Inclusive analysis of a nonlocal fractional hybrid differential problem via differential feedback control

Volume 37, Issue 4, pp 449--464 https://dx.doi.org/10.22436/jmcs.037.04.07

Publication Date: November 14, 2024 Submission Date: July 21, 2024 Revision Date: August 27, 2024 Accteptance Date: September 14, 2024

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Authors

A. M. A. El-Sayed - Faculty of Sciences, Department of Mathematics, Alexandria University, Alexandria, Egypt. Sh. M. Al-Issa - Faculty of Arts and Sciences, Department of Mathematics, Lebanese International University, Saida, Lebanon. - Faculty of Arts and Sciences, Department of Mathematics, The International University of Beirut, Beirut, Lebanon. H. H. G. Hashem - Faculty of Sciences, Department of Mathematics, Alexandria University, Alexandria, Egypt. I. H. Kaddoura - Faculty of Arts and Sciences, Department of Mathematics, Lebanese International University, Saida, Lebanon. - Faculty of Arts and Sciences, Department of Mathematics, The International University of Beirut, Beirut, Lebanon. A. A. El-Bakari - Faculty of Arts and Sciences, Department of Mathematics, Lebanese International University, Saida, Lebanon.

Abstract

In this paper, we investigate the existence of continuous solutions for a fractional-order hybrid non-local differential problem that involves differential feedback control. The problem is formulated with a fractional-order derivative, which adds complexity to the system’s dynamics. Our approach relies on key mathematical tools, including the Schauder fixed point theorem and the contraction mapping theorem, to establish the existence of solutions. Furthermore, we analyze the continuous dependence of the solutions on various parameters, ensuring robustness in the behavior of the system under perturbations. We present a comprehensive discussion on the solvability of the problem within the space $~L_1(I)$, providing detailed remarks on the conditions that guarantee the existence of solutions. To demonstrate the practical applicability of our results, we include an illustrative example that elucidates the theoretical results and supports the validity of our approach.

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Keywords

• Hybrid differential equation
• continuous dependence
• control variable
• feedback control

MSC

•  26A33
•  34A60

References

• [1] B. Ahmad, S. K. Ntouyas, Initial-value problems for hybrid Hadamard fractional differential equations, Electron. J. Differ. Equ., 2014 (2014), 8 pages
  • View Article
  • Google Scholar
  • MATH


• [2] S. M. Al-Issa, A. M. A. El-Sayed, H. H. G. Hashem, An Outlook on Hybrid Fractional Modeling of a Heat Controller with Multi-Valued Feedback Control, Fractal Fract., 7 (2023), 19 pages
  • View Article
  • Google Scholar


• [3] D. Baleanu, S. Etemad, S. Pourrazi, S. Rezapour, On the new fractional hybrid boundary value problems with three-point integral hybrid conditions, Adv. Differ. Equ., 2019 (2019), 21 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] D. Baleanu, S. Etemad, S. Rezapour, A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions, Bound. Value Probl., 2020 (2020), 16 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] J. Bana´s, M. Lecko, W. G. El-Sayed, Existence theorems for some quadratic integral equations, J. Math. Anal. Appl., 222 (1998), 276–285
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] J. Bana´s, T. Zaja¸c, A new approach to the theory of functional integral equations of fractional order, J. Math. Anal. Appl., 375 (2011), 375–387
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] F. Chen, The permanence and global attractivity of Lotka-Volterra competition system with feedback controls, Nonlinear Anal. Real World Appl., 7 (2006), 133–143
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] M. Cicho´ n, Differential inclusions and abstract control problems, Bull. Austral. Math. Soc., 53 (1996), 109–122
  • View Article
  • MathSciNet
  • Google Scholar


• [9] R. F. Curtain, A. J. Pritchard, Functional analysis in modern applied mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York (1977)
  • View Article
  • Google Scholar


• [10] K. Deimling, Nonlinear functional Analysis, Springer-Verlag, Berlin (1985)
  • View Article
  • MathSciNet
  • Google Scholar


