Nonlinear Functional Analysis and Applications

The Basic Sciences Research Institute, Kyungnam University (경남대학교 기초과학연구소)
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ON A TYPE OF DIFFERENTIAL CALCULUS IN THE FRAME OF GENERALIZED HILFER INTEGRO-DIFFERENTIAL EQUATION

  • Mohammed N. Alkord (Department of Mathematics, Maulana Azad College research center, Dr.Babasaheb Ambedkar Marathwada University) ;
  • Sadikali L. Shaikh (Department of Mathematics, Maulana Azad College of arts, Science and Commerce) ;
  • Mohammed B. M. Altalla (Department of Mathematics, PET Research Foundation, University of Mysore)
  • Received : 2023.05.12
  • Accepted : 2023.09.04
  • Published : 2024.03.15
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Abstract

In this paper, we investigate the existence and uniqueness of solutions to a new class of integro-differential equation boundary value problems (BVPs) with ㄒ-Hilfer operator. Our problem is converted into an equivalent fixed-point problem by introducing an operator whose fixed points coincide with the solutions to the given problem. Using Banach's and Schauder's fixed point techniques, the uniqueness and existence result for the given problem are demonstrated. The stability results for solutions of the given problem are also discussed. In the end. One example is provided to demonstrate the obtained results

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References

  1. B.N. Abood, S.S. Redhwan, O. Bazighifan and K. Nonlaopon, Investigating a Generalized Fractional Quadratic Integral Equation, Fractal and Fractional, 6(5) (2022), 251. 
  2. R.P. Agarwal, M. Benchohra and S.A. Hamani, Survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109(3) (2010), 973-1033.  https://doi.org/10.1007/s10440-008-9356-6
  3. M.A. Almalahi, S.K. Panchal and F. Jarad, Multipoint BVP for the Langevin Equation under-Hilfer Fractional Operator, J. Funct. Spaces, 2022 (2022), Article ID 2798514. 
  4. S. Asawasamrit, A. Kijjathanakorn, S.K. Ntouyas and J. Tariboon, Nonlocal boundary value problems for Hilfer fractional differential equations, Bull. Kor. Math. Soc., 55(6) (2018), 1639-1657. 
  5. A. Atangana, Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system, Chaos Solitons Fractals, 102 (2017), 396-406.  https://doi.org/10.1016/j.chaos.2017.04.027
  6. A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer, Model. Therm. Sci., 20(2) (2016), 763-769.  https://doi.org/10.2298/TSCI160111018A
  7. M. Benchohra and S. Bouriah, Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order, Moroccan J. Pure Appl. Anal., 1 (2015), 22-37.  https://doi.org/10.7603/s40956-015-0002-9
  8. T.A. Burton and U.C. Kirk, A fixed point theorem of Krasnoselskii Schaefer type, Math. Nachrichten, 189 (1998), 23-31.  https://doi.org/10.1002/mana.19981890103
  9. K.M. Furati, M.D. Kassim and N.E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64(6) (2012), 1616-1626.  https://doi.org/10.1016/j.camwa.2012.01.009
  10. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. 
  11. F. Jarad, T. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos Solitons Fractals, 117 (2018), 16-20.  https://doi.org/10.1016/j.chaos.2018.10.006
  12. M.B. Jeelani, A.S. Alnahdi, M.A. Almalahi, M.S. Abdo, H.A. Wahash and N.H. Alharthi, Qualitative Analyses of Fractional Integrodifferential Equations with a Variable Order under the Mittag-Leffler Power Law, J. Funct., Spaces, (2022), Article ID 6387351. 
  13. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, 204, Elsevier Science, Amsterdam, 2006. 
  14. A.D. Mali and K.D. Kucche, Nonlocal boundary value problem for generalized Hilfer implicit fractional differential equations, Math. Meth. Appl. Sci., 43(15) (2020), 8608-8631.  https://doi.org/10.1002/mma.6521
  15. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. 
  16. S.S. Redhwan, M.S. Abdo, K. Shah, T. Abdeljawad, S. Dawood, H.A. Abdo and S. L.Shaikh, Mathematical modeling for the outbreak of the coronavirus (COVID-19) under fractional nonlocal operator, Results in Phys., 19 (2020), 103610. 
  17. S.S. Redhwan and S.L. Shaikh, Implicit fractional differential equation with nonlocal integral-multipoint boundary conditions in the frame of Hilfer fractional derivative, J. Math. Anal. Mod., 2(1) (2021), 62-71.  https://doi.org/10.48185/jmam.v2i1.176
  18. S.S. Redhwan, S.L. Shaikh and M.S. Abdo, Caputo-Katugampola-type implicit fractional differential equation with anti-periodic boundary conditions, Results in Nonlinear Anal., 5(1) (2022), 12-28.  https://doi.org/10.53006/rna.974148
  19. S.S. Redhwan, S.L. Shaikh, M.S. Abdo, W. Shatanawi, K. Abodayeh, M.A. Almalahi and T. Aljaaidi, Investigating a generalized Hilfer-type fractional differential equation with two-point and integral boundary conditions, AIMS Mathematics, 7(2) (2022), 1856-1872.  https://doi.org/10.3934/math.2022107
  20. S.S. Redhwan, A.M. Suad, S.L. Shaikh and M.S. Abdo, A coupled non-separated system of Hadamard-type fractional differential equations. Adv. Theory Nonlinear Anal. Appl., 6(1) (2021), 33-44.  https://doi.org/10.31197/atnaa.925365
  21. I.A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpath. J. Math., 26 (2010), 103-107. 
  22. S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Switzerland, 1993. 
  23. J.V.D.C. Sousa and E.C.de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator, Diff. Equ. Appl., 11(1) (2019), 87-106.  https://doi.org/10.7153/dea-2019-11-02
  24. S.T.M. Thabet, B. Ahmad and R.P. Agarwal, On abstract Hilfer fractional integrodifferential equations with boundary conditions, Arab J. Math. Sci., 26(1/2) (2020), 107-125.  https://doi.org/10.1016/j.ajmsc.2019.03.001
  25. C.da. Vanterler, J. Sousa and E. Capelas de Oliveira, On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72-91.  https://doi.org/10.1016/j.cnsns.2018.01.005
  26. Y. Zhou, Basic Theory of Fractional Differential Equations, 6, World Scientific, Singapore, 2014.