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ON CONTROLLABILITY FOR FRACTIONAL VOLTERRA-FREDHOLM SYSTEM

  • Ahmed A. Hamoud (Department of Mathematics, Taiz University) ;
  • Saif Aldeen M. Jameel (Department of Computer Systems, Middle Technical University) ;
  • Nedal M. Mohammed (Department of Computer Science & IT, Taiz University) ;
  • Homan Emadifar (Department of Mathematics, Islamic Azad University) ;
  • Foroud Parvaneh (Department of Mathematics, Kermanshah Branch, Islamic Azad University) ;
  • Masoumeh Khademi (Department of Mathematics, Islamic Azad University)
  • Received : 2022.08.04
  • Accepted : 2022.10.17
  • Published : 2023.06.15

Abstract

In this manuscript, we study the sufficient conditions for controllability of Volterra-Fredholm type fractional integro-differential systems in a Banach space. Fractional calculus and the fixed point theorem are used to derive the findings. Some examples are provided to illustrate the obtained results.

Keywords

References

  1. R.P. Agarwal, M. Benchohra and S. Hamanani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010), 973-1033.  https://doi.org/10.1007/s10440-008-9356-6
  2. H.M. Ahmed, Controllability for Sobolev type fractional integro-differential systems in a Banach space, Advances Diff. Equa., 2012(1) (2012), 1-10.  https://doi.org/10.1186/1687-1847-2012-1
  3. A. Abed, M. Younis and A. Hamoud, Numerical solutions of nonlinear Volterra-Fredholm integro-differential equations by using MADM and VIM, Nonlinear Funct. Anal. Appl., 27(1) (2022), 189-201. 
  4. H.M. Ahmed, Approximate controllability via resolvent operators of Sobolev-type fractional stochastic integrodifferential equations with fractional Brownian motion and Poisson jumps, Bull. Iranian Math. Soc., 45(4) (2019), 1045-1059.  https://doi.org/10.1007/s41980-018-0183-x
  5. H.M. Ahmed, M.M. El-Borai, H.M. El-Owaidy and A.S. Ghanem, Existence solution and controllability of Sobolev type delay nonlinear fractional integro-differential system, Mathematics, 7(1) (2019), 79. 
  6. H.M. Ahmed, M.M. El-Borai, AS Okb El Bab and M.E. Ramadan, Controllability and constrained controllability for nonlocal Hilfer fractional differential systems with Clarkes subdifferential, J. Ineq. Appl., 2019(1) (2019), 1-23.  https://doi.org/10.1186/s13660-019-1955-4
  7. W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems. Birkhuser, Berlin (2011). 
  8. A. Anguraj, P. Karthikeyan and G.M. N'Guerekata, Nonlocal Cauchy problem for some fractional abstract integrodifferential equations in Banach space, Commun. Math. Anal., 6 (2009), 31-35. 
  9. K. Balachandran and J.P. Dauer, Controllability of nonlinear systems via fixed point theorems, J. Opti. Theory Appl., 53 (1987), 345-352.  https://doi.org/10.1007/BF00938943
  10. K. Balachandran and J.P. Dauer, Controllability of nonlinear systems in Banach spaces: A survey, J. Opti. Theory Appl., 115 (2002), 7-28.  https://doi.org/10.1023/A:1019668728098
  11. K. Balachandran and J.Y. Park, Controllability of fractional integrodifferential systems in Banach spaces, Nonlinear Anal. : Hybrid Systems, 3 (2009), 363-367.  https://doi.org/10.1016/j.nahs.2009.01.014
  12. B. Bonilla, M. Rivero, L. Rodriguez-Germa and J.J. Trujillo, Fractional differential equations as alternative models to nonlinear differential equations, Appl. Math. Comput., 187 (2007), 79-88. 
  13. K. Diethelm and A.D. Freed, On the solution of nonlinear fractional-order differential equation used in the modeling of viscoelasticity. In: F.keil, W.Mackens, H.Voss, J. Werther (Eds), Scientific Computing in Chemical Engineering IIComputational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer-Verlag, Heidelberg (1999), 217-224. 
  14. A. Hamoud and K. Ghadle, Existence and uniqueness of solutions for fractional mixed Volterra-Fredholm integro-differential equations, Indian J. Math., 60(3) (2018), 375-395.  https://doi.org/10.12732/ijam.v31i3.3
  15. A. Hamoud, K. Ghadle, M. Bani Issa and Giniswamy, Existence and uniqueness theorems for fractional Volterra-Fredholm integro-differential equations, Int. J. Appl. Math., 31(3) (2018), 333-348.  https://doi.org/10.12732/ijam.v31i3.3
  16. A. Hamoud and K. Ghadle, Some new existence, uniqueness and convergence results for fractional Volterra-Fredholm integro-differential equations, J. Appl. Comput. Mech., 5(1) (2019), 58-69. 
