Abstract
This work presents the results of the global existence for fractional differential equations involving generalized Caputo derivative with the case of the fractional order derivative α∈1,2. In addition, the Ulam–Hyers–Mittag-Leffler stability of the given problems is also established.
Highlights
There are a vast number of various concepts for fractional integrals and derivatives, such as Riemann–Liouville, Riesz, Grunwald–Letnikov, Hadamard, and Caputo derivatives and/or integrals
For more details on fractional calculus theory and interesting applications, one can see the monographs and the interesting papers in [1,2,3,4,5,6] and the references cited therein. Both of the definitions of Hadamard and Riemann–Liouville fractional derivatives have their own disadvantages as well; one of which is that the derivative of a constant is not equal to zero. en, to overcome the disadvantage of two types of these fractional derivatives, the Caputo and Caputo–Hadamard fractional derivatives were proposed
In [7, 8], Katagampola has proposed a new generalized concept of the fractional derivative, the so-called Caputo–Katugampola, that unifies the definitions of Caputo and Caputo-Hadamard fractional derivatives into a single form. e parameter family ρ of Caputo–Katugampola fractional derivative, CDαa+,ρ, of the noninteger order α allows one to interpolate two types of the Caputo and Caputo–Hadamard fractional derivatives
Summary
There are a vast number of various concepts for fractional integrals and derivatives, such as Riemann–Liouville, Riesz, Grunwald–Letnikov, Hadamard, and Caputo derivatives and/or integrals. For more details on fractional calculus theory and interesting applications, one can see the monographs and the interesting papers in [1,2,3,4,5,6] and the references cited therein Both of the definitions of Hadamard and Riemann–Liouville fractional derivatives have their own disadvantages as well; one of which is that the derivative of a constant is not equal to zero. The motivation for the elaboration of this paper is the investigation of some kinds of the Ulam–Hyers stability for the following problem involving the concept of Caputo–Katugampola fractional derivative with the case of the α ∈ (1, 2): CDαa+,ρψ(t) f(t, ψ(t)), ψ(a) ψ1,.
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