In:
Abstract and Applied Analysis, Hindawi Limited, Vol. 2013 ( 2013), p. 1-11
Abstract:
We investigate mild solutions of the fractional order nonhomogeneous Cauchy problem D t α u ( t ) = A u ( t ) + f ( t ) , t 〉 0 , where 0 〈 α 〈 1 . When A is the generator of a C 0 -semigroup ( T ( t ) ) t ≥ 0 on a Banach space X , we obtain an explicit representation of mild solutions of the above problem in terms of the semigroup. We then prove that this problem under the boundary condition u ( 0 ) = u ( 1 ) admits a unique mild solution for each f ∈ C ( [ 0,1 ] ; X ) if and only if the operator I - S α ( 1 ) is invertible. Here, we use the representation S α ( t ) x = ∫ 0 ∞ Φ α ( s ) T ( s t α ) x d s , t 〉 0 in which Φ α is a Wright type function. For the first order case, that is, α = 1 , the corresponding result was proved by Prüss in 1984. In case X is a Banach lattice and the semigroup ( T ( t ) ) t ≥ 0 is positive, we obtain existence of solutions of the semilinear problem D t α u ( t ) = A u ( t ) + f ( t , u ( t ) ) , t 〉 0 , 0 〈 α 〈 1 .
Type of Medium:
Online Resource
ISSN:
1085-3375
,
1687-0409
Language:
English
Publisher:
Hindawi Limited
Publication Date:
2013
detail.hit.zdb_id:
2064801-7
SSG:
17,1
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