Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations. (13th November 2013)

• Record Type: Journal Article
• Authors: Keyantuo, Valentin; Lizama, Carlos; Warma, Mahamadi
• Other Names: Xie Fuding Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2013(2013)
• Publication Date: 2013-11-13
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2013/614328
• Languages: English
• ISSNs: 1085-3375

Abstract

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 .