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On the Boundary Value Problems of Ψ-Hilfer Fractional Differential Equations

Ashwini Dhondiram Mali|Kishor Deoman Kucche
경북대학교 자연과학대학 수학과|2024.12|Kyungpook Mathematical Journal|제64권 제4호|pp. 669-702|DOI 10.5666/KMJ.2024.64.4.669

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키워드

Existence and uniquenessBoundary value problemsΨ-Hilfer fractional derivativeUpper and lower solutionsExtremal solutionsMonotone iterative technique
주제분류수학

초록

In the present paper, we derive comparison results for the linear initial value problem (IVP) of Ψ-Hilfer fractional differential equations. With the assistance of the comparison results and the location of roots theorem the existence and uniqueness of the solution for the linear Ψ-Hilfer boundary value problem (BVP) is obtained via linear IVP. We prove there exist minimal and maximal solutions of a nonlinear Ψ-Hilfer BVP in the line segment joining the lower solution and upper solution. For nonlinear Ψ-Hilfer BVP, the existence of extremal and exact solutions are acquired through iterative Picard type sequences. Examples are provided in support of the obtained results.

Abstract

In the present paper, we derive comparison results for the linear initial value problem (IVP) of Ψ-Hilfer fractional differential equations. With the assistance of the comparison results and the location of roots theorem the existence and uniqueness of the solution for the linear Ψ-Hilfer boundary value problem (BVP) is obtained via linear IVP. We prove there exist minimal and maximal solutions of a nonlinear Ψ-Hilfer BVP in the line segment joining the lower solution and upper solution. For nonlinear Ψ-Hilfer BVP, the existence of extremal and exact solutions are acquired through iterative Picard type sequences. Examples are provided in support of the obtained results.

참고문헌 (30)

  1. X. Su, "Positive solutions to singular boundary value problems for fractional functional differential equations with changing sign nonlinearity", Computers & Mathematics with Applications, 64, pp.3425~3435 (2012)
  2. Y. Arioua, "New class of boundary value problem for nonlinear fractional differential equations involving Erd´elyi-Kober derivative", Communications in Mathematics, 27, pp.113~141 (2019)
  3. R. Ma, "Positive solutions for nonlinear m-point boundary value problems of Dirichlet type via fixed-point index theory", Applied Mathematics Letters, 16, pp.863~869 (2003)
  4. Z. Bai, "On positive solutions of a nonlocal fractional boundary value problem", Nonlinear Analysis: Theory, Methods & Applications, 72, pp.916~924 (2010)
  5. X. Hao, "Existence and multiplicity results for nonlinear periodic boundary value problems", Nonlinear Analysis: Theory, Methods & Applications, 72, pp.3635~3642 (2010)
  6. H. Gou, "Existence of mild solutions for Sobolev-type Hilfer fractional evolution equations with boundary conditions", Boundary Value Problems, 2018, pp.48~ (2018)
  7. Z. Baitiche, "Weak solutions for nonlinear fractional differential equations with integral and multi-point boundary conditions", PanAmerican Mathematical Journal, 29, pp.86~100 (2019)
  8. H. Gou, "Study on the mild solution of Sobolev type Hilfer fractional evolution equations with boundary conditions", Chaos, Solitons & Fractals, 112, pp.168~179 (2018)
  9. X. Liu, "Multiple solutions for fractional differential equations with nonlinear boundary conditions", Computers & Mathematics with Applications, 59, pp.2880~2886 (2010)
  10. Y. Zhao, "The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations", Commun. Nonlinear Sci. Numer. Simulat, 16, pp.2086~2097 (2011)
  11. T. Long, "Existence of positive solutions for period BVPs with Hilfer derivative", Journal of Applied Mathematics and Computing, 60, pp.223~236 (2019)
  12. G. S. Ladde, "Monotone iterative techniques for nonlinear differential equations" (1985), Pitman
  13. F. Li, "Existence and uniqueness of solutions of secend-order three-point boundary value problems with upper and lower solutions in the reverse order", Nonlinear Analysis: Theory, Methods & Applications, 68, pp.2381~2388 (2008)
  14. F. Li, "Monotone iterative method for the second-order three-point boundary value problem with upper and lower solutions in the reversed order", Applied Mathematics and Computation, 217, pp.4840~4847 (2011)
  15. G. Wang, "Monotone iterative technique for boundary value problems of a nonlinear fractional differential equation with deviating arguments", Journal of Computational and Applied Mathematics, 236, pp.2425~2430 (2012)
  16. L. Lin, "Method of upper and lower solutions for fractional differential equations", Electronic Journal of Differential Equations, 2012, pp.1~13 (2012)
  17. R. Hilfer, "Applications of fractional calculus in Physics" (2000), World Scientific
  18. K. M. Furati, "Existence and uniqueness for a problem involving Hilfer fractional derivative", Computers & Mathematics with Applications, 64(6), pp.1616~1626 (2012)
  19. J. Wang, "Nonlocal initial value problems for differential equations with Hilfer fractional derivative", Applied Mathematics and Computation, 266, pp.850~859 (2015)
  20. A. A. Kilbas, "Theory and applications of fractional differential equations" (2006), Elsevier
  21. R. Almeida, "A Caputo fractional derivative of a function with respect to another function", Commun. Nonlinear Sci. Numer. Simulat., 44, pp.460~481 (2017)
  22. J. Vanterler da C. Sousa, "On the ψ-Hilfer fractional derivative", Commun. Nonlinear Sci. Numer. Simul., 60, pp.72~91 (2018)
  23. J. Vanterler da C. Sousa, "A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator", Diff. Equ. Appl., 11, pp.87~106 (2019)
  24. K. D. Kucche, "On the nonlinear Ψ-Hilfer fractional differential equations", Computational and Applied Mathematics, 38, pp.73~ (2019)
  25. K. D. Kucche, "Mali, Initial time difference quasilinearization method for fractional differential equations involving generalized Hilfer fractional derivative", Computational and Applied Mathematics, 39, pp.31~ (2020)
  26. A. D. Mali, "Nonlocal boundary value problem for generalized Hilfer implicit fractional differential equations", Mathematical Methods in the Applied Sciences, 43, pp.8608~8631 (2020)
  27. K. D. Kucche, "On the nonlinear Ψ-Hilfer hybrid fractional differential equations", Computational and Applied Mathematics, 41, pp.86~ (2022)
  28. R. G. Bartle, "Introduction to real analysis" (2000), Wiley
  29. U. N. Katugampola, "A new approach to generalized fractional derivatives", Applied Mathematics and Computation, 218, pp.860~865 (2011)
  30. K. Diethelm, "The analysis of fractional differential equations" (2010), Springer-verlag

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