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Study of a sequential \(\psi \)-Hilfer fractional integro-differential equations with nonlocal BCs

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Abstract

This paper deals with the existence and uniqueness of solutions for a nonlinear boundary value problem involving a sequential \(\psi \)-Hilfer fractional integro-differential equations with nonlocal boundary conditions. The existence and uniqueness of solutions are established for the considered problem by using the Banach contraction principle, Sadovski’s fixed point theorem, and Krasnoselskii-Schaefer fixed point theorem due to Burton and Kirk. In addition, the Ulam-Hyers stability of solutions is discussed. Finally, the obtained results are illustrated by examples.

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References

  1. Lazopoulos, K.A., Lazopoulos, A.K.: On the fractional deformation of a linearly elastic bar. J. Mech. Behav. Mater. 29(1), 9–18 (2020). https://doi.org/10.1515/jmbm-2020-0002

    Article  Google Scholar 

  2. Boutiara, A., Benbachir, M., Alzabut, J., Samei, M.E.: Monotone iterative and upper-lower solutions techniques for solving nonlinear \(\uppsi -\)Caputo fractional boundary value problem. Fractal Fract. 5, 194 (2021). https://doi.org/10.3390/fractalfract5040194

    Article  Google Scholar 

  3. Hilfer, R.: Experimental evidence for fractional time evolution in glass forming materials. Chem. Phys. 284(1–2), 399–408 (2002). https://doi.org/10.1016/S0301-0104(02)00670-5

    Article  Google Scholar 

  4. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204., Elsevier Science B.V. Amsterdam (2006)

  5. Yue, X.-G., Samei, M.E., Fathipour, A., Kaabar, M.K.A., Kashuri, A.: Using Krasnoselskii’s theorem to investigate the Cauchy and neutral fractional \(q\)-integro-differential equation via numerical technique. Nonlinear Eng. Model. Appl. 11, 207–227 (2022). https://doi.org/10.1515/nleng-2022-0023

    Article  Google Scholar 

  6. Podlubny, L.: Fractional Differential Equations. Academic Press, New York (1999)

    Google Scholar 

  7. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Switzerland (1993)

    Google Scholar 

  8. Tarasov, V.E.: Fractional mechanics of elastic solids: continuum aspects. J. Eng. Mech. 143(5), 4016001 (2017). https://doi.org/10.1061/(ASCE)EM.1943-7889.0001074

    Article  Google Scholar 

  9. Silva Costa, F., Contharteze Grigoletto, E., Vaz Jr, J., Capelas de Oliveira, E.: Slowing-down of neutrons: a fractional model. Commun. Appl. Ind. Math 6(2), 538 (2015). https://doi.org/10.1685/journal.caim.538

    Article  MathSciNet  Google Scholar 

  10. Herrmann, R.: Fractional Calculus: An Introduction for Physicists. World Scientific Publishing Company, Singapore (2011)

    Book  Google Scholar 

  11. Atanackovic, T.M., Pilipovic, S., Stankovic, B., Zorica, D.: Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes. Wiley-ISTE, London (2014)

    Book  Google Scholar 

  12. Sousa, J., da Vanterler, C., de Oliveira, E.C.: On the \(\psi \)-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018). https://doi.org/10.1016/j.cnsns.2018.01.005

    Article  MathSciNet  Google Scholar 

  13. Khan, A., Syam, M.I., Zada, A., Khan, H.: Stability analysis of nonlinear fractional differential equations with Caputo and Riemann-Liouville derivatives. European Phys. J. Plus 133, 264 (2018). https://doi.org/10.1140/epjp/i2018-12119-6

    Article  Google Scholar 

  14. Khan, A., Khan, H., Gómez-Aguilar, J.F., Abdeljawad, T.: Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel. Chaos Solitons Fractals 127, 422–427 (2019). https://doi.org/10.1016/j.chaos.2019.07.026

    Article  MathSciNet  Google Scholar 

  15. Khan, A., Alshehri, H.M., Gómez-Aguilar, J.F., Khan, Z.A., Fernández-Anaya, G.: A predator-prey model involving variable-order fractional differential equations with Mittag-Leffler kernel. Adv. Diff. Equ. 2021, 183 (2021). https://doi.org/10.1186/s13662-021-03340-w

