Fractional tempered differential equations depending on arbitrary kernels

Ricardo Almeida; Natália Martins; J. Vanterler da C. Sousa; Ricardo Almeida; Natália Martins; J. Vanterler da C. Sousa

doi:10.3934/math.2024443

AIMS Mathematics

2024, Volume 9, Issue 4: 9107-9127. doi: 10.3934/math.2024443

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Fractional tempered differential equations depending on arbitrary kernels

1.
Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro, Portugal
2.
Aerospace Engineering, PPGEA-UEMA, Department of Mathematics, DEMATI-UEMA, São Luís, MA 65054, Brazil

Received: 18 December 2023 Revised: 21 February 2024 Accepted: 29 February 2024 Published: 05 March 2024
MSC : 26A33, 34D20

In this paper, we expanded the concept of tempered fractional derivatives within both the Riemann-Liouville and Caputo frameworks, introducing a novel class of fractional operators. These operators are characterized by their dependence on a specific arbitrary smooth function. We then investigated the existence and uniqueness of solutions for a particular class of fractional differential equations, subject to specified initial conditions. To aid our analysis, we introduced and demonstrated the application of Picard's iteration method. Additionally, we utilized the Gronwall inequality to explore the stability of the system under examination. Finally, we studied the attractivity of the solutions, establishing the existence of at least one attractive solution for the system. Throughout the paper, we provide examples and remarks to support and reinforce our findings.
- fractional differential equations,
- tempered fractional derivatives,
- existence,
- uniqueness,
- attractivity
Citation: Ricardo Almeida, Natália Martins, J. Vanterler da C. Sousa. Fractional tempered differential equations depending on arbitrary kernels[J]. AIMS Mathematics, 2024, 9(4): 9107-9127. doi: 10.3934/math.2024443

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Abstract

In this paper, we expanded the concept of tempered fractional derivatives within both the Riemann-Liouville and Caputo frameworks, introducing a novel class of fractional operators. These operators are characterized by their dependence on a specific arbitrary smooth function. We then investigated the existence and uniqueness of solutions for a particular class of fractional differential equations, subject to specified initial conditions. To aid our analysis, we introduced and demonstrated the application of Picard's iteration method. Additionally, we utilized the Gronwall inequality to explore the stability of the system under examination. Finally, we studied the attractivity of the solutions, establishing the existence of at least one attractive solution for the system. Throughout the paper, we provide examples and remarks to support and reinforce our findings.

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