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The Abstract Cauchy Problem with Caputo–Fabrizio Fractional Derivative.
- Source :
-
Mathematics (2227-7390) . Oct2022, Vol. 10 Issue 19, p3540. 20p. - Publication Year :
- 2022
-
Abstract
- Given an injective closed linear operator A defined in a Banach space X , and writing C F D t α the Caputo–Fabrizio fractional derivative of order α ∈ (0 , 1) , we show that the unique solution of the abstract Cauchy problem (∗) C F D t α u (t) = A u (t) + f (t) , t ≥ 0 , where f is continuously differentiable, is given by the unique solution of the first order abstract Cauchy problem u ′ (t) = B α u (t) + F α (t) , t ≥ 0 ; u (0) = − A − 1 f (0) , where the family of bounded linear operators B α constitutes a Yosida approximation of A and F α (t) → f (t) as α → 1. Moreover, if 1 1 − α ∈ ρ (A) and the spectrum of A is contained outside the closed disk of center and radius equal to 1 2 (1 − α) then the solution of (∗) converges to zero as t → ∞ , in the norm of X, provided f and f ′ have exponential decay. Finally, assuming a Lipchitz-type condition on f = f (t , x) (and its time-derivative) that depends on α , we prove the existence and uniqueness of mild solutions for the respective semilinear problem, for all initial conditions in the set S : = { x ∈ D (A) : x = A − 1 f (0 , x) }. [ABSTRACT FROM AUTHOR]
- Subjects :
- *BANACH spaces
*COMMERCIAL space ventures
*LINEAR operators
*CAUCHY problem
Subjects
Details
- Language :
- English
- ISSN :
- 22277390
- Volume :
- 10
- Issue :
- 19
- Database :
- Academic Search Index
- Journal :
- Mathematics (2227-7390)
- Publication Type :
- Academic Journal
- Accession number :
- 159673870
- Full Text :
- https://doi.org/10.3390/math10193540