On tempered Prabhakar fractional calculus
Keywords:
fixed point theory, successive approximation, tempered fractional calculus, Prabhakar fractional calculus, representation of solution, Ulam-type stabilityAbstract
The prime objective of the current paper is to continue improving the theory of tempered fractional integral and derivatives of a function. This theory combines the tempered fractional calculus with the Prabhakar calculus. Subsequent to working on the semigroup property for the tempered Prabhakar integrals, we investigate some ties and inverse properties for the tempered Prabhakar fractional derivatives of both Riemann-Liouville and Caputo types with the tempered Prabhakar integrals. Moreover, we examine the Cauchy problem for fractional differential equations with the Caputo type of the Prabhakar derivative, showing the existence uniqueness results by using both the Schaefer fixed point theorem and the Banach fixed point theorem as well as the method of successive approximations, making stability analysis in the Ulam-Hyers'sense. A couple of numerical examples are presented to illustrate the results.
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