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\(\psi \)-Mittag–Leffler pseudo-fractional operators

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Abstract

In this paper we present new types of pseudo-fractional integral and derivatives on a semiring \(([a,b],\oplus ,\odot )\). These operators are called the \(\psi \)-Riemann–Liouville–Mittag–Leffler pseudo-fractional integrals and derivatives and \(\psi \)-Caputo–Mittag–Leffler pseudo-fractional derivatives. Some properties, like the semigroup law, composition relations between pseudo-fractional differentiation and pseudo-fractional integration operators, composition relations between pseudo-fractional derivatives, among others, are discussed. Finally, as an application, we proved two versions of the fundamental theorem of fractional calculus.

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Acknowledgements

The author is grateful to Prof. E. Capelas de Oliveira for useful and fruitful discussions.

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  1. Universidade Tecnológica Federal do Paraná, Campus Guarapuava, Guarapuava, Paraná, 85053-525, Brazil

    D. S. Oliveira

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Oliveira, D.S. \(\psi \)-Mittag–Leffler pseudo-fractional operators. J. Pseudo-Differ. Oper. Appl. 12, 40 (2021). https://doi.org/10.1007/s11868-021-00412-z

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