Abstract
A new and general existence and uniqueness theorem of almost automorphic solutions is obtained for the semilinear fractional differential equation , in complex Banach spaces, with Stepanov-like almost automorphic coefficients. Moreover, an application to a fractional relaxation-oscillation equation is given.
Highlights
We investigate the existence and uniqueness of almost automorphic solutions to the following semilinear abstract fractional differential equation: Dtαu t Au t Dtα−1f t, u t, t ∈ R, 1.1 where 1 < α < 2, A : D A ⊂ X → X is a sectorial operator of type ω in a Banach space X, and f : R × X → X is Stepanov-like almost automorphic in t ∈ R satisfying some kind of Lipschitz conditions in x ∈ X
The Stepanov-like almost automorphic problems have been studied by many authors cf., e.g., 9, 10 and references therein. Stimulated by these works, in this paper, we study the almost automorphy of solutions to the fractional differential equation 1.1 with Stepanov-like almost automorphic coefficients
A new and general existence and uniqueness theorem of almost automorphic solutions to the equation is established
Summary
We investigate the existence and uniqueness of almost automorphic solutions to the following semilinear abstract fractional differential equation: Dtαu t Au t Dtα−1f t, u t , t ∈ R, 1.1 where 1 < α < 2, A : D A ⊂ X → X is a sectorial operator of type ω in a Banach space X, and f : R × X → X is Stepanov-like almost automorphic in t ∈ R satisfying some kind of Lipschitz conditions in x ∈ X. The Stepanov-like almost automorphic problems have been studied by many authors cf., e.g., 9, 10 and references therein Stimulated by these works, in this paper, we study the almost automorphy of solutions to the fractional differential equation 1.1 with Stepanov-like almost automorphic coefficients
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