The Abstract Cauchy Problem with Caputo–Fabrizio Fractional Derivative
نویسندگان
چکیده
Given an injective closed linear operator A defined in a Banach space X, and writing CFDtα the Caputo–Fabrizio fractional derivative of order α∈(0,1), we show that unique solution abstract Cauchy problem (∗)CFDtαu(t)=Au(t)+f(t),t≥0, where f is continuously differentiable, given by first u′(t)=Bαu(t)+Fα(t),t≥0;u(0)=−A−1f(0), family bounded operators Bα constitutes Yosida approximation Fα(t)→f(t) as α→1. Moreover, if 11−α∈ρ(A) spectrum contained outside disk center radius equal to 12(1−α) then (∗) converges zero t→∞, norm provided f′ have exponential decay. Finally, assuming Lipchitz-type condition on f=f(t,x) (and its time-derivative) depends α, prove existence uniqueness mild solutions for respective semilinear problem, all initial conditions set S:={x∈D(A):x=A−1f(0,x)}.
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ژورنال
عنوان ژورنال: Mathematics
سال: 2022
ISSN: ['2227-7390']
DOI: https://doi.org/10.3390/math10193540