Abstract and Applied Analysis

Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations

Valentin Keyantuo, Carlos Lizama, and Mahamadi Warma

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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 .

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 614328, 11 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393512131

Digital Object Identifier
doi:10.1155/2013/614328

Mathematical Reviews number (MathSciNet)
MR3129328

Zentralblatt MATH identifier
07095167

Citation

Keyantuo, Valentin; Lizama, Carlos; Warma, Mahamadi. Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations. Abstr. Appl. Anal. 2013 (2013), Article ID 614328, 11 pages. doi:10.1155/2013/614328. https://projecteuclid.org/euclid.aaa/1393512131


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