On the \(\Psi \)-fractional integral and applications

  • J. Vanterler da C. SousaEmail author
  • E. Capelas de Oliveira


Motivated by the \(\Psi \)-Riemann–Liouville \((\Psi -\mathrm{RL})\) fractional derivative and by the \(\Psi \)-Hilfer \((\Psi -\mathrm{H})\) fractional derivative, we introduced a new fractional operator the so-called \(\Psi \)-fractional integral. We present some important results by means of theorems and in particular, that the \(\Psi \)-fractional integration operator is limited. In this sense, we discuss some examples, in particular, involving the Mittag–Leffler \((\mathrm{M-L})\) function, of paramount importance in the solution of population growth problem, as approached. On the other hand, we realize a brief discussion on the uniqueness of nonlinear \(\Psi \)-fractional Volterra integral equation (\(\mathrm{VIE}\)) using \(\beta \)-distance functions.


Fractional calculus \(\Psi \)-fractional integral Population growth model Fractional Volterra integral equation \(\beta \)-distance 

Mathematics Subject Classification

26A33 33BXX 34A08 34K60 45D05 



We are grateful to the editor and anonymous referee for the suggestions that have improved the manuscript.


  1. Abbas S, Benchohra M, Mouffak, Graef JR, Henderson J (2011) Generalized double-integral Ostrowski type inequalities on time scales. Appl Math Lett 24(8):1461–1467MathSciNetCrossRefGoogle Scholar
  2. Abbas S, Benchohra Mouffak M, Graef JR, Henderson J (2018) Implicit fractional differential and integral equations: existence and stability, vol 26. Walter de Gruyter GmbH, MunichCrossRefGoogle Scholar
  3. Almeida R (2017a) A Caputo fractional derivative of a function with respect to another function. Commun Nonlinear Sci Numer Simul 44:460–481MathSciNetCrossRefGoogle Scholar
  4. Almeida R (2017b) Caputo-Hadamard fractional derivatives of variable order. Numer Funct Anal Optim 38(1):1–19MathSciNetCrossRefGoogle Scholar
  5. Almeida R, Bastos NR, Monteiro MTT (2016a) Modeling some real phenomena by fractional differential equations. Math Methods Appl Sci 39(16):4846–4855MathSciNetCrossRefGoogle Scholar
  6. Almeida R, Bastos N, Monteiro MTT (2016b) A fractional Malthusian growth model with variable order using an optimization approach. In: Proceedings da CMMSE, pp 51–54Google Scholar
  7. Crow JF, Kimura M (1970) An introduction to population genetics theory. Harper and Row, New YorkzbMATHGoogle Scholar
  8. Furati KM, Kassim M (2012) Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput. Math. Appl. 64(6):1616–1626MathSciNetCrossRefGoogle Scholar
  9. Gordji ME, Baghani H, Baghani O (2011) On existence and uniqueness of solutions of a nonlinear integral equation. J Appl Math 2011:743923MathSciNetzbMATHGoogle Scholar
  10. Gorenflo R, Kilbas AA, Mainardi F, Rogosin SV (2014) \({\rm M-L}\) functions, related topics and functions. Springer, BerlinzbMATHGoogle Scholar
  11. Herrmann R (2011) Fractional calculus: an introduction for physicists. World Scientific, SingaporeCrossRefGoogle Scholar
  12. Jumarie G (2006) New stochastic fractional models for malthusian growth, the poissonian birth process and optimal management of populations. Math. Comput. Model. 44(3):231–254MathSciNetCrossRefGoogle Scholar
  13. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. North-Holland Mathematics Studies, vol 207. Elsevier, AmsterdamGoogle Scholar
  14. Magin RL (2012) Fractional calculus in bioengineering. In: 2012 13th International Carpathian control conference, ICCC 2012Google Scholar
  15. Mainardi F (2010) Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. World Scientific, SingaporeCrossRefGoogle Scholar
  16. Maleknejad K, Nouri K, Mollapourasl R (2009) Existence of solutions for some nonlinear integral equations. Commun Nonlinear Sci Numer Simul 14(6):2559–2564MathSciNetCrossRefGoogle Scholar