    Article  MathSciNet  Google Scholar 

  16. Aslam, M., Murtaza, R., Abdeljawad, T., Rahman, Gu., Khan, A., Khan, H., Gulzar, H.: A fractional order HIV/AIDS epidemic model with Mittag-Leffler kernel. Adv. Differ. Equ. 2021, 107 (2021). https://doi.org/10.1186/s13662-021-03264-5

    Article  MathSciNet  Google Scholar 

  17. Khan, A., Alshehri, H.M., Abdeljawad, T., Al-Mdallal, Q.M., Khan, H.: Stability analysis of fractional nabla difference COVID-19 model. Results Phys. 22, 103888 (2021). https://doi.org/10.1016/j.rinp.2021.103888

    Article  Google Scholar 

  18. Gómez-Aguilar, J.F., Cordova-Fraga, T., Abdeljawad, T., Khan, A.: Analysis of fractal-fractional malaria transmission model. Math. Methods Appl. Sci. 28(08), 2040041 (2020). https://doi.org/10.1142/S0218348X20400411

    Article  Google Scholar 

  19. Rezapour, S., Mohammadi, H., Samei, M.E.: SEIR epidemic model for COVID-19 transmission by Caputo derivative of fractional order. Adv. Differ. Equ. 2020, 490 (2020). https://doi.org/10.1186/s13662-020-02952-y

    Article  MathSciNet  Google Scholar 

  20. Khan, H., Gómez-Aguilar, J.F., Alkhazzan, A., Khan, A.: A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler law. Math. Methods Appl. Sci. 43(6), 3786–3806 (2020). https://doi.org/10.1002/mma.6155

    Article  MathSciNet  Google Scholar 

  21. Ahmad, S., Ullah, A., Al-Mdallal, Q.M., Khan, H., Shah, K., Khan, A.: Fractional order mathematical modeling of COVID-19 transmission. Chaos Solitons Fractals 139, 110256 (2020). https://doi.org/10.1016/j.chaos.2020.110256

    Article  MathSciNet  Google Scholar 

  22. Khan, A., Gómez-Aguilar, J.F., Khan, T.S., Khan, H.: Stability analysis and numerical solutions of fractional order HIV/AIDS model. Chaos Solitons Fractals 122, 119–128 (2019). https://doi.org/10.1016/j.chaos.2019.03.022

    Article  MathSciNet  Google Scholar 

  23. Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 44, 460–481 (2017). https://doi.org/10.1016/j.cnsns.2016.09.006

    Article  MathSciNet  Google Scholar 

  24. Abdo, M.S., Shah, K., Panchal, S.K., Wahash, H.A.: Existence and Ulam stability results of a coupled system for terminal value problems involving \(\psi \)-Hilfer fractional operator. Adv. Differ. Equ. 2020, 316 (2020). https://doi.org/10.1186/s13662-020-02775-x

    Article  MathSciNet  Google Scholar 

  25. Etemad, S., Rezapour, S., Samei, M.E.: \(\alpha \)-\(\psi \)-contractions and solutions of a \(q\)-fractional differential inclusion with three-point boundary value conditions via computational results. Adv. Differ. Equ. 2020, 218 (2020). https://doi.org/10.1186/s13662-020-02679-w

    Article  MathSciNet  Google Scholar 

  26. Haddouchi, F.: Positive solutions of nonlocal fractional boundary value problem involving Riemann-Stieltjes integral condition. J. Appl. Math. Comput. 64(1–2), 487–502 (2020). https://doi.org/10.1007/s12190-020-01365-0

    Article  MathSciNet  Google Scholar 

  27. Haddouchi, F.: On the existence and uniqueness of solution for fractional differential equations with nonlocal multi-point boundary conditions. Differ. Equ. Appl. 13(3), 227–242 (2021). https://doi.org/10.7153/dea-2021-13-13

    Article  MathSciNet  Google Scholar 

  28. Djaout, A., Benbachir, M., Lakrib, M., Matar, M.M., Khan, A., Abdeljawad, T.: Solvability and stability analysis of a coupled system involving generalized fractional derivatives. AIMS Math. 8(4), 7817–7839 (2023). https://doi.org/10.3934/math.2023393