  17. A. Hamoud, Existence and uniqueness of solutions for fractional neutral Volterra-Fredholm integro-differential equations, Advan. Theory Nonlinear Anal. Appl., 4(4) (2020), 321-331.  https://doi.org/10.31197/atnaa.799854
  18. A. Hamoud, M. SH. Bani Issa and K. Ghadle, Existence and uniqueness results for nonlinear Volterra-Fredholm integro-differential equations, Nonlinear Funct. Anal. Appl., 23(4) (2018), 797-805.  https://doi.org/10.7862/rf.2018.9
  19. K. Hussain, A. Hamoud and N. Mohammed, Some new uniqueness results for fractional integro-differential equations, Nonlinear Funct. Anal. Appl., 24(4) (2019), 827-836. 
  20. A. Hamoud and K. Ghadle, Some new results on nonlinear fractional iterative Volterra-Fredholm integro differential equations, TWMS J. Appl. Eng. Math., 12(4) (2022), 1283-1294.  https://doi.org/10.5890/DNC.2023.12.013
  21. J.W. Hanneken, D.M. Vaught and B.N. Narahari Achar, Enumeration of the real zeros of the Mittag-Leffler function Eα(z), 1 < α < 2. In: Sabatier, J., Agrawal, O.P., Machado, J.A.T. (eds.) Advances in Fractional Calculus, pp. 15-26. Springer, Dordrecht (2007). https://doi.org/10.1007/978-1-4020-6042-7 2. 
  22. P. Karthikeyan, Some results for boundary value problem of an integrodifferential equations with fractional order, Dynamic Syst. Appl., 20 (2011), 17-24. 
  23. K. Karthikeyan and J.J. Trujillo, Existence and uniqueness results for fractional integrodifferential equations with boundary value conditions, Commu. Nonlinear Sci. Numer. Simu., 17 (2012), 4037-4043.  https://doi.org/10.1016/j.cnsns.2011.11.036
  24. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential equations, Elsevier, Amsterdam, 2006. 
  25. J. Klamka, Controllability of Dynamical Systems, Kluwer Academic, Dordrecht, 1991. 
  26. V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal., 69 (2008), 3337-3343.  https://doi.org/10.1016/j.na.2007.09.025
  27. V. Lakshmikantham and A.S. Vatsala, Basic theory of fractional differential equation, Nonlinear Anal., 69 (2008), 2677-2682.  https://doi.org/10.1016/j.na.2007.08.042
  28. T. Li, N. Pintus and G. Viglialoro, Properties of solutions to porous medium problems with different sources and boundary conditions, Z. Angew. Math. Phys., 70 (2019), 86-94.  https://doi.org/10.1007/s00033-019-1130-2
  29. T. Li and G. Viglialoro, Analysis and explicit solvability of degenerate tensorial problems, Bound. Value Probl., 2018 (2018), 21-34.  https://doi.org/10.1186/s13661-018-0939-5
  30. W. Lin, Global existence theory and chaos control of fractional differential equation, J. Math. Anal. Appl., 332 (2007), 709-726.  https://doi.org/10.1016/j.jmaa.2006.10.040
  31. S. Magar, A. Hamoud, A. Khandagale and K. Ghadle, Generalized Shehu transform to Ψ-Hilfer-Prabhakar fractional derivative and its regularized version, Advances in the Theory of Nonlinear Analysis and its Application, 6(3) (2022), 364-379.  https://doi.org/10.31197/atnaa.1032207
  32. A. Pazy, Semigroups of Linear operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. 
  33. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. 
  34. M. Raja, V. Vijayakumar, A. Shukla, K. Sooppy Nisar and S. Rezapour, New discussion on nonlocal controllability for fractional evolution system of order 1 < r < 2, Advances in Difference Equations, 2021 (2021), 1-19.  https://doi.org/10.1186/s13662-020-03162-2
  35. A. Sharif, and A. Hamoud, On ψ-Caputo fractional nonlinear Volterra-Fredholm integrodifferential equations, Disconti. Nonlinearity and Complexity, 11(1) (2022), 97-106.  https://doi.org/10.5890/DNC.2022.03.008
  36. D.R. Smart, Fixed point Theorems, Cambridge University Press, Cambridge 66 1980. 
  37. S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993. 
  38. D. Tamizharasan and K. Karthikeyan, Controllability results for fractional integrodifferential systems with boundary conditions, Indian J. Pure Appl. Math., 52 (2021), 39-45.  https://doi.org/10.1007/s13226-021-00129-1
  39. P. Wang, Y. Wang, C. Jiang and T. Li, Convergence of solutions for functional integrodifferential equations with nonlinear boundary conditions, Adv. Diff. Equ. 2019 (2019), 521. 
  40. S. Zhang, Positive solutions for boundary value problems for nonlinear fractional differential equations, Elec. J. Diff. Equ., 36 (2006), 1-12.