Abstract
The generalized Caputo fractional derivative is a name attributed to the Caputo version of the generalized fractional derivative introduced in Jarad et al. (J. Nonlinear Sci. Appl. 10:2607–2619, 2017). Depending on the value of ρ in the limiting case, the generality of the derivative is that it gives birth to two different fractional derivatives. However, the existence and uniqueness of solutions to fractional differential equations with generalized Caputo fractional derivatives have not been proven. In this paper, Cauchy problems for differential equations with the above derivative in the space of continuously differentiable functions are studied. Nonlinear Volterra type integral equations of the second kind corresponding to the Cauchy problem are presented. Using Banach fixed point theorem, the existence and uniqueness of solution to the considered Cauchy problem is proven based on the results obtained.
Highlights
1 Introduction The fractional calculus is the branch of mathematics that studies the integration and differentiation of real or complex orders
The most interesting speciality of the fractional operators is that there are many of these operators
This enables a researcher to choose the most suitable operator in order to describe the dynamics in a real world problem
Summary
The fractional calculus is the branch of mathematics that studies the integration and differentiation of real or complex orders. This paper studies fractional Cauchy problems with left generalized Caputo fractional derivatives in the space of continuously differentiable functions and proves the existence and uniqueness of solutions to these problems. The right Caputo fractional derivative of order α, (α) ≥ 0 reads as follows: CDαb f (t) = Ibn–α(–1)nf (n) (t). The right Hadamard fractional derivative of order α, (α) ≥ 0 reads as follows: Dbαf (t) =. The generalized left and right fractional integrals of order α, (α) > 0 are defined in [10] as aIα,ρ f (t) =. The left and right generalized fractional derivatives of order α, (α) ≥ 0 are defined by (see [11]). Caputo fractional derivatives of order α, (α) > 0 are given respectively as in [1] as follows: Dα,ρ f (n – α) a tρ – uρ ρ n–α–1.
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