Abstract
Let and A be the generator of an -times resolvent family on a Banach space X. It is shown that the fractional Cauchy problem has maximal regularity on if and only if is of bounded semivariation on .
Highlights
Many initial and boundary value problems can be reduced to an abstract Cauchy problem of the form u′(t ) = Au (t ) + f (t ), t ∈[0, r]
Travis [1] proved that the maximal regularity is equivalent to the C0 -semigroup generated by A being of bounded semivariation on [0, r]
Where α ∈ (1, 2), A is the generator of an α -times resolvent family and Dαt u is understood in the Caputo sense
Summary
Let 1 < α < 2 and A be the generator of an α -times resolvent family {Sα (t }) t≥0 on a Banach space X. It is shown that the fractional Cauchy problem Dαt u (t ) = Au (t ) + f (t ) , t ∈ (0, r] ; u (0),u′(0) ∈ D ( A) has maximal regularity on C ([0, r]; X ) if and only if Sα (⋅) is of bounded semivariation on [0, r] . Where α ∈ (1, 2) , A is the generator of an α -times resolvent family (see Definition 2.2) and Dαt u is understood in the Caputo sense.
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