    Article  MathSciNet  Google Scholar 

  29. Haddouchi, F.: Existence of positive solutions for a class of conformable fractional differential equations with parameterized integral boundary conditions. Kyungpook Math. J. 61(1), 139–153 (2021). https://doi.org/10.5666/KMJ.2021.61.1.139

    Article  MathSciNet  Google Scholar 

  30. Rezapour, S., Boulfoul, A., Tellab, B., Samei, M.E., Etemad, S., George, R.: Fixed point theory and the Caputo-Liouville integro-differential FBVP with multiple nonlinear terms. J. Funct. Space 2022, 18 (2022). https://doi.org/10.1155/2022/6713533

    Article  Google Scholar 

  31. Haddouchi, F.: Positive solutions of \(p\)-Laplacian fractional differential equations with fractional derivative boundary condition. Siberian Electron. Math. Rep. 18(2), 1596–1614 (2021). https://doi.org/10.33048/semi.2021.18.118

    Article  MathSciNet  Google Scholar 

  32. Bhairat, S.P., Samei, M.E.: Non-existence of a global solution for Hilfer-Katugampola fractional differential problem. Partial Diff. Equ. Appl. Math. 2023, 100495 (2023). https://doi.org/10.1016/j.padiff.2023.100495

    Article  Google Scholar 

  33. Ismail, M., Saeed, U., Alzabut, J., Rehman, M.: Approximate solutions for fractional boundary value problems via Green-cas wavelet method. Mathematics 7(12), 1164 (2021). https://doi.org/10.3390/math7121164

    Article  Google Scholar 

  34. Kumar, A., Chauhan, H.V.S., Ravichandran, C., Nisar, K.S., Baleanu, D.: Existence of solutions of non-autonomous fractional differential equations with integral impulse condition. Adv. Difference Equ. 2020, 434 (2020). https://doi.org/10.1186/s13662-020-02888-3

    Article  MathSciNet  Google Scholar 

  35. Vivek, D., Kanagarajan, K., Elsayed, E.M.: Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions. Mediterr. J. Math. 15, 15 (2018). https://doi.org/10.1007/s00009-017-1061-0

    Article  MathSciNet  Google Scholar 

  36. Nieto, J.J., Chen, F., Zhou, Y.: Global attractivity for nonlinear fractional differential equations. Nonlinear Anal. Real World Appl. 13(1), 287–298 (2012). https://doi.org/10.1016/j.nonrwa.2011.07.034

    Article  MathSciNet  Google Scholar 

  37. Furati, K.M., Kassim, N.D., Tatar, N.E.: Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput. Math. Appl. 64(6), 1616–1626 (2012). https://doi.org/10.1016/j.camwa.2012.01.009

    Article  MathSciNet  Google Scholar 

  38. Hajiseyedazizi, S.N., Samei, M.E., Alzabut, J., Chu, Y.: On multi-step methods for singular fractional \(q\)-integro-differential equations. Open Math. 19, 1378–1405 (2021). https://doi.org/10.1515/math-2021-0093

    Article  MathSciNet  Google Scholar 

  39. Harikrishman, S., Elsayed, E., Kanagarajan, K.: Existence and uniqueness results for fractional pantograph equations involving \(\psi \)-Hilfer fractional derivative. Dyn. Contin. Discrete Impuls. Syst. A Math. Anal. 25(5), 319–328 (2018)

    MathSciNet  Google Scholar 

  40. Rezapour, S., Etemad, S., Tellab, B., Agarwal, P., Guirao, J.L.G.: Numerical solutions caused by DGJIM and ADM methods for multiterm fractional BVP involving the generalized \(\psi \)-RL-operators. Symmetry 13(4), 532 (2021). https://doi.org/10.3390/sym13040532

    Article  Google Scholar 

  41. Thaiprayoon, C., Sudsutad, W., Alzabut, J., Etemad, S., Rezapour, S.: On the qualitative analysis of the fractional boundary value problem describing thermostat control model via \(\psi \)-Hilfer fractional operator. Adv. Differ. Equ. 2021, 201 (2021). https://doi.org/10.1186/s13662-021-03359-z

