).
  • [13]  V. Hedayati and M. E. Samei, Positive solutions of fractional differential equation with two pieces in chain interval and simultaneous Dirichlet boundary conditions, Bound Value Probl., 141 (2019).
  • [14]  M. Houas and M. E. Samei, Existence and Mittag-Leffler-Ulam-stability results for Duffing type problem involving sequential fractional derivatives,Int. J. Appl. Comput. Math 8, 185 (2022).
  • [15]  T. Kherraz, M. Benbachir, M. Lakrib, M. E. Samei, M. K. A. Kaabar, and S. A. Bhanotar, Existence and uniqueness results for fractional boundary value problems with multiple orders of fractional derivatives and integrals, Chaos Solitons & Fractals. 166(1) (2023), 113007.
  • [16]  A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North Holland Mathematics Studies, Elsevier Science B. V., Amsterdam, 204 (2006).
  • [17]  D. Kumar and D. Baleanu, Fractional calculus and its applications in physics Frontiers in Physics, 7(6) 81 (2019).
  • [18]  M. Rahaman, S. P. Mondal, A. El Allaoui, S. Alam, and A. Ahmadian, and S. Salahshour, Solution strategy for fuzzy fractional order linear homogeneous differential equation by Caputo-H differentiability and its application in fuzzy EOQ model. Advances in Fuzzy Integral and Differential Equations, (2022), 143–157.
  • [19]  S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, (Yverdon: Gordon and Breach), 1993.
  • [20]  E. Shivanian and A. Dinmohammadi, On the solution of a nonlinear fractional integro-differential equation with non-local boundary condition, International Journal of Nonlinear Analysis and Applications, (2023), 1–12.
  • [21]  M. E. Samei, L. Karimi, and M. K. A. Kaabar, To investigate a class of multi-singular pointwise defined fractional integro-differential equation with applications. AIMS Mathematics, 7(5) (2022), 7781–7816.
  • [22]  M. E. Samei, H. Zanganeh, and S. M. Aydogan, Investigation of a Class of the Singular Fractional Integrodifferential Quantum Equations with Multi-Step Methods, Journal of Mathematical Extension, 15 (2021), 1–54.
  • [23]  M. Sangi and S. Saiedinezhad, and M. B. Ghaemi, A System of High-Order Fractional Differential Equations with Integral Boundary Conditions, J Nonlinear Math Phys, 30 (2023), 699–718.
  • [24]  H. Sun, A. Chang, Y. Zhang, and W. Chen, A review on variable-order fractional differential equations: Mathematical foundations, physical models, numerical methods and applications, Fractional Calculus and Applied Analysis, 22(1) (2019), 27–59.
  • [25]  J. A. Tenreiro Machado, Handbook of Fractional Calculus with Applications (Berlin-Boston: Walter de Gruyter GmbH), (2019).
  • [26]  J. Wang, Y. Zhou, and W. Wei1, Study in fractional differential equations by means of topological degree methods, Numer. Funct. Anal. Optim., 33(2) (2012), 216–238.
Computational Methods for Differential Equations
Volume 12, Issue 3
July 2024
Pages 438-453

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History

  • Receive Date: 09 May 2023
  • Revise Date: 04 November 2023
  • Accept Date: 22 November 2023

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