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Fractional boundary value problem with \(\varvec{\psi }\)-Caputo fractional derivative

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Abstract

This paper is concerned with a boundary value problem for a nonlinear fractional differential equation involving a general form of Caputo fractional derivative operator with respect to new function \(\psi \). The existence and uniqueness results of solutions are obtained. Our analysis relies on a variety of tools of fractional calculus together with fixed point theorems of Banach and Schaefer. The investigation of the results will be illustrated by providing a suitable example.

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References

  1. Abdo M S and Panchal S K, Existence and continuous dependence for fractional neutral functional differential equations, J. Mathematical Model.  5 (2017) 153–170

    MathSciNet  MATH  Google Scholar 

  2. Abdo M S and Panchal S K, Fractional integro-differential equations involving \(\psi \)-Hilfer fractional derivative, Adv. Appl. Math. Mech.  11 (2019) 338–359

    Article  MathSciNet  Google Scholar 

  3. Abdo M S, Ibrahim A G and Panchal S K, Nonlinear implicit fractional differential equation involving \(\psi \)-Caputo fractional derivative, Proc. Jangjeon Math. Soc. 22(3) (2019) (will appear soon)

  4. Abdo M S and Panchal S K, Weighted fractional neutral functional differential equations, J. Sib. Fed. Univ. Math. Phys.  11 (2018) 535–549

    Article  Google Scholar 

  5. Agarwal R, Hristova S and O’Regan D, A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations, Fract. Calc. Appl. Anal.  19 (2016) 290–318

    Article  MathSciNet  Google Scholar 

  6. Almeida R, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul.  44 (2017) 460–481

    Article  MathSciNet  Google Scholar 

  7. Almeida R, Fractional differential equations with mixed boundary conditions, Bull. Malays. Math. Sci. Soc. 42(4) (2019) 1687–1697

    Article  MathSciNet  Google Scholar 

  8. Almeida R, Malinowska A B and Monteiro M T, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Meth. Appl. Sci.  41 (2018) 336–352

    Article  MathSciNet  Google Scholar 

  9. Atanackovic T M, Pilipovic S, Stankovic B and Zorica D, Fractional calculus with applications in mechanics: wave propagation, impact and variational principles (2014) (London: Wiley)

    Book  Google Scholar 

  10. Al-Saqabi B and Kiryakova V S, Explicit solutions of fractional integral and differential equations involving Erdelyi–Kober operators, Appl. Math. Comput.  95 (1998) 1–13

    Article  MathSciNet  Google Scholar 

  11. Delboso D and Rodino L, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl.  204 (1996) 609–625

    Article  MathSciNet  Google Scholar 

  12. Gaul L, Klein P and Kempfle S, Damping description involving fractional operators, Mech. Systems Signal Processing  5 (1991) 81–88

    Article  Google Scholar 

  13. Garra R, Gorenflo R, Polito F and Tomovski Z, Hilfer–Prabhakar derivatives and some applications, Appl. Math. Comput.  242 (2014) 576–589

    MathSciNet  MATH  Google Scholar 

  14. Glockle W G and Nonnenmacher T F, A fractional calculus approach of self-similar protein dynamics, Biophys J.  68 (1995) 46–53

    Article  Google Scholar 

  15. Hilfer R, Applications of Fractional Calculus in Physics (2000) (Singapore: World Scientific)

    Book  Google Scholar 

  16. Katugampola U N, New fractional integral unifying six existing fractional integrals, arXiv preprint arXiv:1612.08596 (2016) 6 pages

  17. Kilbas A A, Srivastava H M and Trujillo J J, Theory and applications of fractional differential equations, North-Holland mathematics studies, vol. 204 (2006) (Amsterdam: Elsevier)

    Google Scholar 

  18. Kou C, Liu J and Ye Y, Existence and uniqueness of solutions for the Cauchy-type problems of fractional differential equations, Discr. Dyn. Nature Soc.  2010 (2010) 1–15

    Article  MathSciNet  Google Scholar 

  19. Kucche K D, Mali A D, Sousa J V, Theory of nonlinear \( \psi \)-Hilfer fractional differential equations, arXiv preprint, arXiv:1808.01608 (2018) 26 pages

  20. Li M and Wang J, Existence of local and global solutions for Hadamard fractional differential equations, Electron. J. Differ. Equ.  2015 (2015) 1–8

    Article  MathSciNet  Google Scholar 

  21. Magin R L, Fractional calculus in bioengineering (2006) (Danbury: Begell House Inc. Publisher)

    Google Scholar 

  22. Oliveira D S and de Oliveira E C, Hilfer–Katugampola fractional derivatives, Comput. Appl. Math. 73(3) (2018) 3672–3690

    Article  MathSciNet  Google Scholar 

  23. Panchal S K, Khandagale A D, Dole P V, \(k\)-Hilfer–Prabhakar fractional derivatives and applications, arXiv preprint, arXiv:1609.05696 (2016) 18 pages

  24. Samko S G, Kilbas A A and Marichev O I, Fractional Integrals and Derivatives, Theory and Applications (1993) (Yverdon: Gordon and Breach)

    MATH  Google Scholar 

  25. Sousa J V and de Oliveira E C, A Gronwall inequality and the Cauchy-type problem by means of \(\psi \)-Hilfer operator, arXiv preprint, arXiv:1709.03634 (2017) 19 pages

  26. Sousa J V and de Oliveira E C, On the Ulam–Hyers–Rassias stability for nonlinear fractional differential equations using the \(\psi \)-Hilfer operator, J. Fix. Point Theory Applic.  20 (2018) 96 pages

    Article  MathSciNet  Google Scholar 

  27. Sousa J V and de Oliveira E C, On two new operators in fractional calculus and application, arXiv preprint, arXiv:1710.03712 (2017)

  28. Sun Y, Zeng Z and Song J, Existence and uniqueness for the boundary value problems of nonlinear fractional differential equation, Appl. Math.  8 (2017) 312

    Article  Google Scholar 

  29. Wang J, Dong X and Zhou Y, Analysis of nonlinear integral equations with Erdelyi–Kober fractional operator, Commun. Nonlinear Sci. Numer. Simul.  17 (2012) 3129–3139

    Article  MathSciNet  Google Scholar 

  30. Wang J, Zhou Y and Medved M, Existence and stability of fractional differential equations with Hadamard derivative, Topol. Methods Nonlinear Anal.  41 (2013) 113–133

    MathSciNet  MATH  Google Scholar 

  31. Xu Y, Fractional boundary value problems with integral and anti-periodic boundary conditions, Bull. Malays. Math. Sci. Soc.  39 (2016) 571–587

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for their careful reading of the manuscript and insightful comments, which helped improve the quality of the paper. The authors would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to the improvement of the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, 431 004, India

    Mohammed S Abdo & Satish K Panchal

  2. Department of Mathematics, College of Science, Qassim University, Buraydah, 51452, Saudi Arabia

    Abdulkafi M Saeed

Authors
  1. Mohammed S Abdo
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  2. Satish K Panchal
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  3. Abdulkafi M Saeed
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Correspondence to Mohammed S Abdo.

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Communicating Editor: Mythily Ramaswamy

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Abdo, M.S., Panchal, S.K. & Saeed, A.M. Fractional boundary value problem with \(\varvec{\psi }\)-Caputo fractional derivative. Proc Math Sci 129, 65 (2019). https://doi.org/10.1007/s12044-019-0514-8

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  • DOI: https://doi.org/10.1007/s12044-019-0514-8

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