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Some new results for \(\psi -\)Hilfer fractional pantograph-type differential equation depending on \(\psi -\)Riemann–Liouville integral

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Abstract

The aim of the present work is to study a large class of \(\psi -\)Hilfer fractional differential equation of Pantograph-type depending on \(\psi -\)Riemann–Liouville fractional integral operator associated with periodic-type fractional integral boundary conditions in a weighted space of continuous functions. We shall prove the existence and uniqueness results by means of Mawhin’s coincidence degree theory. At the end, an illustrative example will be constructed to approve our findings.

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References

  1. Abbas, S., M. Benchohra, and G.M. N’Guérékata. 2012. Topics in fractional differential equations. New York: Springer-Verlag.

  2. Abbas, S., M. Benchohra, and G.M. N’Guérékata. 2014. Advanced fractional differential and integral equations. New York: Nova Science Publishers.

  3. Almalahi, M.A., and S.K. Panchal. 2020. Existence results of \(\psi -\)Hilfer integro-differential equations with fractional order in Banach space. Ann. Univ. Paedagog. Crac. Stud. Math. 19: 171–192.

    MathSciNet  Google Scholar 

  4. Abdeljawad, A., R.P. Agarwal, E. Karapinar, and P.S. Kumari. 2019. Solutions of he Nonlinear Integral Equation and Fractional Differential Equation Using the Technique of a Fixed Point with a Numerical Experiment in Extended b-Metric Space. Symmetry 11: 686.

    Article  Google Scholar 

  5. Adiguzel, R.S., U. Aksoy, E. Karapinar and ý.M. Erhan. On the solution of a boundary value problem associated with a fractional differential equation mathematical methods in the applied sciences https://doi.org/10.1002/mma.665.

  6. R.S. Adiguzel, Aksoy, U., Karapinar, E. and ý.M. Erhan. 2021. On The solutions of fractional differential equations via geraghty type hybrid contractions. Appl. Comput. Math., 20(2), .

  7. Afshari, H. 2020. E, Karapınar, A discussion on the existence of positive solutions of the boundary value problems via ?-Hilfer fractional derivative on b-metric spaces advances in difference equations volume 2020. Article number 616.

  8. Afshari, H., S. Kalantari, and E. Karapinar. 2015. Solution of fractional differential equations via coupled fixed point. Electronic Journal of Differential Equations 2015 (286): 1–12.

    MathSciNet  MATH  Google Scholar 

  9. Alqahtani, B., H. Aydi, E. Karapınar, and V. Rakocevic. 2019. A solution for volterra fractional integral equations by hybrid contractions. Mathematics 7: 694.

    Article  Google Scholar 

  10. Benchohra, M., S. Bouriah, and J. Henderson. 2015. Existence and stability results for nonlinear implicit neutral fractional differential equations with finite delay and impulses. Comm. Appl. Nonlinear Anal. 22 (1): 46–67.

    MathSciNet  MATH  Google Scholar 

  11. Benchohra, M., and S. Bouriah. 2015. Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order. Moroccan J. Pure. Appl. Anal. 1 (1): 22–36.

    Article  Google Scholar 

  12. Benchohra, M., S. Bouriah, and J.. J. Nieto. 2018. Existence of periodic solutions for nonlinear implicit Hadamard fractional differential equations. Rev. R. Acad. Cienc. Exactas Fsí. Nat. Ser. A Mat. RACSAM 112 (1): 25–35.

    Article  MathSciNet  Google Scholar 

  13. Benchohra, M., S. Bouriah, and J.. R. Graef. 2016. Nonlinear implicit differential equation of fractional order at resonance. Electron. J. Differential Equations 2016 (324): 1–10.

    MathSciNet  MATH  Google Scholar 

  14. Bouguetof, K., and D. Foukrach. 2020. On local existence and blow-up solutions for a time-space fractional variable order superdiffusion equation with exponential nonlinearity. PanAmer. Math. J. 30 (3): 21–34.

    Google Scholar 

  15. Foukrach, D., T. Moussaoui, and S.K. Ntouyas. 2013. Boundary value problems for a class of fractional differential equations depending on first derivative. Commun. Math. Anal. 15 (2): 15–28.

    MathSciNet  MATH  Google Scholar 

  16. Foukrach, D., T. Moussaoui, and S.K. Ntouyas. 2015. Existence and uniqueness results for a class of BVPs for nonlinear fractional differential equations. Georgian Math. J. 22 (1): 45–55.

    Article  MathSciNet  Google Scholar 

  17. Gaines, R.E., and J. Mawhin. 1977. Coincidence degree and nonlinear differential equations, vol. 568. Lecture Notes in Math. Berlin: Springer-Verlag.

  18. Herrmann, R. 2011. Fractional calculus: an introduction for physicists. Singapore: World Scientific Publishing Company.