  17. Malthus TR (1798) An essay on the principle of population. J. Johnson in St Paul’s Church-yard, LondonGoogle Scholar
  18. Manam SR (2011) Multiple integral equations arising in the theory of water waves. Appl Math Lett 24(8):1369–1373MathSciNetCrossRefGoogle Scholar
  19. Mittag-Leffler GM (1903) Sur la nouvelle fonction \(\mathbf{E}_{\mu }(x)\). C R Acad Sci Paris 137:554–558zbMATHGoogle Scholar
  20. Moradi S, Anjedani MM, Analoei E (2015) On existence and uniqueness of solutions of a nonlinear Volterra–Fredholm integral equation. Int J Nonlinear Anal Appl 6(1):62–68zbMATHGoogle Scholar
  21. O’Regan D, Meehan M (2012) Existence theory for nonlinear integral and integrodifferential equations, vol 445. Springer, New YorkzbMATHGoogle Scholar
  22. Oldham K, Spanier J (1974) The fractional calculus. Theory and applications of differentiation and integration to arbitrary order, vol 111. Academic Press, New YorkzbMATHGoogle Scholar
  23. Peng S, Wang J (2015) Existence and Ulam–Hyers stability of ODES involving two Caputo fractional derivatives. Electron. J. Qual. Theor. Differ. Equ. 2015(52):1–16MathSciNetCrossRefGoogle Scholar
  24. Ray SS (2015) Fractional calculus with applications for nuclear reactor dynamics. CRC Press, Boca RatonCrossRefGoogle Scholar
  25. Salim TO, Faraj AW (2012) A generalization of Mittag-Leffler function and integral operator associated with fractional calculus. J. Fract. Calc. Appl. 3:1–13CrossRefGoogle Scholar
  26. Samko SG (1995) Fractional integration and differentiation of variable order. Anal. Math. 21(3):213–236MathSciNetCrossRefGoogle Scholar
  27. Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives, theory and functions. Gordon and Breach, YverdonzbMATHGoogle Scholar
  28. Teerawat Wongyat MR (2016) The existence of solutions for Fredholm/Volterra equations and fractional differential equations via fixed point theorems using a \(\omega \)-distance functions, Thesis Mastership. Thammasat UniversityGoogle Scholar
  29. Teerawat Wongyat MR, Sintunavarat W (2017a) The existence and uniqueness of the solution for nonlinear Fredholm and \({\rm VIE}\)s together with nonlinear fractional differential equations via \(\omega \)-distances. Adv Differ Equ 1:211CrossRefGoogle Scholar
  30. Teerawat Wongyat MR, Sintunavarat W (2017b) The existence and uniqueness of the solution for nonlinear Fredholm and \({\rm VIE}\) via adapting-ceiling distances. J Math Anal 8(5):105–118MathSciNetzbMATHGoogle Scholar
  31. Vanterler da C. Sousa J, Capelas de Oliveira E (2017) A Gronwall inequality and the Cauchy-type problem by means of \(\psi \)-Hilfer operator. arXiv:1709.03634
  32. Vanterler da C. Sousa J, Capelas de Oliveira E (2018) On the \(\psi \)-Hilfer fractional derivative. Commun Nonlinear Sci Numer Simul 60:72–91Google Scholar
  33. Vanterler da C. Sousa J, Capelas de Oliveira E, Magna LA (2017) Fractional calculus and the ESR test. AIMS Math., pp 1–15Google Scholar
  34. Vanterler da C. Sousa J, dos Santos MNN, Magna LA, Capelas de Oliveira E (2018) Validation of a fractional model for erythrocyte sedimentation rate. Comput. Appl. Math.
  35. Wiman A (1905) Uber den fundamental satz in der theorie der funktionen \(\mathbf{E}_{\mu }(x)\). Acta Math 29:191–201MathSciNetCrossRefGoogle Scholar
  36. Xu Y, Agrawal OP (2016) New fractional operators and function to fractional variational problem. Comput. Math. with Appl.
  37. Zhou Y (2018) Attractivity for fractional differential equations in Banach space. Appl. Math. Lett. 75:1–6MathSciNetCrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  • J. Vanterler da C. Sousa
    • 1
    Email author
  • E. Capelas de Oliveira
    • 1
  1. 1.Department of Applied Mathematics, Institute of Mathematics, Statistics and Scientific ComputationImecc-Unicamp, CampinasCampinasBrazil

Personalised recommendations