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

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Abstract

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.

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Applied Mathematics, Institute of Mathematics, Statistics and Scientific Computation, Imecc-Unicamp, Campinas, rua Sérgio Buarque de Holanda 651, Campinas, SP, 13083-859, Brazil

    J. Vanterler da C. Sousa & E. Capelas de Oliveira

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  1. J. Vanterler da C. Sousa
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  2. E. Capelas de Oliveira
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Correspondence to J. Vanterler da C. Sousa.

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Communicated by Vasily E. Tarasov.

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Vanterler da C. Sousa, J., Capelas de Oliveira, E. On the \(\Psi \)-fractional integral and applications. Comp. Appl. Math. 38, 4 (2019). https://doi.org/10.1007/s40314-019-0774-z

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