• [11] B. C. Dhage, N. S. Jadhav, Basic results in the theory of hybrid differential equations with linear perturbations of second type, Tamkang J. Math., 44 (2013), 171–186
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] B. C. Dhage, V. Lakshmikantham, Basic results on hybrid differential equation, Nonlinear Anal. Hybrid Syst., 4 (2010), 414–424
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] A. M. A. El-Sayed, H. H. G. Hashem, S. M. Al-Issa, New Aspects on the Solvability of a Multidimensional Functional Integral Equation with Multivalued Feedback Control, Axioms, 12 (2023), 15 pages
  • View Article
  • Google Scholar


• [14] A. M. A. El-Sayed, H. H. G. Hashem, Sh. M. Al-Issa, A comprehensive view of the solvability of non-local fractional orders pantograph equation with a fractal-fractional feedback control, AIMS Math., 9 (2024), 19276–19298
  • View Article
  • MathSciNet
  • Google Scholar


• [15] J. R. Graef, S. Heidarkhani, L. Kong, S. Moradi, Three solutions for impulsive fractional boundary value problems with p-Laplacian, Bull. Iranian Math. Soc., 48 (2022), 1413–1433
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] A. K. Gupta, R. K. Patel, Theoretical Analysis of Fractional-Order Hybrid Differential Equations via Measure of Noncompactness, Mathematics, (),


• [17] S. Heidarkhani, Infinitely many solutions for nonlinear perturbed fractional boundary value problems, An. Univ. Craiova Ser. Mat. Inform., 41 (2014), 88–103
  • View Article
  • Google Scholar
  • MATH


• [18] S. Heidarkhani, Multiple solutions for a nonlinear perturbed fractional boundary value problem, Dynam. Systems Appl., 23 (2014), 317–331
  • View Article
  • Google Scholar


• [19] S. Heidarkhani, A. Cabada, G. A. Afrouzi, S. Moradi, G. Caristi, A variational approach to perturbed impulsive fractional differential equations, J. Comput. Appl. Math., 341 (2018), 42–60
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] S. Heidarkhani, A. Salari, Nontrivial solutions for impulsive fractional differential systems through variational methods, Math. Methods Appl. Sci., 43 (2020), 6529–6541
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] S. Heidarkhani, A. Salari, G. Caristi, Infinitely many solutions for impulsive nonlinear fractional boundary value problems, Adv. Differ. Equ., 2016 (2016), 19 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] M. A. E. Herzallah, D. Baleanu, On fractional order hybrid differential equations, Abstr. Appl. Anal., 2014 (2014), 7 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [23] S. Melliani, K. Hilal, M. Hannabou, Existence results in the theory of hybrid fractional integro-differential equations, J. Univers. Math., 1 (2018), 166–179
  • View Article
  • Google Scholar


• [24] P. Nasertayoob, Solvability and asymptotic stability of a class of nonlinear functional-integral equation with feedback control, Commun. Nonlinear Anal., 5 (2018), 19–27
  • View Article
  • Google Scholar


• [25] P. Nasertayoob, S. M. Vaezpour, Positive periodic solution for a nonlinear neutral delay population equation with feedback control, J. Nonlinear Sci. Appl., 6 (2013), 152–161
  • View Article
  • Google Scholar


• [26] A. Platzer, Differential dynamic logic for hybrid systems, J. Automat. Reason., 41 (2008), 143–189
  • View Article
  • MathSciNet
  • Google Scholar


• [27] W. Quaghebeur, E. Torfs, B. De Baets, I. Nopens, Hybrid differential equations: Integrating mechanistic and data-driven techniques for modelling of water systems, Water Res., 2013 (2022),
  • View Article
  • Google Scholar


• [28] S. Sitho, S. K. Ntouyas, J. Tariboon, Existence results for hybrid fractional integro-differential equations, Bound. Value Probl., 2015 (2015), 13 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] H. M. Srivastava, A. M. A. El-Sayed, H. H. G. Hashem, Sh. M. Al-Issa, Analytical investigation of nonlinear hybrid implicit functional differential inclusions of arbitrary fractional orders, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 116 (2022), 19 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] C. Swartz, Measure, integration and function spaces, World Scientific Publishing Co., River Edge (1994)
  • View Article
  • MathSciNet
  • Google Scholar