Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations
Joint Authors
Warma, Mahamadi
Lizama, Carlos
Keyantuo, Valentin
Source
Issue
Vol. 2013, Issue 2013 (31 Dec. 2013), pp.1-11, 11 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2013-11-13
Country of Publication
Egypt
No. of Pages
11
Main Subjects
Abstract EN
We investigate mild solutions of the fractional order nonhomogeneous Cauchy problem Dtαu(t)=Au(t)+f(t), t>0, where 0<α<1.
When A is the generator of a C0-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-Sα(1) is invertible.
Here, we use the representation Sα(t)x=∫0∞Φα(s)T(stα)x ds, t>0 in which Φα 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 Dtαu(t)=Au(t)+f(t,u(t)), t>0, 0<α<1.
American Psychological Association (APA)
Keyantuo, Valentin& Lizama, Carlos& Warma, Mahamadi. 2013. Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations. Abstract and Applied Analysis،Vol. 2013, no. 2013, pp.1-11.
https://search.emarefa.net/detail/BIM-485225
Modern Language Association (MLA)
Keyantuo, Valentin…[et al.]. Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations. Abstract and Applied Analysis No. 2013 (2013), pp.1-11.
https://search.emarefa.net/detail/BIM-485225
American Medical Association (AMA)
Keyantuo, Valentin& Lizama, Carlos& Warma, Mahamadi. Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations. Abstract and Applied Analysis. 2013. Vol. 2013, no. 2013, pp.1-11.
https://search.emarefa.net/detail/BIM-485225
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-485225