    Book  Google Scholar 

  19. Hilfer, R. 2000. Applications of fractional calculus in physics. Singapore: World Scientific.

    Book  Google Scholar 

  20. Jalilian, Y., and M. Ghasmi. 2017. On the solutions of a nonlinear fractional integro-differential equation of pantograph type. Mediter. J. Math. 14: 194.

    Article  MathSciNet  Google Scholar 

  21. Karapinar, E., A. Fulga, M. Rashid, L. Shahid, and H. Aydi. 2019. Large contractions on quasi-metric spaces with an application to nonlinear fractional differential-equations. Mathematics 7: 444.

    Article  Google Scholar 

  22. Karapinar, E. 2021. Ho Duy Binh, Nguyen Hoang Luc, and Nguyen Huu Can On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems. Advances in difference equations 2021: 70. https://doi.org/10.1186/s13662-021-03232-z.

  23. Karapinar, E., T. Abdeljawad, and F. Jarad. 2019. Applying new fixed point theorems on fractional and ordinary differential equations. Advances in Difference Equations 2019: 421.

    Article  MathSciNet  Google Scholar 

  24. Kilbas, A.. A., H.. M. Srivastava, and Juan J.. Trujillo. 2006. Theory and applications of fractional differential equations. North-Holland mathematics studies, vol. 204. Amsterdam: Elsevier Science B.V.

    Google Scholar 

  25. Lazreg, Jamal Eddine. 2021. Saïd Abbas. Mouffak Benchohra, and Erdal Karapinar Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces, Open Mathematics 19: 363–372. https://doi.org/10.1515/math-2021-0040.

    Article  MathSciNet  Google Scholar 

  26. Mawhin, J. 1979. Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics, vol. 40. Providence, R.I.: American Mathematical Society.

    Book  Google Scholar 

  27. Ockendon, J.R., and A.B. Taylor. 1971. The dynamics of a current collection system for an electric locomotive. Proc RSoc London Ser. A 322: 447–468.

    Google Scholar 

  28. O’Regan, D., Y.J. Chao, and Y.Q. Chen. 2006. Topological Degree Theory and Application. Boca Raton, London, New York: Taylor and Francis Group.

  29. Samko, S.G., A.A. Kilbas, and O.I. Marichev. 1993. Fractional integrals and derivatives. Gordon and breach, Yverdon: Theory and applications.

  30. Salim, A., B. Benchohra, E. Karapinar, and J.E. Lazreg. 2020. Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations. Adv Differ Equ 2020: 601.

    Article  MathSciNet  Google Scholar 

  31. Shah, K., D. Vivek, and K. Kanagarajan. 2021. Dynamics and stability of \(\psi \)-fractional pantograph equations with boundary conditions. Bol. Soc. Paran. Mat. 39 (5): 43–55.

    Article  MathSciNet  Google Scholar 

  32. Sudsutad, W., C. Thaiprayoon, and S.K. Ntouyas. Existence and stability results for \(\psi -\)Hilfer fractional integro-differential equation with mixed nonlocal boundary conditions. AIMS Mathematics 6 (4): 4119–4141.

  33. C. Thaiprayoon1, Sudsutad, W., Alzabut, J., Etemad, S., and S. Rezapour. 2021. On the qualitative analysis of the fractional boundary value problem describing thermostat control model via \(\psi -\)Hilfer fractional operator, Adv. Differ. Equ. 2021:201.

  34. Vivek, D., E.. M.. Elsayed, and K. Kanagarajan. 2020. Existence and uniqueness results for \(\psi -\)fractional integro-differential equations with boundary conditions. Publ. Inst. Math. (Beograd) (N.S.) 107 (121): 145–155.

    Article  MathSciNet  Google Scholar 

  35. Vanterler, J., C. da Sousa, and E. Capelas de Oliveira. 2018. On the \(\psi -\)Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simulat. 60: 72–91.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Exact Sciences and Informatics, University Hassiba Benbouali of Chlef, Chlef, Algeria

    Djamal Foukrach & Soufyane Bouriah

  2. Laboratory of Mathematics, University of Sidi Bel-Abbes, P.O. Box 89, 22000, Sidi Bel-Abbes, Algeria

    Mouffak Benchohra

  3. Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Ho Chi Minh, Vietnam

    Erdal Karapinar

  4. Department of Mathematics, Çankaya University, 06790, Etimesgut, Ankara, Turkey

    Erdal Karapinar

  5. Department of Medical Research, China Medical University Hospital, China Medical University, 40402, Taichung, Taiwan

    Erdal Karapinar

Authors
  1. Djamal Foukrach
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  2. Soufyane Bouriah
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  3. Mouffak Benchohra
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  4. Erdal Karapinar
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Correspondence to Erdal Karapinar.

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Foukrach, D., Bouriah, S., Benchohra, M. et al. Some new results for \(\psi -\)Hilfer fractional pantograph-type differential equation depending on \(\psi -\)Riemann–Liouville integral. J Anal 30, 195–219 (2022). https://doi.org/10.1007/s41478-021-00339-0

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  • DOI: https://doi.org/10.1007/s41478-021-00339-0

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