Distributed State-Dependent with Conjugate Feedback Control

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DOI: https://doi.org/10.28924/2291-8639-22-2024-229
Published: Dec 9, 2024
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Sh. A. Abd El-Salam, A. M. A. El-Sayed, M. E. I. El-Gendy, Distributed State-Dependent with Conjugate Feedback Control, Int. J. Anal. Appl., 22 (2024), 229.

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Sh. A. Abd El-Salam, A. M. A. El-Sayed, M. E. I. El-Gendy

Abstract

Our goal in this work is to study the existence of solution of the Distributed state-dependent integral equation \[x(\ell)~=~a_1(\ell)~+~\int_{0}^{\varphi_1(x(\ell))}f_1(s,y(s))ds,\] with conjugate feedback control \[y(\ell)~=~a_2(\ell)~+~\int_{0}^{\varphi_2(y(\ell))}f_2(s,x(s))ds.\] Then some properties of this solution will be studied like uniqueness, continuous dependence and Hyers-Ulam stability.

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References

  1. B. Ahmad, A. Alsaedi, Existence and Uniqueness of Solutions for Coupled Systems of Higher-Order Nonlinear Fractional Differential Equations, Fixed Point Theory Appl. 2010 (2010), 364560. https://doi.org/10.1155/2010/364560.
  2. B. Ahmad, S.K. Ntouyas, A. Alsaedi, On a Coupled System of Fractional Differential Equations with Coupled Nonlocal and Integral Boundary Conditions, Chaos Solitons Fractals 83 (2016), 234–241. https://doi.org/10.1016/j.chaos.2015.12.014.
  3. K.L. Baluja, P.G. Burke, L.A.Morgan, R-Matrix Propagation Program for Solving Coupled Second-Order Differential Equations, Comput. Phys. Commun. 27 (1982), 299-307.
  4. K. Deimling, Nonlinear Functional Analysis, Dover publications, Mineola, 2010.
  5. E. Eder, The Functional Differential Equation x 0 (t) = x(x(t)), J. Differ. Equ. 54 (1984), 390–400. https://doi.org/10.1016/0022-0396(84)90150-5.
  6. A.M.A. El-Sayed, S.A. Abd El-Salam, Coupled System of a Fractional Order Differential Equations with Weighted Initial Conditions, Open Math. 17 (2019), 1737–1749. https://doi.org/10.1515/math-2019-0120.
  7. A.M.A. El-Sayed, H. El-Owaidy, R. Gamal Ahmed, Solvability of a Boundary Value Problem of Self-Reference Functional Differential Equation with Infinite Point and Integral Conditions, J. Math. Comput. Sci. 21 (2020), 296–308. https://doi.org/10.22436/jmcs.021.04.03.
  8. A.M.A. El-Sayed, R. Gamal Aahmed, Solvability of the Functional Integro-Differential Equation with Self-Reference and State-Dependence, J. Nonlinear Sci. Appl. 13 (2020), 1–8. https://doi.org/10.22436/jnsa.013.01.01.
  9. H.R. Ebead, Self-Reference (State-Dependence) Quadratic Integral Equation of Fractional Order, J. Fract. Calc. Appl. 15 (2024), 1-15.
  10. F. Hartung, T. Krisztin, H.-O. Walther, J. Wu, Chapter 5 Functional Differential Equations with State-Dependent Delays: Theory and Applications, in: Handbook of Differential Equations: Ordinary Differential Equations, Elsevier, 2006: pp. 435–545. https://doi.org/10.1016/S1874-5725(06)80009-X.
  11. H.H.G. Hashem, Continuous Dependence of Solutions of Coupled Systems of State Dependent Functional Equations, Adv. Differ. Equ. Control Process. 22 (2020), 121–135. https://doi.org/10.17654/DE022020121.
  12. U.V. Lê, E. Pascali, An Existence Theorem for Self-Referred and Hereditary Differential Equations, Adv. Differ. Equ. Control Process. 1 (2008), 25-32.
  13. N.T.T. Lan, P. Eduardo, A Two-Point Boundary Value Problem for a Differential Equation With Self-Reference, Electron. J. Math. Anal. Appl. 6 (2018), 25–30.
  14. M. Feckan, On a Certain Type of Functional Differential Equations, Math. Slovaca 43 (1993), 39–43. http://dml.cz/dmlcz/130391.
  15. M. Miranda Jr., E. Pascali, On a Type of Evolution of Self-Referred and Hereditary Phenomena, Aequat. Math. 71 (2006), 253–268. https://doi.org/10.1007/s00010-005-2821-7.
  16. I. Talib, N.A. Asif. C. Tunc, Existence of Solutions to Second-Order Nonlinear Coupled Systems With Nonlinear Coupled Boundary Conditions, Electron. J. Differ. Equ. 2015 (2015), 313.
  17. R. Tian, Z. Zhang, Existence and Bifurcation of Solutions for a Double Coupled System of Schrödinger Equations, Sci. China Math. 58 (2015), 1607–1620. https://doi.org/10.1007/s11425-015-5028-y.
  18. S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publishers, New York, 1960.
  19. S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1964.