    Article  MathSciNet  Google Scholar 

  42. Kotsamran, K., Sudsutad, W., Thaiprayoon, C., Kongson, J., Alzabut, J.: Analysis of a nonlinear \(\psi \)-Hilfer fractional integro-differential equation describing cantilever beam model with nonlinear boundary conditions. Fractal Fract. 5(4), 177 (2021). https://doi.org/10.3390/fractalfract5040177

    Article  Google Scholar 

  43. Ntouyas, S.K., Vivek, D.: Existence and uniqueness results for sequential \(\psi \)-Hilfer fractional differential equations with multi-point boundary conditions. Acta Math. Univ. Comenianae XC(2), 171–185 (2021)

  44. Sousa, J.V.C., de Oliveira, E.C.: On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the \(\psi \)-Hilfer operator. J. Fixed Point Theory Appl. 20(3), 96 (2018)

    Article  MathSciNet  Google Scholar 

  45. Lima, K.B., Sousa, J.V.C., Oliveira, E.C.: Ulam-Hyers type stability for \(\psi \)-Hilfer fractional differential equations with impulses and delay. Comput. Appl. Math. 40(8), 293 (2021). https://doi.org/10.1007/s40314-021-01686-1

    Article  MathSciNet  Google Scholar 

  46. Khan, A., Khan, Z., Abdeljawad, T., Khan, H.: Analytical analysis of fractional-order sequential hybrid system with numerical application. Adv. Contin. Discrete Models 2022, 12 (2022). https://doi.org/10.1186/s13662-022-03685-w

    Article  MathSciNet  Google Scholar 

  47. Vanterler, J., de Oliveira, E.C.: On a fixed point principle. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018). https://doi.org/10.1016/j.cnsns.2018.01.005

    Article  MathSciNet  Google Scholar 

  48. Smart, D.R.: Fixed Point Theorems. Cambridge University Press, London-New York (1974)

    Google Scholar 

  49. Kuratowski, K.: Sur les espaces complets. Fund. Math. 15, 301–309 (1930)

    Article  Google Scholar 

  50. Banas, J., Goebel, K.: Measure of Noncompactness in Banach Spaces. Marcel Dekker, New York (1980)

    Google Scholar 

  51. Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2005)

    Google Scholar 

  52. Zeidler, E.: Nonlinear Functional Analysis and its Applications. Springer, Springer (1986)

    Book  Google Scholar 

  53. Sadovskii, B.N.: On a fixed point principle. Funct. Anal. Appl. 1967(1), 151–153 (1967). https://doi.org/10.1007/BF01076087

    Article  Google Scholar 

  54. Burton, T.A., Kirk, C.: A fixed point theorem of Krasnoselskii-Schaefer type. Math. Nachr. 189(1), 23–31 (1998). https://doi.org/10.1002/mana.19981890103

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, University of Sciences and Technology of Oran-MB, Oran, Algeria

    Faouzi Haddouchi

  2. Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Oran, Algeria

    Faouzi Haddouchi

  3. Department of Mathematics, Faculty of Basic Science, Bu-Ali Sina University, Hamedan, 65178-38695, Iran

    Mohammad Esmael Samei

  4. Department of Mathematics, Kyuing Hee University, 26 Kyungheedae-ro Dongdaemun-gu, Seoul, Republic of Korea

    Shahram Rezapour

  5. Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

    Shahram Rezapour

  6. Department of Medical Research, China Medical University Hospital China Medical University, Taichung, Taiwan

    Shahram Rezapour

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  1. Faouzi Haddouchi
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  2. Mohammad Esmael Samei
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  3. Shahram Rezapour
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Contributions

FH: Actualization, methodology, formal analysis, validation, investigation, initial draft and was a major contributor in writing the manuscript. MES: Actualization, methodology, formal analysis, validation, investigation, software, simulation, initial draft and was a major contributor in writing the manuscript. SR: Actualization, methodology, formal analysis, validation, investigation and initial draft. All authors read and approved the final manuscript.

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Correspondence to Mohammad Esmael Samei or Shahram Rezapour.

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Haddouchi, F., Samei, M.E. & Rezapour, S. Study of a sequential \(\psi \)-Hilfer fractional integro-differential equations with nonlocal BCs. J. Pseudo-Differ. Oper. Appl. 14, 61 (2023). https://doi.org/10.1007/s11868-023-00555-1

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